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Mathematics - Course Outline
The University of Cambridge offers an undergraduate course in Mathematics. The course is well built and incorporates theories as well as practical exercises. The Mathematics course outlines are given as follows.
Part First
In the first two years, students concentrate on the basic tools they need to mathematics at deeper levels with an equal combination of applied and pure mathematics.
Part 'A'
Year First
The first year of the course introduces students to the basic of higher mathematics, which may include:
Students who take Mathematics with Physics may also cover, for instance, electromagnetism, kinetic theory and Fourier analysis.
Part First 'B'
Year Second
In the part first 'B', students choose, under the close supervision of their Director of Studies, from a wide range of seventeen papers options available to them. In this phase of the course, the topics become deeper and broader and are classified as applicable, applied and pure; however, there are strong associations between the different areas. For example:
The pure side is divided into analysis and algebra.
Algebra does not mean tedious manipulation of letters of the alphabet. Algebra is the study of systems of objects like groups that obey certain rules. These rules explain the symmetries that underline most areas of physics and mathematics.
In analysis, the foundation of calculus is assessed comprising the theory of functions of a complex variable
The applied side includes theoretical physics and mathematics methods
The purpose of the mathematics methods is to establish techniques for solving a series of problems without much focus upon rigorous justification
Theoretical Physics introduces students to the pillars of contemporary physics: quantum electromagnetism. There is a paper on fluid dynamics that is studied for physical importance as well as its mathematical elegance.
Applicable mathematics consists of optimization and statistics, for example, choosing the best route through a network. The fundamental ideas of data analysis are introduced. Here, students need to take a paper in Optimization because of its tidy mathematical treatment of familiar problems.
There are some optional computational projects, evaluated by programmes and note books submitted before the exam in the summer. Students use computers to solve mathematical problems.
Part Second
Year Third
The third year offers students an opportunity to unfold their mathematical interests and use the skills they have achieved. Students have a broad choice of topics including:
Waves
Cosmology
Number Theory
Algebraic Topology
Stochastic Financial Models
Coding and Cryptography
Principles of Quantum Mechanics
Logic and Set Theory
Students also have option of studying computational projects.
Part Third
Year Forth
In the forth year, part third, students have more than eighty courses to choose from including all areas of theoretical physics and mathematical and are motivated to complete a mathematical project or essay chosen from a similar wide range of topics. | 677.169 | 1 |
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this file type before downloading and/or purchasing.
3 MB|11 pages
Product Description
This is an Algebra 1 Common Core Lesson on Histograms. Students will create a histogram and cumulative frequency histogram based upon a survey in the class of how many text messages are sent per day by each student. This lesson is meant to follow a lesson on Frequency Tables. After a few teacher led examples, students will work alone or with a partner to practice. | 677.169 | 1 |
Algebra I Module 3, Topic B, Lesson 14
Student Outcomes
Students compare linear and exponential models by focusing on how the models change over intervals of equal length. Students observe from tables that a function that grows exponentially will eventually exceed a function that grows linearly. | 677.169 | 1 |
This innovative text features a graphing calculator approach, incorporating real-life applications and such technology as graphing utilities and Excel spreadsheets to help students learn mathematical skills that they will use in their lives and careers. The texts overall goal is to improve learning of basic calculus concepts by involving students with new material in a way that is different from traditional practice. The development of conceptual understanding coupled with a commitment to make calculus meaningful to the student are guiding forces. Targeted toward students majoring in liberal arts, economics, business, management, and the life and social sciences, the text's focus on technology along with its use of real data and situations make it a sound choice to help you develop an intuitive, practical understanding of concepts.
Book Description Book Condition: Brand New. New. SoftCover International edition. Different ISBN and Cover image but contents are same as US edition. Customer Satisfaction guaranteed!!. Bookseller Inventory # SHUB41579
Book Description Book Condition: Brand New. New, SoftCover International edition. Different ISBN and Cover image but contents are same as US edition. Excellent Customer Service. Bookseller Inventory # ABEUSA-41579
Book Description Brooks/Cole 43300362 | 677.169 | 1 |
Test Prep for 7th Grade Math Round 2
Word Document File
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0.22 MB | 12 pages
PRODUCT DESCRIPTION
This is an end-of-the-year test prep Word Document for 7th Grade Math. There are 5 standards included. There are 4 worksheets, 1 test, and keys for each one. Each page has 10 problems which includes 2 practice problems per standard. The standards included with this particular test prep are as follows:
*Solving contextual problems that involve operations with integers
*Evaluating algebraic expressions involving rational values for coefficients and or variables
*Solving linear equations with rational coefficients symbolically
*Translating between verbal and symbolic representations of real-world phenomena involving linear equations
*Solving contextual problems involving two-step linear equations | 677.169 | 1 |
VSB Math
The Mathematics VSB Course will discuss topics in Algebra. The first lesson will lay the groundwork for all succeeding lessons, by familiarizing the user with definitions and notations that will be used throughout the course | 677.169 | 1 |
RELATED INFORMATIONS
Learn why Common Core is important your child. What parents should know; Myths vs. facts. This timetable has been updated with changes made up and including: May 16, 2017 Changes are red; additions are blue.. Logarithm and exponential questions with solutions and answers grade 12.. Edmentum is leading provider of online learning programs designed drive student achievement academic and career success.. Cheat Sheets & Tables Algebra, Trigonometry and Calculus cheat sheets and variety of tables. Class Notes Each class has notes available. Most of classes have .... . | 677.169 | 1 |
Advanced Maths is a command line tool for calculating advanced maths functions, it diplays every step it calculates and provides you with as full an answer as it can. To download the pre-compiled binary packages go to the project | 677.169 | 1 |
How to Ace Calculus: The Streetwise Guide
Written by three gifted-and funny-teachers, How to Ace Calculus provides humorous and readable explanations of the key topics of calculus without the technical details and fine print that would be found in a more formal text. Capturing the tone of students exchanging ideas among themselves, this unique guide also explains how calculus is taught, how to get the best teachers, what to study, and what is likely to be on exams-all the tricks of the trade that will make learning the material of first-semester calculus a piece of cake. Funny, irreverent, and flexible, How to Ace Calculus shows why learning calculus can be not only a mind-expanding experience but also fantastic | 677.169 | 1 |
Showing 1 to 27 of 27
Math 30 College Algebra
March 31, 2013
Quiz #12
Name:
David Arnold
Instructions. (10 points) This quiz is open notes, open book. This includes any supplementary texts or online documents. You are not allowed to work in groups or pairs on the
quiz. You are
Math 30 College Algebra
March 19, 2013
Name:
David Arnold
Quiz #6
Instructions. (10 points) Place all of your work on graph paper, then staple this quiz on top of your graph
paper. For each item, show the required work using the methods demonstrated in cl
Math 30 College Algebra
February 5, 2013
Dierence Quotient I
Name:
Math Department
Instructions. (8 points) The following exercises are designed to help students in College Algebra prepare
for their rst course in calculus, whose emphasis during the rst se
Math 30 College Algebra
February 7, 2013
Quiz #3 3, 2013
Name:
David Arnold
Quiz #4
Instructions. (20 points) Place all of your work on graph paper, then staple this quiz on top of your graph
paper. For each item, show the required work using the methods demonstrated in cla
Math 30 College Algebra
March 3, 2013
Name: Answer Key
David Arnold
Midterm #1B
Instructions. (60 points) For each of the following questions, select the best answer and darken the
corresponding circle on your scantron.
(5pts )
1. Which of the following b
Math 30 College Algebra
January 30, 2013
Name: Answer Key
David Arnold
Quiz #2
Instructions. (20 points) This quiz is open notes, open book. This includes any supplementary texts or
online documents. You must answer all of the exercises on your own. You a
Math 30 College Algebra
February 13, 2013
Name: Answer Key
David Arnold
Midterm #1A
Instructions. (60 points) For each of the following questions, select the best answer and darken the
corresponding circle on your scantron.
(5pts )
1. Which of the followi
Math 30 College Algebra
January 24, 2013
Quiz #1 20, 2013
Name: Answer Key
David Arnold
Quiz #7
Instructions. (20 points) Perform each of the following tasks in the space provided. The quiz is open book,
open notes. All work must be your own. You are not allowed to work wit
Math 30 College Algebra
March 31, 2013
Quiz #11
Name:
David Arnold
Instructions. (10 points) Place your solution to the given question in the space provided.
(10pts )
1. Initially, there are 1000 milligrams of a radioactive substance present. After 23 day
Math 30 College Algebra
March 31, 2013
Quiz #13
Name:
David ArnoldApril 25, 2013
Name:
David Arnold
Quiz #14March 26, 2013
Name: Answer Key
David Arnold
Quiz #8
Instructions. (15 points) This quiz is open notes, open book. This includes any supplementary texts or
online documents. You are not allowed to work in groups or pairs on the qui
Math 30 College Algebra
March 28, 2013
Name: Answer Key
David Arnold
Quiz #9
Instructions. (20 points) This quiz is open notes, open book. This includes any supplementary texts or
online documents. You are not allowed to work in groups or pairs on the qui
Math 30 College Algebra
April 16, 2013
Name: Answer Key
David Arnold
Midterm #2B
Instructions. (50 points) For each of the following questions, select the best answer and darken the
corresponding circle on your scantron.
(5pts )
1. Which of the following
Math 30 College Algebra
April 25, 2013
Name: Answer Key
David Arnold
Midterm #2A
Instructions. (50 points) For each of the following questions, select the best answer and darken the
corresponding circle on your scantron.
(5pts )
1. Which of the following | 677.169 | 1 |
Need Help?
Intermediate Counting & Probability Online Book
An intermediate textbook in counting and probability for students in grades 9-12, containing topics such as inclusion-exclusion, recursion, conditional probability, generating functions, graph theory, and more.
Overview
Continue your exploration of more advanced counting and probability topics from former USA Mathematical Olympiad winner David Patrick. This book is the follow-up to the acclaimed Introduction to Counting & Probability textbook.
As with all of the books in Art of Problem Solving's Introduction and Intermediate series, the text is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help before proceeding. The text then includes solutions to these problems, through which counting and probability techniques are taught. Important facts and powerful problem solving approaches are highlighted throughout the text. In addition to the instructional material, the book contains over 650 problems.Full solutions to all of the problems, not just answers, are built into the book. | 677.169 | 1 |
This textbook represents an extensive and easily understood introduction to tensor analysis, which is to be construed here as the generic term for classical tensor analysis and tensor algebra, and which is a requirement in many physics applications and in engineering sciences.
Tensors in symbolic notation and in Cartesian and curvilinear co-ordinates are introduced, amongst other things, as well as the algebra of second stage tensors.
The book is primarily directed at students on various engineering study courses. It imparts the required algebraic aids and contains numerous exercises with answers, making it eminently suitable for self study | 677.169 | 1 |
Showing 1 to 7 of 7
1.2 Gauss-Jordan elimination
Definition 1. A matrix is in reduced row-echelon form if it satisfies all of the following conditions:
(1) If a row has nonzero entries, then the first nonzero entry is a 1, called a leading 1, or pivot.
(2) If a column contai
1.3 vector equations
We have already seen that a system of linear equations may be solved by solving a matrix equation of the
form Ax = b. But why is that the case?
Theorem 1 (Lay, Ch. 1, Th. 3). If A = [aij ] is an m n matrix with columns a1 , . . . , an
1.4 The equation Ax = b
Theorem (Lay, Ch. 1, Th. 4). Let A be an m n matrix. Then the following are equivalent (either all
true or all false)
a. For each b Rm , the equation Ax = b has a solution.
b. Each b Rm is a linear combination of the columns of A.
1.1 Introduction to linear systems
Linear algebra finds its origins in the study of linear systems of equations. Some familiar examples
of linear equations are
ax + by = c
and
ax + by + cz = d.
There is no limit to the number of variables a linear equatio | 677.169 | 1 |
Book Algebra 1 Textbook Online
We can help you with middle school, high school, or even college algebra, and we have math lessons in.
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Algebra worksheets contain translating phrases, simplifying and evaluating algebraic expressions, equations, inequalities, polynomials, matrices and more.Intended to be used by teachers and students in middle school and high school in years 6, 7 8 or 9, learning pre algebra, or.
Functions include math expressions, plots, unit converter, equation solver, complex numbers and calculation history.Access world-class Algebra content based on college intro-level Algebra content.Coolmath Algebra has hundreds of really easy to follow lessons and examples. Algebra 1.It is suitable for high-school Algebra I, as a refresher for college students who need help.
Online Math Games
Free Online Math Games
TutorService.com provides live interactive online tutoring service tutoring you in maths, biology, chemistry, physics, as well online tutoring service for SAT, ACT.Math League Adaptive Learning System contains the actual math contests given to students participating in Math League Contests, Grades 4, 5, 6, 7, 8, Algebra Course 1.Learn about algebra from solving equations and sketching graphs to complex numbers and logarithms.Practice math online with unlimited questions in more than 200 Algebra 1 math skills.It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of.
This free online course offers a comprehensive introduction to algebra and carefully explains the concepts of algebraic fractions.
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Free intermediate and college algebra questions and problems are presented along with answers and explanations.Beginner to Intermediate Algebra is intended for students who need to gather a basic understanding of how to perform Algebra operations.Online algebra video lessons to help students with the concepts, equations and calculator use, to improve their math problem solving skills while they study their.Algebra online in the form of interactive quizzes enables young learners to gain access to free materials at all times of the day.Get help with Algebra homework and solving Algebra problems in Algebra I and Algebra II.Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in.
Dummies has always stood for taking on complex concepts and making them easy to understand.Dummies helps everyone be more knowledgeable and confident in applying.Free math lessons and math homework help from basic math to algebra, geometry and beyond.Boundless Algebra readings, quizzes, and PowerPoints and free to edit, share, and use.Get a jump start on your college degree with this College Algebra course. | 677.169 | 1 |
Introduction to Precise Numerical Methods.* Clearer, simpler descriptions and explanations ofthe various numerical methods* Two new types of numerical problems; accurately solving partial differential equations with the included software and computing line integrals in the complex plane | 677.169 | 1 |
Solving Equations on the Smartboard
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this file type before downloading and/or purchasing.
563 KB|9 pages
Product Description
This Smartboard Notebook file lesson is designed for a Pre-Algebra or Algebra class that are learning to solve equations. There are four pages of Algebra balances and Algebra tiles that are used to model progressively more difficult equations, beginning with one-step and ending with equations that have variables on both sides. Two pages of notes (one for Pre-Algebra and a more advanced version for Algebra) are also included. | 677.169 | 1 |
class 10 maths ncert solutions
The National Council for Educational Research and Training (NCERT) is an educational body, established by the Government of India.The body was established with an objective of providing assistance to the government at the centre and at state levels on the academic field. The body also strives to achieve the aim of enhancing school education in the country. Various independent bodies are helping NCERT in carrying out this task. The bodies are National Institute of Education(NIE), New Delhi, Central Institute of Education Technology(CIET), New Delhi, Pandit Sunderlal Sharma Central Institute of Vocational Education (PSSCIVE), Bhopal, Regional Institute of Education(RIE), Ajmer, Regional Institute of Education(RIE), Bhopal, Regional Institute of Education(RIE), Bhubaneshwar, Regional Institute of Education(RIE), Mysore, North Eastern-Regional Institute of Education(NE-RIE), Shillong.
The NCERT undertakes publishing work and publishes textbooks for students studying in classes from I to XII std. on various subjects. The textbooks are published in three different languages-English, Hindi and Urdu. The educational body has started a new initiative of making its textbooks available on the internet for the benefit of teachers and students. But it has not covered all classes. Online textbooks are available for 5 classes- I, III, IV, IX and XI. All these textbooks are based on the National Curriculum Framework of 2005. The educational body has a special manual too 'Inclusive Education' whose objective is to guide and emphasize schools for inclusion of special needs students into the mainstream education.
Programs – NCERT
NCERT undertakes three main activities. They are
Research
National Council for Educational Research and Training (NCERT) in association with other independent bodies conducts research in different aspects of school and teacher education. The other institutions also take its support and help in carrying out their research work. The international institutes, for their inter-country research and projects work also look forward to it countries.
Training
It provides training supports to all those who either aspire to be a teacher or those who have already taken plunge in the teaching profession. The training for service or regular teachers is provided at various levels- pre primary, secondary and higher secondary.
The pre service or aspiring teachers, on the other hand, are taught the features of integrating content and the teaching techniques. They are also given training in the actual classroom setting by the assigning them the position of trainee for a fairly long period of time. It also trains key personnel of the states through its Regional Institute of Educations (RIEs).
Developmental Activities
Development is another activity that NCERT undertakes. It deals with the framing of course curriculum of various classes. Thus, the apex body mainly develops the curriculum and instruction material for the various courses and subjects. The course curriculum is not restricted to just students textbooks and it is also for the vocational course and for teachers training course, it also updates the courses as and when required.
NCERT members periodically select, guide and contribute to the educational research projects. A good example is the past project, the NCERT National Reading Initiative. It held two highly successful best practices reading and literacy conferences, "Every Child a Proficient Reader (June 2003)," and "Keys to Closing the Achievement Gap (June 2004). Vision of NCERT is to be a magnet for education's foremost thinkers and best resources to foster the development of world class practices in education.
Eligibility
In-service teachers – graduates with teaching degree.
Candidates 'presently not working' – graduation with teaching degree and at least two years of teaching or related experience.
Postgraduates in psychology/education/social work/child development/special education. Preference will be given to those with at least one year of teaching or related experience.
Candidates from outside India who are not graduates but have teaching degree are required to have three years of work experience in the relevant area.
Minimum % of marks for all target groups is 50% (5% relaxation for SC/ST).
Duration
One year, from 3.1.2011 to 2.1.2012
Guided Self-Learning Six Months (Distance/ Online)
Intensive Practicum* Three Months (Face-to-Face)
Internship Three Months (in Home Town)
Selection Procedure
Screening of candidates will be done by a committee following a selection criteria. Shortlisted candidates will be called at the study centre in their region for selection test which includes essay writing and interview.
Please Click the link given below to download the NCERT Class 10th Question Book | 677.169 | 1 |
User reviews
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Calculus Help
Calculus reveals the beauty of math and the agony of math. Calculus communicates topics in a graceful, brain-bending method. Darwin's Theory of Evolution: once understood, you start seeing Nature in terms of existence. Math isn't the hard part of math; motivation is. Math is about ideas, not robotically manipulating the formulas that express them. Let's define calculus as "the branch of mathematics that deals with limits and the differentiation and integration of functions of one or more variables".
Integral calculus and differential calculus are akin by the fundamental theorem of calculus, which describes that differentiation is the reverse process to integration. Calculus is about the very big, the very small, and how everything changes. The surprise is that something apparently so theoretical ends up clarifying the real world. Calculus portrays a showcasing role in the physical, biological, and social sciences. By concentrating outside of the classroom, we will see examples of calculus appearing in our day-to-day life.
Calculus is the study of change, with the basic focus being on
Rate of change
Accumulation
Calculus for Kids
TutorVista gives a friendly introduction to calculus, apt for someone who has never seen the subject before, or for somebody who has seen some calculus but needs to assess the theories, concepts and practice relating and applying those concepts to solve problems.
A classic calculus course covers the following topics:
How to find the prompt immediate change (called the "derivative") of various functions. (The process of doing so is called "differentiation")
How to go back from the derivative of a function to the function itself. (This process is called "integration".)
How to use derivatives to resolve various kinds of problems
Study of thorough methods for integrating functions of definite kinds.
How to use integration to solve numerous geometric problems, such as computations of areas and volumes of certain regions.
There are a few additional standard topics in calculus. These comprise description of functions in terms of power series, and the study of when an infinite series "converges" to a number.
So what do kids study and learn about calculus?
Single variable calculus, deals with motion of an object along a fixed path and multivariable calculus handles when motion can take place on a surface, or in space. We study the latter by finding astute tricks for using the one dimensional methods and ideas to handle more generalized problems. Hence single variable calculus is vital for general problems.
When we deal with an object moving along a path, its position differs with time we can define its position at any time by a single number, which can be the distance in some units from some fixed point on that path, called the "origin."
The motion of the object is then categorized by the set of its numerical situations at applicable points in time.
The set of situations and times that we use to define motion is what is called a function. And interrelated purposes are used to term the quantities of interest wherever calculus is applied.
Calculus starts with a review of numbers and functions and their properties.
With TutorVista kids will systemically solve problems. These basic principles apply to a wide array of real world's problems dealing with physics, biology, chemistry, engineering, medicine, computer science, business, astronomy and other routine problems that could not have been solved without Calculus. | 677.169 | 1 |
Statistics And Probability Archive | September 22, 2015 | Chegg.com
Algebra is often taught abstractly with little or no emphasis on what algebra is or how it can be used to solve real problems.Just because that experience was developedin a different class.8 My agent will be an.
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Course 3, Lesson 40, Area of a Circle
Rating:
Description:
1. Make sense of problems and persevere in solving them. 2. Look for and express regularity in repeated reasoning. 3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. 4. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 5. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 6. Understand | 677.169 | 1 |
Über dieses Buch
Beschreibung:
Please note that at 1.1Kg. this book will require additional postage outside of the UK.| 528pp. | Buying this book means my Jack Russells get their supper! Condition :: Buchnummer des Verkäufers
Buchbeschreibung Springer-Verlag Gmbh Sep 2012, 2012. Taschenbuch. Buchzustand: Neu. 242x159x28 mm. Neuware - An understanding of discrete mathematics is essential for students of computer science wishing to improve their programming competence.Fundamentals of Discrete Math for Computer Science provides an engaging and motivational introduction to traditional topics in discrete mathematics, in a manner specifically designed to appeal to computer science students. The text empowers students to think critically, to be effective problem solvers, to integrate theory and practice, and to recognize the importance of abstraction. Clearly structured and interactive in nature, the book presents detailed walkthroughs of several algorithms, stimulating a conversation with the reader through informal commentary and provocative questions. Topics and features: highly accessible and easy to read, introducing concepts in discrete mathematics without requiring a university-level background in mathematics; ideally structured for classroom-use and self-study, with modular chapters following ACM curriculum recommendations; describes mathematical processes in an algorithmic manner, often including a walk-through demonstrating how the algorithm performs the desired task as expected; contains examples and exercises throughout the text, and highlights the most important concepts in each section; selects examples that demonstrate a practical use for the concept in question. This easy-to-understand and fun-to-read textbook is ideal for an introductory discrete mathematics course for computer science students at the beginning of their studies. The book assumes no prior mathematical knowledge, and discusses concepts in programming as needed, allowing it to be used in a mathematics course taken concurrently with a student s first programming course. 416 pp. Englisch. Artikel-Nr. 9781447140689 | 677.169 | 1 |
4. The student connects algebraic and geometric representations of functions.
5. The student knows the relationship between the geometric and algebraic
descriptions of conic sections.
Exponential and Logarithmic Functions
Objectives:
11. The student formulates equations and inequalities based
on exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes
the solutions in terms of the situation.
Foundation for Functions
Objectives:
1. The student uses properties and attributes of functions and applies
functions to problem situations.
2. The student understands the importance of the skills required to
manipulate symbols in order to solve problems and uses the necessary algebraic skills required to
simplify algebraic expressions and solve equations and inequalities in problem situations.
3. The student formulates systems of equations and inequalities from
problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of
the situations.
Quadratic and Square Root Functions
Objectives:
6. The student understands that quadratic functions can be
represented in different ways and translates among their various representations. | 677.169 | 1 |
This book presents the state of the art in tackling differential equations using advanced methods and software tools of symbolic computation. It focuses on the symbolic-computational aspects of three kinds of fundamental problems in differential equations: transforming the equations, solving the equations, and studying the structure and properties of their solutions. The 20 chapters are written by leading experts and are structured into three parts.
The book is worth reading for researchers and students working on this interdisciplinary subject but may also serve as a valuable reference for everyone interested in differential equations, symbolic computation, and their | 677.169 | 1 |
Saturday, August 31, 2013
This year, I have resolved to do a much better job at the interactive notebook in Algebra 2 than last year. Last year, we had 12 students in my entire school who were enrolled in Algebra 2. This year, that number is just under 40. This is both exciting and kinda terrifying. First of all, it means that more students are prepared and willing to take Algebra 2. At the same time, my Algebra 2 students this year are greater in number and much more varied in level. This presents many challenges. But, these are challenges I am excited to attempt to meet.
Our first unit in Algebra 2 is an introduction to functions, function notation, domain and range, intercepts, maximums and minimums, intervals of increasing and decreasing, finding solutions, and transformations. My goal is to create a foundation which I can build off of once we start linear functions. I am also working hard to prove to my students that they are capable of doing Algebra 2 level work. Many of my students have extremely low confidence. We are also learning how to use the graphing calculator. This is the first experience any of my students have had with a graphing calculator, and I am working hard to make it a positive one.
So far, my Algebra 2 students are loving our interactive notebooks. They thank me on an (almost) daily basis for making Algebra 2 visual, fun, and easy. I have some students who are complaining right now that Algebra 2 is too easy. I told them that they just had to wait. Before they knew it, we would be exploring logarithms, exponentials, conic sections, and all kinds of other exciting mathematical relations. After going on and on about how excited I was about everything we were going to be learning and studying this year, one student asked, "Do you like math?" I was a bit taken aback by this question. Are there math teachers who don't like math? Of course, I like math. I love math! I eat, breath, and sleep math.
You'll notice that I love printing my foldables on colored paper. I think that notes taken on colored paper are just more memorable than notes taken on plain, white copy paper. My students often remember which page in their notebooks they need to reference by what color the foldable was printed on. The paper I use the most is Fireworx brand colored paper. It's available from Amazon, comes in loads of colors, and is priced very reasonably. Astrobrights paper makes for the prettiest, brightest foldables, but it's a bit more pricey. So, I save it for printing posters.
Link to download these files is at the end of the post as always. :)
My Algebra 2 Interactive Notebook
Algebra 2 Unit 1 Table of Contents (Thus Far)
I have already blogged about the NAGS foldable I had my Algebra 2 students create here.
NAGS Foldable - Outside
NAGS Foldable - Inside
NAGS Chart
I still haven't found a better way to practice differentiating between function/not a function than this card sort. I blogged about this last year.
Function / Not a Function Card Sort
We also created a Frayer Model for the word "function."
Function / Vertical Line Test Frayer Model
I stole this coordinate plane foldable from Ms. Haley and her wonder Journal Wizard blog! I think this is a big improvement over the coordinate plane foldable I did with my students last year. I created a template for this foldable which I have linked to at the end of this post.
Parts of the Coordinate Plane Foldable
Parts of the Coordinate Plane Foldable
Parts of the Coordinate Plane Foldable
Our notes over independent and dependent variables were less than exciting. Maybe next year I will come up with a card sort or something. Hmm...
Independent and Dependent Variable Notes
Last year, my students had a TERRIBLE time remembering the difference between domain and range. This summer, at the amazing Common Core Training I received from the Oklahoma Geometry and Algebra Project (OGAP), I was introduced to an amazing resource--Shmoop. They have amazing commentary for each and every high school common core math standard! I learned about the DIXROY acronym from their commentary on F-IF.1.
I was able to re-use the domain/range notation foldable that I created last year for my Algebra 2 students. My students were VERY confused by the different notations. I haven't yet figured out a way to introduce these notations without overwhelming my students. They recovered, eventually.
Domain and Range Notation Foldable - Outside
Domain and Range Notation Foldable - Inside
I downloaded the domain and range cards from this blog post. There are 32 cards which give my students 32 opportunities to practice finding the domain and range!
Domain and Range Foldable
We made a tiny envelope to hold our 32 cards. Let me just say - having the students cut out all 32 cards took WAY too much time. I was about ready to pull my hair out. I think we might of spent half of a fifty minute class period just cutting these cards out. But, we used them a lot, so I think it was worth it.
I LOVED the envelope template that Kathryn (iisanumber.blogspot.com) posted earlier this summer. I downsized her template to the exact size needed to fit the domain and range cards I linked to earlier.
As you can see, this foldable perfectly holds the domain and range practice cards from our handy-dandy envelope!
The foldable is made to perfectly hold our domain and range practice cards that are housed in the envelope.
Students fold over the domain tabs to help them determine the left-most and right-most points on the graph. If the graph goes approaches negative or positive infinity, the students leave the flap open where it reads positive or negative infinity. I wanted my treatment of domain and range to be much more hands-on this year, and I think this foldable does the trick!
After doing many, many cards together, I had students find the domain and range of all 32 cards as homework. They had to write the domain and range in both interval and algebraic notation. (And, the discrete graphs had to have their domain written in set notation.) The next day, I gave them an answer key to use the check their work.
The Domain and Range Foldable in Action
One of the main thing my students need to be able to do on their Algebra 2 EOI is to describe graphs. This foldable is an attempt to introduce my students to the concepts of x-intercepts, y-intercepts, relative maximums, relative minimums, increasing intervals, decreasing intervals, roots, solutions, and zeros.
Describing Characteristics of Graphs Foldable - Outside
Because there is so much information on this one foldable, this was a perfect opportunity to use COLOR WITH A PURPOSE. Each term was marked with a different color. And the corresponding part of the graph was marked with the same color. This is one of my favorite foldables that we have done this year!
I've posted some close-ups of the flaps if you'd like to see what I had my students write.
Close-up of Right Flaps
Close-Up of Left Flaps
Last year, my Algebra 2 students really struggled with the concept of an inverse. So, this year, I decided to start talking about inverses very early in the school year. This will allow us to revisit the concept over and over as we explore different types of function in a much more in depth manner. By the time the EOI rolls around, my students should no longer be scared when they see the word inverse!
This foldable was inspired by @druinok's post from February.
Inverse of a Function Foldable - Outside
I want my students to be able to find the inverse if they are given a set of points, a graph, or an equation. Since we have only just started exploring functions in general, the examples we went through were quite simplistic. We will explore much more complicated inverses as the year progresses!
A lot of my students were terrified when I told them that we would be learning about inverses. By the end of the lesson, they were amazed that inverses were actually quite easy.
Inverse of a Function Foldable - Inside
Inverse of a Function - Important Fact!
I still have to figure out how I want to introduce transformations to my Algebra 2 students. That topic should end our first unit. Hmm...
Friday, August 30, 2013
My Algebra 1 students just finished up their first unit on Thursday. This year, I am attempting to model my Algebra 1 class on the Kagan Cooperative Learning Algebra 1 curriculum. As I'm getting to know my students, I have made some adjustments. Okay. I've made a ton of adjustments. My Algebra 1 students this year seem to be at a lower level than last year's students. Let's just say it's going to be an interesting year...
I'm so excited to share with you what my students' interactive notebooks consist of thus far! Link to download the files is at the end of the post. Hope these can be of some use to you!
Algebra 1 Interactive Notebook
Unit 1 Table of Contents
3 Requirements for a Good Definition
The Kagan curriculum I am using focuses on having students develop their own definitions for the vocabulary words for each unit. This is an entirely new approach to me. I'm used to just giving the students the definition and having them write it down. I'm kinda obsessed with the Frayer Model.
I'm enjoying the approach, however, of having students examine examples and counterexamples to determine what a word means for themselves. Time will tell if this leads to a higher level of understanding and recall. The first lesson of the year focused on what a good definition consists of. Our three requirements for a good definition were: 1) states the term, 2) states the nearest classification, and 3) states those items that make it unique.
We practiced a lot with this. I used some examples provided in the book to give my students the opportunity to practice writing a definition that had absolutely nothing to do with mathematics. Students were given a minute or two to examine the zingers and thingmabobs. Then, each student wrote their own definition based on the examples and counterexamples. Next, each person in the group shared their definition with the rest of the group. Each group discussed everybody's definition and combined the best parts of each definition into one group definition. Finally, we discussed the group definitions as a class and wrote one class definition. The process is very time consuming. But, it produces AMAZING discussions. So, I think it's worth it.
Writing Good Definitions Practice
Since I used images from this book, I cannot post these files to download. If you are interested in these resources, you may purchase the book here.
Unit 1 Vocabulary Foldables
My students repeated this process for our 10 vocabulary words for unit 1. They completed the "My Definition" section as homework. And, the entire next class period was spent creating group and class definitions. These were formatted as two book-type foldables that students glued in their interactive notebooks. I love that these foldables capture the process that students went through to write their own definition.
The Real Number System Graphic Organizer
We summarized the real number system using the graphic organizer that I created last year. It felt good to be able to reuse something from last year instead of creating something entirely new from scratch!
Exploring Rational and Irrational Numbers Foldable - Outside
This is a picture of a foldable that I created for my students to use as they walked through an exploration activity. This activity did not go as planned. I thought it would take one class period. After one class period, we had only started to scratch the surface. I ended up never finishing this activity with my students because I felt a need to move on. Still, I want to share this activity with you in the hopes that someone might be inspired to take this and make it better or at least give me insight on why it didn't quite work.
This activity was created with N-RN.3 (3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.) in mind. I had some blank sticker name badges in my desk. So, I took a Sharpie and wrote various rational and irrational numbers on each name badge. As students came in the classroom, they got to pick a random number from the pile. After a brief introduction to the TI-30 Scientific Calculator, students were instructed to partner up and fill out the first line of this chart. They would fill in their number, the classification of their number (rational or irrational), their partner's number, their partner's number's classification, and the operation of addition. Using the scientific calculator, students would add the two numbers together, record the result, and then determine whether the result was rational or irrational. Find a new partner. Repeat until the first five lines have been filled in.
Exploring Rational and Irrational Numbers Foldable - Inside
After completing five addition operations, we would have a class discussion. I was hoping that students would arrive at the fact (on their own) that a rational plus a rational is always rational, an irrational plus a rational is always irrational, etc. This went semi-well. Some of my classes struggled with this way more than others. None of my classes made it to division. One class barely made it through subtraction.
Rational and Irrational Number Name Badges
What I didn't account for was the sheer amount of time that it took for my students to partner up, do the calculations, and fill out this chart. This ended up feeling like an entire waste of a class period because it was a lot of work with very little to show for it. I ended up doing most of the talking and discovering during our discovery period which was frustrating.
I still want to put this idea out there, though. I think it's good to blog about the lessons that go well, the lessons that are just mediocre, and the lessons that don't go as planned.
And, yes, I'm the crazy teacher who wore a sticker around ALL DAY that read 3/8. I put it on during first hour because I had an odd number of students. So, I actually went through the activity with my students. I didn't take off my sticker because I figured I would just have to make another one to wear for third hour and sixth hour. Third hour, we had a fire drill. Of course, I got asked by everyone I saw why I was wearing a number on my shirt. The math teacher definitely came out in my answer as I explained that I wasn't just wearing any number, but a rational number. Conversation ensued about what a rational number was. Yes, I'm that teacher who takes every opportunity possible to teach my kiddos something mathematical!
Me with My Rational Number Badge
Later that morning, we had our tornado drill. As the students huddled in the stairwell, the teachers stood in the hall. The history teacher looked at me and said, "So, you're less than half?" Confused, I asked her to repeat the question. "So, you're less than half?" Still confused, I decided that I would just agree with the history teacher and go on down the road even though I had no idea what she was talking about. Luckily, she motioned toward my sticker, and I realized that she was referring to the fraction I was proudly wearing. The other teachers in my building are used to my crazy methods that I use to teach math by now, so they weren't that surprised.
What did surprise me was my statistics students. I have a class of 5 juniors who are taking statistics as their upper level math elective. Statistics is the only class that we offer our students above Algebra 2. These are our best and brightest math students who have chosen to take their math class at our school instead of our local Career Tech center. Of course, they had to ask about my lovely sticker that I was wearing. I explained why I was wearing it, and I told them about all the crazy conversations I had had that morning as a result. What do my students decide to do? They decide that they want to make their own stickers with numbers on them to wear around the school for the rest of the day. There are kids who love math, and I'm so thankful that I get the chance to teach them! (I'm thankful for my students who don't love math, too. But, that's another post for another day!)
Integer Operations Foldable - Outside
After using two-colored counters to derive the rules for adding, subtracting, multiplying, and dividing integers, I had my students create a four-door foldable to summarize the results of their findings.
Integer Operations Foldable - Inside
Inside, we wrote the rules for each type of problem and included several examples of each for students to refer to.
Order of Operations Graphic Organizer
The last topic covered in Unit 1 was the order of operations. I did not follow the approach laid out in the Kagan curriculum for this topic. I attempted to adapt last year's interactive notebook set-up for this topic. But, I added new activities and songs as an attempt to reach students that I felt like I had not reached during our work with integer operations. I'm hoping that I will have time this weekend to create a post of the resources I used from other blogs to teach each major topic for this unit. Next year, I'd love to be able to come back and find the links of all the resources I used the previous year.
Tuesday, August 27, 2013
I've been keeping my students super busy with their interactive notebooks. So far, my Algebra 2 students are loving the interactive notebooks. My Algebra 1 students HATE them. I still think they'll come around eventually...
Since I left both my camera and my interactive notebook at school, I'm going to share some photos of the interactive notebook pages that my Algebra 1 students created last year for their semester project. Yes, this post has been setting in my drafts since May...
I know that I am continually inspired by foldables and other creative creations I see online. It is my hope that one of these might serve useful as inspiration to you. Remember - these were created by students. Make sure you check these very carefully for errors before using with students | 677.169 | 1 |
4
4 Reading...... Use the time to read, reading is the basic policy (Hisham al-Talib) The book, like a vitamin that mempu enhance our intellectual and emotional (Clare Boothe Luce) I always imagined that Paradise is like a library (Jorge Luis Borges: 1899 to 1985) All civilized nations are guided by the literature (Voltaire) There are few worse crimes than burning books. One of them is not reading the book (joseph Brodsky) I have found my religion, nothing is more important than books. I looked at the library as a place of worship (Jean Paul Sartre)
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8 UNDERSTANDING By language: -Mathema = Knowledge -Mathematics: Studies on how to learn knowledge -Mathematics: The study and classification of various structures and patterns -The science of numbers and shapes as well Mathematics is "the language of science"
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10 THE ROLE OF MATHEMATICS IN ECONOMICS Explain the relationship between variables Explaining the change in the form of tables and diagrams Making the definitions and assumptions Deduce
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11 UNDERSTANDING THE BASIC MATH Symbols: expression of an idea (numbers, symbols) Constant: the symbol of a particular object Variables: symbols of objects in a certain set Parameters: constant that has not been rated yet Model: a structure or form of relationship as the embodiment of the minds on an issue | 677.169 | 1 |
Presentation on theme: "SPLIT UP SYLLABUS FOR THE YEAR 2012-2013 CLASS IX."— Presentation transcript:
3You are aware that the Central Board of Secondary Education has introduced Continuous and Comprehensive Evaluation [CCE] in its affiliated schools. The total school year is divided into first and second term. Each term contains two Formative Assessment and one Summative Assessment.Formative assessment will comprise of Projects, Assignments, Activities and Class Tests/Periodic tests for which board has already issued guidelines to schools.
4The summative assessment will comprise of theory paper as per the prescribed design of Question Paper. Whether there is sufficient knowledge about any area of study is usually known by conducting a summative test at the end of each term. This text covers a sample of the entire course and is divided into difficult and easy questions.The prescribed syllabus will be assessed using Formative and Summative assessments in the following manner.
17CH-3: Coordinate Geometry MONTHSEXAMCONTENTS (CHAPTER REF TEXT BOOK)SUGGESTED ACTIVITIESJULY-AUGTERM-IF A -IICH-6: Lines and anglesCH -7: TrianglesCH-12: Heron's FormulaCH-3: Coordinate GeometryTo find area of a triangle by paper cutting and pasting.To verify the inequalities in a triangle by pasting sticks of different lengths.To explore criteria for congruency of triangles using a set of triangular cut outs.Activities for FA –II:Seminar/ Pen paper test/Assignment.
18MONTHEXAMCONTENTSSEP-OCTSA IREVISION FOR SA-ISA I will be conducted and the syllabus is inclusive of FA I & FA II
19CH-9:Areas Of Parallelograms And Triangles MONTHSEXAMCONTENTS (CHAPTER REF TEXT BOOK)SUGGESTED ACTIVITIESNOV-DECTERM IIF A - IIICH-8: QuadrilateralsCH-9:Areas Of Parallelograms And TrianglesCH-4:Linear Equation In Two VariablesCH: 10 CirclesTo verify the mid-point theorem for a triangle by paper cutting and pastingTo Verify that the quad. Obtained by joining the mid point of a quad. is a parallelogram.Activities for FA-III:Power point Presentation/field survey.
20CH-13: Surface areas and Volumes MONTHEXAMCONTENTS (CHAPTER REF TEXT BOOK)SUGGESTED ACTIVITIESJAN-FEBTERM- IIF A - IVCH-14: StatisticsCH-15: ProbabilityCH-13: Surface areas and VolumesCH- 11: ConstructionTo explore similarities and differences in the properties with respect to diagonals of quadrilaterals.To verify experimentally that angle formed in the same segment of a circle are equalTo verify experimentally that the opposite angles of a cyclic quad are supplementary.Activities for FA – IV:Maths Quiz/Puzzles/crossword
21MONTHSEXAMCONTENTSMARCHTERM- IISA- IIREVISION FOR SA-IISA II to be conducted and the syllabus is inclusive of FA-III & FA-IVDECLARATION OF RESULTS
24APR –MAY . MONTHS EXAM CONTENTS (CHAPTER REF TEXT BOOK) SUGGESTED ACTIVITIESAPR –MAY .TERM – IFA - IChem :UNIT: MATTER – NATURE AND BEHAVIOURCh. 1 Matter in our surroundingsDefinition of matter, solid, liquid and gas; characteristics – shape, volume, density; change of state-melting(absorption of heat), freezing, evaporation(cooling by evaporation), condensation, sublimationTo show that matter is made up of very small particles.To show that gases can be compressed more easily than liquid.Lab Activity: To determine the melting point and the boiling point of waterTo study the effect of temp. on solids & liquids.
39PRESCRIBED BOOKS – History N. C. E. R PRESCRIBED BOOKS – History N.C.E.R.T textbook : India and the Contemporary World Part-1Note: According to the C.B.S.E guidelines two chapters each will be taken from Unit 1 and any one from Unit 2 of the textbook. One chapter will be taken from Unit3 of the textbook.Political Science: N.C.E.R.T textbook: Democratic Politics Part-1Economics: Economics.Geo/D.M. : (NCERT) Contemporary India-II / Together with safer IndiaMONTHSEXAMCONTENTS (CHAPTER REF TEXT BOOK)SUGGESTED ACTIVITIESAPR-MAYTERM- IFA-IHistory : Chapter 1- The French Revolution(Compulsory Chapter)Geo :Ch.1 India –Size and LocationCh.2 India- Physical FeaturesPrepare a wall paper on the rights demanded by the French people.Prepare a map of India on a chart and paste pictures of items that India exported to the far-east in ancient IndiaProject on India's trade relation with neighbouring countries
40APR - MAY MONTHS EXAM CONTENTS (CHAPTER REF TEXT BOOK) SUGGESTED ACTIVITIESAPR - MAYTERM- IFA-IPol .Sc: Ch 1 Democracy in the contemporary world.Chapter-2 What is democracy? Why democracy?Make a collage of struggle for democracy in MyanmarCartoon making depicting the non democratic practices in different countries of the world ..
41JUL - AUG MONTHS EXAM CONTENTS (CHAPTER REF TEXT BOOK) SUGGESTED ACTIVITIESJUL - AUGTERM -IFA-IIHist : Ch3 Nazism & the rise of HitlerGeo:Ch.2.India-Physical FeaturesCh.3 DrainageEco : Ch.1-The story of village PalampurCh.2 People as a ResourceMake posters depicting Nazi policies as anti-democratic.Clay Models of the Physical Features of India.Comparative analysis on the formation of Himalayas & the Peninsular regionMake a collage depicting how the rivers are getting polluted.Powerpoint Presn. On the state of Indian agri. & problems faced by Indian Farmers.Make banners publicizing the pulse polio prog.
42JUL- AUG MONTHS EXAM CONTENTS (CHAPTER REF TEXT BOOK) SUGGESTED ACTIVITIESJUL- AUGTERM- IFA-IIPol. Sc : Ch. 3- Constitutional designDiscussions on the main points of the constitution of USA, India & South Africa.Trace out the designing of the Indian Constitution.
43SEP - OCT NOV-DEC MONTHS EXAMS CONTENTS (CHAPTER REF TEXT BOOK) SUGGESTED ACTIVITIESSEP - OCTSA-IPol.Sc. :RevisionHist: RevisionEco : Ch.2:People as a ResourceSA I will be conducted and the syllabus is inclusive of FA I & FA IINOV-DECTERM- IIFA-IIIHist: Ch 4:Forest, Society & colonialism*(*Any one of the theme from Unit II to be taken)Geo : Ch.4:ClimateCh.5 Natural Vegetation and WildlifeFind out how the Govt is helping to conserve the forest & wildlife in your native place.Students talk on climate change & its effect on Indian Monsoon.Collect pictures,travel writing ,brochure etc to describe the natural resources of India.
45JAN- FEB MONTHS EXAMS CONTENTS (CHAPTER REF TEXT BOOK) SUGGESTED ACTIVITIESJAN- FEBFA-IVPol.Sc. :Ch 5. Working of InstitutionCh. 6 Democratic RightsEco: Ch 3: Poverty as a challengeCh 4 Food Security in IndiaConduct a Mock formation of govt to understand the working of Executive & Legislature in India.Field trip to Municipal Corp to find how city is administeredPresentation on violation of Human rights in different parts of the world.Illustrate with the help of graphical representation & map of India of the effectiveness of NREGA in states of India.
46JAN -FEB MAR MONTHS EXAMS CONTENTS (CHAPTER REF TEXT BOOK) SUGGESTED ACTIVITIESJAN -FEBFA-IVHist. Ch.8 Clothing a Social HistoryGeo: Ch.6 PopulationPrepare a presentation on clothing. Can include movies, slides, posters, newspaper, clippings etc.Conduct a class census interviewing five familiesVisit a ration shop & report on its functioning in the class.MARSA-IIREVISION FOR SA-IISA II will be conducted and the syllabus is inclusive of FA III and FA IVDECLARATION OF RESULTS | 677.169 | 1 |
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The 2014 2ND EDITION of this book is NOW AVAILABLE!! Search for ISBN 978-0983055846.
If you need to prepare for Florida's Postsecondary Education Readiness Math Test (PERT), this book is the perfect guide!
Prep for Success: Mastering Florida's Postsecondary Education Readiness Test Math teaches skills crucial to obtaining a good score on the PERT. The PERT is a placement test used by colleges, universities and other educational institutions in Florida to determine the appropriate course level of newly enrolled students.
This book teaches PERT math in the most effective way: by example. Follow along with the step-by-step solutions to the provided examples then enforce the concepts by practicing with exercises at the end of each chapter. The workbook section at the end of the book provides over 300 practice questions and full solutions.
This book covers all three math sub-sections of the Postsecondary Education Readiness Test: Arithmetic, Elementary Algebra and College Level Math. Use this book as a refresher course if you are returning to academic endeavors after postponing your education.
Specifications
Country
USA
Author
Stacey Francis
Binding
Spiral-bound
Brand
Brand: Track 2 Success, Inc.
EAN
9780983055822
Feature
Used Book in Good Condition
ISBN
0983055823
Label
Track 2 Success, Inc.
Manufacturer
Track 2 Success, Inc.
NumberOfPages
416
PublicationDate
2011
Publisher
Track 2 Success, Inc.
Studio
Track 2 Success | 677.169 | 1 |
You will learn calculus through this. It is not the best tutorial, but feel free to ask questions.
I think a site called particleadventure.com is very good and informative about the sub-atomic world. I was lost before I saw that site, now I'm good.
You should also consider books, from the library or store. Some libraries have Physics/Math textbooks. Photocopy the pages and study them at home. :)
Have FUN!
Note: Don't be afraid to pick up books in the Junior section. I found two books that didn't belong there. (Introduction to Relativity, and The Basic Elements of Chemistry) I don't know why it is there, but I highly doubt a junior would pick that up. | 677.169 | 1 |
Algebra 2 - Exponential and Logarithmic Equations
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Product Description
Lesson 8 – Exponential and Logarithmic Functions is a lesson for Algebra 2. This lesson covers how transform was one to the other, and solve equations using these forms. . In this bundle, you get the SmartBoard notes for the lesson, the SmartBoard filled in notes, the note guide for the students to go along with the lesson, and the worksheet for the lesson. | 677.169 | 1 |
Introduction to Computing Electives. Choose one or more.
Discover the world of computing, learn software design and development while solving puzzles with world renowned lecturer Richard Buckland. UNSW Computing 1 is presented by OpenLearning with original content derived from UNSW COMPUTING's first yearSemester 2Intermediate Programming Choose one of alternatives:
In the first unit, we will learn the mechanics of editing and compiling a simple program written in C++. We will begin with a discussion of the essential elements of C++ programming: variables, loops, expressions, functions, and string class. Nex…
If you are a student wanting to learn C programming, or an adult learner simply researching C programming courses, this free introductory course is for you.The C programming language is one of the most popular and widely used programming languages. I…
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If you invest in financial markets, you may want to predict the price of a stock in six months from now on the basis of company performance measures and other economic factors. As a college student, you may be interested in knowing the dependence of…
Statistics is about extracting meaning from data. In this class, we will introduce techniques for visualizing relationships in data and systematic techniques for understanding the relationships using mathematics.
We are surrounded by information, much of it numerical, and it is important to know how to make sense of it. Stat2x is an introduction to the fundamental concepts and methods of statistics, the science of drawing conclusions from data. The course is…
Introduction to statistics. We start with the basics of reading and interpretting data and then build into descriptive and inferential statistics that are typically covered in an introductory course on the subject. Overview of Khan Academy statistic…
Discrete Math Choose one of alternatives:
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Calculus can be thought of as the mathematics of CHANGE. Because everything in the world is changing, calculus helps us track those changes. Algebra, by contrast, can be thought of as dealing with a large set of numbers that are inherently CONSTANT.
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Study physics abroad in Europe -- virtually! Learn the basics of physics on location in Italy, the Netherlands and the UK, by answering some of the disciplineSemester 3
Principles of Management Choose one of alternatives:
Management refers to the organization and coordination of work to produce a desired result. A manager is a person who practices management by working with and through people in order to accomplish his or her organization's goals. When you thinkIntro to Networks Choose one of alternatives: w…
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What images come to mind when you think of the term professional? Do you picture an executive in a fancy suit strutting into a boardroom? Or, perhaps you envision a supervisor walking among cubicles and issuing orders to employees. While it is…
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Software Development II Electives. Choose one or more.
This course is about learning to program well: building programs that are elegant, well tested and easy to maintain. The course is designed for students with no programming experience at all. Nonetheless, former students who already knew how to prog…
Understanding how to approach programming problems and devise a solution is an essential skill for any Python developer. In this course, you'll learn new concepts, patterns, and methods that will expand your coding abilities from programming expert…
For anyone who would like to apply their technical skills to creative work ranging from video games to art installations to interactive music, and also for artists who would like to use programming in their artistic practice.
User dependency on the Internet increases every day; nowadays, everyday tasks like paying bills, communicating with others, and applying for jobs are all routinely carried out via the Internet. While the Internet represents a huge network, it is me…
Compilers Choose one of alternatives:
Because we have compiler programs, software developers often take the process of compilation for granted. However, as a software developer, you should cultivate a solid understanding of how compilers work in order to develop the strongest code poss…
This course will discuss the major ideas used today in the implementation of programming language compilers. You will learn how a program written in a high-level language designed for humans is systematically translated into a program written in low-…
Parallel Programming Choose one of alternatives:
This course introduces concepts, languages, techniques, and patterns for programming heterogeneous, massively parallel processors. Its contents and structure have been significantly revised based on the experience gained from its initial offering in…
Learn the fundamentals of parallel computing with the GPU and the CUDA programming environment! In this class, you'll learn about parallel programming by coding a series of image processing algorithms, such as you might find in Photoshop or Instagram…
For best oracle apps technical training with job assistance... for all graduates ... in India , U.S.A , U.K ... many more countries....
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Missing courses in Algebra and especially linear algebra.
A math course on logic wouldn't go astray.
Needs to have courses on the analysis of algorithms.
If the compiler course is taken then there needs to be a course on automata taken prior to the compiler course.
Each semester should have at least one course which has programming in it from a mix of languages. Why no Java? Coursera's Scala course is a must for the later semesters.
No operating system theory/design course.
Later options should include electives in Robotics, Combitorial Analysis, NoSQL databases (eg MongoDB), DataScience
Way too many non-math/comp-sci electives. Emphasis on US History electives does not sit well with the global nature of MOOCs.
Missing courses in Algebra and especially linear algebra.
A math course on logic wouldn't go astray.
Needs to have courses on the analysis of algorithms.
If the compiler course is taken then there needs to be a course on automata taken prior to the compiler course.
Each semester should have at least one course which has programming in it from a mix of languages. Why no Java? Coursera's Scala course is a must for the later semesters. | 677.169 | 1 |
Gauss's law for electric fields, Gauss's law for magnetic fields, Faraday's law, and the Ampere-Vectors and tensors are among the most powerful problem-solving tools available, with applications ranging from mechanics and electromagnetics to general relativity. Understanding the nature and application of vectors and tensors is critically important to students of physics and engineering. Adopting the same approach used in his highly popular A Student's Guide to Maxwell's Equations, Fleisch explains vectors and tensors in plain language. Written for undergraduate and beginning graduate students, the book provides a thorough grounding in vectors and vector calculus before transitioning through contra and covariant components to tensors and their applications. Matrices and their algebra are reviewed on the book's supporting website, which also features interactive solutions to every problem in the text where students can work through a series of hints or choose to see the entire solution at once. Audio podcasts give students the opportunity to hear important concepts in the book explained by the author.
DownloadRead Online
A concise but rigorous treatment of variational techniques, focussing primarily on Lagrangian and Hamiltonian systems, this book is ideal for physics, engineering and mathematics students. The book begins by applying Lagrange's equations to a number of mechanical systems. It introduces the concepts of generalized coordinates and generalized momentum. Following this the book turns to the calculus of variations to derive the Euler–Lagrange equations. It introduces Hamilton's principle and uses this throughout the book to derive further results. The Hamiltonian, Hamilton's equations, canonical transformations, Poisson brackets and Hamilton–Jacobi theory are considered next. The book concludes by discussing continuous Lagrangians and Hamiltonians and how they are related to field theory. Written in clear, simple language and featuring numerous worked examples and exercises to help students master the material, this book is a valuable supplement to courses in mechanics.
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The 10th edition of Elementary Differential Equations and Boundary Value Problems, like its predecessors, is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be quite theoretical, sometimes intensely practical, and often somewhere in between. The authors have sought to combine a sound and accurate exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. While the general structure of the book remains unchanged, some notable changes have been made to improve the clarity and readability of basic material about differential equations and their applications. In addition to expanded explanations, the 10th edition includes new problems, updated figures and examples to help motivate students. The book is written primarily for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. WileyPLUS sold separately from text.
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Striving to explore the subject in as simple a manner as possible, this book helps readers understand the elusive concept of entropy. Innovative aspects of the book include the construction of statistical entropy from desired properties, the derivation of the entropy of classical systems from purely classical assumptions, and a statistical thermodynamics approach to the ideal Fermi and ideal Bose gases. Derivations are worked through step-by-step and important applications are highlighted in over 20 worked examples. Around 50 end-of-chapter exercises test readers' understanding. The book also features a glossary giving definitions for all essential terms, a time line showing important developments, and list of books for further study. It is an ideal supplement to undergraduate courses in physics, engineering, chemistry and mathematics. | 677.169 | 1 |
2
This Module concerns the CIRCULAR FUNCTIONS, so named because they were originally derived from the circle. One way to think of these functions is of functions where the variable is an ANGLE. It turns out that these functions are extremely common - and are good approximations for many phenomena: for example waves, orbits, swings, pendulums and springs. Their study is rewarding, and provides many more applications than the familiar right- angled triangle problems of school days.
3
Oh yes,..... this is what you might have called TRIGONOMETRY in the past.
4
The Module finishes with a few lectures introducing you to two fascinating and very important types of mathematical objects. The first are MATRICES. These are simply rows and columns of numbers, but they can be used to describe whole sets of equations, or geometrical transformations like reflections. The second type of object are COMPLEX NUMBERS. These are the numbers you get if you pretend that it is possible to take the square root of a negative number. They have many, many important uses in mathematics and its applications.
9
Angles as Variables All kinds of "objects" can be variables. Usually we think of variables as numbers: f(x) = 3x 2 – 2x + 1 f(-2) = 12 + 4 + 1 = 17 But last lecture, for example, we had another function as a variable: f(x) = g(h(x)), e.g. f(x) = 3e 2t^3
10
Angles as Variables We can make up other kinds of functions: E.g. a function which determines the distance of a point from the origin: D(3,4) = √(3 2 + 4 2 ) So the variable is a point (3,4)
11
Angles as Variables And we can make up functions where the variable is an angle: E.g. Full(ø) = the number of angle ø's which are needed to make a full turn. E.g. Ch(ø) = the length of the chord of a circle of radius 1, which is generated by an angle ø at the centre.
14
Degrees, Mils, Radians Degrees are a well-known unit of angle. There are 90° in a quarter turn Grads are a surveyors measure, based on 100grads in a quarter turn. Mils are an old military measure, used for artillery calculations.
15
Other Measures We can make up other angle measures: e.g. let us define a "hand" as the angle subtended by the width of our hand at arm's length. How many degrees in a hand?
16
Radians A mathematical measure of angle is defined using the radius of a circle.
32
Lecture 4/1 – Summary There are many functions where the variable can be regarded as an ANGLE. One way of measuring an angle is that derived from the radius of the circle. This is called RADIAN measure. From the UNIT CIRCLE, we can see that the SINE of an angle is the height of a triangle drawn inside the circle. Sine(ø) then becomes a function depending on the size of the angle ø.
33
445.102 Lecture 4/1 Going Round Again Before the next lecture........ Go over Lecture 4/1 in your notes Do the Post-Lecture exercise Do the Preliminary Exercise See you tomorrow........ | 677.169 | 1 |
Problem solving and mathematical modeling are introduced early and reinforced throughout, providing students with a solid foundation in the principles of mathematical thinking. Comprehensive and evenly paced, the book provides complete coverage of the function concept, and integrates a significant amount of graphing calculator material to help students develop insight into mathematical ideas. | 677.169 | 1 |
This master may only be reproduced by the
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publisher prohibits the loaning or onselling of this
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FOREWORD
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Books A–G of Problem-solving in mathematics have been developed to provide a rich resource for teachers
of pupils from the early years to the end of primary school and into secondary school. The series of problems,
discussions of ways to understand what is being asked and means of obtaining solutions have been built up to
improve the problem-solving performance and persistence of all pupils. It is a fundamental belief of the authors
that it is critical that pupils and teachers engage with a few complex problems over an extended period rather than
spend a short time on many straightforward 'problems' or exercises. In particular, it is essential to allow pupils
time to review and discuss what is required in the problem-solving process before moving to another and different
problem. This book includes extensive ideas for extending problems and solution strategies to assist teachers in
implementing this vital aspect of mathematics in their classrooms. Also, the problems have been constructed and
selected over many years' experience with pupils at all levels of mathematical talent and persistence, as well as in
discussions with teachers in classrooms and professional learning and university settings.
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Problem-solving does not come easily to most people,
so learners need many experiences engaging with
problems if they are to develop this crucial ability. As
they grapple with problem meaning and find solutions,
pupils will learn a great deal about mathematics
and mathematical reasoning—for instance, how to
organise information to uncover meanings and allow
connections among the various facets of a problem
to become more apparent, leading to a focus on
organising what needs to be done rather than simply
looking to apply one or more strategies. In turn, this
extended thinking will help pupils make informed
choices about events that affect their lives and to
interpret and respond to the decisions made by others
at school, in everyday life and in further study.
Pupil and teacher pages
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The pupil pages present problems chosen with a
particular problem-solving focus and draw on a range
of mathematical understandings and processes.
For each set of related problems, teacher notes and
discussion are provided, as well as indications of
how particular problems can be examined and solved.
Answers to the more straightforward problems and
detailed solutions to the more complex problems
Prim-Ed Publishing®
ensure appropriate explanations, and suggest
ways in which problems can be extended. Related
problems occur on one or more pages that extend the
problem's ideas, the solution processes and pupils'
understanding of the range of ways to come to terms
with what the problems are asking.
At the top of each teacher page, a statement highlights
the particular thinking that the problems will demand,
together with an indication of the mathematics that
might be needed and a list of materials that can be
used in seeking a solution. A particular focus for the
page or set of three pages of problems then expands
on these aspects. Each book is organised so that when
a problem requires complicated strategic thinking, two
or three problems occur on one page (supported by a
teacher page with detailed discussion) to encourage
pupils to find a solution together with a range of
means that can be followed. More often, problems
are grouped as a series of three interrelated pages
where the level of complexity gradually increases,
while the associated teacher page examines one or
two of the problems in depth and highlights how the
other problems might be solved in a similar manner.
Problem-solving in mathematics
iii
FOREWORD
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the various year levels, although problem-solving both
challenges at the point of the mathematics that is being
learned and provides insights and motivation for what
might be learned next. For example, the computation
required gradually builds from additive thinking, using
addition and subtraction separately and together,
to multiplicative thinking, where multiplication and
division are connected conceptions. More complex
interactions of these operations build up over the
series as the operations are used to both come to terms
with problems' meanings and to achieve solutions.
Similarly, two-dimensional geometry is used at first
but extended to more complex uses over the range
of problems, then joined by interaction with threedimensional ideas. Measurement, including chance
and data, also extends over the series from length to
perimeter, and from area to surface area and volume,
drawing on the relationships among these concepts to
organise solutions as well as give an understanding
of the metric system. Time concepts range from
interpreting timetables using 12-hour and 24-hour
clocks, while investigations related to mass rely on
both the concept itself and practical measurements.
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Each teacher page concludes with two further aspects
critical to the successful teaching of problem-solving.
A section on likely difficulties points to reasoning and
content inadequacies that experience has shown may
well impede pupils' success. In this way, teachers can
be on the lookout for difficulties and be prepared to
guide pupils past these potential pitfalls. The final
section suggests extensions to the problems to enable
teachers to provide several related experiences with
problems of these kinds in order to build a rich array
of experiences with particular solution methods; for
example, the numbers, shapes or measurements in
the original problems might change but leave the
means to a solution essentially the same, or the
context may change while the numbers, shapes or
measurements remain the same. Then numbers,
shapes or measurements and the context could be
changed to see how the pupils handle situations that
appear different but are essentially the same as those
already met and solved.
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Other suggestions ask pupils to make and pose their
own problems, investigate and present background
to the problems or topics to the class, or consider
solutions at a more general level (possibly involving
verbal descriptions and eventually pictorial or symbolic
arguments). In this way, not only are pupils' ways of
thinking extended but the problems written on one
page are used to produce several more problems that
utilise the same approach.
Mathematics and language
The difficulty of the mathematics gradually increases
over the series, largely in line with what is taught at
iv
Problem-solving in mathematics
The language in which the problems are expressed is
relatively straightforward, although this too increases
in complexity and length of expression across the
books in terms of both the context in which the
problems are set and the mathematical content that is
required. It will always be a challenge for some pupils
to 'unpack' the meaning from a worded problem,
particularly as the problems' context, information and
meanings expand. This ability is fundamental to the
nature of mathematical problem-solving and needs to
be built up with time and experiences rather than be
Prim-Ed Publishing®
FOREWORD
Analyse
Try
the problem
an approach
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Explore
means to a solution
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The careful, gradual development of an ability to
analyse problems for meaning, organising information
to make it meaningful and to make the connections
among the problems more meaningful in order to
suggest a way forward to a solution is fundamental
to the approach taken with this series, from the first
book to the last. At first, materials are used explicitly
to aid these meanings and connections; however, in
time they give way to diagrams, tables and symbols
as understanding and experience of solving complex,
engaging problems increases. As the problem forms
expand, the range of methods to solve problems
is carefully extended, not only to allow pupils to
Prim-Ed Publishing®
e
Not only is this model for the problem-solving process
helpful in solving problems, it also provides a basis
for pupils to discuss their progress and solutions and
determine whether or not they have fully answered
a question. At the same time, it guides teachers'
questions of pupils and provides a means of seeing
underlying mathematical difficulties and ways in
which problems can be adapted to suit particular
needs and extensions. Above all, it provides a common
framework for discussions between a teacher and
group or whole class to focus on the problem-solving
process rather than simply on the solution of particular
problems. Indeed, as Alan Schoenfeld, in Steen L. (Ed)
Mathematics and democracy (2001), states so well, in
problem-solving:
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An approach to solving problems
successfully solve the many types of problems, but also
to give them a repertoire of solution processes that
they can consider and draw on when new situations
are encountered. In turn, this allows them to explore
one or another of these approaches to see whether
each might furnish a likely result. In this way, when
they try a particular method to solve a new problem,
experience and analysis of the particular situation
assists them in developing a full solution.
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diminished or left out of the problems' situations. One
reason for the suggestion that pupils work in groups
is to allow them to share and assist each other with
the tasks of discerning meanings and ways to tackle
the ideas in complex problems through discussion,
rather than simply leaping into the first ideas that
come to mind (leaving the full extent of the problem
unrealised).
getting the answer is only the beginning rather than
the end … an ability to communicate thinking is
equally important.
We wish all teachers and pupils who use these books
success in fostering engagement with problem-solving
and building a greater capacity to come to terms with
and solve mathematical problems at all levels.
Problem-solving lies at the heart of mathematics.
New mathematical concepts and processes have
always grown out of problem situations and pupils'
problem-solving capabilities develop from the very
beginning of mathematics learning. A need to solve
a problem can motivate pupils to acquire new ways
of thinking as well as come to terms with concepts
and processes that might not have been adequately
learned when first introduced. Even those who can
calculate efficiently and accurately are ill prepared for
a world where new and adaptable ways of thinking
are essential if they are unable to identify which
information or processes are needed.
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By learning problem-solving in
mathematics, pupils should acquire ways
of thinking, habits of persistence and
curiosity, and confidence in unfamiliar
situations that will serve them well
outside the mathematics classroom. In
everyday life and in the workplace, being
a good problem solver can lead to great
advantages.
Well-chosen problems encourage deeper exploration
of mathematical ideas, build persistence and highlight
the need to understand thinking strategies, properties
and relationships. They also reveal the central role of
sense making in mathematical thinking—not only to
evaluate the need for assessing the reasonableness
of an answer or solution, but also the need to consider
the interrelationships among the information provided
with a problem situation. This may take the form of
number sense, allowing numbers to be represented
in various ways and operations to be interconnected;
through spatial sense that allows the visualisation of
a problem in both its parts and whole; to a sense of
measurement across length, area, volume and chance
and data.
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Problem-solving and mathematical thinking
A problem is a task or situation for which there is
no immediate or obvious solution, so that problemsolving refers to the processes used when engaging
with this task. When problem-solving, pupils engage
with situations for which a solution strategy is not
immediately obvious, drawing on their understanding
of concepts and processes they have already met, and
will often develop new understandings and ways of
thinking as they move towards a solution. It follows
that a task that is a problem for one pupil may not
be a problem for another and that a situation that
is a problem at one level will only be an exercise or
routine application of a known means to a solution at
a later time.
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On the other hand, pupils who can analyse the
meaning of problems, explore means to a solution
and carry out a plan to solve mathematical problems
have acquired deeper and more useful knowledge
than simply being able to complete calculations,
name shapes, use formulas to make measurements or
determine measures of chance and data. It is critical
that mathematics teaching focuses on enabling all
pupils to become both able and willing to engage with
and solve mathematical problems.
Problem-solving
For a pupil aged 5–6 years, sorting out the information
about being on the lily pad and being in the water
may take some consideration and require counters
to represent the numbers and find the answer. For
Prim-Ed Publishing®
Problem-solving in mathematics
vii
INTRODUCTION
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However, many pupils feel inadequate when they
encounter problem-solving questions. They seem to
have no idea of how to go about finding a solution and
are unable to draw on the competencies they have
learned in number, space and measurement. Often
these difficulties stem from underdeveloped concepts
for the operations, spatial thinking and measurement
processes. They may also involve an underdeveloped
capacity to read problems for meaning and a tendency
to be led astray by the wording or numbers in a problem
situation. Their approach may then simply be to try a
series of guesses or calculations rather than consider
using a diagram or materials to come to terms with
what the problem is asking and using a systematic
approach to organise the information given and
required in the task. It is this ability to analyse problems
that is the key to problem-solving, enabling decisions
to be made about which mathematical processes to
use, which information is needed and which ways of
proceeding are likely to lead to a solution.
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As the world in which we live becomes ever more
complex, the level of mathematical thinking and
problem-solving needed in life and in the workplace
has increased considerably. Those who understand
and can use the mathematics they have learned will
have opportunities opened to them that those who do
not develop these ways of thinking will not. To enable
pupils to thrive in this changing world, attitudes
and ways of knowing that enable them to deal with
new or unfamiliar tasks are now as essential as the
procedures that have always been used to handle
familiar operations readily and efficiently. Such
an attitude needs to develop from the beginning of
mathematics learning as pupils form beliefs about
meaning, the notion of taking control over the activities
they engage with and the results they obtain, and as
they build an inclination to try different approaches. In
other words, pupils need to see mathematics as a way
of thinking rather than a means of providing answers
to be judged right or wrong by a teacher, textbook
or some other external authority. They must be led
to focus on means of solving problems rather than
on particular answers so that they understand the
need to determine the meaning of a problem before
beginning to work on a solution.
of him. When another 6 cars passed him, there were
now 9 ahead of him. If he is to win, he needs to pass
all 9 cars. The 4 and 6 implied in the problem were not
used at all! Rather, a diagram or the use of materials
is needed first to interpret the situation and then see
how a solution can be obtained.
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children in the middle primary years, understanding
of the addition concept and knowledge of the addition
facts would lead them immediately to think about the
sum of 3 and 4 and come up with the solution of 7
frogs.
In a car race, Jordan started in fourth place. During
the race, he was passed by six cars. How many
cars does he need to pass to win the race?
In order to solve this problem, it is not enough to
simply use the numbers that are given. Rather, an
analysis of the race situation is needed first to see
that when Jordan started, there were 3 cars ahead
viii
Problem-solving in mathematics
Making sense in mathematics
Making sense of the mathematics being developed
and used must be seen as the central concern of
learning. This is important, not only in coming to
terms with problems and means to solutions, but also
in terms of bringing meaning, representations and
relationships in mathematical ideas to the forefront of
thinking about and dealing with mathematics. Making
sensible interpretations of any results and determining
which of several possibilities is more or equally likely
is critical in problem-solving.
Number sense, which involves being able to
work with numbers comfortably and competently,
is important in many aspects of problem-solving,
in making judgments, interpreting information and
communicating ways of thinking. It is based on a
full understanding of numeration concepts such
Prim-Ed Publishing速
INTRODUCTION
Number sense requires:
• understanding relationships among numbers
• a capacity to calculate and estimate
mentally
• fluent processes for larger numbers and
adaptive use of calculators
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• an inclination to use understanding and
facility with numeration and computation in
flexible ways.
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The following problem highlights the importance of
these understandings.
There were 317 people at the New Year's Eve
party on 31 December. If each table could seat 5
couples, how many tables were needed?
Reading the problem carefully shows that each table
seats five couples or 10 people. At first glance, this
problem might be solved using division; however, this
would result in a decimal fraction, which is not useful
in dealing with people seated at tables:
10 317 is 31.7
Prim-Ed Publishing®
3 1 7
3 1
tens
7
ones
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This provides for all the people at the party and
analysis of the number 317 shows that there have
to be at least 32 tables for everyone to have a seat
and allow partygoers to move around and sit with
others during the evening. Understanding how to
rename a number has provided a direct solution
without any need for computation. It highlights how
coming to terms with a problem and integrating this
with number sense provides a means of solving the
problem more directly and allows an appreciation of
what the solution might mean.
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• appreciating the relative size of numbers
In contrast, a full understanding of numbers allows
317 to be renamed as 31 tens and 7 ones:
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as zero, place value and the renaming of numbers
in equivalent forms, so that 207 can be seen as 20
tens and 7 ones as well as 2 hundreds and 7 ones (or
that 52, 2.5 and 2 12 are all names for the same fraction
amount). Automatic, accurate access to basic facts
also underpins number sense, not as an end in itself,
but rather as a means of combining with numeration
concepts to allow manageable mental strategies and
fluent processes for larger numbers. Well-understood
concepts for the operations are essential in allowing
relationships within a problem to be revealed and
taken into account when framing a solution.
Spatial sense is equally important, as information
is frequently presented in visual formats that need
to be interpreted and processed, while the use of
diagrams is often essential in developing conceptual
understanding across all aspects of mathematics.
Using diagrams, placing information in tables or
depicting a systematic way of dealing with the various
possibilities in a problem assist in visualising what is
happening. It can be a very powerful tool in coming
to terms with the information in a problem, and it
provides insight into ways to proceed to a solution.
Spatial sense involves:
• a capacity to visualise shapes and their
properties
• determining relationships among shapes
and their properties
• linking two-dimensional and threedimensional representations
• presenting and interpreting information in
tables and lists
• an inclination to use diagrams and models
to visualise problem situations and
applications in flexible ways.
The following problem shows how these
understandings can be used.
Problem-solving in mathematics
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INTRODUCTION
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Measurement sense is dependent on both number
sense and spatial sense, since attributes that are
one‑, two- or three-dimensional are quantified to
provide both exact and approximate measures and
allow comparison. Many measurements use aspects
of space (length, area, volume), while others use
numbers on a scale (time, mass, temperature).
Money can be viewed as a measure of value and
uses numbers more directly, while practical activities
such as map reading and determining angles require
a sense of direction as well as gauging measurement.
The coordination of the thinking for number and
space, along with an understanding of how the metric
system builds on place value, zero and renaming, is
critical in both building measurement understanding
and using it to come to terms with and solve many
practical problems and applications.
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Reading the problem carefully shows that only two
spaces in the box can be used each time and that no
use of the spaces can be duplicated. A systematic
approach, placing one chocolate in a fixed position
and varying the other spaces will provide a solution;
however, care will be needed to see that the same
placement has not already occurred:
How many cubes are needed to make this shape?
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Cathy has 2 chocolates and 1 box. In how many
different ways can she place the chocolates in the
box?
There are six possible arrangements. The placement
of objects on the diagram has provided a solution,
highlighting how coming to terms with a problem and
integrating this with spatial sense allows a systematic
analysis of all the possibilities.
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Similar thinking is used with arrangements of twodimensional and three-dimensional shapes and in
visualising how they can fit together or be taken
apart.
Measurement sense includes:
• understanding how numeration and
computation underpin measurement
• extending relationships from number
understanding to the metric system
• appreciating the relative size of measurements
• a capacity to use calculators, mental or
written processes for exact and approximate
calculations
• an inclination to use understanding and facility
with measurements in flexible ways.
The following problem shows how these
understandings can be used.
Which of these shapes can be made using all of
the tangram pieces?
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Problem-solving in mathematics
Prim-Ed Publishing®
INTRODUCTION
Data sense involves:
• a capacity to use calculators or mental
and written processes for exact and
approximate calculations
• presenting and interpreting data in tables
and graphs
• an inclination to use understanding and
facility with number combinations and
arrangements in flexible ways.
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Carefully reading the problem shows that the snail
will travel 45 cm as it moves along each side of
the square. In order to come to terms with what is
needed, 3 m 15 cm needs to be renamed as 315 cm.
The distances the snail travels along each side can
then be totalled until 315 cm is reached. It can also be
inferred that it will travel along some sides more than
once as the distance around the outside of the square
is 180 cm. At this point, the snail will be back at A.
Travelling a further 45 cm will take it to B, a distance
of 225 cm. At C it will have travelled 270 cm and it will
have travelled 315 cm (or 3 m 15 cm) when it reaches
D for the second time.
• appreciating the relative likelihood of
outcomes
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At which corner
will it stop?
• understanding how numeration and
computation underpin the analysis of data
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A snail crawls
3 m 15 cm around a
square garden.
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By using an understanding of the problem situation,
a diagram has been integrated with a knowledge of
metres and centimetres and a capacity to calculate
mentally using addition and multiplication to provide
an appropriate solution. Both spatial sense and number
sense have been used to understand the problem and
suggest a means to a solution.
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Data sense is an outgrowth of measurement sense
and refers to an understanding of the way number
sense, spatial sense and a sense of measurement
work together to deal with situations where patterns
need to be discerned among data or when likely
outcomes need to be analysed. This can occur among
frequencies in data or possibilities in chance.
Prim-Ed Publishing®
The following problem shows how these
understandings can be used.
You are allowed 3
scoops of ice-cream:
1 chocolate, 1 vanilla
and 1 strawberry.
How many different
ways can the scoops
be placed on a
cone?
There are six possibilities for placing the scoops of icecream on a cone. Systematically treating the possible
placements one at a time highlights how the use of a
diagram can account for all possible arrangements.
This problem also shows how patterning is another
aspect of sense-making in mathematics. Often a
problem calls on discerning a pattern in the placement
of materials, the numbers involved in the situation or
the possible arrangements of data or outcomes to
determine a likely solution. Being able to see patterns
is also very helpful in getting an immediate solution or
understanding whether or not a solution is complete.
Allied to patterning are notions of symmetry, repetition
and extending ideas to more general cases. All of
these aspects of mathematical sense-making are
critical to developing the thinking on which problemsolving depends, as well as solving problems per se.
Problem-solving in mathematics
xi
INTRODUCTION
As more experience in solving problems is gained, an
ability to see patterns in what is occurring will also
allow solutions to be obtained more directly and help
in seeing the relationship between a new problem and
one that has been solved previously. It is this ability to
relate problem types, even when the context appears
to be quite different, that often distinguishes a good
problem solver from one who is more hesitant.
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On Saturday, Peta went to the shopping
centre to buy a new outfit to wear at her
friend's birthday party. She spent half of her
money on a dress and then one-third of what
she had left on a pair of sandals. After her
purchases, she had £60.00 left in her purse.
How much money did she have to start with?
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While the teaching of problem-solving has often
centred on the use of particular strategies that could
apply to various classes of problems, many pupils are
unable to access and use these strategies to solve
problems outside of the teaching situations in which
they were introduced. Rather than acquire a process
for solving problems, they may attempt to memorise
a set of procedures and view mathematics as a set of
learned rules where success follows the use of the
right procedure to the numbers given in the problem.
Any use of strategies may be based on familiarity,
personal preference or recent exposure rather than
through a consideration of the problem to be solved.
A pupil may even feel it is sufficient to have only one
strategy and that the strategy should work all of the
time—and if it doesn't, then the problem can't be
solved.
e
Building a problem-solving process
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in
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By reading the problem carefully, it can be determined
that Peta had an original amount of money to spend.
She spent some on a dress and some on shoes and
then had £60.00 left. All of the information required
to solve the problem is available and no further
information is needed. The question at the end asks
how much money did she start with, but really the
problem is how much did she spend on the dress and
then on the sandals.
Vi
In contrast, observation of successful problem-solvers
shows that their success depends more on an analysis
of the problem itself—what is being asked, what
information might be used, what answer might be
likely and so on—so that a particular approach is used
only after the intent of the problem is determined.
Establishing the meaning of the problem before any
plan is drawn up or work on a solution begins is critical.
Pupils need to see that discussion about the problem's
meaning, and the ways of obtaining a solution, must
take precedence over a focus on the answer. Using
collaborative groups when problem-solving, rather
than tasks assigned individually, is an approach that
helps to develop this disposition.
Looking at a problem and working through what is
needed to solve it will shed light on the problemsolving process.
xii
Problem-solving in mathematics
The discussion of this problem has served to identify
the key element within the problem-solving process;
it is necessary to analyse the problem to unfold its
meanings and discover what needs to be considered.
What the problem is asking is rarely found in the
question in the problem statement. Instead, it is
necessary to look below the surface level of the
problem and come to terms with the problem's
structure. Reading the problem aloud, thinking of
previous problems and other similar problems,
selecting important information from the problem
that may be useful, and discussion of the problem's
meaning are all essential.
The next step is to explore possible ways to solve the
problem. If the analysis stage has been completed,
Prim-Ed Publishing®
INTRODUCTION
Ways that may come to mind during the analysis
include:
She spent half of her money on a dress.
She then spent one-third of what she had left on
sandals, which has minimised and simplified the
calculations.
Sa
m
• Materials – Base 10 materials could be used to
represent the money spent and to help the pupil
work backwards through the problem from when
Peta had £60.00 left.
Total amount available to spend:
e
It is here that strategies, and how they might be
useful to solving a problem, can arise. However,
most problems can be solved in a variety of ways,
using different approaches, and pupils need to be
encouraged to select a method that makes sense and
appears achievable.
Another way to solve the problem is with a diagram.
If we use a rectangle to represent how much money
Peta took with her, we can show by shading how much
she spent on a dress and sandals:
pl
then ways in which the problem might be solved will
emerge.
• Try and adjust – Select an amount that Peta
might have taken shopping, try it in the context of
the question, examine the resulting amounts, and
then adjust them, if necessary, until £60.00 is the
result.
ew
in
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• Backtrack using the numbers – The sandals
were one-third of what was left after the dress,
so the £60.00 would be two-thirds of what was
left. Together, these two amounts would match
the cost of the dress.
• Use a diagram to represent the information in
the problem.
• Think of a similar problem – For example, it
is like the car race problem in that the relative
portions (places) are known and the final result
(money left, winning position) are given.
Vi
Now one of the possible means to a solution can be
selected to try. Backtracking shows that £60 was twothirds of what she had left, so the sandals (which are
one-third of what she had left) must have cost £30.
Together, these are half of what Peta took, which is
also the cost of the dress. As the dress cost £90, Peta
took £180 to spend.
Materials could also have been used with which to
work backwards: 6 tens represent the £60 left, so the
sandals would cost 3 tens and the dress 9 tens—she
took 18 tens or £180 shopping.
Prim-Ed Publishing®
At this point she had £60 left, so the twounshaded parts must be worth £60 or £30 per
part—which has again minimised and simplified
the calculations.
£30
£30
Each of the six equal parts represents £30, so
Peta took £180 to spend.
Having tried an idea, an answer needs to be analysed
in the light of the problem in case another solution is
required. It is essential to compare an answer to the
original analysis of the problem to determine whether
the solution obtained is reasonable and answers the
problem. It will also raise the question as to whether
other answers exist and even whether there might be
other solution strategies. In this way the process is
cyclic and should the answer be unreasonable, then
the process would need to begin again.
We believe that Peta took £180 to shop with. She
spent half (or £90) on a dress, leaving £90. She spent
one-third of the £90 on sandals (£30), leaving £60.
Looking again at the problem, we see that this is
correct and the diagram has provided a direct means
Problem-solving in mathematics
xiii
INTRODUCTION
Analyse
the problem
Try
a solution strategy
Explore
means to a solution
e
Managing a problem-solving programme
Teaching problem-solving differs from many other
aspects of mathematics in that collaborative work
can be more productive than individual work. Pupils
who may be tempted to quickly give up when working
on their own can be encouraged to see ways of
proceeding when discussing a problem in a group;
therefore building greater confidence in their capacity
to solve problems and learning the value of persisting
with a problem in order to tease out what is required.
What is discussed with their peers is more likely to
be recalled when other problems are met, while the
observations made in the group increase the range of
approaches that a pupil can access. Thus, time has
to be allowed for discussion and exploration rather
than insisting that pupils spend 'time on task' as for
routine activities.
Sa
m
Thinking about the various ways this problem
was solved highlights the key elements within the
problem-solving process. When starting the process,
it is necessary to analyse the problem to unfold its
layers, discover its structure and understand what
the problem is really asking. Next, all possible ways
to solve the problem are explored before one, or a
combination of ways, are selected to try. Finally,
once something is tried, it is important to check
the solution in relation to the problem to see if the
solution is reasonable. This process highlights the
cyclic nature of problem-solving and brings to the fore
the importance of understanding the problem (and
its structure) before proceeding. This process can be
summarised as:
solving another problem at a later stage. It allows the
thinking to be carried over to the new situation in a
way that simply trying to think of the strategy used
often fails to reveal. Analysing problems in this way
also highlights that a problem is not solved until the
answer obtained can be justified. Learning to reflect
on the whole process leads to the development of a
deeper understanding of problem-solving, and time
must be allowed for reflection and discussion to fully
build mathematical thinking.
pl
to the solution that has minimised and simplified the
calculations.
A plan to manage
problem-solving
Vi
ew
in
g
This model for problem-solving provides pupils with a
means of talking about the steps they take whenever
they have a problem to solve: Discussing how they
initially analysed the problem, explored various ways
that might provide a solution, and then tried one or
more possible solution paths to obtain a solution—
which they then analysed for completeness and
sense-making—reinforces the very methods that will
give them success on future problems. This process
brings to the fore the importance of understanding the
problem and its structure before proceeding.
Further, returning to an analysis of any answers
and solution strategies highlights the importance
of reflecting on what has been done. Taking time to
reflect on any plans drawn up, processes followed and
strategies used brings out the significance of coming
to terms with the nature of the problem, as well as
the value and applicability of particular approaches
that might be used with other problems. Thinking of
how a related problem was solved is often the key to
xiv
Problem-solving in mathematics
Correct answers that fully solve a problem are
always important, but developing a capacity to use
an effective problem-solving process needs to be the
highest priority. A pupil who has an answer should be
encouraged to discuss his or her solution with others
who believe they have a solution, rather than tell his
or her answer to another pupil or simply move on to
another problem. In particular, explaining to others
why he or she believes an
answer is reasonable, as well
as why it provides a solution,
gets other pupils to focus
on the entire problemsolving process rather
than just quickly
getting an answer.
Prim-Ed Publishing®
INTRODUCTION
pl
e
too few may cause pupils to become frustrated with
the task and think that it is beyond them. Pupils need
to experience the challenge of problem-solving and
gain pleasure from working through the process
that leads to a full solution. Taking time to listen to
pupils as they try out their ideas, without comment
or without directing them to a particular strategy, is
also important. Listening provides a sense of how
pupils' problem-solving is developing, as assessing
this aspect of mathematics can be difficult. After
all, solving one problem will not necessarily lead to
success on the next problem, nor will difficulty with a
particular problem mean that the problems that follow
will also be as challenging.
A teacher also may need to extend or adapt a given
problem to ensure the problem-solving process is
understood and can be used in other situations,
instead of moving on to a different problem in another
area of mathematics learning. This can help pupils to
understand the significance of asking questions of a
problem, as well as seeing how a way of thinking can
be adapted to other related problems. Having pupils
engage in this process of problem posing is another
way of both assessing them and bringing them to
terms with the overall process of solving problems.
Sa
m
Expressing an answer in a sentence that relates to
the question stated in the problem also encourages
reflection on what was done and ensures that
the focus is on solving the problem rather than
providing an answer. These aspects of the teaching
of problem-solving should then be taken further, as
particular groups discuss their solutions with the
whole class and all pupils
are able to participate in
the discussion of the
problem. In this way,
problem-solving as a
way of thinking comes to
the fore, rather than focusing
on the answers as the main
aim of their mathematical
activities.
Vi
ew
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Questions must encourage pupils to explore possible
means to a solution and try one or more of them, rather
than point to a particular procedure. It can also help
pupils to see how to progress in their thinking, rather
than get into a loop where the same steps are repeated
over and over. While having too many questions that
focus on the way to a solution may end up removing
the problem-solving aspect from the question, having
Prim-Ed PublishingÂŽ
Problem-solving in mathematics
xv
INTRODUCTION
Building a problem-solving process
The cyclical model Analyse–Explore–Try provides a very helpful means of organising and discussing possible
solutions. However, care must be taken that it is not seen simply as a procedure to be memorised and then applied
in a routine manner to every new problem. Rather, it needs to be carefully developed over a range of different
problems, highlighting the components that are developed with each new problem.
Explore
• When a problem is being explored, some problems
will require the use of materials to think through the
whole of the problem's context. Others will demand
the use of diagrams to show what is needed.
Another will show how systematic analysis of the
situation using a sequence of diagrams, on a list or
table, is helpful. As these ways of thinking about
the problem are understood, they can be included
in the cycle of steps.
Sa
m
• Further reading will be needed to sort out which
information is needed and whether some is
not needed or if other information needs to
be gathered from the problem's context (e.g.
data presented within the illustration or table
accompanying the problem), or whether the pupils'
mathematical understandings need to be used to
find other relationships among the information. As
the form of the problems becomes more complex,
this thinking will be extended to incorporate
further ways of dealing with the information; for
example, measurement units, fractions and larger
numbers might need to be renamed to the same
mathematical form.
e
• As pupils read a problem, the need to first read
for the meaning of the problem can be stressed.
This may require reading more than once and can
be helped by asking pupils to state in their own
words what the problem is asking them to do.
• Developing a capacity to see 'through' the problem's
expression—or context to see similarities between
new problems and others that might already have
been met—is a critical way of building expertise
in coming to terms with and solving problems.
pl
Analyse
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Try
• Thinking about any processes that might be
needed and the order in which they are used, as
well as the type of answer that could occur, should
also be developed in the context of new levels of
problem structure.
Expanding the problem-solving process
Vi
• Put the solution back into the problem.
• Does the answer make sense?
• Does it solve the problem?
• Is it the only answer?
• Could there be another way?
• Use materials or a model.
• Use a calculator.
• Use pencil and paper.
• Look for a pattern.
xvi
Analyse
the problem
Try
a solution strategy
Problem-solving in mathematics
• Many pupils often try to guess a result. This can
even be encouraged by talking about 'guess and
check' as a means to solve problems. Changing to
'try and adjust' is more helpful
in building a way of thinking
and can lead to a very powerful
way of finding solutions.
• Read carefully.
• What is the problem asking?
• What is the meaning of the information? Is it all
needed? Is there too little? Too much?
• Which operations will be needed and in what order?
• What sort of answer is likely?
• Have I seen a problem like this before?
Explore
means to a solution
• Use a diagram or materials.
• Work backwards or backtrack.
• Put the information into a table.
• Try and adjust.
Prim-Ed Publishing®
INTRODUCTION
• The point in the cycle where an answer is assessed
for reasonableness (e.g. whether it provides
a solution, is only one of several solutions or
whether there may be another way to solve the
problem) also needs to be brought to the fore as
different problems are met.
Sa
m
The role of calculators
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When calculators are used, pupils devote less time
to basic calculations, providing time that might be
needed to either explore a solution or find an answer
to a problem. In this way, attention is shifted from
computation, which the calculator can do, to thinking
about the problem and its solution—work that the
calculator cannot do. It also allows more problems (and
more realistic problems) to be addressed in problemsolving sessions. In these situations, a calculator
serves as a tool rather than a crutch, requiring pupils
to think through the problem's solution in order to
know how to use the calculator appropriately. It
also underpins the need to make sense of the steps
along the way and any answers that result, as keying
incorrect numbers, operations or order of operations
quickly leads to results that are not appropriate.
Vi
Choosing, adapting and extending
problems
When problems are selected, they need to be examined
to see if pupils already have an understanding of
the underlying mathematics required and that the
problem's expression can be meaningfully read by the
group of pupils who will be attempting the solution—
though not necessarily by all pupils in the group. The
problem itself should be neither too easy (so that it
is just an exercise, repeating something readily done
before), nor too difficult (thus beyond the capabilities
of most or all in the group). A problem should engage
Prim-Ed Publishing®
As a problem and its solution is reviewed, posing
similar questions—where the numbers, shapes or
measurements are changed—focuses attention back
on what was entailed in analysing the problem and
in exploring the means to a solution. Extending these
processes to more complex situations shows how
the particular approach can be extended to other
situations and how patterns can be analysed to obtain
more general methods or results. It also highlights the
importance of a systematic approach when conceiving
and discussing a solution and can lead to pupils asking
themselves further questions about the situation and
pose problems of their own as the significance of the
problem's structure is uncovered.
e
Analyse
the interests of the pupils and also be able to be
solved in more than one way.
pl
• When materials, a diagram or table have been
used, another means to a solution is to look for a
pattern in the results. When these have revealed
what is needed to try for a solution, it may also
be reasonable to use pencil and paper or a
calculator.
Problem structure and expression
When analysing a problem, it is also possible to
discern critical aspects of the problem's form and
relate this to an appropriate level of mathematics
and problem expression when choosing or extending
problems. A problem of first-level complexity uses
simple mathematics and simple language. A secondlevel problem may have simple language and more
difficult mathematics or more difficult language and
simple mathematics, while a third-level problem has
yet more difficult language and mathematics. Within
a problem, the processes that must be used may be
more or less obvious, the information that is required
for a solution may be too much or too little, and
strategic thinking may be needed in order to come to
terms with what the problem is asking.
Level
(ii) The processes required are not immediately
obvious, as these problems contain all the
information necessary to find a solution but demand
further analysis to sort out what is wanted and
pupils may need to reverse what initially seemed
to be required.
Sa
m
(iii) The problem contains more information than is
needed for a solution, as these problems contain
not only all the information needed to find a
solution but also additional information in the form
of times, numbers, shapes or measurements.
e
(i) The processes to be used are relatively obvious, as
these problems are comparatively straightforward
and contain all the information necessary to find a
solution.
Assessment of problem-solving requires careful and
close observation of pupils working in a problemsolving setting. These observations can reveal the
range of problem forms and the level of complexity
in the expression and underlying mathematics that a
pupil is able to confidently deal with. Further analysis
of these observations can show to what extent the
pupil is able to analyse the question, explore ways to
a solution, select one or more methods to try and then
analyse any results obtained. It is the combination
of two fundamental aspects—the types of problem
that can be solved and the manner in which solutions
are carried out—that will give a measure of a pupil's
developing problem-solving abilities, rather than a
one-off test in which some problems are solved and
others are not.
pl
The varying levels of problem structure and
expression
How? • Analyse
• Explore
• Try
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(iv) Further information must be gathered and applied
to the problem in order to obtain a solution.
These problems do not contain all the information
necessary to find a solution but do contain a means
to obtain the required information. The problem's
setting, the pupil's mathematical understanding or
the problem's wording need to be searched for the
additional material.
(v) Strategic thinking is required to analyse the
question in order to determine a solution strategy.
Deeper analysis, often aided by the use of diagrams
or tables, is needed to come to terms with what
the problem is asking so a means to a solution can
be determined.
Vi
This analysis of the nature of problems can also serve
as a means of evaluating the provision of problems
within a mathematics programme. In particular, it can
lead to the development of a full range of problems,
ensuring they are included across all problem forms,
with the mathematics and expression suited to the
level of the pupils.
xviii
Problem-solving in mathematics
Observations based on this analysis have led to a
categorisation of many of the possible difficulties that
pupils experience with problem-solving as a whole,
rather than the misconceptions they may have about
particular problems.These often involve inappropriate
attempts at a solution based on little understanding
of the problem.
A major cause of possible difficulties is the lack of
a well-developed plan of attack, leading pupils
to focus on the surface level of problems. In such
cases, pupils:
• locate and manipulate numbers with little or no
thought about their relevance to the problem
• try a succession of different operations if the first
ones attempted do not yield a (likely) result
• focus on keywords for an indication of what might
be done without considering their significance
within the problem as a whole
• read problems quickly and cursorily to locate the
numbers to be used
Prim-Ed Publishing®
INTRODUCTION
Problem
Likely causes
Pupil is unable to make any
attempt at a solution.
•
•
•
•
Pupil has no means of linking
the situation to the implicit
mathematical meaning.
• needs to create diagram or use materials
• needs to consider separate parts of question and then bring parts
together
• cannot see interactions between operations
• lack of understanding means he/she unable to reverse situations
• data may need to be used in an order not evident in the problem
statement or in an order contrary to that in which it is presented
Sa
m
pl
e
not interested
feels overwhelmed
cannot think of how to start to answer question
needs to reconsider complexity of steps and information
• use the first available word cue to suggest the
operation that might be needed.
g
Other possible difficulties result from a focus on being
quick, which leads to:
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in
• no attempt to assess the reasonableness of an
answer
• little perseverance if an answer is not obtained
using the first approach tried
• not being able to access strategies to which they
have been introduced.
Vi
When the approaches to problem processing
developed in this series are followed and the specific
suggestions for solving particular problems or types of
problems are discussed with pupils, these difficulties
can be minimised, if not entirely avoided. Analysing
the problem before starting leads to an understanding
of the problem's meanings. The cycle of steps within
the model means that nothing is tried before the intent
of the problem is clear and the means to a solution
have been considered. Focusing on a problem's
meaning and discussing what needs to be done builds
perseverance. Making sense of the steps that must
be followed and any answers that result are central
to the problem-solving process. These difficulties are
unlikely to occur among those who have built up an
understanding of this way of thinking.
A final comment
If an approach to problem-solving can be built up using the ideas developed here and the problems in the
investigations on the pages that follow, pupils will develop a way of thinking about and with mathematics that will
allow them to readily solve problems and generalise from what they already know to understand new mathematical
ideas. They will engage with these emerging mathematical conceptions from their very beginnings, be prepared to
debate and discuss their own ideas, and develop attitudes that will allow them to tackle new problems and topics.
Mathematics can then be a subject that is readily engaged with and can become one in which the pupil feels in
control, instead of one in which many rules devoid of meaning have to be memorised and applied at the right time.
This early enthusiasm for learning and the ability to think mathematically will lead to a search for meaning in new
situations and processes that will allow mathematical ideas to be used across a range of applications in school
and everyday life.
Prim-Ed Publishing®
Problem-solving in mathematics
xix
TEACHER NOTES
e
Page 5
Pattern-making and numbers are combined in these
activities. When lining up their blocks in groups of two
and three, some pupils will have blocks left over. This
encourages discussion about who can line up their blocks
with none left over and who have some left over. The last
question involves pupils looking at their blocks to see if
they can make a pattern. They should have enough blocks
to make some sort of pattern.
Page 4
The six different arrangements now need to be continued
to form a pattern using more blocks in the same repeating
order. The pupils are not told how many blocks to get
and some may not have enough blocks to make a pattern
and may need to get more to complete their pattern. This
requires pupils to problem-solve how to make a pattern
and what to do if they don't have enough blocks.
pl
Problem-solving
To reason logically, and to identify, create and describe
patterns.
Materials
Blocks such as Unifix™ cubes, counters, teddies or plastic
animals in different colours
Extension
• Pupils can be encouraged to make and describe more
complex patterns of their own.
g
Focus
To explore making patterns, changing patterns and using
patterns and numbers. To have pupils analyse what
makes a pattern and make predictions based on their
experiences.
Possible difficulties
• Indiscriminately moving blocks around
• Inability to keep track of what they are trying.
• Difficulty in repeating a consistent pattern.
• Content to find only one or two possibilities.
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Discussion
Page 3
The blocks can be lined up in six ways. For example, a
pupil might put the same block first and then find two
ways by swapping the other two blocks. Then the pupil
changes the colour of the first block and swaps the other
two colours again. This can be done a third time and gives
the following six arrangements:
red blue yellow
yellow blue red
blue yellow red
red yellow blue
yellow red blue
Vi
blue red yellow
The next activity involves taking four blocks in different
colours and lining them up in different groups of two.
Possible arrangements are:
blue
2
red
red
blue
blue yellow
red
blue
red
green
yellow blue
green
blue
yellow
yellow
green
red
green
yellow green
red
green yellow
Problem-solving in mathematics
Prim-Ed Publishing®
BLOCK TIME
Take 3 blocks, each a different colour.
How many ways can you line them up?
Sa
m
pl
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Draw a picture of all the ways they can be lined up.
g
Take 4 blocks, each a different colour.
How many ways can you line them up in groups of two?
Vi
ew
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Draw the different groups of two you made.
How many different groups can you make?
Prim-Ed Publishing速
Problem-solving in mathematics
3
BLOCK PATTERNS
Sa
m
pl
e
Choose some blocks in 3 different colours and make a
pattern.
Draw your pattern.
g
Did you have enough to make a pattern or did
you have to get some more blocks?
Vi
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Think about other patterns you can make using the same blocks.
Draw the different patterns.
How many different patterns can you make using the blocks?
4
Problem-solving in mathematics
Prim-Ed Publishing速
MORE BLOCKS
How many blocks can you hold in one hand?
Guess and then count them.
How many blocks can you hold in two hands?
How many groups did you make?
Sa
m
Can you line them up in groups of 3?
pl
Can you line up your blocks in groups of 2?
e
Guess and then count them.
How many groups did you make?
Look at your blocks and see if they make a pattern.
Vi
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Draw your pattern below.
Prim-Ed Publishing速
Problem-solving in mathematics
5
TEACHER NOTES
Vi
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6
Problem-solving in mathematics
e
1 green house
1 red house
1 blue house
g
Focus
To analyse problems with a three-dimensional aspect. This
extends the previous work with coloured blocks on page
3. There are a number of solutions to this problem, and
pupils should be encouraged to explore and try a number
of different possibilities.
Discussion
Page 7
Individually or in pairs, pupils use the 10 blocks and
arrange them on their street according to the street map.
For example, the five red people could either be five
houses or one apartment block and two houses, while the
green people can be only houses. Possible arrangements
could include:
1 red house
Problem-solving
To reason logically and use patterns to represent and
solve problems.
Possible difficulties
• Only making houses
• Mixing the colours in the apartments
• Not realising that a different order gives a different
solution
Extension
• Use the 'Block Street' copymaster and make up other
criteria for pupils to make streets.
• Take a digital photo of each block street and ask the
pairs of pupils to write about their street, listing how
many houses and apartments there are.
Prim-Ed Publishing®
Prim-Ed Publishing速
Problem-solving in mathematics
7
apartment = 3 blocks of the same colour
pl
Sa
m
Use the same blocks to make a different-looking street.
g
e
10 people live on Block Street:
5 red people, 3 blue people and 2 green people.
Make your street with houses and apartments to show where the people live.
house = 1 block
BLOCK STREET
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Put blocks on the street to show:
Vi
TEACHER NOTES
e
Possible difficulties
• An inability to compare numbers
• Combining all numbers present, rather than seeing
that some information is not needed
• Using the cue 'more' to find an answer by adding
rather than comparing
• Combining the information about bugs and beetles
• Not determining which information to use first when
graphing
Page 11
The information in the graph is analysed to sort out
which of the entries on the graph belong to which animal.
Understanding comparison enables pupils to work out
which label goes with the largest number (dogs) and
smallest number (fish). The statement that the number
of cats added to the number of fish is the same as the
number of dogs is not needed, but it does show if the
columns have been labelled correctly. This leaves only
guinea pigs and budgies. There must be seven guinea
pigs and six budgies to match the criteria of there being
one less budgie than guinea pigs.
pl
Problem-solving
To identify and use information in a problem.
Focus
These pages explore the reading and interpretation of
information to solve problems involving numeration. No
addition or subtraction is needed. Pupils analyse the
problem to locate the required information, decide what
information is not needed and then use comparison, rather
than addition or subtraction, to obtain solutions.
ew
in
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Discussion
Page 9
These problems state information that has to be analysed
in order to answer a series of questions. The first problems
involve reading and identifying the information needed.
The next problem requires several comparisons among
the number of marbles the children have.
Extension
• In pairs, have the pupils use the problem structure on
page 9 to write problems of their own to give to other
pupils to solve.
Vi
The last problem is about the red cars, with the blue cars
included only as additional information. Although Ashley
counted the largest number of cars, she, in fact, counted
more blue cars than red cars, so Rob counted the largest
number of red cars.
Page 10
Analysis of the first problem reveals that Kim skipped more
times than Josh, because she skipped as many times as
he did and then a further six times.
The second problem involves the two types of insects
being counted by Dillon. The questions ask the pupils to
sort out on what day of the week a certain number of bugs
were counted and when a number of beetles were counted
and to understand that at no time are the numbers of bugs
and beetles combined.
On Monday, Dillon counted 24 bugs and 4 beetles.
On Tuesday, he counted 16 bugs and 9 beetles.
The next day, he counted 19 bugs and 7 beetles.
On which day did he count the most bugs?
How many beetles did he count on Wednesday?
On which day did he count the most beetles?
10
Problem-solving in mathematics
Prim-Ed Publishing速
ANIMAL PETS
Sonia's class has collected information about pets.
Animal pets
12
Sa
m
11
10
9
8
7
6
5
g
Number of pets
pl
e
Class members keep five different kinds of animals as pets.
Below is a graph showing how many there are of each pet.
The names of the types of animals have not been put on the
graph.
Cut out the animals' names and glue them onto the graph. Use
the information below to help you.
ew
in
4
3
2
1
0
Vi
Clues:
• There are 3 fewer cats than dogs.
• The smallest number of pets kept are fish.
• The number of cats added to the number of fish
is the same as the number of dogs.
• Most pupils have dogs as pets.
• There is one fewer budgie than guinea pigs.
cats
Problem-solving
To analyse and sort data according to one or more
criteria.
g
Focus
These pages explore sets of objects to make decisions
about how and what criteria can be used to organise
them. Identifying, analysing and writing about the specific
criteria to sort objects is needed for pupils to find solutions.
Pupils should be encouraged to explore and try a number
of different possibilities.
the pairs of clowns. Pupils might keep the hat on the first
clown the same and change the hat on the second clown.
This activity extends the previous work with the blocks
(pages 2–5). Putting a hat on a clown, rather than just
lining up hats, means that having Hat A and Hat B on one
pair is different from having Hat B and Hat A on another
pair. In the first example, Hat A is on the first clown, and
in the second example, Hat B is on the first clown.
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Discussion
Page 13
There are a number of ways the animals can be sorted.
Criteria such as the number of legs, flying or not flying,
insects or not insects, and living in colonies or not, are
possibilities that could be used.
Vi
Page 14
Again, there are a number of ways the clothes can be
sorted. One possibility would be size, so the small things
(such as underwear and socks) could go in the first drawer,
the larger, bulky things in the bottom drawer, and the shirt
and T-shirts in the other drawers. Pupils would need to be
able to explain and justify their solutions.
12
Problem-solving in mathematics
Possible difficulties
• Needing help to think of how the objects can be
sorted
• Sorting according to only one criteria
• Placing the clothes into drawers but not according to
any criteria
• Using two sets of clowns and drawing each of the four
hats only once
Extension
• Make an A3-sized display of the animals and write
how they have been sorted.
• Discuss whether some of the clothes would be best
hung in a cupboard rather than using the chest of
drawers.
• Look at a bedroom cupboard and sort what could be
hung up and what could go into the drawers.
• Explore what would happen if there were three clowns
with four types of hats.
Prim-Ed Publishing®
ANIMAL ANTICS
Frances looked for and found lots of animals in her garden.
Help her to sort and display them. Cut out the pictures and sort
them into groups.
How did you sort them?
spider
butterfly
butterfly
pl
spider
spider
butterfly
dragonfly
dragonfly
grasshopper
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g
spider
Sa
m
How many ways can you sort them?
e
Now choose another way to sort them.
grasshopper
bee
bee
bee
bee
ladybird
ladybird
ladybird
ant
ant
ant
ant
ant
ant
ant
ant
frog
frog
frog
Vi
grasshopper
Prim-Ed Publishing速
Problem-solving in mathematics
13
SORTING CLOTHES
Nick has a new chest of drawers.
He needs to put his socks, shorts, underwear, shirt, jumper,
tracksuit trousers and T-shirts into the four drawers.
Vi
ew
in
g
Sa
m
pl
e
Can you help him to sort out what clothes go into each drawer?
Now sort the clothes another way.
14
Problem-solving in mathematics
Prim-Ed Publishing速
CLOWN HATS
The clowns have 4 types of hats.
Draw 2 different hats on each pair of clowns.
Hat B
Hat C
Hat D
e
Hat A
Vi
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in
g
Sa
m
pl
Did all of the clowns get a hat?
Prim-Ed Publishing速
Problem-solving in mathematics
15
TEACHER NOTES
e
Extension
• Pupils think of a number and make up criteria to
match, using similar criteria of between, greater than
and lower than.
• The new problem is given to other pupils to try and
solve.
• Pupils could start with a one-digit number and then try
it with a two-digit number.
Sa
m
Curriculum linksPossible difficulties
• Selecting a number only based on the first criteria
• Not using all of the criteria
• Selecting a number only based on the last criteria
pl
Problem-solving
To read, interpret and analyse information.
Focus
This page explores relationships among numbers and
uses this analysis to find a number that matches specific
criteria. This process encourages pupils to disregard
numbers that are not possible rather than simply look for
ones that are likely to work.
ew
in
g
Some pupils will use the information provided to discard
numbers until only the correct number remains. Other
pupils may prefer to try each number in turn against
all of the criteria until only one number suits all of the
conditions.
Vi
Discussion
Page 17
The criteria listed allow numbers to not only be selected
but also to be ruled out; for example, when a number
has to be between three and eight, then two and nine
can be ruled out. Some pupils may work down the list
of conditions, while others pupils might read all the
conditions and then decide where to start.
One way in which problem-solving differs from work with
computation, measurement and other direct applications
is that finding what is not likely is often more important
than simply using a known method to obtain a definite
result.
16
Problem-solving in mathematics
Prim-Ed Publishing®
WHAT'S MY NUMBER? 1
7
2
5
9
3
My number is:
e
• between 3 and 8
pl
• greater than 4
• an odd number
Sa
m
• greater than 5.
My number is
.
g
WHAT'S MY NUMBER? 2
8
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3
6
9
5
My number is:
Vi
• between 0 and 7
• lower than 6
• an odd number
• greater than 4.
My number is
Prim-Ed Publishing®
.
Problem-solving in mathematics
17
TEACHER NOTES
e
Possible difficulties
• Putting the numbers in the correct order in the story
• Not considering the sense of the story.
• Inability to see that there can be more than one
answer.
• Not taking into account the context of the story when
selecting numbers.
• Difficulty thinking up their own numbers for a story for
it to make sense.
Sa
m
Curriculum LinksPage 21
These problems require the pupils to think of their own
numbers; however, this time there is no model of what
makes sense, as in the problem on page 19. There is
considerable leeway as to what numbers could be used
for the story to make sense. Often it is what does not
make sense as opposed to what does.
pl
Problem-solving
To interpret and organise information in a series of
interrelated problem statements.
g
Focus
These pages explore ways to put given numbers into
problem situations so that the resulting story makes
sense. Pupils need to read the stories carefully to work
out which number goes where. The numbers are not listed
in the order they are used in the story. This thinking is then
extended so that pupils think of their own numbers to fit
problem situations.
Extension
• Pupils can be encouraged to write further stories of
their own and swap them with others in the class to
solve.
• Pupils could write stories with and without numbers.
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Discussion
Page 19
The first two problems involve pupils reading the story
and putting numbers into the story. Pupils need to make
decisions about who is the older child and who is the
younger based on the story and put the numbers in
accordingly. The last problem uses the story structure of
the previous problem and requires pupils to think of their
own numbers so the story makes sense.
Vi
Page 20
Reading the stories and working out the relationships
among the number of stickers and counters is needed to
solve the problem. Pupils need to select numbers based
on the information given. The questions do not ask for
the numbers in order, so pupils are required to read and
scan to find a solution. The last problem has two possible
answers. This introduces the understanding that there
may be more than one answer to a problem.
18
Problem-solving in mathematics
Prim-Ed Publishing®
MISSING NUMBERS
Put in the numbers so the story makes sense.
7
9
Jason is younger than Rose.
e
years old.
Rose is
years old.
pl
Jason is
Sa
m
Put in the numbers so the story makes sense.
15
19
13
Mason is older than Will. Will is older than Bella.
years old.
g
Mason is
years old.
ew
in
Will is
years old.
Bella is
Vi
Now use your own numbers so the story makes sense.
Mason is older than Will. Will is older than Bella.
Mason is
years old.
Will is
years old.
Bella is
Prim-Ed Publishing速
years old.
Problem-solving in mathematics
19
NUMBER SENSE
Put in the numbers so the story makes sense.
27
29
26
28
Jason has one more sticker than Rose.
e
Rose has one more sticker than Harry.
pl
Harry has one more sticker than Bree.
stickers.
Jason has
stickers.
Sa
m
Bree has
Harry has
stickers.
Rose has
stickers.
41
42
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in
44
g
Put in the numbers so the story makes sense.
45
43
The red box has one more counter than the blue box.
The blue box has one more counter than the green box.
Vi
The green box has one more counter than the yellow box.
The red box has
counters.
The yellow box has
counters.
The green box has
counters.
The blue box has
20
Problem-solving in mathematics
counters.
Prim-Ed Publishing速
NUMBER STORIES
Make up your own numbers to go in the stories below.
Jason has
stickers. Rose has more
stickers.
pl
e
stickers than Jason. She has
Sa
m
stickers. Rose
Harry has
has fewer stickers than Harry.
Rose has
stickers.
The red box has one more counter than the blue box.
g
The blue box has one more counter than the green box.
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in
The green box has one more counter than the yellow box.
The red box has
counters.
counters.
The green box has
counters.
Vi
The yellow box has
The blue box has
Prim-Ed Publishing速
counters.
Problem-solving in mathematics
21
TEACHER NOTES
Problem-solving
added.
Possible difficulties
• Inability to identify the need to add to find a solution.
• The need to add two different items to get a solution.
• Adding all of the numbers written rather than just the
numbers required.
Extension
• Using the problems on page 23 as a model, have pupils
write simple problems for other pupils to solve.
Page 25
Addition of two unlike items is again used in these
problems but this time with larger numbers. The idea
of holes being dug to plant the trees is more complex
than just asking how many trees have been planted. The
last problem involves more reading then the previous
problems.
pl
To analyse and use information in addition problems.
Materials
Counters or blocks
Focus
g
These pages explore word problems that require addition.
Pupils need to determine what the problem is asking in
order to find a solution. Analysis of the problems reveals
that more information may be needed.
ew
in
Counters or blocks can be used to assist with these
problems. Pupils are not simply concerned with basic
facts but about reading for information and determining
what the problems are asking.
Discussion
Vi
Page 23
Each problem involves adding like items together—e.g.
bugs—and has the clearly identified question of 'How
many altogether?' at the end, which suggests use of
addition. The last problem has additional information
about geese, which is not needed.
These are simple problems. Pupils need to read the
problem, identify that they need to add to solve it and
then correctly select the numbers to be added.
Page 24
These problems involve the addition of two unlike items
to get a total—e.g. boys and girls, to get the number of
children. This is more complex than adding two like items
together. Accordingly, the language has been kept fairly
simple in each problem.
Again, pupils need to read the problem, identify that they
need to add to solve it and then select the numbers to be
22
Problem-solving in mathematics
Prim-Ed Publishing®
pl
3 frogs on the lily pad.
4 frogs in the water.
How many frogs altogether?
Vi
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g
Sa
m
4 bugs on a bush.
6 bugs on another bush.
How many bugs altogether?
e
NUMBER PROBLEMS 1
9 ducks on the lake.
4 geese in the sky.
5 ducks on the bank.
How many ducks altogether?
Prim-Ed Publishing速
Problem-solving in mathematics
23
NUMBER PROBLEMS 2
Sa
m
pl
e
7 girls riding bikes.
5 boys on the grass.
How many children altogether?
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in
g
The parrot lost 8 yellow
feathers and 9 blue
feathers.
How many feathers did
the parrot lose?
Vi
Bert made 16 bags of toffee for the market. Later, he made
21 bags of fudge. How many bags did he make to sell at
the market?
24
Problem-solving in mathematics
Prim-Ed Publishing速
NUMBER PROBLEMS 3
The gardener planted 15
apple trees and 27 pear trees.
How many holes did he need
to dig?
The shopping bag had a hole
and 27 oranges fell out. Then 18
apples fell out.
How many pieces of fruit fell
out of the shopping bag?
Prim-Ed Publishing速
Problem-solving in mathematics
25
TEACHER NOTES
Problem-solving
Discussion
Curriculum links
Page 27
Pupils read the items for sale and note how much each
costs. Pupils who are not familiar with money can still do
the activity with a calculator. This investigation involves
the pupils obtaining information not only from the question
but also from another source—the pictures.
e
They need to remember what they are buying and then
work out how much it is, and in some cases add on to
compare amounts, to see if they have enough money.
The last two questions have a number of possible
solutions. Pupils might choose three items they would
like and then add and compare only to discover they don't
have enough money, while others may just choose the
three cheapest items. Either way, they need to compare
money amounts and make decisions accordingly.
To solve problems involving money and to make decisions
based on particular criteria.
Possible difficulties
Some pupils may need counters, play money or a
calculator.
• Confusion with the £ (pound) symbol.
• The concept of 'enough money' as opposed to an
'exact amount'.
• Not buying different things when necessary.
• Thinking the exact amount of £10 has to be spent as
opposed to not spending all that is available.
Focus
Extension
• Make a list of all the different possibilities of how
pupils could spend their £10.
• In pairs, have pupils write other questions about the
toy animals.
g
Materials
ew
in
This page explores the concepts of reading for information,
obtaining information from another source (the picture)
and using both to find solutions. The problems are about
using money, making decisions based on money and
comparing amounts of money, rather than addition or
mental facts.
Vi
Solutions can be obtained using materials and comparison
of amounts. The item amounts have been kept small to
assist with the problem-solving. Counters, blocks,
play money or a calculator can be used if needed. This
investigation lends itself to using a calculator and could
be used to introduce this tool or to extend work previously
completed on a calculator.
26
Problem-solving in mathematics
Prim-Ed Publishing®
TOY ANIMALS
cat £2
sheep £3
Sa
m
pl
e
chicken £1
goat £3
g
cow £2
dog £2
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in
horse £4
pig £3
Nancy has £4. Can she buy a chicken and a cat?
Maria has £5. Does she have enough
to buy the cow and the sheep?
Vi
Daniel has bought a goat, a dog and a pig.
How much did he spend?
Mike has £8. He wants to buy 3 toy animals. Choose 3
different things he can buy with his money.
If you had £10, what would you buy?
Prim-Ed Publishing®
Problem-solving in mathematics
27
TEACHER NOTES
Curriculum links
Possible difficulties
• Moving blocks around indiscriminately
• Focusing on the particular positions; e.g. forming a line
where the fourth block is red and the seventh block is
blue, but without a pattern in between—blocks are
just in a line and no pattern is evident
• Difficulty in repeating a consistent pattern
Page 31
The car race problem provides a very engaging way of
consolidating the use of patterns and the manner in which
ordinal numbers are used to describe them. If pupils have
difficulty organising the data, have them use coloured
cubes to represent the cars and then transfer the results
to the drawn cars. They may also need to do this first
when planning their own race.
e
To investigate patterns and order and make predictions
based on these.
pl
Problem-solving
Focus
• Have pupils make patterns in which three blocks are
specified.
• In pairs, have one pupil decide the criterion for a
pattern while the other pupil makes the pattern.
• In pairs, play a game where one pupil forms a pattern
behind a barrier that stops the other from seeing the
pattern. The first pupil then describes the pattern, using
colours and ordinal numbers, to the other pupil. The
second pupil forms this pattern, as he/she understands
it to be, on his/her side of the barrier. When the second
pattern has been completed, the barrier is removed
and the two patterns are compared.
• Have pupils use a table to find the results of their car
races.
ew
in
g
These pages build and extend the earlier activities of
using coloured blocks, and introduce ordinal numbers as
an aspect of forming and describing patterns. Pupils are
required to manipulate items (both real and drawn) to fit
particular criteria and determined patterns, including how
they relate to ordinal place.
Extension
Discussion
Page 29
Some pupils may have to put additional blocks out and
count them to find their solutions, while others may be
able to predict the block's colour from the pattern they
can see. Similarly, some pupils may be able to think of
a pattern in their mind for the second problem and then
arrange the blocks accordingly.
Vi
Page 30
These activities require pupils to work backwards to
form different patterns. Ask pupils to check the blocks
other classmates have arranged. Pupils have to see if the
arrangements do show a pattern and then come up with a
way of describing the pattern.
What colour is the 8th block?
What colour would the 16th block be?
What about the 24th block?
Prim-Ed Publishing速
Problem-solving in mathematics
29
ORDINAL PATTERNS
Sa
m
pl
e
Make a pattern where the 6th block is green.
Make and draw other patterns where the 6th block is green.
Vi
ew
in
g
How many patterns did you make?
Make a pattern where the 9th block is red.
Make five patterns where the 9th block is red.
Make a pattern where the 8th and 9th blocks are red.
Make a pattern where the 4th block is red and the 7th block is
green.
Make and draw other patterns where the 4th block is red and
the 7th block is green.
30
Problem-solving in mathematics
Prim-Ed Publishing速
CAR RACE
37
The yellow car is placed second.
Sa
m
The 6th car is black.
45
e
37
pl
FINISH
Use the clues to colour the cars.
The red car is in front of the yellow car.
The orange car is between the black car and the blue car.
The green car is behind the yellow car.
1st
g
Use the drawing to fill in which place each car is positioned.
4th
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2nd
5th
3rd
6th
Vi
Draw your own car race and write clues to match.
Prim-Ed Publishing速
Problem-solving in mathematics
31
TEACHER NOTES
Curriculum links
Materials
Some pupils may need materials to assist them.
Focus
Possible difficulties
• Not using a table or list to manage the data.
• Inability to see that a ham and cheese sandwich is the
same as a cheese and ham sandwich.
Page 35
In this scenario, there is now the choice of three types of
bread (white, brown or multi-grain) with three different
fillings. Again pupils may be able to predict how many
sandwiches can be made and what will happen if two
fillings are used, based on experience with the previous
problems. The questions about more, fewer or the same
number of sandwiches is designed to have pupils think
about their experiences and make a prediction, which is
then investigated.
e
To organise data and make predictions
pl
Problem-solving
• Make a class table showing the possible combinations
using one type of bread, two types and three types
with the three fillings.
• Explore what would happen with four fillings and the
possible combinations.
• Pupils could list the sandwiches they would choose to
make and take to a picnic.
ew
in
g
These pages explore the various ways sandwiches can
be made using two different types of bread and one of
three fillings. The variations are then recorded. These
explorations are then extended to include a larger
number of bread types and fillings. In order to manage the
possibilities, it is important to organise the data in a list
or table or to use materials.
Discussion
Vi
Page 33
There are three possible white bread sandwiches—ham,
cheese or tomato. The question about using two fillings is
designed to make pupils think about whether there will be
more sandwiches, fewer sandwiches or the same number
of sandwiches. In this case it is the same number.
Page 34
This problem builds on the problem on page 33 and adds
a second choice of bread. Now sandwiches can be made
using brown or white bread with the three different fillings.
Pupils may be able to predict how many sandwiches can
be made and what will happen if two filling are used,
based on their experience with the previous problem.
32
Problem-solving in mathematics
Prim-Ed Publishing®
MAKING SANDWICHES
Zac and Rachel are making sandwiches for a picnic. They
have decided to make each sandwich different.
We have
ham, cheese
and tomato.
Sa
m
pl
e
We have
white bread.
Write all the different sandwiches they can make if they use
one filling in each sandwich.
BREAD
FILLING
white
g
white
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in
white
How many sandwiches can they make?
Vi
Write all the different ways they can make sandwiches if
they use two fillings.
BREAD
FILLINGS
white
white
white
How many sandwiches can they make?
Prim-Ed Publishing速
Problem-solving in mathematics
33
SANDWICH CHOICES
Zac and Rachel are making sandwiches for a picnic. They
decide to make each sandwich different.
We have ham,
cheese and
tomato.
pl
e
We have white
bread and
brown bread.
Sa
m
Write all the different sandwiches they can make
using one filling.
WHITE
BROWN
g
ew
in
How many sandwiches can they make?
Write all the different ways they can make sandwiches if
they use two fillings.
Vi
WHITE
BROWN
What if they use all three fillings?
Altogether, Zac and Rachel can make
sandwiches using all three fillings.
34
Problem-solving in mathematics
Prim-Ed Publishing速
MORE CHOICES
Zac and Rachel are making sandwiches for a picnic. They
have decided to make each sandwich different.
We have white,
brown and multigrain bread.
pl
e
We have ham,
cheese and
tomato.
WHITE
Sa
m
Write all the different sandwiches they can make if they use
one filling.
BROWN
g
MULTI-GRAIN
ew
in
How many sandwiches can they make?
What happens if they use two fillings? Will they have more,
fewer or the same number of sandwiches?
Write all the different ways they can make sandwiches if they
use two fillings.
Vi
WHITE
BROWN
MULTI-GRAIN
How many sandwiches can they make?
Prim-Ed Publishing速
Problem-solving in mathematics
35
TEACHER NOTES
Problem-solving
Discussion
Curriculum links
Page 37
Pupils read the items for sale and note how much each
one costs. Pupils who are not familiar with money can
still do the activity with a calculator. This investigation
involves pupils' reading for information but also getting
information from another source—the illustration.
Materials
Some pupils may need counters, play money or a
calculator.
e
Possible difficulties
• Confusion with the £ (pound) symbol.
• The concept of 'enough money' as opposed to an
'exact amount'.
• Not buying different things when necessary.
• Thinking the exact amount of £8, £7 or £5 has to
be spent as opposed to not spending all that is
available.
Extension
g
Focus
They need to remember what they are buying and then
work out how much it is—and in some cases add and
in others compare amounts—to see if they have enough
money. The last three questions have a number of possible
solutions. Pupils might choose three items they would
like and then add and compare only to discover they don't
have enough money, while others may choose the three
cheapest items. Either way, they need to compare money
amounts and make decisions accordingly.
To solve problems involving money and to make decisions
based on particular criteria.
• Make a list of all the different possibilities for the last
three questions.
• In pairs, ask pupils to write other questions about the
fruit shop and give them to another pair to solve.
Vi
ew
in
This page explores reading for information, obtaining
information from another source (a drawing) and using it
to find solutions. The problems are about using money,
making decisions based on money and comparing amounts
of money, rather than addition or mental facts. Solutions
can be obtained by using materials and comparing
amounts. The item amounts have been kept small to assist
with the problem-solving. Counters, blocks, play money
or a calculator can be used if needed. This investigation
builds on the earlier 'Toy animals' investigation on page
27. It has the added dimension of a number of items for
a particular price while the 'Toy animals' investigation
involved only one item for a particular price.
36
Problem-solving in mathematics
Prim-Ed Publishing®
AT THE TOYSHOP
£1
£3
Sa
m
£1
pl
e
£2
£2
£4
Kelly has £6. Can she buy a koala and a teddy bear?
g
Mandy has £3. Does she have enough to buy a helicopter and a
ew
in
boat?
Davey bought a whistle, a koala and a car. How much did he
spend?
Vi
Mark has £8. Choose 3 different things he can buy with his
money.
Jane spent £7. List what she could have bought.
What would you buy if you had £5 to spend?
Prim-Ed Publishing®
Problem-solving in mathematics
37
TEACHER NOTES
Curriculum links
Materials
coloured pencils; counting materials, if needed
Focus
Possible difficulties
• Colouring all of the cones or boxes, whether or not
they are all needed.
• Not organising the data and just randomly positioning
any combination.
• Using the same combination more than once.
Page 41
This problem is based on a similar idea, exploring the
various positions in which an item can be placed in a box.
There are six possible ways to position the two chocolates
in the box. Some pupils might think that, because there
are two chocolates, the solution will be the same as the
two scoops of ice-cream. As with the previous problems,
there are more boxes than required.
e
To organise data and make predictions.
pl
Problem-solving
• Revisit the problem on page 39 and explore the
possibilities of using any combination of flavours,
rather than only one scoop of each flavour; e.g. two
chocolate and one vanilla.
ew
in
g
These pages explore the recording of different ways
ice-creams and chocolates can be organised according
to various criteria. In each investigation there are more
cones or boxes than needed. This requires the pupils to
carefully analyse their solutions and to begin to be able
to justify their responses.
Discussion
Vi
Page 39
There are six possible ways the scoops can be placed
on the cones. In this activity eight cones are drawn on
the page and some pupils may simply repeat a previous
combination in order to fill all the cones. The question
states that pupils choose one of each flavour; therefore
each cone needs one chocolate scoop, one vanilla scoop
and one strawberry scoop.
Page 40
Pupils can choose two scoops of ice-cream from three
possibilities. This problem does not state that they must
choose different flavours, so there are nine possibilities,
since they could have two scoops of the same flavour
if they wished. Again, there are more cones on the
page than needed and some pupils may simply repeat a
previous combination in order to fill all the cones.
Colour the scoops to show the different ways each ice-cream
flavour could be placed on the cone.
Did you need all of the cones?
Prim-Ed Publishing速
Problem-solving in mathematics
39
ICE-CREAMS 2
pl
e
You are allowed 2 scoops of ice-cream.
You can choose from
vanilla, chocolate and strawberry.
Vi
ew
in
g
Sa
m
Colour the scoops to show the different ways the ice-cream
flavours could be placed on the cone.
Did you need all of the cones?
40
Problem-solving in mathematics
Prim-Ed Publishing速
CHOCOLATES
Cathy's box has spaces for 4
chocolates.
Sa
m
pl
e
There are 4 ways she can put 1 chocolate into the box:
Cathy has 2 chocolates and 1 box.
Vi
ew
in
g
Draw the different ways she can place the chocolates in the
box. The first one has been done for you.
Did you use all the boxes?
What if there were 3 chocolates? There are
Prim-Ed PublishingÂŽ
ways.
Problem-solving in mathematics
41
TEACHER NOTES
Curriculum links
Materials
Drawing materials, including coloured pencils
Focus
Page 45
Using the road map, pupils track a path from one point
to another without backtracking along the way. Each
route can be used only once. Some pupils may need to
use different coloured pencils to help keep track of the
different paths. There are at least 10 possibilities. Discuss
the various paths available by using an enlarged version
of the picture to show all of the different possibilities.
Page 44
This investigation continues from the previous page. Now
the towels are joined together so that, rather than four
pegs for two towels, there are three pegs for two towels.
In the case of three pegs for every two towels, pupils could
draw the towels and then arrange the pegs to count them.
The same diagram could be used to solve the problem for
10 towels instead of pupils having to draw a new diagram.
Again, some pupils may be able to predict the number of
pegs needed by using the previous activity and will only
need to draw a new diagram to confirm this.
e
To use diagrams, make predictions and reason logically.
pl
Problem-solving
• Putting two pegs on each towel when there should
be three pegs per two towels.
• Taking only the most direct route and not thinking
about all the possibilities.
Extension
• Make a table listing how many towels and pegs are
used when there are two pegs per towel and how
many are used when there are three pegs per two
towels.
• Use several enlarged pictures to show all of the
different paths.
ew
in
g
These pages explore how many pegs are needed to hang
towels in various combinations, as well as exploring the
different paths between two points. Pupils are required
to make predictions and use diagrams to gather the
information needed to find solutions. The problems could
be completed using multiplication; however, they can also
be done using a diagram and counting, using addition with
counters, calculating doubles or with a calculator.
Possible difficulties
Discussion
Vi
Page 43
Pupils have to look at the diagram of the towels and pegs.
Using the diagram, they need to figure out how many
pegs will be needed if there are five towels to hang. Some
pupils may be able to look at the diagram and use it to
solve the problem, while others may need to draw the
five towels and count the pegs. This information can then
be used to figure out how many pegs are needed for nine
towels. Some pupils may be able to predict the number of
pegs from the previous activity and will only need to draw
a new diagram to confirm this.
42
Problem-solving in mathematics
Prim-Ed Publishing®
HANGING OUT THE WASHING
He uses 2 pegs for each towel.
pl
e
Mr Smith is hanging out his towels to dry.
ew
in
g
Sa
m
How many pegs does he use if he hangs out 5 towels?
Draw the towels and pegs.
Vi
What about for 9 towels?
Draw the towels and pegs.
Prim-Ed Publishing速
Problem-solving in mathematics
43
WASHING DAY
pl
e
Mr Smith has noticed that many of his pegs are broken and
that he does not have enough pegs to use for each towel.
He decides to join the towels together and use 3 pegs for 2
towels.
ew
in
g
Draw the towels and pegs.
Sa
m
He wants to hang out 7 towels. How many pegs
would he need if he was to peg the corners
together, using only 3 pegs for 2 towels?
Vi
What about 10 towels?
Draw the towels and pegs.
44
Problem-solving in mathematics
Prim-Ed Publishing速
DRY-CLEANING
Mr Smith has to walk to get to the drycleaners.
He likes to walk a different way each time.
Vi
ew
in
g
Sa
m
pl
e
He can only walk along each road once on his way to the
drycleaners.
Using a different colour each time, draw the different ways he
can walk to the drycleaners.
Mr Smith's home
Prim-Ed Publishing®
Problem-solving in mathematics
45
TEACHER NOTES
Curriculum links
Possible difficulties
• Not joining squares along a side.
• Making only one or two possible shapes.
• Inability to visualise the pattern needed to grow a
shape.
• Only rotating or flipping existing shapes, to create
duplicates.
Extension
• Investigate other shape patterns that 'grow' using
different numbers of squares or shapes.
Page 49
Pupils will need to physically manipulate squares or draw
squares on grid paper to see how the pattern 'grows'.
Encourage pupils to make predictions and give a verbal
description of their findings.
e
To visualise relationships among two-dimensional
shapes.
pl
Problem-solving
Materials
Square tiles, and grid paper or cut-out squares and
shapes
Focus
ew
in
g
These pages explore arrangements of squares and other
shapes. Spatial thinking, as well as logical thinking and
organisation, is involved as pupils investigate all possible
arrangements and extensions. Being able to visualise
patterning of this form will assist pupils in solving many
other problems, including number, measurement, and
chance and data, as well as other spatial situations.
Discussion
Vi
Page 47
A number of shapes can be made using the four squares,
all of which are commonly called tetrominoes. Some
pupils may need to physically manipulate the squares
in order to find the combinations and analyse whether
the shapes are the same or different. This might involve
rotating or flipping.
Page 48
This activity extends from the previous problem. Again,
some pupils may need to physically manipulate the
squares in order to find the combinations and analyse
whether shapes are the same or different. This might
involve rotating or flipping. Shapes made with five squares
are called pentominoes.
46
Problem-solving in mathematics
Prim-Ed Publishing®
USING SQUARES
This shape is a square
. Find 4 squares.
Make a larger square using 4
.
Sa
m
pl
e
What other shapes can you make using 4 squares?
Here is one.
How many other shapes can you make?
Vi
ew
in
g
Draw each one.
How many shapes did you make?
Prim-Ed Publishing速
Problem-solving in mathematics
47
USING 5 SQUARES
This shape is a square.
Find 5 squares.
What shapes can you make using 5
?
Sa
m
pl
e
Here is one.
Vi
ew
in
g
Draw each shape.
How many shapes did you make?
48
Problem-solving in mathematics
Prim-Ed Publishing速
GROWING SHAPES
Make each shape. How many squares are in each shape?
Shape 1
Shape 3
g
Look for a pattern.
Sa
m
pl
e
Shape 2
ew
in
Make Shape 4. Draw it.
Vi
How many squares are in it?
Make Shape 5. Draw it.
How many squares are in it?
Prim-Ed Publishing速
Problem-solving in mathematics
49
TEACHER NOTES
Discussion
Curriculum links
Possible difficulties
• Figuring out how far was really travelled when
moving forward and back
Page 51
The first problem involves the character climbing three
metres and then slipping down one metre so that, each
day, the distance travelled is two metres. As the tree is
eight metres high, it will take exactly four days to reach
the top. The next problem involves a distance travelled of
three metres, as the snail climbs five metres but also slips
down two metres. As the rock wall is 16 metres high, it
will take six days of travel, with not all of the last day
needed to complete the climb. On the sixth day, only one
metre of travel is needed to reach the top, so there will
be time to spare.
e
To use visualisation to understand measurement.
pl
Problem-solving
Materials
Drawing materials, counters
Focus
Extension
• Make a table showing how far each animal travelled
each day.
Vi
ew
in
g
This page explores the concept of reading and interpreting
information to solve problems involving distance travelled.
Analysis of the problems reveals that the distance entails
movement back and forth; for example, climbing forward
three metres and then slipping back one metre is an
actual travelling distance of two metres. Some pupils
many need to draw a diagram or act out each problem to
fully grasp the concept.
50
Problem-solving in mathematics
Prim-Ed Publishing®
HOW LONG?
Cathy Caterpillar wants to
climb to the top of an 8-metre
high tree.
Vi
ew
in
g
pl
Sa
m
How long will it take her to
reach the top of the tree?
e
Each day she climbs forward
3 metres, but slips back 1
metre overnight.
Prim-Ed Publishing速
Suzy Snail is climbing a
steep 16-metre high rock
wall.
Each day she climbs
forward 5 metres but slips
back 2 metres overnight.
How long will it take her to
get to the top of the rock
wall?
Problem-solving in mathematics
51
SOLUTIONS
Note: Many solutions are written statements rather than simply numbers. This is to encourage teachers and
students to solve problems in this way.
BLOCK TIME ....................................................... page 3
YBR
YRB
SOLUTIONS
Note: Many solutions are written statements rather than simply numbers. This is to encourage teachers and
students to solve problems in this way.
WASHING DAY . ............................................... page 44
1. 8 pegs
2. 11 pegs | 677.169 | 1 |
Course Catalog
Courses
Algebra 1 A/B
A comprehensive study of all of the concepts of Algebra I required to meet state and Common Core standards. With multiple opportunities for practice and review, students easily master skills including variables, linear equations, quadratic equations, function notation, and exponential functions.
Algebra 2 A/B
Algebra 2 expands on the algebraic functions learned in Algebra I by bringing in concepts of linear, quadratic, and simultaneous equations; laws of exponents; progression; binomial theorems; and logarithms. The course units are competency-based. Learners experience new situations which they practice in a real-world environment and match to previous learning.
Consumer Mathematics
This course explains how four basic mathematical operations – addition, subtraction, multiplication, and division – can be used to solve real-life problems. It addresses practical applications for math, such as wages, taxes, money management, and interest and credit. Projects for the Real World activities are included that promote cross-curricular learning and higher-order thinking and problem-solving skills.
Geometry A/B
A comprehensive examination of geometric concepts, each lesson provides thorough explanations and builds on prior lessons. Step-by-step instruction and multiple opportunities for self-check practice develop skills and confidence in students as they progress through the course. The course features animations, which allow students to manipulate angles or create shapes, such as triangles, engage students in learning and enhance mastery. Labs extend comprehension by giving students hand-on experiences.
Integrated Math 1 A/B
These two semester-long courses are designed to enable all students at the high-school level to develop a deep understanding of the math objectives covered and leave them ready for their next steps in mathematics. The courses are built to the Common Core State Standards. The three units in Semester A advance students through the study of single-variable expressions to systems of equations, while Semester B covers functions, advanced functions, and concludes with a practical look at the uses of geometry and trigonometry.
Integrated Math 2 A/B
Building on the concepts covered in Integrated Math 1, these courses are based on proven pedagogical principles and employ sound course design to effectively help students master rules of exponents and polynomials, advanced single-variable quadratic equations, independent and conditional probability, and more. Online and offline activities combine to create an engaging learning experience that prepares high school learners for their next step in their studies of mathematics.
Integrated Math 3 A/B
Beginning with the simplification of rational and polynomial expressions, Semester A takes students through the next steps in mastering the principles of integrated math. These two semester-long courses focus on meeting Common Core objectives with engaging and interactive content. Semester B begins with the derivation of the trigonometric formula for the area of a triangle, and proceeds through the use of functions and on developing the critical thinking skills necessary to make logical and meaningful inferences from data.
Math 6 A/B
This semester-long middle school course will provide students with a deep understanding and mastery of the objectives that will prepare them for algebra. It is aligned to Common Core State Standards, and is based on best practices in the teaching of mathematics and the disciplines of STEM learning. Students will develop 21st century skills as they master ratios and proportional relationships; the number system; and number visualization. The course is highly engaging while being easy for teachers to customize and manage.
Math 7 A/B
Math 7 builds on material learned in earlier grades, including fractions, decimals, and percentages and introduces students to concepts they will continue to use throughout their study of mathematics. Among these are surface area, volume, and probability. Real-world applications facilitate understanding, and students are provided multiple opportunities to master these skills through practice problems within lessons, homework drills, and graded assignments.
Math 8 A/B
This course is designed to enable all students at the middle school level to develop a deep understanding of math objectives and leaves students ready for algebra. The first semester covers objectives in transformations, linear equations, systems of equations, and functions. The second semester focuses on scientific notation, roots, the Pythagorean Theorem and volume, and statistics and probability. The course is based on the Common Core State Standards Initiative and on a modern understanding of student learning in mathematics.
Precalculus A/B
Precalculus builds on algebraic concepts to prepare students for calculus. The course begins with a review of basic algebraic concepts and moves into operations with functions, where students manipulate functions and their graphs. Precalculus also provides a detailed look at trigonometric functions, their graphs, the trigonometric identities, and the unit circle. Finally, students are introduced to polar coordinates, parametric equations, and limits.
Probability & Statistics
This course is designed for students in grades 11 and 12 who may not have attained a deep and integrated understanding of the topics in earlier grades. Students acquire a comprehensive understanding of how to represent and interpret data; how to relate data sets; independent and conditional probability; applying probability; making relevant inferences and conclusions; and how to use probability to make decisions.
Courses
English 06 A/B
This course provides a strong foundation in grammar and the writing process. It emphasizes simple but useful composition and language mechanics strategies with multiple opportunities for modeling practical, real-world writing situations that will enable students to improve their written communication skills quickly. Through a variety of grade-appropriate reading selections, students develop a clear understanding of key literary genres and their distinguishing characteristics.
English 07 A/B
English 7 Integrates the study of writing and literature through the examination of a variety of genres. Students identify the elements of composition in the reading selections to understand their function and effect on the reader. Practice is provided in narrative and expository writing. Topics include comparison and contrast, persuasion, and cause and effect essays, as well as descriptive and figurative language. Lessons are supplemented with vocabulary development, grammar, and syntax exercises, along with an introduction to verbal phrases and research tools.
English 08 A/B
Extends the skills developed in English 7 through detailed study of parts of sentences and paragraphs to understand their importance to good writing. Students also acquire study skills such as time management and improved test-taking strategies. Other topics include punctuation, word choice, syntax, varying of sentence structure, subordination and coordination, detail and elaboration, effective use of reference materials, and proofreading.
English 09 A/B
English 9 introduces the elements of writing poems, short stories, plays, and essays. Grammar skills are enhanced by the study of sentence structure and style and by student composition of paragraphs and short essays. Topics include narration, exposition, description, argumentation, punctuation, usage, spelling, and sentence and paragraph structure.
English 10 A/B
This course focuses on using personal experiences, opinions, and interests as a foundation for developing effective writing skills. Skills acquired in English I are reinforced and refined. Literary models demonstrate paragraph unity and more sophisticated word choice. A research paper is required for completion of course. Topics include grammar, sentence and paragraph structure, organizing compositions, and the research paper.
English 11 A/B
English 11A explores the relation between American history and literature from the colonial period through the realism and naturalism eras. English 11B explores the relation between American history and literature from the modernist period through the contemporary era, and presents learners with relevant cultural and political history. Readings are scaffolded with pre-reading information, interactions, and activities to actively engage learners in the content. The lessons in both semesters focus on developing grammar, vocabulary, speech, and writing skills.
English 12 A/B
In keeping with the model established in English 11, these courses emphasize the study of literature in the context of specific historical periods, beginning with the Anglo-Saxon and medieval periods in Britain. Each lesson includes tutorials and embedded lesson activities that provide for a more engaging and effective learning experience. Semester B covers the romantic, Victorian, and modern eras. End of unit tests ensure mastery of the concepts taught in each unit, and exemptive pretests allow students to focus on content that they have yet to master.
Courses
Civics A/B
Interactive, problem-centered, and inquiry-based, each unit in Civics emphasizes the acquisition, mastery, and processing of information. Every unit features both factual and conceptual study questions, Instructional strategies include Socratic instruction, student-centered learning, and experiential learning. Topics covered range from Basic Concepts of Power and Authority and National Institutions of Government to analyses of society and citizenship.
Economics
This course covers basic economic problems such as scarcity, choice, and effective use of resources. It also covers topics on a larger scale such as market structures and international trade. It particularly focuses on the US economy and analyzes the role of the government and the Federal Reserve System.
Michigan World History and Geography A/B
Michigan high school students taking this course will get a true survey of world history. Beginning with the study of early human societies and the invention of agriculture, this course takes the students on a journey through time, from ancient societies up through the modern era. This course employs many interactive features like maps and images with clickable hot spots that students can explore to get more information about things such as regions, cities, and geographical features on a map and artistic techniques and features in famous works of art. Best of all, this course is aligned to the Michigan state standards of learning and to the English Language Arts (ELA) Standards for History and Social Studies.
Middle School U.S. History A/B
In Middle School U.S. History, learners will explore historical American events with the help of innovative videos, timelines, and interactive maps and images. The course covers colonial America through the Reconstruction period. Learners will develop historical thinking and geography skills, which they will use throughout the course to heighten their understanding of the material. Specific topics of study include the U.S. Constitution, the administrations of George Washington and John Adams, the War of 1812, and the Civil War.
Middle School World History A/B
In Middle School World History, learners will study major historical world events from early human societies through to the present day. Multimedia tools including custom videos as well as videos from the BBC, custom maps, and interactive timelines will help engage learners as they complete this year-long course. They will explore the development of early humans and early civilizations. They will be introduced to the origins of major world religions, such as Hinduism and Buddhism. Also, learners will study the medieval period. Historical thinking and geography skills will be taught and utilized throughout the course.
U.S. Government
The interactive, problem-centered, and inquiry-based units in U.S. Government emphasize the acquisition, mastery, and processing of information. Semester A units include study of the foundations of American government and the American political culture, with units 2 and 3 covering the U.S. constitution, including its roots in Greek and English law, and the various institutions that impact American politics.
U.S. History A/B
This course not only introduces students to early U.S. History, but it also provides them with an essential understanding of how to read, understand, and interpret history. For example, the first unit, The Historical Process, teaches reading and writing about history; gathering and interpreting historical sources; and analyzing historical information. While covering historical events from the founding events and principles of the United States through contemporary events, the course also promotes a cross-disciplinary understanding that promotes a holistic perspective of U.S. History.
World Geography A/B
In an increasingly interconnected world, equipping students to develop a better understanding of our global neighbors is critical to ensuring that they are college and career ready. These semester-long courses empower students to increase their knowledge of the world in which they live and how its diverse geographies shape the international community. Semester A units begin with an overview of the physical world and the tools necessary to exploring it effectively. Subsequent units survey each continent and its physical characteristics and engage students and encourage them to develop a global perspective.
World History A/B
In World History, learners will explore historical world events with the help of innovative videos, timelines, and interactive maps and images. Learners will develop historical thinking skills and apply them to their study of European exploration, the Renaissance the Reformation, and major world revolutions. They will also study World War I, World War II, the Cold War, and the benefits and challenges of living in the modern world.
World History Survey A/B
In World History Survey, learners will study major historical events from early human societies through to the present day. Multimedia tools including custom videos as well as videos from the BBC, custom maps, and interactive timelines will help engage learners as they complete this year-long course. Topics of study include early civilizations, world religions, the Renaissance, the World Wars, and the globalized world of today.
Courses
Biology A/B
Students develop a clear understanding of the sometimes complex concepts at the root of life science. Course units cover genetics and evolution, cell structure, multiple units on the diversity of life and on plant structure and function. For example, the unit on cell structure and specialization drills down into mitosis, meiosis, and cancer and carcinogens.
Chemistry A/B
The course surveys chemical theory, descriptive chemistry, and changes in matter and its properties. Students learn how to classify different states of matter as well as how atoms and compounds are structured. Additional areas of discussion include chemical energetics, measurements, bonding, stoichiometry, ionization, hydrocarbons, oxidation and reduction. A variety of simple lab experiments are included.
Earth & Space Science A/B
This course takes an in-depth look at the materials and processes that continuously shape the Earth and the Universe. It explores the effects that a growing human population has on Earth's natural resources and how scientific inquiry, technology, and environmental awareness can help to sustain our planet.
Integrated Physics & Chemistry A/B
The lessons in this course employ direct-instruction approaches. They include application and Inquiry-oriented activities that facilitate the development of higher-order cognitive skills, such as logical reasoning, sense-making, and problem solving.
Life Science A/B
This course discusses living organisms, including microorganisms, plants, animals, and humans. It looks at their relationship to one another and to the ecosystems they call home. Using scientific inquiry, students will investigate life processes, including growth and natural selection, and devise methods to promote biodiversity and sustenance of life on Earth.
Physical Science A/B
This course is all about matter and energy. It discusses the atomic and molecular structure of substances and how chemical reactions lead to changes in properties of substances. The course also models how forces affect the motion of objects, including fields of force such as gravity, electricity, and magnetism. Students will see practical applications of forces and energy as they investigate simple machines, motors, generators, and electromagnets. They will also experience how sound, light, and heat interact with different forms of matter.
Physics A/B
Physics introduces students to the physics of motion, properties of matter, force, heat, vector, light, and sound. Students learn the history of physics from the discoveries of Galileo and Newton to those of contemporary physicists. The course focuses more on explanation than calculation and prepares students for introductory quantitative physics at the college level. Additional areas of discussion include gases and liquids, atoms, electricity, magnetism, and nuclear physics.
Science 6 A/B
Science 6 is an integrated science course that covers topics selected from Earth and space science, life science, and physical science. The course discusses the structure and properties of matter, force interactions between objects, and the role gravity plays in Earth and space systems. The course also takes a look at Earth's history, the physical and biological elements of its ecosystems, and how the uneven heating of Earth from the Sun leads to its various climates and weather patterns.
Science 7 A/B
Science 7 is an integrated science course that covers topics selected from Earth and space science, life science, and physical science. The course discusses cell function and the major life processes of organisms, including nutrition, growth and development, and reproduction. The course also takes a look at chemical changes that occur in matter and energy transformations in both natural and human-made systems and how to identify them. It also investigates the factors that affect the strength of gravitational, electric, and magnetic force fields.
Science 8 A/B
Science 8 is an integrated science course that covers topics selected from Earth and space science, life science, and physical science. The course discusses genes and inheritance, the evolution of species as evidenced by fossils and rock strata, and managing energy resources on Earth. The course also takes a look at climate change and methods for confronting it, features of waves and wave technology, and the positive and negative ways that humans and technology affect the Earth and its ecosystems.
Courses
Accounting A/B
The Bureau of Labor Statistics identifies accounting as one of the best careers for job growth in the next decade. This course empowers high school students with the essential skills they need to understand accounting basics. Lessons include Account Types (assets, liabilities, expenses, etc.), Fundamentals of Bookkeeping, Financial Statements, and Careers in Accounting. Engaging and relevant, this course particularly helps both those students with an accounting career orientation, and those in need of an overview of essential accounting principles.
Applied Medical Terminology A/B
Built on the same sound pedagogy and proven course design methodologies as all of our courses, Medical Terminology helps students understand the structure and meaning of medical terms and identify medical terminology associated with various body systems. As the health care industry becomes more and more complex, developing expertise in accurately and efficiently identifying medical terms and their specific application is essential to a growing variety of health care careers. This course begins to prepare your students for those careers.
Audio Video Production 1 A/B
This course is designed to enable all students at the high school level to learn the basics of audio video production. The course will help the students develop an understanding of the industry with a focus on pre-production, production, and post-production audio and video activities. The course is based on Career and Technical Education (CTE) standards designed to help students develop technical knowledge and skills needed for success in the audio video production industry.
Audio Video Production 2 A/B
This course is designed to enable students at high school level to develop the knowledge and skills related to audio video techniques that they can use in their careers. This course discusses the elements of audio video production, preproduction activities, media production techniques, and postproduction activities. The course is based on Career Technical Education (CTE) standards designed to help students develop technical knowledge and skills needed for success in the audio video production industry.
Audio Video Production 3 A/B
This course is designed to enable all students at the high school level to students understand the basic concepts in audio video manufacturing. Students will learn about preproduction techniques, advanced production techniques, advanced post-production techniques, mastering production techniques, special effects and animation, careers, and audio video production laws. The course is based on Career Technical Education (CTE) standards designed to help students prepare for entry into a wide range of careers in audio video production.
Business Information Management A/B
This course is designed to enable students at high school level to develop information management skills that they can use during in their careers in business organizations. This course discusses career opportunities available in Business Information Management, computing technology for business, connecting through the internet, working with documents, working with spreadsheets, working with a presentation program, working with databases, web page design, and project management. The course is based on Career Technical Education (CTE) standards designed to help students develop technical knowledge and skills needed for success in the business information management industry.
Career Explorations
The 21 lessons and additional activities in this one-semester course are fundamental to ensuring career readiness on the part of your students. Covering such essentials as developing and practicing a strong work ethic, time management, communication, teamwork, and the fundamentals of workplace organizations, Career Explorations develops not just essential skills, but the confidence in themselves and their abilities to present themselves that your students need as they prepare to embark on their chosen careers.
Child Development & Parenting A/B
As adulthood and its accompanying responsibilities become closer for many of your students, this one-semester course with 12 lessons introduces them to the basics of parenting. Students will learn the nuances of parenting including learning about prenatal and postnatal care and gain insights on the nurture of children. Students will also learn about the importance of positive parenting skills, parent-child communication, and ways to use community resources for effective parenting. Activities will help your students connect leading research to real-life experience.
Computer Programming 1 A/B
Part of the Plato Courseware Career and Technical Education (CTE) Library, Computer Programming combines engaging online and offline activities in a rigorous one semester course for your high school students who may be aspiring to technical careers. Building on lessons covering the software development lifecycle and software development methodologies, the course uses online discussions, activities, and lessons to lead your students through additional key topics such as quality control, system implementation and maintenance and the increasingly important issue of system security.
Computing for College & Careers A/B
This course is designed to enable students at the high school level to develop basic computer skills that they can use during their college education and also in their careers. This course is designed to enable all students at the high school level to develop the critical skills and knowledge that they will need to be successful in careers throughout their lives. The course is based on Career and Technical Education (CTE) standards designed to help students prepare for entry into a wide range of careers and/or into postsecondary education.
Culinary Arts A/B
This course is designed to enable all students at the high school level to learn the basics of culinary arts. Students will trace the origin and development of the culinary arts. They will also discuss important contributions made by chefs, notable culinary figures, and entrepreneurs. They'll analyze how trends in society influence trends in the food service industry. In addition, they'll examine the social and economic significance of the food service industry. This course also covers topics in health, sanitation, and sanitation, culinary skills, and more. The course is based on Career and Technical Education (CTE) standards designed to help students prepare for entry into a wide range of careers in the culinary industry.
Digital & Interactive Media A/B
This is an effective and comprehensive introduction to careers in the rapidly expanding world of digital art. The course covers creative and practical aspects of digital art in 15 lessons that are enhanced with online discussions and a variety of activities. Beginning with a history of digital art, the course goes on to issues of design, color, and layout. While students will experience creation of digital art, they will also learn about converting traditional art to digital formats.
Drafting & Design A/B
From the history of drafting and design to a look at the latest in the industry's latest computer-aided tools, this course gives your students a comprehensive look at a dynamic and in-demand career. With 14 effective lessons and five engaging activities that lead to mastery of the course content, the course review and end of course assessment help ensure that mastery. The course features skill-embedded content that connects student learning to real-life experiences.
Electronic Communication Skills
This semester-long course is based on Career and Technical Education (CTE) standards to help students prepare for entry into a wide range of careers and/or into postsecondary education. It is designed to enable students at high school level to develop electronic communication skills that they can use in their careers.
Entrepreneurship A/B
This course is based on Career Technical Education (CTE) standards designed to help students understand the roles and attributes of an entrepreneur, marketing and its components, selling process, and operations management. This course discusses entrepreneurship and the economy, marketing fundamentals, managing customers, production and operations management, money, and business law and taxation.
Essential Career Skills
This course helps students understand and practice critical life and workplace readiness skills identified by employers, state boards of education, and Advance CTE. These skills include personal characteristics, such as positive work ethic, integrity, self-representation, and resourcefulness, as well as key people skills, communication skills, and broadly-applicable professional and technical skills. These skills are universally valuable but sometimes assumed or glossed over in more career-specific courses. For that reason, this provides students with a solid foundation in their career studies.
Game Development
Are any of your students gamers? That's what we thought. In this course, they'll learn the ins and outs of game development to prepare them for a career in the field. Whether it is the history of video games, character development, mobile game design, user interface design, social gaming, or the principles of development design and methodologies, this 20-lesson course covers it all. As you might guess, games are included in the course to enhance the learning experience and help assess student progress. While fun and highly engaging, the course focuses on laying a strong foundation for a career in game development.
Graphic Design & Illustration A/B
This course will help students develop an understanding of the industry with a focus on topics such as history of graphic design, types of digital images, graphic design tools, storing and manipulating images, design elements and principles, copyright laws, and printing images. The course is based on Career Technical Education (CTE) standards designed to help students develop technical knowledge and skills needed for success in the graphic design industry.
Health Science 1 A/B
The course is based on Career and Technical Education (CTE) standards to help students develop technical knowledge and skills needed for success in the health science industry. Semester A is designed to enable all students at the high-school level to understand the basic structure and function of the human body and it will help the students identify and analyze the diseases and medical procedures related to each body system. Semester B will help the students develop an understanding of biomolecules such as proteins, carbohydrates, and lipids; biological and chemical processes; and various diseases that affect the body.
Health Science 2 A/B
This course is designed to enable all students at the high-school level to learn the basics of health science. The course will help the students develop an understanding of the academic qualifications, personal skills, training, and use of healthcare tools required to work in the healthcare industry. The course is based on Career and Technical Education (CTE) standards to help students develop technical knowledge and skills needed for success in the healthcare industry.
Introduction to Android Mobile App Development
This course is designed to introduce students to the process involved in creating a mobile platforms for developing Android mobile apps. Further, they learn about the Android development environment. Finally, they create the user interface of an app and make it interactive in Android Studio.
Introduction to Criminology
Introduction to Criminology is a one-semester course with 14 lessons that cover the theories related to criminology. The target audience for this course is high school students. This course covers subject areas such as: classical theory, positivist theory, punishing offenders, routine activity theory, labeling theory, social disorganization theory, peacemaking criminology, and many more.
Introduction to Cybersecurity
This Elective course introduces students to the field of cybersecurity, focusing primarily on personal computer use and vulnerabilities while also highlighting the wider scope of cybersecurity from a societal and career perspective. Specific topics include computer security, VPN and wireless security, risk management, and laws, standards, and ethics related to cybersecurity.
Introduction to Finance
This course is designed to enable students at high school level to develop financial skills that they can use during in their careers in business organizations. Financial The course is based on Career Technical Education (CTE) standards designed to help students develop technical knowledge and skills needed for success in the finance industry.
Introduction to iOS Mobile App Development
This course is designed to introduce students to the process involved in creating an various platforms for developing iOS mobile apps. Further, they learn about the iOS development environment. Finally, they create the user interface of an app and make it interactive in Xcode.
Marketing, Advertising, & Sales
Issues in marketing, advertising, and sales promotion are evolving rapidly in an increasingly digital environment. This course effectively helps your students prepare for a career in that environment through a comprehensive look at essential marketing principles, interactive tools and channels, and the growing impact of data in marketing and advertising. Simple to manage and easy to customize, the course provides an overview of all of the fundamental topics necessary to effectively put your students on a career path that unleashes their creativity and develops and leverages their critical thinking skills.
Principles of Agriculture, Food, & Natural Resources A/B
Throughout this course, your students will learn about various career options in the agriculture, food, and natural resources industries. They will learn about technology, safety, and regulatory issues in agricultural science. They will also learn about some topics related to agriculture, such as international agriculture and world trade, sustainability, environmental management, research, development, and future trends in the industry. The course helps students navigate the rising demand for sustainable food sources while also meeting the challenge of producing higher yields to feed a growing world.
Principles of Architecture & Construction A/B
This interactive course empowers students with the knowledge to appreciate and evaluate career opportunities in architecture and construction. With an emphasis on developing critical thinking skills, this course includes a variety of activities as students learn about structures and loads, materials and costs, urban design, and other aspects of these fascinating career opportunities. This easy-to-manage course will help build a solid foundation for their career options.
Principles of Arts, Audio/Video Technology, & Communications A/B
This course appeals to your students' familiarity with a variety of sensory inputs and stimulus. With an emphasis on visual arts, the 14 lessons introduce learners to careers in design, photography, performing arts, fashion, and journalism, among others. This engaging course covers inherently engaging topics that will stimulate your students as they consider careers in which the arts, technology, and communications intersect.
Principles of Business, Marketing, & Finance A/B
This course has a broad application for almost every career path that your students might choose. This course supplies both essential career skills and life skills. Designed for early high school students, the course offers you the flexibility to customize it to the unique needs of your program and your students. Interactive games and other engaging online and offline activities make practical real-life application of essential business principles understandable useful in the daily lives of your students and in the careers that they choose.
Principles of Education & Training A/B
This course is designed to enable all students at the high school level to learn the basics of education and training. Students will learn about the various trends and factors that influence the education industry. This course introduces various career opportunites in the field of education. The units in this course include personal and professional skills needed in various education careers, child growth and development, child health, delivering instruction, and technolgy in education.The course is based on Career Technical Education (CTE) standards designed to help students develop technical knowledge and skills needed for success in the education industry.
Principles of Engineering & Technology A/B
This easy-to-manage course provides students with essential STEM knowledge and an effective overview of STEM careers. The course's 15 lessons are interspersed with activities and online discussions that engage learners and promote understanding and achievement. Topics covered include biotechnology, mechanics, and fluid and thermal systems. The concluding lesson provides a valuable overview of the overall engineering design process.
Principles of Government & Public Administration A/B
This course is designed to enable all students at the high school level to learn the basics of government and public administration. Students explore career opportunities in the field of government and public administration. They also learn about the career-related skills, such as job acquisition skills, reading and writing, and mathematics they need to possess as professionals in this field. They learn about the safe and healthy working conditions necessary in the field of government and public administration. This course covers topics such as: the influence of geography and technology, and networking and communication as they relate to government and public administration. The course is based on Career and Technical Education (CTE) standards designed to help students prepare for entry into a wide range of careers in government and public administration industry.
Principles of Health Science A/B
With an engaging and interactive instructional approach, this rigorous course provides your students with a comprehensive overview of health science topics and careers. Health science professionals are in increasing demand and of increasing interest, and this semester-long course is an effective way to introduce students to the wide array of health science careers. Beginning with medical terminology, the course includes an overview of physiology and human homeostasis and more.
Principles of Hospitality & Tourism A/B
The hospitality and tourism industry offers a dynamic career path that will pique the interest of many of your students. This course emphasizes learning the practical aspects of the industry and the development of critical-thinking skills that lead to real-world solutions. This 14-lesson course will introduce your students to an exciting industry and will help them evaluate and prepare for a career in this growing and exciting industry.
Principles of Human Services A/B
This course is designed to enable all students at the high school level to develop the critical skills and knowledge necessary in the human services industry. Students will learn about various personal characteristics that they need to demonstrate in the workplace, such as integrity, and positive work ethics. This course covers topics such as employability skills, counseling and mental health services, and consumer services. The course is based on Career Technical Education (CTE) standards designed to help students prepare for entry into a wide range of careers in the human services field.
Principles of Information Technology A/B
Building on the fundamentals learned in Information Technology 1A, this course takes the next steps in preparing learners for a career in information technology. Covering software, hardware, and implementation topics, the course also addresses the security and ethical issues that your students will face in an IT career. Combining lessons, online and offline activities, and interactive discussions, the course will provide a practical yet cutting edge look at the issues faced by leading IT professionals today and in the future.
Principles of Law, Public Safety, Corrections, & Security A/B
For many reasons, high school students are drawn to learning about the careers addressed in this course. This course includes 15 lessons that help students learn about careers that make a powerful impact in all of our lives. From criminal law to every phase of the trial process, the course moves on to include lessons on the correctional system and the implications of legal ethics and the constitution.
Principles of Manufacturing A/B
Principles of Manufacturing is a course comprising of 15 lessons to help your students understand various manufacturing processes, concepts, and systems, and to introduce them to the various career paths available to them in manufacturing. This course emphasizes STEM principles while also covering practical aspects of manufacturing such as marketing and regulatory issues, as well as issues related to launching and managing a manufacturing business.
Principles of Transportation, Distribution, & Logistics A/B
In an increasingly interconnected world, this course will introduce your students to an industry that delivers what people want, when and how they want it. The TDL industry is essential to creating global economic growth through increasingly more efficient delivery of goods and services. This course will help to develop both the quantitative and qualitative skills and knowledge required for students to prepare themselves for a successful TDL career. The course addresses the relevant logistical and geopolitical issues that impact global trade.
Professional Communications
This course is designed to enable all students at the high school level to develop communication skills they will need to be successful in a profession. Students learn about the key aspects of the communication process. They learn to apply communication protocol and appropriate language skills in professional and social communication. Students also explore effective strategies to address diversity in communication. Finally, students familiarize themselves with reading, writing, speaking, and listening skills. This course covers topics such as commination in business organizations and technology for communication. The course is based on Career Technical Education (CTE) standards designed to help students prepare for communication in a wide range of professions.
Professional Photography A/B
Few recent technical innovations have changed an industry as fundamentally as digital photography has changed everything about the way we capture our lives in the way we take, edit, store, and share pictures. Digital Photography provides you with the flexibility to not only use it as an independent individual course or as a group or class course, but to also easily customize the course to the unique needs of your situation. The course combines 15 lessons with online discussions that promote the development of critical thinking skills as your students explore digital photography as an enriching activity or a career.
Sports & Entertainment Marketing
This course is designed to enable all students at the high school level to develop skills they will need to be successful in sports, entertainment, and recreational marketing professions. Students learn about the structure of a business firm and financial statements. Students also learn about the basics of sports, entertainment, and recreation marketing. Finally, students explore essential career skills, such as teamwork and time management. This course covers topics such as marketing staples, mapping markets, marketing communication, and making the sale. The course is based on Career Technical Education (CTE) standards designed to help students prepare for entry into a wide range of careers in sports, entertainment, and recreational marketing field.
Web Technologies A/B
Whether they know it or not, almost all of your students have an interest in web design. This course takes them inside the essentials of web design and helps them discover what makes a site truly engaging and interactive. Lessons such as Elements of Design, Effects of Color, and Typography help them understand the elements of effective and dynamic web design. The course covers the basics of HTML, CSS, and how to organize content, and helps to prepare them for a career in web design.
Courses
Academic Success
As in other areas of life, success in academics results from learning and practicing positive habits. This one-semester elective provides practical, hands-on guidance on developing and improving study habits and skills, regardless of a student's level of accomplishment. Academic Success includes five lessons and two course activities in a flexible structure that is adaptable to the needs and circumstances of individual students. The course can also be used for college-level developmental education.
African American Studies
This semester-long course traces the experiences of Africans in the Americas from 1500 to the present day. In this course, students will explore history, politics, and culture. Although the course proceeds in chronological order, lessons are also grouped by themes and trends in African American history. Therefore, some time periods and important people are featured in more than one lesson.
Art History & Appreciation
This course explores the main concepts of art, expression, and creativity as it helps students answer questions such as what is art; what is creativity; and how and why people respond to art. It covers essential design principles such as emphasis, balance, and unity. Units include: Art, History, and Culture; Western and World Art Appreciation; and Art and the Modern World.
Creative Writing
This course is designed to get students to pursue creative writing as a vocation or as a hobby. To that purpose, it exposes them to different genres and techniques of creative writing, as also the key elements (such as plot and characterization in fiction) in each genre. Great creative writing does not come merely by reading about the craft—one also needs ideas; a process for planning, drafting and revising; and the opportunity to experiment with different forms and genres. The lesson tutorials in this course familiarize students with the basic structure and elements of different types or genres of writing. The course is based on Career and Technical Education (CTE) standards designed to help students prepare for entry into a wide range of careers in creative writing fields.
Environmental Science A/B
This course is designed to introduce students to the history of environmental science in the United States, ecological interactions and succession , environmental change , adaptation, and biogeochemical cycles. Students will learn about the importance of environmental science as an interdisciplinary field. They will describe the importance of biodiversity to the survival of organisms, and learn about ecological pyramids . They will discuss the effects of climate change an d explore different types of adaptation . They will describe the steps of the water cycle, and discuss how carbon, oxygen, nitrogen, and phosphorous cycle in the global environment.
Gothic Literature
Gothic Literature is a one-semester course with 14 lessons that analyze the conventions, elements, themes, and other characteristics of Gothic literature. This course covers subject areas such as: morality and spirituality in gothic poetry, Dr. Jekyll and Mr. Hyde, dual personalities, Edgar Allan Poe, Dracula, gothic conventions across time, and many more.
Introduction to Anthropology
Introduction to Anthropology is a one-semester course with 14 lessons that introduce students to the field of anthropology. Students will explore the evolution of anthropology as a distinct discipline, learn about anthropological terms, concepts and theories, and discuss the evolution of humans and human society and culture. Students will also learn about social institutions, such as marriage, economy, religion, and polity. The target audience for this course is high school students.
Introduction to Archaeology
Introduction to Archaeology is a one-semester course with 14 lessons that discuss the work and techniques involved in archaeology, and the prospects of an archaeologist. This course covers subject areas such as: history of modern archaeology, discoveries in arhaeology, careers in archaeology, research tecnhiques, evidence, site excavation, and many more.
Introduction to Astronomy
Introduction to Astronomy is a one-semester course with 17 lessons that cover a wide range of topics, such as the solar system, planets, stars, asteroids, comets, galaxies, space exploration, and theories of cosmology. The target audience for this course is high school students.
Introduction to Fashion Design
From Components of Fashion to Haute Couture to Production, this course is focused on the practical aspects of career preparation in the fashion design industry. The 17 lessons in the course provide students with both breadth and depth, as they explore the full gamut of relevant topics in fashion design. Online discussions and course activities require students to develop and apply critical thinking skills while the included games appeal to a variety of learning styles and keep students engaged. Fascinating and practical, Introduction to Fashion design will appeal to, and enrich, many of your students.
Introduction to Forensic Science
This course is designed to introduce students to the importance and limitations of forensic science and explore different career options in this field. They also learn to process a crime scene, collect and preserve evidence, and analyze biological evidence such as fingerprints, blood spatter, and DNA samples. Moreover, they learn to determine the time and cause of death in homicides and analyze ballistic evidence and human remains in a crime scene. Finally, they learn about forensic investigative methods related to arson, computer crimes, financial crimes, frauds, and forgeries.
Introduction to Marine Biology
This course is designed to introduce students to oceanic features and processes, ocean habitats and ecosystems, life forms in the ocean, and different types of interactions in the ocean. Students will learn about the formation and characteristic features of the oceans. They will learn about the scientific method and explore careers available in marine biology. They will learn about the characteristic features of different taxonomic groups found in the ocean. They will learn about the different habitats, life forms, and ecosystems that exist in the oceans and explore the different types of adaptation s marine creatures possess to survive in the ocean. They will learn about succession and the flow of energy in marine ecosystems. They will also learn about the resources that the oceans provide and the threats that the oceans face from human activities.
Introduction to Philosophy
This Elective course provides students an introduction to the field of philosophy and its great, timeless questions. Students explore the origin and evolution of philosophy as a discipline and learn about the times, lives, and intellectual contributions of essential philosophers.
Introduction to Social Media
This cutting-edge course develops social media skills and knowledge that will have a practical and positive impact in helping your high school students succeed in today's economy. Of course they already engage in social media, but this course enhances their skills and knowledge in order to apply them in a practical way in their careers. Online discussions are a critical aspect of creating a collaborative learning environment, while games and other interactions ensure engagement and promote a strong career orientation.
Introduction to Veterinary Science
This course is designed to introduce all students at the high school level to the fundamentals of veterinary science, measures to control diseases in animals, and the impact of toxins and poisons on animal health . The students will explore the history of veterinary science and the skills and requirements for a successful career in the veterinary industry. They will also explore the physiology and anatomy of animals , learn how to evaluate animal health and determine effective treatments for infectious and noninfectious diseases in animals . Additionally, they will learn about zoonotic diseases , and the mapct od toxins and poisons on animal health.
Introduction to Visual Arts
This course is designed to enable all students at the high school level to familiarize themselves with different types of visual arts. The students will explore units in: Creativity and Expression in Art, Elements of Art, History of Art, Cultural Heritage of Art, Drawing, Printing, Painting, Graphic Design and Illustration, and Multimedia.
Introduction to World Religions
Introduction to World Religions is a one-semester course with 14 lessons that discuss the origins, beliefs, and practices related to various world religions. The target audience for this course is high school students. This course covers subject areas such as: primal religious traditions, sacred stories, hinduism, buddhism, judaism, christianity, islam, contemporary religious movements, and many more.
Music Appreciation
In a time of an increasing emphasis on STEM courses and skills, it remains essential to provide your students with opportunities to explore the arts from both an informational and career-oriented perspective. In Music Appreciation, students will explore the history and evolution of music, learn the elements of music and musical notations, and the contributions of popular music artists and composers. A variety of lessons, activities, and discussions will help to develop an awareness and appreciation of music that will develop not only critical thinking skills, but life enriching skills as well.
Mythology & Folklore
Introduction to Mythology and Folklore is a one-semester course with 15 lessons that discuss myths, legends, and folklore from around the world. This course covers subjects such as Mythology, Legend, Folklore, Gods and the Goddesses, natural events, and wonders of the world.
Native American Studies: Contemporary Perspectives
This course complements Native American Studies: Historical Perspectives. It explores Native American worldviews, art, media perspectives on Native Americans, and contemporary perspectives and organizations. It concludes by providing a global perspective by examining issues face by indigenous peoples throughout the world.
Native American Studies: Historical Perspectives
By providing historical perspectives, this course provides a comprehensive understanding of the roots of Native American culture. The topics addressed include an exploration of the Native American history in the arctic and subarctic, various regions of the U.S., and the development of Native American life.
Nutrition & Wellness
This course is designed to enable all students at the high school level to develop the critical skills and knowledge that they will need to be successful in careers throughout their lives. The course is based on Career and Technical Education (CTE) standards to help students prepare for entry into a wide range of careers and/or into postsecondary education.
Personal Finance
Financial
Psychology A/B
This flexible, customizable course gives your students an overview of the history of psychology while also giving them the resources to explore career opportunities in the field. Students will learn how psychologists develop and validate theories and will examine how hereditary, social, and cultural factors help form an individual's behavior and attitudes. Students will also evaluate the effectiveness of different types of psychological counseling and therapy. Highly interactive content includes online discussions that help develop critical thinking skills.
Revolutionary Ideas in Science
Revolutionary Ideas in Science is a one-semester course with 15 lessons that cover the discoveries and inventions in science from pre-historic to present times. This course covers subject areas such as: prehistoric science, technology, ancient and medieval science, the scientific revolution, thermodynamics and electricity, and many more.
Social Issues
Because the specifics of social issues change rapidly, this course is designed to have students discover contemporary and relevant perspectives on issues that may have been around for centuries. Students engage in significant research and each lesson ends with an essay assignment that encourages students to express their opinions. Topics include media, government, civil liberties, poverty, terrorism, crime, the environment, and many more.
Sociology
In this course, students will explore the evolution of sociology as a distinct discipline while learning about sociological concepts and processes. They will learn how the individual relates to and impacts society. Students will also learn about the influence of culture, social structure, socialization, and social change on themselves and others. The course combines a variety of content types, including lessons, activities, discussions, and games to engage learners as the discover sociology as a subject and as a career.
Structure of Writing
This semester-long course focuses on building good sentences. Students will learn how to put words, phrases, and clauses together and how to punctuate correctly. They will start using sentences in short compositions. As an extra bonus, students will add some new words to their vocabulary, and they will practice spelling difficult words. Near the end of the course, students are to submit a book report. Early in the course, encourage students to start looking for the books they want to read for the book report. They might also preview the introduction to that lesson so they know what will be expected.
Courses
Advanced French A/B (EdOptions Academy Only)
Our online AP French Language & Culture course is an advanced language course in which students acquire proficiencies that expand their cognitive, analytical and communicative skills. The AP French Language course prepares them for the AP French exam. Its foundation is the three modes of communication (Interpersonal, Interpretive and Presentational) as defined in the Standards for Foreign Language Learning in the 21st Century.
Advanced Spanish A/B (EdOptions Academy Only)
The AP® Spanish Language and Culture course is an advanced language course in which students are directly prepared for the AP® Spanish Language and Culture test. It uses as its foundation the three modes of communication: interpersonal, interpretive and presentational. The course is conducted almost exclusively in Spanish. The course is based on the six themes required by the College Board: (1) global challenges, (2) science and technology, (3) contemporary life, (4) personal and public identities, (5) families and communities, and (6) beauty and aesthetics. The course teaches language structures in context and focuses on the development of fluency to convey meaning. Students explore culture in both contemporary and historical contexts to develop an awareness and appreciation of cultural products, practices, and perspectives. Students should expect to listen to, read, and understand a wide-variety of authentic Spanish-language materials and sources, demonstrate proficiency in interpersonal, interpretive, and presentational communication using Spanish, gain knowledge and understanding of the cultures of Spanish speaking areas of the world, use Spanish to connect with other disciplines and expand knowledge in a wide-variety of contexts, develop insight into the nature of the Spanish language and its culture, and use Spanish to participate in communities at home and around the world. The AP® Spanish Language and Culture course is a college level course. The intensity, quality, and amount of course material can be compared to that of a third-year college course.
Chinese 1 A/B (EdOptions Academy Only)
Students beginChinese 2 A/B (EdOptions Academy Only)
Students continueFrench 1 A/B
These courses are based on a researched scope and sequence that covers the essential concepts of French. Class discussions provide an opportunity for discourse on specific topics in French. A key support tool is the Audio Recording Tool that enables students to learn a critical skill for French: listening and speaking. Beginning with learning personal greetings and continuing through practical communications exchanges, French 1B introduces students to the skills necessary to make the most of traveling to French-speaking countries.
French 2 A/B
Each of these semesters is designed to build on the principles mastered in French 1 and use a combination of online curriculum, electronic learning activities, and supporting interactive activities to fully engage learners. Unit pretests, post-tests, and end-of-semester tests identify strengths and weaknesses, helping to create a more personalized and effective learning experience. As with French 1, these 90-day courses emphasize practical communication skills while also building intercultural awareness and sensitivity.
French 3 A/B (EdOptions Academy Only)
In this expanding engagement with French, students deepen their focus on four key skills in foreign language acquisition: listening comprehension, speaking, reading, and writing. In addition, students read significant works of literature in French, and respond orally or in writing to these works Continuing the pattern, and building on what students encountered in the first two years, each week consists of a new vocabulary theme and grammar concept, numerous interactive games reinforcing vocabulary and grammar, reading and listening comprehension activities, speaking and writing activities, and multimedia cultural presentations covering major French-speaking areas in Europe and the Americas. The course has been carefully aligned to national standards as set forth by ACTFL (the American Council on the Teaching of Foreign Languages).
German 1 A/B
As with all Edmentum world language courses, German 1 A and B address two primary issues: providing a meaningful context that encourages learners to think in the target language as much as possible; and introducing grammatical concepts without over reliance on grammatical analysis. German 1A focuses on communicating basic and practical greetings and personal information. German 1B consists of five units over about 14 weeks, with an emphasis on a variety of practice types throughout the course.
German 2 A/B
According to The Economist and the Census Bureau, German-American is America's largest single ethnic group, with over 46 million Americans claiming German Ancestry. German 2 A and B tap into learners' latent interest in their cultural past, present, and future. These courses employ direct-instruction approaches, including application of the target language through activities. Each unit in the course includes a predefined discussion topic. These discussions provide an opportunity for discourse on specific topics in German.
Latin 1 A/B (EdOptions Academy Only)
Students beginLatin 2 A/B (EdOptions Academy Only)
Students continue a notable ancient myth in LatinSpanish 1 A/B
Spanish is the most spoken non-English language in U.S. homes, even among non-Hispanics, according to the Pew Research Center. There are overwhelming cultural, economic, and demographic reasons for students to achieve mastery of Spanish. Spanish 1A and B engage students and use a variety of activities to ensure student engagement and to promote personalized learning. These courses can be delivered completely online, or implemented as blended courses, according to the unique needs of the teacher and the students.
Spanish 2 A/B
Spanish 2A and B utilize three assessment tools that are designed specifically to address communication using the target language: Lesson Activities, Unit Activities, and Discussions. These tools help ensure language and concept mastery as students grow in their understanding and use of Spanish. Learning games specifically designed for language learning are used and can be accessed on a wide variety of devices.
Spanish 3 A/B
Spanish 3A and B take a unique approach by setting the lessons in each unit in a specific Spanish-speaking locale, immersing students in the language and in a variety of Hispanic cultures and issues. For example, Unit 5 in Semester B includes a discussion of the environmental issues in Argentina. Concluding the three-year cycle of Spanish courses, Spanish 3A and B effectively combine group and individual learning and offer activities and assessments to keep students engaged an on track.
Courses
Adaptive Physical Education
This course is designed specifically for students with physical limitations. The content is similar to Fitness Fundamentals 1, but additional modification resources are provided to allow for customized exercise requirements based on a student's situation. In addition, students learn the basic skills and information needed to begin a personalized exercise program and maintain an active and healthy lifestyle. Students research the benefits of physical activity, as well as the techniques, components, principles, and guidelines of exercise to keep them safe and healthy.
Advanced Physical Education 1
This course guides students through an in-depth examination of the effects of exercise on the body. Students learn how to exercise efficiently and properly, while participating in physical activities and applying principles they've learned. Basic anatomy, biomechanics, physiology, and sports nutrition are all integral parts of this course. Throughout this course students participate in a weekly fitness program involving elements of cardio, strength, and flexibility.
Advanced Physical Education 2
This course gives the student an in-depth view of physical fitness by studying subjects such as: biomechanics, nutrition, exercise programming, and exercise psychology. Students will apply what they learn by participating in a more challenging exercise requirement. Throughout this course students participate in a weekly fitness program involving elements of cardio, strength, and flexibility.
Anatomy
In this course students will explore the anatomy or structure of t he human body. In addition to learning anatomical terminology, students will study and the main systems of the body- including integumentary, skeletal, muscular, circulatory, respiratory, digestive, reproductive, and nervous systems. In addition to identifying the bones, muscles, and organs, students will study the structure of cells and tissues within the body.
Comprehensive Physical Education
In this course students will explore concepts involving personal fitness, team sports, dual sports, and individual and lifetime sports. Students will focus on health-related fitness as they set goals and develop a program to improve their fitness level through cardio, strength, and flexibility training. In addition, they will learn about biomechanics and movement concepts, as they enhance their level of skill-related fitness. Students will learn about game play concepts and specifically investigate t he rules, guidelines, and skills pertaining to soccer, softball, volleyball, tennis, walking and running, dance, and yoga. Throughout this course students will also participate in a weekly fitness program involving elements of cardio, strength, and flexibility training.
Credit Recovery Health
Credit Recovery Health is ideal for students who have had prior exposure to health, yet were unable to receive credit for their previous work by demonstrating mastery of the material. The course contains all the essential content with reduced coursework. Students learn to define mental, social, physical, and reproductive health as well as learning about drugs and safety.
Credit Recovery Physical Education 1Credit Recovery Physical Education 2Drugs & Alcohol
This course delves into the types and effects of drugs, including alcohol, tobacco, steroids, over the counter drugs, marijuana, barbiturates, stimulants, narcotics, and hallucinogens. Students learn about the physiological and psychological effects of drugs, as well as the rules, laws, and regulations surrounding them. The difference between appropriate and inappropriate drug use will also be discussed. In addition, students will learn about coping strategies, healthy behaviors, and refusal skills to help them avoid and prevent substance abuse, as well as available resources where they can seek help.
Elementary Health 1 A/B
Elementary Health 1 Health 2 A/B
Elementary Health 2 3 A/B
Elementary Health 3 4 A/B
Elementary Health 4 helps young learners establish a basic understanding of the aspects of health. Students focus on the various aspects of their health and how they can make healthy choices. Topics of study include personal safety, reducing illness, avoiding bullying, nutrition, healthy friends hips, emergency situations, and the human body.
Elementary Health 5
Elementary Health 5 helps young learners establish a basic understanding of the aspects of health. Students focus on the various aspects of their health and how they can make healthy choices. Topics of study include personal safety, reducing illness, avoiding bullying, nutrition, healthy friendships, emergency situations, and the human body.
Elementary Health Physical Education 1 A/B
Elementary PE 1 helps young learners establish a basic understanding of health and fitness. Students focus on health-related fitness and learn how to become more fit and healthy. Topics of study include exercise safety, making healthy choices, nutrition, the benefits, components and principles of fitness, basic anatomy and physiology, and values of cooperation and teamwork. In addition, students learn age-appropriate motor, non-locomotor, and manipulative skills. Student s are required to participate in regular physical activity.
Elementary Physical Education 2 A/B
Elementary PE 2 3 A/B
Elementary PE 3 4 A/B
Elementary PE 4 5 A/B
Elementary PE 5Exercise Science
This course takes an in-depth examination of the effects of exercise on the body. Through this course, students will learn basic anatomy, biomechanics, and physiology, as well as proper principles and techniques to designing an effective exercise program. The study of nutrition and human behavior will also be integrated into the course to enhance the students' comprehension of this multifaceted subject.
Family & Consumer Science
Family & Consumer Science prepares students with a variety of skills for independent or family living. Topics covered include child care, home maintenance, food preparation, money management, medical management, clothing care, and more. They also focus on household, personal, and consumer health and safety. In addition, students learn goal setting and decision-making skills, as well as explore possible career options.
Family Living & Healthy Relationships
In this course, students examine the family unit and characteristics of healthy and unhealthy relationships at different phases of life-- including information on self- discovery, family, friendships, dating and abstinence, marriage, pregnancy, and parenthood. Students learn about the life cycle and the different stages of development from infancy to adulthood. They also focus on a variety of skills to improve relationships and family living, including coping skills, communication skills, refusal skills, babysitting, parenting, and healthy living and disease prevention habits.
First Aid & Safety
In this course, students learn and practice first aid procedures for a variety of common conditions, including muscular, skeletal, and soft tissue injuries. In addition, students learn how to appropriately respond to a variety of emergency situations. They also learn the procedures for choking and CPR for inf ants, children, and adults. In addition to emergency response, students will explore personal, household, and outdoor safety, and disaster preparedness.
Fitness Basics 1
This course provides students with a basic understanding of fitness and nutrition. Students will learn about exercise safety, team and individual sports, nutrition, and the importance of staying active throughout t heir lifetime. Students conduct fitness assessments, set goals, develop their own fitness program, and participate in weekly physical activity.
Fitness Basics 2
This course provides students with a basic understanding of fitness and nutrition. Students will learn about exercise safety, team and individual sports, nutrition, and the importance of staying active throughout their lifetime. Students conduct fitness assessments and participate in weekly physical activity.
Fitness Fundamentals 1
This course is designed to provide students with the basic skills and information needed to begin a personalized exercise program and maintain an active and healthy lifestyle. Students participate in pre- and post fitness assessments in which they measure and analyze their own levels of fitness based on the five components of physical fitness: muscular strength, endurance, cardiovascular fitness, flexibility, and body composition. In this course, students research the benefits of physical activity, as well as the techniques, principles, and guidelines of exercise to keep them safe and healthy. Throughout this course students participate in a weekly fitness program involving elements of cardio, strength, and flexibility training.
Fitness Fundamentals 2
This course takes a more in-depth look at the five components of physical fitness touched on in Fitness Fundamentals 1: muscular strength, endurance, cardiovascular health, flexibility, and body composition. Th is course allows students to discover new interests as they experiment with a variety of exercises in a non-competitive atmosphere. By targeting different areas of fitness, students increase their understanding of health habits and practices and improve their overall fitness level. Students take a pre- and post-fitness assessment. Throughout this course students also participate in a weekly fitness program involving elements of cardio, strength, and flexibility.
Flexibility Training
This course focuses on the often-neglected fitness component of flexibility. Students establish their fitness level, set goals, and design their own flexibility training program. They study muscular anatomy and learn specific exercises to stretch each muscle or muscle group. Students focus on proper posture and technique while training. They also gain an understanding of how to apply the FITT principles to flexibility training. This course explores aspects of static, isometric, and dynamic stretching, as well as touch on aspects of yoga and Pilates. This course also discusses good nutrition and effective cross-training. Students take a pre- and post fitness assessment. Throughout this course students also participate in a weekly fitness program involving flexibility training, as well as elements of cardio and strength training.
Group Sports
This course provides students with an overview of group sports. Students learn about a variety of sports, yet do an in-depth study of soccer, basketball, baseball/softball, and volleyball. Students learn not only the history, rules, and guidelines of each sport, but practice specific skills related to each sport. Students also learn about sportsmanship and teamwork. In addition, students study elements of personal fitness, goal setting, sport safety, and sports nutrition. Students conduct fitness assessments and participate in regular weekly physical activity.
Health & Personal Wellness
This comprehensive health semester. Other topics of study include substance abuse, safety and injury prevention, environmental health, and consumer health.
Health Careers
In this course, students explore a variety of career options related to the health care field, including medicine, nursing, physical therapy, pharmacy, dental careers, sports medicine, personal training, social work, psychology, and more. Students will learn about various options within each field, what each of these jobs entails, and the education and knowledge required to be successful. In addition, they will focus on basic job skills and information that would aid them in health care and other career paths.
Hope 1Hope 2Individual Sports
This course provides students with an overview of individual sports. Students learn about a variety of sports, yet do an in-depth study of running, walking, hiking, yoga, dance, swimming, biking, and cross-training. Students learn not only the history, rules, and guidelines of each sport, but practice specific skills related to each sport. Students also learn about the components of fitness, the FITT principles, benefits of fitness, safety and technique, and good nutrition. Students conduct fitness assessments and participate in weekly physical activity.
Intro to Coaching
This course focuses on the various responsibilities of a coach and the skills needed to successfully fill this important position. Throughout the course, students will explore various coaching models and leadership styles, sports nutrition and sports psychology, as well as safety, conditioning, and cross-training. Students will learn effective communication, problem-solving, and decision making skills. The course will also introduce students to game strategy, tactical strategy, skills-based training, and coaching ethics.
Intro to Group Sports 1
This course provides students with an overview of group sports. Students learn about a variety of sports, and an in-depth study of soccer of basketball. Students learn not only the history, rules, and guidelines of each sport, but practice specific skills related to each sport. Students also learn about game strategy and the benefits of sports. In addition, students study elements of personal fitness, goal setting, sport safety, and sports nutrition. Students conduct a pre- and post-fitness assessment, as well as participate in regular weekly physical activity.
Intro to Group Sports 2
This course provides students with an overview of group sports. Students learn about a variety of sports and do an in-depth study of baseball/softball, and volleyball. Students learn the history, rules, and guidelines of each sport, as well as practice specific skills related to each sport. Students also learn about sportsmanship and teamwork. In addition, students study elements of personal fitness, goal setting, sport safety, and sports nutrition. Students conduct a pre- and post-fitness assessment, as well as participate in regular weekly physical activity.
Intro to Individual Sports 1 Individual Sports 2 Nursing 1
This two semester course introduces students crisis management will be included.
Intro to Nursing 2
This two semester course introduces student s and crisis management will be included.
Life Skills
This course allows students to explore their personality type and interests, as well as refine important skills that will benefit them throughout their lives, including personal nutrition and fitness skills, time & stress management, communication & healthy relationships, goal setting, study skills, leadership and service, environmental and consumer health, and personal finances. In addition, students will explore possible colleges and careers that match their needs, interests, and talents.
Lifetime & Leisure Sports
This course provides students with an overview of dual and individual sports. Students learn about a variety of sports, and do an in-dept h study of martial arts, Pilates, fencing, gymnastics, and water sports. Students learn not only the history, rules, and guidelines of each sport, but practice specific skills related to many of these sports. Students also learn the components of fitness, benefits of fitness, safety and technique, and good nutrition. Students conduct fitness assessments, set goals, and participate in weekly physical activity.
Medical Terminology
In this course students will be introduced to basic medical language and terminology that they would need to enter a health care field. Emphasis will be placed on definitions, proper usage, spelling, and pronunciation. They will study word structure and parts, including roots, prefixes, and suffixes, as well as symbols and abbreviations. They will examine medical terms from each of the body's main systems, including skeletal, muscular, cardiovascular, respiratory, digestive, urinary, nervous, endocrine, reproductive, and lymphatic systems, and sensory organs. In addition, students will learn proper terminology for common tests, procedures, pharmacology, disease, and conditions.
Middle School Health
Middle School Health aids students in creating a foundation of personal health. Beginning with properly defining health, this course then builds upon basic health practices to emphasize the importance of balance. Attention is given to each of the six dimensions of wellness; namely, physical, intellectual, emotional, spiritual, social, and environmental. Students are taught the skills necessary to improve every aspect of health. They are also encouraged to reflect upon their own personal wellness each week.
Nutrition
This course takes students through a comprehensive study of nutritional principles and guidelines. Students will learn about world- wide views of nutrition, nutrient requirements, physiological processes, food labeling, healthy weight management, diet-related diseases, food handling, nutrition for different populations, and more. Students will gain important knowledge and skills to aid them in attaining and maintaining a healthy and nutritious lifestyle.
Outdoor Sports
This course provides students with an overview of dual and individual sports. Students learn about a variety of sports, and do an in- depth study of hiking and orienteering, golf, and dual volleyball. Students learn not only the history, rules, and guidelines of each sport, but practice specific skills related to many of these sports. Students also learn the FITT principles, benefits of fitness, and safety and technique. Students conduct fitness assessments, set goals, and participate in weekly physical activity.
Personal Health & Fitness
This combined health and PE course prov idesPersonal Training Career Prep
This course examines the role and responsibilities of a personal trainer. Students will learn the steps to become a personal trainer, including performing fitness assessments, designing safe and effective workouts, and proper nutrition principles. Concepts of communication and motivation will be discussed, as well as exercise modifications and adaptations for special populations. Students will also examine certification requirements, business and marketing procedures, and concerns about liability and ethics. In addition, throughout the course students will be able to explore various exercises, equipment, and tools that can be used for successful personal training.
Personal Training Concepts
This course examines basic concepts in fitness that are important for personal fitness, as well as necessary foundational information for any health or exercise career field. Areas of study include musculoskeletal anatomy and physiology, terms of movement, basic biomechanics, health related components of fitness, FITT principles, functional fitness skills, safety and injury prevention, posture and technique, nutrition, and weight management.
Physiology
In this course, students will examine the f unctions of the body's biological systems--including skeletal, muscular, circulatory, respiratory, digestive, nervous, and reproductive systems. In addition to understanding the function of each system, students will learn the function of cells, blood, and sensory organs, as well as study DNA, immunity, and metabolic systems.
Running
This course is appropriate for beginning, intermediate, and advanced runners and offers a variety of training schedules for each. In addition to reviewing the fundamental principles of fitness, students learn about goals and motivation, levels of training, running mechanics, safety and injury prevent ion, appropriate attire, running in the elements, good nutrition and hydration, and effective cross-training. While this course focuses mainly on running for fun and fitness, it also briefly explores the realm of competitive racing. Students conduct fitness assessments and participate in weekly physical activity.
Sports Officiating
In this course, students will learn the rules, game play, and guidelines for a variety of sports, including soccer, baseball, softball, basketball, volleyball, football, and tennis. In addition, they will learn the officiating calls and hand signals for each sport, as well as the role a sport official plays in maintaining fair play.
Strength Training
This one-semester course by Carone Fitness focuses on the fitness components of muscular strength and endurance. Throughout this course students establish their fitness level, set goals, and design their own resistance training program. They study muscular anatomy and learn specific exercises to strengthen each muscle or muscle group. Students focus on proper posture and technique while training. They also gain an understanding of how to apply the FITT principles and other fundamental exercise principles, such as progression and overload, to strength training.
Walking Fitness
This course helps students establish a regular walking program for health and fitness. Walking is appropriate for students of all fitness levels and is a great way to maintain a moderately active lifestyle. In addition to re viewing fundamental principles of fitness, students learn about goals and motivation, levels of training, walking mechanics, safety and injury prevention, appropriate attire, walking in the elements, good nutrition and hydration, and effective cross-training. Students take a pre- and post-fitness assessment. Throughout this course students also participate in a weekly fitness program involving walking, as well as elements of resistance training and flexibility.
Courses
Health
This course is based on a rigorously researched scope and sequence that covers the essential concepts of health. Students are provided with a variety of health concepts and demonstrate their understanding of those concepts through problem solving. The five units explore a wide variety of topics that include nutrition and fitness, disease and injury, development and sexuality, substance abuse, and mental and community health.
Physical Education
This course's three units include Getting Active, Improving Performance, and Lifestyle. Unit activities elevate students' self-awareness of their health and well-being while examining topics such as diet and mental health and exploring websites and other resources. In addition to being effective as a stand-alone course, the components can be easily integrated into other health and wellness courses.
ACT® English
The ACT assesses high school students' general educational development and their ability to complete college-level work. Our course prepares students to take the test by learning the content ideas they will be tested on.
ACT® Mathematics
The ACT assesses high school students' general educational development and their ability to complete college-level work. Our course prepares students to take the test by learning the content ideas they will be tested on.
ACT® Reading
The ACT assesses high school students' general educational development and their ability to complete college-level work. Our course prepares students to take the test by learning the content ideas they will be tested on.
ACT® Science Reasoning
The ACT assesses high school students' general educational development and their ability to complete college-level work. Our course prepares students to take the test by learning the content ideas they will be tested on.
ACT® WORKKEYS
WorkKeys is a job skills assessment system that helps employers select, hire, train, and retain a high-performance workforce. WorkKeys scores help compare a learner's skills to the skills real jobs require.
Advanced Biology A/B
To generate skills for lifelong learning, 25 percent of the lessons in Advanced Biology use student-driven, constructivist approaches for concept development. The remaining lessons employ direct-instruction approaches. In both cases, the lessons incorporate multimedia-rich, interactive resources to make learning an engaging experience. The AP approach to advanced biology topics helps students achieve mastery of abstract concepts and their application in everyday life and in STEM-related professions.
Advanced Calculus A/B
This course grounds the study of calculus in real-world scenarios and integrates it with the four STEM disciplines. The first semester covers functions, limits, derivatives and the application of derivatives. The course goes on to cover differentiation and antidifferentiation, applications of integration, inverse functions, and techniques of integration.
Advanced Chemistry A/B
Advanced Chemistry includes most of the 22 laboratory experiments recommended by the College Board to provide a complete advanced experience in a blended environment. More than 25 percent of the online lesson modules are inquiry-based and employ online simulations, data-based analysis, online data-based tools, and ―kitchen sink labs that require no specialized equipment or supervision. Many of the lessons include significant practice in stoichiometry and other critical, advanced chemistry skills.
Advanced Computer Science A
This course is designed to introduce students to the basic concepts of computer programming. Students learn how to compile and run a Java program. They learn to use arithmetic, relational, and logical operators. They learn to use different decision-making and loop statements. They learn to create classes, methods, String objects, and an ArrayList object. They learn to perform sequential search, binary search, selection sort, and insertion sort on an array. They learn to implement object-oriented programming design. They learn to implement inheritance, polymorphism, and abstraction. Further, they describe privacy and legality in the context of computing.
Advanced English Lit & Comp A/B
Each unit of Advanced English Literature and Composition is based on a researched scope and sequence that covers the essential concepts of literature at an AP level. Students engage in in-depth analysis of literary works in order to provide both depth and breadth of coverage of the readings. Units include Close Analysis and Interpretation of Fiction, Short Fiction, the Novel, and Poetic Form and Content. Writing activities reinforce the reading activities and include writing arguments, analysis, interpretation, evaluation, and college application essays.
Advanced U.S. History A/B
This course develops critical thinking skills by encouraging multiple views as students realized that there are often multiple accounts of a single historical event that may not be entirely consistent. Electronic discussion groups encourage collaboration, and a variety of practice activities are provided, from multiple choice actions to advanced interactions. Units include: The Historical Process; Early America; Revolutionary America; The Civil War; Populism and Progressivism; the emergence of the U.S. as a world power; and contemporary themes.
SAT Reading Language Arts MathematicsTASC Preparation - Social Studies Part 2
Courses
Anthropology 1: Uncovering Human Mysteries
Anthropology uses a broad approach to give students an understanding of our past, present, and future, and also addresses the problems humans face in biological, social, and cultural life. This course explores the evolution, similarity, and diversity of humankind through time. It looks at how we have evolved from a biologically and culturally weak species to one that has the ability to cause catastrophic change. Exciting online video journeys are just one of the powerful learning tools utilized in this course.
Anthropology 2: More Human Mysteries Uncovered
This course continues the study of global cultures and the ways that humans have made sense of their world. It examines ways that cultures have understood and given meaning to different stages of life and death. The course also examines the creation of art within cultures and how cultures evolve and change over time. Finally, students apply the concepts and insights learned from the study of anthropology to several cultures found in the world today.
Archaeology: Detectives of the Past
The field of archaeology helps us better understand the events and societies of the past that have helped to shape theArt in World Cultures
Who is the greatest artist of all time? Is it Leonardo daVinci? Claude Monet? Michelangelo? Pablo Picasso? Is the greatest artist of all time someone whose name has been lost to history? You will learn about some of the greatest artists while also creating art of your own, including digital art. We will explore the basic principles and elements of art, learn how to critique art, and examine some of the traditional art of the Americas, Africa, and Oceania in addition to the development of Western art.
Astronomy: Exploring the Universe
Why do stars twinkle? Is it possible to fall into a black hole? Will the sun ever stop shining? Since the first glimpse of the night sky, humans have been fascinated with the stars, planets, and universe that surrounds us. This course will introduce students to the study of astronomy, including its history and development, basic scientific laws of motion and gravity, the concepts of modern astronomy, and the methods used by astronomers to learn more about the universe. Additional topics include the solar system, the Milky Way and other galaxies, and the sun and stars. Using online tools, students will examine the life cycle of stars, the properties of planets, and the exploration of space.
Biotechnology: Unlocking Nature's Secrets
In today's world, biotechnology helps us grow food, fight diseases, and create alternative fuels. In this course, students will explore the science behind biotechnology and how this science is being used to solve medical and environmental problems.
Careers in Criminal Justice
The criminal justice system offers a wide range of career opportunities. In this course, students will explore different areas of the criminal justice system, including the trial process, the juvenile justice system, and the correctional system.
Cosmetology: Cutting Edge Styles
Interested in a career in cosmetology? This course provides an introduction to the basics of cosmetology. Students will explore career options in the field of cosmetology, learn about the common equipment and technologies used by cosmetologists, and examine the skills and characteristics that make someone a good cosmetologist. Students will also learn more about some of the common techniques used in caring for hair, nails, and skin in salons, spas, and other cosmetology related businesses.
Criminology: Inside the Criminal Mind
Crime and deviant behavior rank at or near the top of many people's concerns. This course looks at possible explanations for crime from the standpoint of psychological, biological and sociological perspectives, explore the categories and social consequences of crime, and investigate how the criminal justice system handles not only criminals, but also their crimes. Why do some individuals commit crimes and others do not? What aspects in our culture and society promote crime and deviance? Why are different punishments given for the same crime? What factors shape the criminal case process?
Digital Photography 1: Creating Images with Impact
Digital Photography I focuses on the basics of photography, including building an understanding of aperture, shutter speed, lighting, and composition. Students will be introduced to the history of photography and basic camera functions. Students use basic techniques of composition and camera functions to build a personal portfolio of images, capturing people, landscapes, close-ups, and action photographs.
Digital Photography 2: Discovering Your Creative Potential
In this course, we examine various aspects of professional photography, including the ethics of the profession, and examine some of the areas in which professional photographers may choose to specialize, such as wedding photography and product photography. Students also learn about some of the most respected professional photographers in history and how to critique photographs in order to better understand what creates an eye-catching photograph.
Early Childhood Education
Children experience enormous changes in the first few years of their lives. They learn to walk, talk, run, jump, read and write, among other milestones. Caregivers can help infants, toddlers, and children grow and develop in positive ways. This course is for students who want to influence the most important years of human development. In the course, students learn how to create fun and educational environments for children; how to keep the environment safe for children; and how to encourage the health and well-being of infants, toddlers, and school-aged children.
Fashion & Interior Design
Do you have a flair for fashion? Are you constantly redecorating your room? If so, the design industry might just be for you! In this course, you'll explore what it is like to work in the industry by exploring career possibilities and the background that you need to pursue them. Get ready to try your hand at designing as you learn the basics of color and design then test your skills through hands-on projects. In addition, you'll develop the essential communication skills that build success in any business. By the end of the course, you'll be well on your way to developing the portfolio you need to get your stylishly clad foot in the door of this exciting field.
Forensic Science 1: Secrets of the Dead
In this unit, students are introduced to forensic science. We discuss what forensic science consists of and how the field developed through history. Topics covered include some of the responsibilities of forensic scientists and about some of the specialty areas that forensic scientists may work in. Objective and critical thinking questions are combined with lab activities to introduce students to analyzing the crime scene, a wide variety of physical evidence such as firearm and explosion evidence, and DNA evidence.
Forensic Science 2: More Secrets of the Dead
Although the crime scene is the first step in solving crimes through forensic science, the crime laboratory plays a critical role in the analysis of evidence. This course focuses on the analysis of evidence and testing that takes place within the lab. It examines some of the basic scientific principles and knowledge that guide forensic laboratory processes, such as those testing DNA, toxicology, and material analysis. Techniques such as microscopy, chromatography, odontology,, mineralogy, and spectroscopy will be examined.
Gothic Literature: Monster Stories
From vampires to ghosts, frightening stories have influenced fiction writers since the 18th century. This course focuses on the major themes found in Gothic literature and demonstrates how core writing drivers produce thrilling psychological environments for the reader. Terror versus horror, the influence of the supernatural, and descriptions of the difference between good and evil are just a few of the themes presented. By the time students have completed this course, they will have gained an understanding of and an appreciation for the complex nature of dark fiction.
Great Minds in Science: Ideas for a New Generation
Is there life on other planets? What extremes can the human body endure? Can we solve the problem of global warming? Today, scientists, explorers, and writers are working to answer all of these questions. Like Edison, Einstein, Curie, and Newton, scientists of today are asking questions and working on problems that may revolutionize our lives and world. This course focuses on 10 of today's greatest scientific minds. Each unit takes an in-depth look at one of these individuals, and shows how their ideas may help to shape tomorrow's world.
History of the Holocaust
Holocaust education requires a comprehensive study of not only times, dates, and places, but also the motivation and ideology that allowed these events. In this course, students will study the history of anti-Semitism; the rise of the Nazi party; and the Holocaust, from its beginnings through liberation and the aftermath of the tragedy. The study of the Holocaust is a multi disciplinary one, integrating world history, geography, American history, and civics. Through this in-depth, semester-long study of the Holocaust, high school students will gain an understanding of the ramifications of prejudice and indifference, the potential for government-supported terror, and they will get glimpses of kindness and humanity in the worst of times.
Hospitality & Tourism: Traveling the Globe
With greater disposable income and more opportunities for business travel, people are traversing the globe in growing numbers. As a result, hospitality and tourism is one of the fastest growing industries in the world. This course will introduce students to the hospitality and tourism industry, including hotel and restaurant management, cruise ships, spas, resorts, theme parks, and other areas. Student will learn about key hospitality issues, the development and management of tourist locations, event planning, marketing, and environmental issues related to leisure and travel. The course also examines some current and future trends in the field.
Human Geography: Our Global Identity
How do language, religion, and landscape affect the physical environment? How do geography, weather, and location affect customs and lifestyle? Students will explore the diverse ways in which people affect the world around them and how they are affected by their surroundings. Students will discover how ideas spread and cultures form, and learn how beliefs and architecture are part of a larger culture complex. In addition to introducing students to the field of Human Geography, this course will teach students how to analyze humans and their environments.
International Business: Global Commerce in the 21st Century
From geography to culture, Global Business is an exciting topic. This course helps students develop the appreciation, knowledge, skills, and abilities needed to live and work in a global marketplace. Business structures, global entrepreneurship, business management, marketing, and the challenges of managing international organizations are all explored in this course. Students cultivate an awareness of how history, geography, language, cultural studies, research skills, and continuing education are important in business activities and the 21st century.
Introduction to Agriscience
Agriculture has played an important role in the lives of humans for thousands of years. It has fed us and given us materials that have helped us survive. Today, scientists and practitioners are working to improve and better understand agriculture and how it can be used to continue to sustain human life. In this course, students learn about the development and maintenance of agriculture, animal systems, natural resources, and other food sources. Students also examine the relationship between agriculture and natural resources and the environment, health, politics, and world trade.
Introduction to Culinary Arts
Food is fundamental to life. Not only does it feed our bodies, but it's often the centerpiece for family gatherings and social functions with friends. In this course, you will learn all about food including food culture, food history, food safety, and current food trends. You'll also learn about the food service industry and try your hand at preparing some culinary delights. Through hands-on activities and in-depth study of the culinary arts field, this course will help you hone your cooking skills and give you the opportunity to explore careers in this exciting industry.
Introduction to Entrepreneurship
Do you dream of owning your own business? This course can give you a head start in learning about what you'll need to own and operate a successful business of your own. Students will explore creating a business plan, financing a business, and pricing products and services. Students will also learn more about the regulations that apply to businesses, marketing products and services, and the legal and ethical guidelines that govern businesses.
Introduction to Manufacturing: Product Design & Innovation
Think about the last time you visited your favorite store. Now picture the infinite number of products you see. Have you ever wondered how all those things actually made it to the shelves? Whether video games, clothing, or sports equipment, the goods we purchase must go through a manufacturing process before they can be marketed and sold. In Introduction to Manufacturing: Product Design and Innovation, you will learn about the different types of manufacturing systems used to create the everyday products we depend on. Discover the various career opportunities in the manufacturing industry, including those for engineers, technicians, and supervisors. As a culminating project, you will plan your own manufacturing process and create an entirely original product! If you thought manufacturing was little more than mundane assembly lines, this course will show you just how exciting, creative, and practical this industry can be.
Introduction to Social Media: Our Connected World
Have a Facebook account? What about Twitter? Whether you've already dipped your toes in the waters of social media or are still standing on the shore wondering what to make of it all, learning how to interact on various social media platforms is crucial in order to survive and thrive in this age of digital communication. In this course, you'll learn the ins and outs of social media platforms such as Facebook, Twitter, Pinterest, Google+, and more. You'll also discover other types of social media you may not have been aware of and how to use them for your benefit—personally, academically, and eventually professionally as well. If you thought social media platforms were just a place to keep track of friends and share personal photos, this course 0will show you how to use these resources in much more powerful ways.
Law & Order: Introduction to Legal Studies
From traffic laws to regulations on how the government operates, laws help provide society with order and structure. Our lives are guided and regulated by our society's legal expectations. Consumer laws help protect us from faulty goods; criminal laws help to protect society from individuals who harm others; and family law handles the arrangements and issues that arise in areas like divorce and child custody. This course focuses on the creation and application of laws in various areas of society.
Middle School Career Explorations
What career are you best suited for? In this course, students will explore career options in many different fields including business, health science, public administration, the arts, and information technology.
Middle School Journalism
Who? What? When? Where? Journalism provides us with the answers to these questions for the events that affect our lives. In this course, students will learn how to gather information, organize ideas, format stories for different forms of news media, and edit their stories for publication. The course will also examine the historical development of journalism and the role of journalism in society.
Middle School Photography: Drawing with Light
"A picture is worth a thousand words." Photographs play an important role in our world today. We photograph to preserve memories, document events, and create artistic works. This course introduces students to the basics of photography, including camera functions and photo composition. Students will learn what it takes to create a good photograph and how to improve photographs of animals, people, and vacations. They will also begin working with their photographs using photo-editing software. Through a variety of assigned projects, students will engage their creativity by photographing a range of subjects and learning to see the world through the lens of their cameras.
Music Appreciation: The Enjoyment of Listening
Music is part of everyday life and reflects the spirit of our human condition. To know and understand music, we distinguish and identify cultures on local and global levels. This course provides students with an aesthetic and historical perspective of music, covering a variety of styles and developments from the Middle Ages through the 21st Century. Students acquire basic knowledge and listening skills, making their future music experiences more informed and enriching.
Peer Counseling
Helping people achieve their goals is one of the most rewarding of human experiences. Peer counselors help individuals reach their goals by offering them support, encouragement, and resource information. This course explains the role of a peer counselor, teaches the observation, listening, and emphatic communication skills that counselors need, and provides basic training in conflict resolution, and group leadership. Not only will this course prepare you for working as apeer counselor, but the skills taught will enhance your ability to communicate effectively in your personal and work relationships.
Personal & Family Finance
How do personal financial habits affect students' financial futures? How can they make smart decisions with money in the areas of saving, spending, and investing? This course introduces students to basic financial habits such as setting financial goals, budgeting, and creating financial plans. Students learn about topics such as taxation, financial institutions, credit, and money management. The course also addresses how occupations and educational choices can influence personal financial planning, and how individuals can protect themselves from identity theft.
Personal Psychology 1: The Road to Self-Discovery
Self-knowledge is the key to self-improvement. More than 800,000 high school students take psychology classes each year. Among the different reasons, there is usually the common theme of self-discovery. Sample topics include the study of infancy, childhood, adolescence, perception and states of consciousness. The course features amazing online psychology experiments dealing with our own personal behavior.
Personal Psychology 2: Living in a Complex World
This course enriches the quality of students' lives by teaching them to understand the actions of others. Topics include the study of memory, intelligence, emotion, health, stress and personality. This courses features exciting online psychology experiments involving the world around us.
Philosophy: The Big Picture
This course is an exciting adventure that covers more than 2,500 years of history. Despite their sometimes odd behavior, philosophers of the Western world are among the most brilliant and influential thinkers of all time. As students learn about these great thinkers, they'll come to see how and where many of the most fundamental ideas of Western Civilization originated. They'll also get a chance to consider some of the same questions these great thinkers pondered.
Principles of Public Service: To Serve & Protect
Ambulances scream along, heading toward those in need. But who makes sure someone is there to answer the 9-1-1 call? When you pick up a prescription or take a pill, who has determined that drug is safe for the public? All of these duties are imperative to our comfort and success as a society and an essential part of public service, a field that focuses on building a safe and healthy world. Principles of Public Service: To Serve and Protect will introduce you to many different careers in this profession and illustrate how they all work together to provide for the common good. The protection of society is one of our greatest challenges, and public service provides a way for people to work together, ensure safety, and provide an indispensable service to those around us. If you've ever contemplated being one of these real-life heroes, now is the time to learn more.
Public Speaking
The art of public speaking is one which underpins the very foundations of Western society. This course examines those foundations in both Aristotle and Cicero's views of rhetoric, and then traces those foundations into the modern world. Students will learn not just the theory, but also the practice of effective public speaking, including how to analyze the speeches of others, build a strong argument, and speak with confidence and flair. By the end of this course, students will know exactly what makes a truly successful speech and will be able to put that knowledge to practical use.
Real World Parenting
What is the best way to care for children and teach them self-confidence and a sense of responsibility? Parenting involves more than having a child and providing food and shelter. Students learn what to prepare for, what to expect, and what vital steps parents can take to create the best environment for their children. Parenting roles and responsibilities, nurturing and protective environments for children, positive parenting strategies, and effective communication in parent/child relationships are other topics covered in this course.
Social Problems 1: A World in Crisis
This course introduces students to the topic of social problems. The initial unit helps students develop an understanding of social problems, some of the characteristics common to many of them, and how those problems evolve. Social Problems 1 makes use of labs, discussions, and other learning modalities to maximize effective learning. The course looks closely at the problem of poverty and its root causes, as well as problems in education. It also examines the problem of crime, what has historically succeeded and failed in addressing it, and how to move society forward in effectively mitigating the problem.
Social Problems 2: Crisis, Conflicts & Challenges
Building on the mastery of basics students acquire in Social Problems 1, this course explores issues such as globalization, alcohol and drug abuse, gangs and cults, and the ever-present and growing issue of personal privacy and its related complexities. It also addresses issues of nutrition and health, and their impact on society's wellbeing. Discussion questions encourage the development of critical thinking skills, and better equips students for college and career by helping them better understand the issues affecting themselves and their world.
Sociology 1: The Study of Human Relationships
The world is becoming more complex. How do beliefs, values and behaviors affect people and the world in which we live? Students examine social problems in our increasingly connected world, and learn how human relationships can strongly influence and impact their lives. Exciting online video journeys are an important component of this relevant and engaging course.
Sociology 2: Your Social Life
Sociology is the study of people, social life, and society. By developing a "sociological imagination" students are able to examine how society itself shapes human action and beliefs, and how in turn these factors re-shape society itself. Fascinating online video journeys will not only inform students, but motivate them to seek more knowledge on their own.
Sports & Entertainment Marketing
Have you ever wished to play sports professionally? Have you dreamed of one day becoming an agent for a celebrity entertainer? If you answered yes to either question, then believe it or not, you've been fantasizing about entering the exciting world of sports and entertainment marketing. Although this particular form of marketing bears some resemblance to traditional marketing, there are many differences as well—including a lot more glitz and glamour! In this course, you'll have the opportunity to explore basic marketing principles and delve deeper into the multi-billion dollar sports and entertainment marketing industry. You'll learn about how professional athletes, sports teams, and well known entertainers are marketed as commodities and how some of them become billionaires as a result. If you've ever wondered about how things work behind the scenes of a major sporting event such as the Super Bowl or even entertained the idea of playing a role in such an event, then this course will introduce you to the fundamentals of such a career.
Veterinary Science: The Care of Animals
As animals play an increasingly important role in our lives, scientists have sought to learn more about their health and well-being. This course examines some of the common diseases and treatments for domestic animals. Toxins, parasites, and infectious diseases impact not only the animals around us, but at times humans as well. Through veterinary medicine and science, the prevention and treatment of diseases and health issues is studied and applied.
World Religions: Exploring Diversity
Throughout the ages, religions have shaped the political, social, and cultural aspects of societies. This course focuses on the major religions that have played a role in human history, including Buddhism, Christianity, Confucianism, Hinduism, Islam, Judaism, Shintoism, and Taosim. Students trace major developments in these religions and explore their relationships with social institutions and culture. The course also discusses some of the similarities and differences among the major religions and examines their related connections and differences. | 677.169 | 1 |
Sunday, October 28, 2012
Versions Available: Free Version $0.99 Scientific Version $0.99 Financial Version
The only difference between the Free Version and the Scientific Version is that the Free Version is ad-supported.
Keyboard
The keyboard is a simple layout of 34 keys. The keys are labeled with one function. To access the shift function, press the gold shift key. The names of the shifted function replace the primary functions. Kudos for simplicity, however, my personal preference is to see both the primary and shift functions at the same time.
RPN and Algebraic Mode
The default mode is RPN (Reverse Polish Notation), but you can always enter an algebraic formula by pressing the "=f" key. The "=f" becomes the "=" to terminate entry of the formula.
Many, many, many calculators are available
Accessing the Focused Calculator list gives many calculators such as Math and Trig (the main calculator), Time Value of Money, Bonds, Statistics, Probability and Conversion calculators. For me the Conversion calculators leave a little to be desired, since there are no direct conversion keys attached to the default keyboard (luckily, that can be remedied!).
In addition to the Focused Calculators, CalcFxC gives formula template calculates. Among these formula calculators you have Percent Change, Distribution Functions, and the Quadratic and Cubic Equations.
The keyboard gives a good response when the "keys" are pressed. The screen is a two line (adjustable) screen which displays the y-stack and x-stack. The stack is four levels.
Help can be accessed by pressing and holding a key.
Real Numbers and Limits
The calculator operates on real numbers only. So, entering √-1 will return an error. As a consequence, the polynomial solvers return only real roots. The numbers range in the order of -10^-308 to 10^308.
There are no fractions or exact values of trig functions.
Math and Trig Keyboard
This keyboard contains functions usually not found a standard scientific calculator:
sqrtpi: takes the square root of the number then multiplies it by π exmp1: e^x - 1, known on Hewlett Packard Calculators as the EXM1 function log1p: ln(x + 1), known on Hewlett Packard Calculators as the LNP1 function jn: Bessel Function of the First Kind, with x on the y-stack and the order n on the x-stack. jn also returns the Bessel Function of the Second Kind. quad: Takes three arguments from the stack (a, b, c of ax^2+bx+c) and returns the real roots.
If you want to access the hyperbolic functions, you will need to call the Math and Hyperbolic calculator.
Memory
The calculator has 27 memories: memories a through z, and a special register ra. Storage and recall arithmetic can be performed on register ra - no idea why (except for maybe programming limitations) CalcFxC decided to restrict this feature to one register.
Programmability in the Form of Customizable Calculators
CalcFxC does not offer "traditional" macro or program capability. Instead, CalcFxC offers the ability to edit and create custom keyboards. You can redefine the key's name and help screen, along with it's formula. You have access to all of the functions. To use the stack arguments, use rgx(), rgy(), rgz(), and rgt() for the x, y, z, and t stacks respectively. This is a fun feature for those who has ever wanted to design their own calculators, and I am one of them!
To emulate the stack operations properly, you will need to define the formulas for each stack.
For example, if I designate the comb function as the Combination function, I would use the following formulas: x Formula: combin(rgy(),rgx()) y Formula: rgz() z Formula: rgt() t Formula: rgt() Last x Formula: rgx()
Functions can be copied and pasted. You will need some experience with RPN calculators to take advantage of this feature.
Final Word
This app is enjoyable to use. I am probably going to spend an afternoon or evening making a custom calculator. RPN fans, this is a good calculator app to get.
Since I am not a fan of ads, I will pay the $0.99 to get the non-ad version.
Wednesday, October 24, 2012
Monday's blog entry (10/22/2012), Factorials and Arrangements of Unique Objects dealt with permutations of arranging a group of objects where all the objects are unique. Today's blog entry looks at three situations where objects can be repeated.
All the Choices are Available all the Time
This is where you make permutations in which all the objects are available for each slot. For example, let's take a five digit zip code. There are five slots and for each slot the 10 digits are available: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
For the first slot there are 10 choices. For each of the 10 choices, the second slot presents another 10 choices. For each of those 10 choices, the third slot presents another 10 choices. And so on.
Remember when calculating permutations, order matters. For our example, 11110, 11101, 11011, 10111, and 01111 are five different arrangements. The general rule is presented below.
Permutations where all the choices are available all the time = n^k
where n = number of objects k = number of slots
Objects Can Repeat, but with Some Restrictions
This is similar to the first situation in the way of calculating the number of permutations (arrangements).
Let's say we are on a very famous game show. One of it's mini games is played for a car valued anywhere from $20,000 to $59,999. We want to know how many possible prices for this game are possible.
Looking at the range from $20,000 to $59,999: we can see the choices for each of the five digits. The first digit must be a 2, 3, 4, or 5. The other four digits can be anything 0 through 9 (10 choices each).
The number of prices possible are 4 × 10 × 10 × 10 × 10 = 40,000.
So using a random guess, the contestant has a 1 in 40,000 choice in getting the price exactly right.
Another mini-game offers cars from $11,111 to $36,666 where the first digit is given to the contestant (1, 2, or 3) and the contestant tries to roll the other four digits using a single die. The die contains the numbers 1, 2, 3, 4, 5, and 6. Our task to find out how many prices are possible.
There are 3 choices for the first digit, and 6 choices for the other four digits.
The number of prices possible are 3 × 6 × 6 × 6 × 6 = 3,888.
This mini-game can be played up to 3,888 times before a price repeats.
Permutations where: 1) Different slots can have restrictions 2) Each choice is independent
In this situation we are finding the number of permutations of a group of objects, except some of the objects repeat.
Let's try to find the number of ways to arrange the letters in the word PHYSICS, removing any restriction that the arrangement has to make a sensible word (HYSSICP would count as arrangement).
In the word PHYSICS, there is 1 "P", 1 "H", 1 "Y", 2 "S"s, 1 "I", and 1 "C", for a total of 7 letters. PHYSICS counts as one permutation, regardless which "S" is used in each slot. We still have 7 letters, which can be arranged 7! = 5,040 ways, but have to account for the 2 "S"s.
The true number of ways to arrange the letters in the word PHYSICS is 2,520 ways.
Let's take another example. Find the number of ways to arrange the letters in the "word" AAABB. In this example, there are 3 "A"s and 2 "B"s for a total of 5 letters.
If all the letters were unique, the number of ways is 5!. But we have to account for the repeats. Divide by 3! for the "A"s and 2! for the "B"s. The result is:
5! / (3! × 2!) = 120 / (6 × 2) = 10
There are only 10 ways to arrange the letters of the "word" AAABB. The ten are:
AAABB AABAB AABBA ABABA ABBAA
BABAA BBAAA ABBAA AABBA BAAAB
Finding the Number of Arrangements of Objects Where Some Objects Repeat
n! / ( (k_1)! × (k_2)! × ... × (k_m)! )
where: n = the total number of objects, including repeated objects m = the number of unique objects k_1 = number of "k_1" objects k_2 = number of "k_2" objects and so on until... k_m = number of "k_m" objects
The expression above is known as a multinomial coefficient.
I hope this day is well for each of you. Thank you for your comments and suggestions. Take care,
Tuesday, October 23, 2012
Last weekend my dad asked me about my blog, and the last entry was about factorials of large numbers. Clicking on this link will take you there. On our way to classic car auto shop in Orange, CA; my dad and I talked about the factorials as I tried to come up with a way of finding applications using factorials. Last weekend became the inspiration for my blog entry. Love you, Dad.
In this section, we consider three common arrangement problems. The task is to find the number of possible permutations a group of objects can be arranged. A permutation is an arrangement where the order of which the objects are placed is important.
In this section, we are going to arrange all the objects.
Arrangement of Unique Objects
Consider a bookshelf that has room for five books. For simplicity, let's call the books A, B, C, D, and E (and not "Combinatorics", "The Irrationals", "Euclid's Number", "Programming", and "Making Soup for Dummies" like I originally planned.)
The for the first slot there are five choices. The second slot provides four choices. Whatever is available for the second slot depends on what book was put in the first slot. For example, if I choose to put book A in the first slot, the books B, C, D, and E are available for the second slot. Instead, if I choose to put book C in the first slot, then A, B, D, and E are available.
For each choice I make on the first slot, I get four choices for the second. Considering the first two slots alone, this gives me a total of 5 × 4 = 20 arrangements.
Continuing in this way, there are three choices for the third slot, two choices for the fourth slot, and whatever is left over gets the fifth slot.
So, the total number of arrangements of five books is:
5 × 4 × 3 × 2 × 1 = 5! = 120
Yes, 120 different arrangements. Since order is important, the arrangement is considered a permutation.
In general, working with n objects and n slots: There are n objects for the first slot, there are n - 1 objects for the second slot, there are n - 2 objects for the third slot, and so on, until we reach 1 slot left for the last object.
Hence, the number of arrangements are:
n × (n - 1) × (n - 2) × ... × 1 = n!
Number of Permutations of n Unique Objects = n!
Arrangement of Unique Slots, Limited Spaces Available
Let's go back to our problem of arranging five books (A, B, C, D, and E) but this time, we only have three slots available.
For the first slot, I have 5 books available to choose from. Depending on what I choose, I will have 4 books for the second slot. Each of those 4 books present a choice of the 3 books for the last slot. Whatever is left either goes somewhere else in the house or gets donated.
The number of arrangements that are available to me has decreased due to the fact I only have three slots available. Hence, 5 × 4 × 3 = 60.
Saturday, October 20, 2012
Hi everyone! I am blogging from Coffee Klatch in San Dimas, CA! I was overdue for a latte. Their pumpkin brownies are to die for.
Factorial of Large Numbers
Calculators have an upper limit when working with numbers. Typically, the scientific calculators allow numbers up to 9.999999999 × 10^99. Anything over 10^100 is considered an overflow. Some of the higher graphing and CAS calculators allow for higher limits.
This means if you need to calculate anything over 69!, most calculators will give an "Overflow" error message.
Factorial:
The factorial of an integer n is the product:
n! = n × (n - 1) × (n - 2) × ... × 3 × 2 × 1
We can remedy this by using the laws of logarithms to calculate factorials of higher integers. Common logarithms (base 10) are used.
Because the accuracy of calculators and computer programs, I recommend that you round 10^(frac(T)) to a set amount of decimal places. I find that 5 decimal places works well. Then:
8! ≈ 4.03200 × 10^4
In fact, 8! = 40,320
Example 2: 111!
1. Calculate the sum ∑( log(k) from k=2 to 111). We find the answer to be 180.2462406.
2. Split the integer and fractional portions. Take the anti-common logarithm of the fractional portion. Hence, 10^0.2462406 ≈ 1.762952551. I round this answer to five decimal places to get 1.76295.
3. Answer: 111! ≈ 1.76295 × 10^180
In fact, using an HP 50g, 111! = 1,762,952,255,109,024,466,387,216,104,710,707,578,876,140,953,602,656,551,604,157,406,334,734,695,508,724,831,643,655,557,459,846,231,577,319,604,766,283,797,891,314,584,749,719,987,162,332,009,625,414,533,120,000,000,000,000,000,000,000,000
And yes, typing out this long number is not as easy it sounds.
Programming
Now we come to the programming portion of calculating n! using the logarithmic method.
This program was done on a HP 39gii, but can easily adapted on most graphing/programming calculators.
! Round Fraction to five places - we have no ROUND function to work with ! 10^x when x<1 has a value less than 10 ! Use string functions to round the answer to 5 decimal places (7 characters: the integer, decimal point, and five decimal places) 30 F = 10^F 32 F = VAL(LEFT(STR(F),7))
I hope this series is helpful. This series has showed how to make various calculations with a simple calculator: arithmetic procedures including fractions, area of a circle, distance between two points, calculating the total shopping bill, solving 2 x 2 simultaneous equations, and solving quadratic equations.
Ever though what your bill would be as you shop? This section will show you what the potential bill will be and hopefully will lead you to make smart shopping decisions, and keep in budget.
Approach: 1. Clear Memory. We will use the memory register to keep track of our purchases. 2. Determine whether the item is subject to sales tax. You can press MR at any time to get a subtotal. 3. Add the total. 4. *If all of your items are subject to sales tax, add sales tax to the total.
I = amount of interest P = amount of the principal R% = annual interest rate (as expressed as a decimal) T = time, in years
The total amount paid is principal and interest, in other words, P + I.
--------------- Example 1: A bank makes a short term loan to Fred and Suzy of $1,000. The bank charges 9.6% interest on short term loans. Fred and Suzy have to pay the loan in two months. If Fred and Suzy wait for the two months, how much interest have they paid?
The total interest paid is $16. --------------- Example 2: Terrell is looking over his credit card bill. The balance is $1,540.29. His credit card charges an annual rate of 15.99%. Terrell is planning to make a $300.00 payment. Assuming Terrell does not use his credit card for the next month, what will be his balance?
Terrell's new balance next month would be $1,256.82. --------------- Example 3: Lita deposited $500 in a Double Your Money CD. The bank will pay her $1,000 when the CD doubles in value. The bank pays 7.5% interest on these deposits. How long will Lita wait?
This time we are looking for T.
Variables: P = $500 R% = 7.5% I = $500
Why is I = $500? Lita deposits $500 and will wait for her account to grow $500, to earn $500 in interest. Solving for T, time:
where r is the radius and π is the constant pi. In this series, I am working with an 8 digit calculator, I will use the approximation π ≈ 3.1415927. I can use less digits, but I want as much accuracy as possible.
--------------- Area of a Circle: A = π r^2
Keystrokes: radius × = × 3.1415927 = --------------- Example: Find an area of a circle with a radius of 14.5 inches.
The Order of Operations says we must multiply the mixed fractions first. However, we only have one memory register, and we don't have the ability to "switch" whatever is in the display with memory. Tackling this problem requires a plan. Here is one way:
First of all, I am home taking care of my dad today. He's doing fine, recovering from surgery.
I also want to thank everyone who reads this blog and leaves comments. I appreciate the input and conversation.
Eddie
---------------
Introduction
Don't have a scientific calculator and only have a simple calculator with you? Interested in maximizing the abilities of a simple calculator? Want to impress your friends and co-workers? This series is for you.
What do I mean by a simple calculator? it is the calculator that you find everywhere, not just produced by the big calculator manufacturers Hewlett Packard, Texas Instruments, Sharp, and Casio, but as a novelty item from companies with almost any color or design you like. If you prefer, there are thousands of calculator apps on almost any cell phone, tablet, or iPod Touch. I refer to this type of calculator as a "four-banger" because it's primary functions are the arithmetic functions (+, -, ×, ÷).
Part 1 will cover:
* Chain Mode * The Memory Keys * Arithmetic Operations
Chain Mode
I estimate that 99% of simple calculators operate on chain mode. That is calculations take place as you enter them, without regard to the Order of Operations. You may remember the expression "My Dear Aunt Sally" or it's expanded version "Please Excuse My Dear Aunt Sally" as a mnemonic for the Order of Operations.
Here is how to know whether your calculator is operating in Chain Mode:
Type, in order: 4 + 2 × 3 =
Now using the proper order of operations, 4 + 2 × 3 = 10. However, calculators in chain mode don't "know" the order of operations. In that case, that calculator completes 4 + 2 first before multiplying the result by 3, giving a result of 18.
So if you type 4 + 2 × 3 = , in that order, and get 18, your calculator operates in Chain Mode. This means we have to manually take the Order of Operations into account to ensure we get the correct answer of 10.
One way to do this is rearrange the expression to 2 × 3 + 4. Typing the expression in this order will give us the correct answer, 10.
For this series, we will work with calculators operating in Chain Mode, which covers about 95% of simple calculators.
The Memory Keys
The simple calculator has four memory operations: M+: Add whatever is in the display to Memory. M-: Subtract whatever is in the display from Memory. MR: Recall the contents of Memory. MC: Clears the contents of Memory.
Often, you will see the key MRC. Press this key once to recall the contents of memory, twice to clear it.
For this series, I will keep MR and MC separate. Just remember if you are working with the MRC key, pressing MRC twice will clear memory.
The memory register is a key to advanced calculations on a simple calculator. One, it can help keep our calculations in proper operation. Second, it can store a number for later use.
Your calculator will give an indicator ("M" or "MEMORY") whenever a number other than zero is stored in memory. The memory is consider cleared when 0 is stored and the calculator does not display a memory indicator.
On to the Arithmetic Section.
Arithmetic Section
Let's tackle some common arithmetic calculations with the simple calculator to get the correct answers. Remember the order you press the keys is critical, since the simple calculator operates in Chain Mode.
For each section, I will give a proper keystroke to tackle the problem. Then I will give an example. Each capital letter (A, B, C, etc.) represents a variable.
1. A × B + C
This is a fairly simple expression. Just keep "Please Excuse My Dear Aunt Sally" in mind and you're gold.
--------------- A × B + C
Keystrokes: A × B + C = --------------- Example: 8 × 6 + 3 = 51
Keystrokes: 8 × 6 + 3 =
The result is 51, which the correct result with the Order of Operations.
2. A × B + C × D
Now we have two multiplications to do before the addition. This is where the memory keys (M+, M-, MR, MC) come in handy. Before you start any operation involving memory, clear it first! Remember if your calculator has a MRC key, press it twice to clear memory.
Tuesday, October 2, 2012
Here are four famous numerical constants, extended to 50 decimal places: π, e, γ, and √2. For each constant, the approximation will be listed and a histogram of each of the 50 decimal places and the integer part (51 digits in total) will be presented | 677.169 | 1 |
Supplemental materials are not guaranteed for used textbooks or rentals (access codes, DVDs, workbooks)....
Show More 2,500 fully worked problems of varying difficulty Clear, concise explanations of arithmetic, algebra, and geometry Outline format supplies a concise guide to the standard college courses in elementary mathematics Appropriate for the following courses: Basic Mathematics, Elementary Mathematics, Introduction to Mathematics, Review of Arithmetic, Elementary Algebra, Review of Algebra, Business Mathematics I, Math for the GED Detailed explanations and practice problems in arithmetic, algebra, and geometry Comprehensive review of specialized topics such as fractions, decimals, percents, ratios, proportions, and rates | 677.169 | 1 |
CLAST MANUAL: Thinking Mathematically
9780131752115
01317521113.48
More Prices
Summary
The CLAST Manual helps build studentsrs" proficiency in 55 mathematical skills in the fields of Arithmetic, Geometry, Algebra, Statistics, and Logic. Written as an additional supplement for students who are required to take the CLAST (students in the State of Florida), the first part of the CLAST Manual focuses on the skill-by-skill description of each of the 55 skills required in the exam, including worked out examples.The second part of this manual is a collection of CLAST-style homework exercises linked to various sections ofThinking Mathematically4e. | 677.169 | 1 |
This text features a combination of unique pedagogical tools–Exercises, Worked Examples with solutions in 2-column format, Active Examples, Conceptual Checkpoints–that provide the right tool at the right time and place. This text employs each tool when and where it can contribute most to developing students' conceptual insight hand-in-hand with developing their problem-solving skills.
"synopsis" may belong to another edition of this title.
Review:
na
From the Back Cover:
Foundation Maths has been written for students taking higher and further education courses who have not specialised in mathematics on post-16 qualifications and need to use mathematical tools in their courses. It is ideally suited to those studying marketing, business studies, management, science, engineering, social science, geography, combined studies and design. It will be useful for those who lack confidence and who need careful, steady guidance in mathematical methods. For those whose mathematical expertise is already established, the book will be a helpful revision and reference guide. The style of the book also makes it suitable for self-study and distance learning.
Features of the book
· Mathematical processes are described in everyday language – mathematical ideas are usually developed by example rather than formal proof, thereby encouraging students' learning.
· Key points highlight important results that need to be referred to easily or remembered.
· Worked examples are included throughout the book to reinforce learning.
· Self-assessment questions are provided at the end of most sections to test understanding of important parts of the section. Answers are given at the back of the book.
· Exercises provide a key opportunity to develop competence and understanding through practice. Answers are given at the back of the book.
· Test and assignment exercises (with answers provided in a separate Lecturers' Manual on the website) allow lecturers and tutors to set regular assignments or tests throughout the course.
· Extra end-of-chapter questions for students (with answers) on the website at
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· PowerPoint slides for lecturers on the website featuring Key Points from the book with their related Worked Examples.
Anthony Croft has taught mathematics in further and higher education institutions for twenty four years. He is currently Director of the Mathematics Education Centre at Loughborough university, which has been designated a Centre for Excellence in Teaching and Learning by the Higher Education Funding Council for England. He teaches mathematics and engineering undergraduates, and has championed mathematics support for students who find the transition from school to university difficult and for students with learning difficulties. He has authored many very successful mathematics textbooks including several for engineering students.
Robert Davison has twenty five years experience teaching mathematics in both further and higher education. He is currently Head of Quality in the Faculty of Computing Sciences and Engineering at De Montfort University, where he also teaches mathematics. He has authored many very successful mathematics textbooks including several for engineering students. | 677.169 | 1 |
This best selling author team explains concepts simply and clearly, without glossing over difficult points. Problem solving and mathematical modeling are introduced early and reinforced throughout, providing students with a solid foundation in the principles of mathematical thinking.
Comprehensive and evenly paced, the book provides complete coverage of the function concept, and integrates a significant amount of graphing calculator material to help students develop insight into mathematical ideas. The authors' attention to detail and clarity, the same as found in James Stewart's market-leading Calculus text, is what makes this text the market leader.
Chapter 3: Functions 3.1: What Is a Function? (83) 3.2: Graphs of a Function (83) 3.3: Getting Information from the Graph of a Function (55) 3.4: Average Rate of Change of a Function (30) 3.5: Transformations of Functions (90) 3.6: Combining Functions (66) 3.7: One-to-One Functions and Their Inverses (89) | 677.169 | 1 |
S o [allws
Name
Math 142, Midterm Exam I
Prof. Sherman
This is a 50 minute exam. There are 5 problems7 each work 12 points, for a total of 60 possible points. Mark all your answers
clearly with a box, and show all relevant work. K J3:
wt-J? I r".
.
Math 142, Calculus II, Winter 2014
Written Homework Example 1
1. Write up the student study guide solution to the following:
Find the values of c such that the area of the region bounded by the parabolas y = x2 c2 and y = c2 x2
is 576.
We have been nding
Math 142, Calculus II, Winter 2014
Written Homework Example 2
1. Write up the student study guide solution to the following:
A bowl is shaped like a hemisphere with diameter 30cm. A ball with diameter 10 cm is placed in the bowl
and water is poured into t
Math 142, Calculus II,
Winter 2014, Quiz 2
Name:
1. Use the shell method to nd the volume of the solid generated by revolving the region bounded by
y = x + 2 and y = x2 about the line y = 4. Explain, of course. That is to say, you need to show your
work,
Calculus II Advice
Showing 1 to 2 of 2
Calculus is a really fun math subject, and the methods of integration that you are taught in Calc 2 seem like really clever tricks to work around with integrals we wouldn't normally be able to solve. It's a breeze with Professor Rawlings and plenty of practice too!
YES! The course itself was extremely intense and rigorous, but the professor was by far the most amazing professor I've ever had. He was a real character and you could never get bored in his class. He was hilarious, but at the same time extremely witty and great at making the complicated simple.
Course highlights:
I learned extremely efficient ways of doing long mathematical problems. I couldn't name any in particular because it's been awhile since I've done any calculus. He's all about making the long crazy math problems as short and easy as possible. And he's great at it.
Hours per week:
12+ hours
Advice for students:
Do the homework. It's almost never mandatory, but trust me, just do it. | 677.169 | 1 |
Saxon Geometry Homeschool: Saxon Teacher CD ROM 1st Edition 2010
Delivery: 10-20 Working Days
Give your Saxon Geometry students support and reinforcement! Comprehensive lesson instructions feature complete solutions to every practice problem, problem set, and test problem with step-by-step explanations and helpful hints. These user-friendly CD-ROMs contain hundreds of hours of instruction, allowing students to see and hear actual textbook problems being worked on a computer whiteboard. A slider button allows students to skip problems they don't need help on, or rewind, pause, or fast-forward to get to the sections they'd like to access. Problem set questions can be watched individually after the being worked by the student; the practice set is one continuous video that allows for easy solution review. For use with the 1st Edition. Four Lesson CDs and 1 Test Solutions CD are included. System Requirements: Windows: # 98, 2000, ME, XP, 7, or Vista with latest updates # 450 MHz or faster # 128 MB RAM # 8x CD-ROM Drive # 800 x 600 display Macintosh: # 10.2 or up # G3, 500 MHz or faster # 128 MB RAM # 8x CD-ROM Drive # 800 x 600 display
Specifications
Country
USA
Author
SAXON PUBLISHERS
Binding
Audio CD
EAN
9780547442563
Edition
1
ISBN
0547442564
Label
SAXON PUBLISHERS
Manufacturer
SAXON PUBLISHERS
NumberOfItems
1
NumberOfPages
8
PublicationDate
2010-04-30
Publisher
SAXON PUBLISHERS
Studio
SAXON PUBLISHERS | 677.169 | 1 |
Teach Yourself Trigonometry
Teach Yourself Trigonometry is suitable for beginners, but it also goes beyond the basics to offer comprehensive coverage of more advanced topics. Each chapter features numerous worked examples and many carefully graded exercises, and full demonstrations of trigonometric proofs are given in the answer key | 677.169 | 1 |
Welcome
This online course is designed to be completed by incoming high school freshman during the summer break. It will ensure that the students understand the math skills necessary to succeed with high school level courses.
Please feel free to contact your school district's point of contact if you have any questions when completing the course.
The categories on the left hand side of the page will take you to the modules that must be completed. | 677.169 | 1 |
Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. The major difference between algebra and arithmetic is the inclusion of variables. While in arithmetic only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra, one also uses symbols such as x and y, or a and b to denote variables.
The purpose of using variables, symbols that denote numbers, is to allow the making of generalizations in mathematics. It allows reference to numbers which are not known. It allows the exploration of mathematical relationships between quantities (such as "if you sell x tickets, then your profit will be 3x − 10 dollars"). | 677.169 | 1 |
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Odyssey Algebra
04/01/05
CompassLearning ( has expanded its entire suite of Odyssey products, including Odyssey Algebra for middle schools and secondary education. The browser-based curriculum will help teachers offer a comprehensive approach to math education, while providing a platform that supports a variety of instructional strategies and learning styles. Odyssey Algebra has 13 chapters and 131 objectives to cover in an entire school year. The curriculum's online features include interactive tutorials that are woven throughout the program and aids such as online calculators, graph paper, number lines, protractors, spreadsheets and rulers. The program also provides additional offline materials for students that are designed to extend learning beyond the classroom.
This article originally appeared in the 04 | 677.169 | 1 |
Free and Easy Math Assignment Help
If you are stuck with algebra homework, the Internet offers a lot of websites with tools to aid you in performing
various maths. There are online calculators, and collections of math rules and formulas, to aid you in
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if you already have so many questions, then you need a human helper instead of reference material.
Mathematicians think a lot – that is what helps them solve difficult problems. But students can't afford to spend
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Whether you lack some knowledge in math or you have other reasons for not being able to cope with an assignment,
we welcome you to use our questions and answers service. Here you can find help from fellow students and
complete all your worksheets on time. Don't procrastinate or you risk lowering your grade. When you are
at home, working on your assignments, and something just doesn't work out, type learnok.com into your
browser and get free help from knowledgeable friends.
Features of Using LearnOk
It's a friendly community of students who want to help each other study and learn new information.
Asking questions is free of charge, unless you want you want help from a tutor.
You can do something good and contribute to our database by answering some questions.
That and more can be found at our question-answering service. So, try it out right away! | 677.169 | 1 |
Modern surveying includes a lot of computations that are complex in nature and are almost all time they are computer aided. To keep up to date the author put different chapter for different types of surveying methods and ways. This book also contains a chapter that is named as computer programs in surveying which will help students to learn about the recent inventions in the field of surveying and applications of those inventions in real life in the most easy manner. | 677.169 | 1 |
Numerical Methods Homework Help - neubiolifitzva
Numerical Methods Homework Help
WAMAPWAMAP is a web based mathematics assessment and course management platform. Its use is provided free to Washington State public educational institution students and [Solution] numerical methods for engineers chapra - …Oct 19, 2012 · [Solution] numerical methods for engineers chapra 1. CHAPTER 22.1 IF x < 10 THEN IF x < 5 THEN x = 5 ELSE Central Differences - HoloborodkoThe most common way of computing numerical derivative of a function at any point is to approximate by some polynomial in the neighborhood of . It is expected that if For chrisbenjamin ONLY - Psychology homework helpGet homework help at HomeworkMarket.com is an on-line marketplace for homework assistance and tutoring. You can ask homework …A+ Answers of the following Questions - Homework MarketGet homework help at HomeworkMarket.com is an on-line marketplace for homework assistance and tutoring. You can ask homework …Numerical Integration -- from Wolfram MathWorldNumerical integration is the approximate computation of an integral using numerical techniques. The numerical computation of an integral is sometimes called quadrature.FREE Mathematics How-to Library - math homework help maths homework helper – algebra help – math software from teachers choice software - math homework help from basic math to algebra, geometry and beyond.Online Math Tutor: Exam preparation & Homework help - …Math tutoring sessions, professional atention and help to solve problems, prepare exams, understand homeworkCalculations and Numerical Methods :: Estimating and The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities Nexcheck - Payment ServicesNexcheck® Payment Solutions provides ACH, Check Guarantee, Check Recovery, Check Verification, and Credit Card processing for merchants across the united states.
numerical methods - Check if a varchar is a number (TSQL
is there an easy way to figure out if a varchar is a number? Examples: abc123 --> no number 123 --> yes, its a number Thanks :)Newton-Cotes Formulas -- from Wolfram MathWorldNewton-Cotes Formulas. The Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques. To integrate a function …Georgia Performance StandardsGeorgiaStandards.Org (GSO) is a free, public website providing information and resources necessary to help meet the educational needs of students.Computing + Mathematical Sciences | Course DescriptionsCourse Descriptions. Courses offered in our department for Applied and Computational Mathematics, Control and Dynamical Systems, and Computer Science are listed below.Homework Answers and Assignments Solutions Online Don't run up against deadlines with your homework. StudyDaddy is the easiest way to complete your homework at any time and score A grades.Normal Distribution - Math is Fun - Maths ResourcesYou can see a normal distribution being created by random chance! It is called the Quincunx and it is an amazing machine. Have a play with it!Foundations of Algorithms, Fifth EditionFoundations of Algorithms, Fifth Edition offers a well-balanced presentation of algorithm design, complexity analysis of algorithms, and computational complexity.Continued Fractions - An introduction - MathematicsUsing jigsaw puzzles to introduce the Continued Fraction, the simplest continued fraction is for Phi - the golden section; how continued fractions arise naturally Mathematics Georgia Standards of Excellence (GSE) 9-12Standards Documents • High School Mathematics Standards • Coordinate Algebra and Algebra I Crosswalk • Analytic Geometry and Geometry CrosswalkWolfram Demonstrations ProjectExplore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.Building Java Programs 3rd Edition, Self-Check SolutionsBuilding Java Programs, 3rd Edition Self-Check Solutions NOTE: Answers to self-check problems are posted publicly on our web site and are accessible to students.History of mathematical notation - WikipediaThe history of mathematical notation includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation Books in the Mathematical SciencesThis site is intended as a resource for university students in the mathematical sciences. Books are recommended on the basis of readability and other pedagogical …Using knowledge of student cognition to differentiate By all accounts, learning is a complex task that requires a student to use and apply a range of cognitive skills. A student's ability to retain information while | 677.169 | 1 |
Abstract
All collections of the familiar types of numbers are sets. You don't really need to think about what a free product or a tensor product actually is ("what it looks like," in my words). Abstract Algebra is a very advanced level of algebra focused on algebraic structures. The operations are usually binary operations that take two arguments, but some are unary. My students have found the easy step by step instructions, and the explanations on how the formula works to be a great help.
It was harder for me to understand Theorem 8.17 and Corollary 8.18. I received an MS degreee in Mathematics from Oregon State Univeristy. For smaller n, we note that a single plane divides 3 into 2 regions, while if n = 0, then 3 is not divided at all: there is 1 region. The APOS theory can be applied to every topic in abstract algebra and at every level. Weekly homework assignments will be given in the class. They were completely on task in both the classroom and the math lab and focused on getting their work done.
Department of Mathematics and Computer Science Hobart and William Smith Colleges Fall, 2004. Elementary algebra, the part of algebra that is usually taught in elementary courses of mathematics. Universal algebra is a related subject that studies the nature and theories of various types of algebraic structures as a whole. Of course this was a time when algebraists, analysts, and topologists talked to each other, were interested in each other's work, were influenced by each other, and in many cases were actually the same individuals.
With his algebra book, you can understand algebra with the help of real-world examples, and realize that mathematics is more than basic facts and memorized procedures. It is entirely theoretical and heavily proofed (there are many proofs). KEYWORDS: Tutorials, Algorithms, Implementations in Computer Algebra System, Factorization Challenges, Research Teams, Applications, Literature ADD. Here are a few different things that you should be aware of: Pre-algebra: You will need to a wide variety of pre-algebraic functions.
Questions on Stack Overflow are expected to relate to programming within the scope defined by the community. Perhaps surprisingly, the course was a huge success and I was immediately sold on the potential impact that IBL can have on a student's learning and character development. It is really surprising that in order to read an essential theorem like Cayley's Theorem (page 126) one is referred back to Exercise 23, and to Exercise 9. This Mathematica-based book does not make any assumptions about the main text being used.
Gain access to members only, premium content that includes past essays, DBQs, practice tests, term papers, homework assignments and other vital resources for your success! Money and Credit is for things like checks, credit cards, loans, etc. My plan is to write a theorem-sequence in the spirit of Nathan Carter's VGT, but with the mathematical rigor of Judson's AATA and Clark's Group Theory. The output matrix has m operation rows for each input, and q inputs, giving a "m x q" matrix.
Nonlinear equations fifth grade, gcse using graphs to solve equations, factor tree to form prime factorization of 86, English Problems 3rd Graders, power basic download, long division ks3, multiplying square root calculator. Furthermore because G is cyclic we also know that there exists an a ∈ G such that ¸a) = G. I do not claim that the next book is useful for investing. So now in later upper division courses I am more comfortable with trying more complex problems, which ultimately lead me to do undergraduate research.
We can add additional constraints on the algebraic structure, such as associativity (to form semigroups ); identity, and inverses (to form groups ); and other more complex structures. But by theorem 8.1, [(g, h)[ = lcm([g[, [h[) = [G[[H[, so it must follow that [g[ = [G[ and [h[ = [H[ by theorem 8.2. At most, include a city and country (no zip codes, postal codes, or street addresses). We're pretending our input exists in a 1-higher dimension, and put a "1" in that dimension. | 677.169 | 1 |
MYTUTOR SUBJECT ANSWERS
What's the point of Maths?
Every subject has a paticular career path. Maths is an exception which is at the forfront of most things. Anytime you saw data, results, surveys, polls, predictions, forecasts, returns & any others that was maths.
When your exam results they are sometimes created by combining marks to create a spread of data. Then the grades are given from the percentage levels in the data | 677.169 | 1 |
Edition details
This paper sets out a method for creating and marking individualised mathematics assessments for students based on their ID numbers. It therefore provides a means for setting assessed coursework questions that give students the incentive to put in the practice needed to master mathematical techniques without the risk of collusion between students. A marking grid can then be constructed using only basic Excel skills. The method is explained here in the context of basic mathematical techniques applied to economics, but it can also be applied to other academic disciplines that involve numerical problems | 677.169 | 1 |
Mathematics for Game Developers (Game Development)
Christopher Termblay
ISBN: 159200038X;
Издательство: Muska & Lipman/Premier-Trade
Mathematics for Game Developers is just thata math book designed specifically for the game developer, not the mathematician. As a game developer, you know that math is a fundamental part of your programming arsenal. In order to program a game that goes beyond the basics, you must first master concepts such as matrices and vectors. In this book, you will find some unique solutions for dealing with real problems youll face when programming many types of 3D games. Not only will you learn how to solve these problems, youll also learn why the solution works, enabling you to apply that solution to other problems. Youll also learn how to leverage software to help solve algebraic equations. Through numerous examples, this book clarifies how mathematical ideas fit together and how they apply to game programming. | 677.169 | 1 |
PRODUCT DESCRIPTION
This lesson and activity bundle includes everything you will need to teach students how to factor quadratic expressions (with the exception of using the quadratic formula). Lessons/notes are all available in a PowerPoint presentation with explanations and examples of different methods of factoring as well as student handouts to go with the presentation. This presentation is designed to take several days, with each strategy being presented and practiced in it's own class. Each method of factoring is complete with notes, examples, practice questions, activities, and exit slips for assessing student understanding at the end of class | 677.169 | 1 |
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
5 years ago
calculus and vectors are differnt things,calculus is all about integration and differentiation...whereas vectors algebra helps to find the magnitude and direction of quantity without any calculus...
anonymous
5 years ago
Vectors usually deal with space. For example, in a 3D game world, you need vectors to determine the position of things. Bob is at [x=3, y=10, z=4]. Bob is moving with a speed of [2km/h, 4km/h, 0km/h].
Calculus would be needed to help with some complicated physics. Sometimes you need to find out how fast an object is moving, or how fast the rate at which it is moving is changing, or how fast that rate is changing, or how fast that range is changing, or how fast that rate is changing... and calculus can help with these complex mathematics.
Advanced functions... like f(x)=zi*sqrt(x+34y-n/345)? Well, they're uh... useful in torturing students.
anonymous
5 years ago
In an advanced functions class, you will likely not use calculus as much. It'll be more like Algebra 2 or Pre-Calculus. Calculus and Vectors will include more analytic geometry and linear algebra.
More answers
ghazi
5 years ago
calculus is basically-- summing up things which is defined by a mathematical function (called as mathematical modelling) , let's say something defined by a variable now to sum up the variation we need integration. we find area of any graph by calculus.
whereas vectors are designed to define direction of the variables involved in that function .
i think this is enough for now :)
ghazi
5 years ago
@waheguru
anonymous
5 years ago
Funny how everyone is trying to define the individual terms instead of describing the curricula of the specific school courses mentioned. . .
ghazi
5 years ago
exactly , i just believe knowing application matters than just solving problem, it's worth less to know unless you know what to do with it | 677.169 | 1 |
The best way to tell is to take our Pre-Algebra placement test. If your child is unable to take the placement test at this time, then you should know that if a student has had some exposure to basic math and can handle the concept of using a letter (usually an x) to represent a missing number in an equation, then he or she is probably ready for Pre-Algebra.
The TT Pre-Algebra program includes a complete review of basic math (whole numbers, fractions, decimals, percents, units of measurement, etc.), but it explains things more conceptually than books for younger students. This approach teaches students why the techniques they've been using for years really work. Knowing the whys is necessary prep-aration for Algebra 1 and other higher math courses. | 677.169 | 1 |
Square Root Word Problems
Be sure that you have an application to open
this file type before downloading and/or purchasing.
302 KB|2 pages
Product Description
Real life situations modeled by the square root function. Students will take a scenario that con be represented with a square root function, and complete a table and/or graph, then answer questions about the data and graph. | 677.169 | 1 |
Teaching Textbooks Homeschool Math Curriculum Pros and Cons
Teaching Textbooks can be used for homeschooling students as a complete homeschool math curriculum. There are special curriculum packages for Geometry, Pre-Algebra, Algebra, and Pre-Calculus, and also there are grade specific packages for third through seventh grades.
Each package includes textbooks, answer keys, tests, and coordinating CDs. Parents may also choose to order only the CDs for some grade levels. The textbooks are known to provide exceptional explanations for all lessons introduced, which makes it easier for homeschooling parents who may not already be math whizzes.
Teaching Textbooks Advantages
The biggest advantage to using Teaching Textbooks is the extra help explaining lessons. Parents are sometimes intimidated by the idea of presenting advanced math concepts to their children, and it doesn't help that some homeschool mathematics programs deliver very little instruction in student textbooks.
Teaching Textbooks deliver a lot more information in each textbook and the answer keys give correct answers for every problem presented in the program. Not only does this make it easier for students to guide themselves through math lessons, but it helps parents who need to learn along with their children to serve as effective teachers.
Parents do not have to be as hands-on when using Teaching Textbooks curriculum. Children can watch the CD presentation of material and work through sample problems on their own, with parents coming in at the end to help with concepts children may have trouble grasping on their own. Parents can also choose to sit with their child through the initial lessons to help in a more hands-on manner.
Children who struggle to understand Singapore Math and other homeschool math curriculums often thrive with Teaching Textbooks. This is because this program starts with extremely basic concepts and gradually progresses. This allows children who do not naturally gravitate toward mathematic concepts to understand basic concepts and build on that knowledge.
Parents enjoy seeing their children thrive in math while managing their own lessons on the computer. Parents are able to check up on their child's progress and even ensure their children are checking problems that are missed the first time around. When children are not managing their lessons appropriately, parents will clearly see that when they check their child's progress. This allows parents to be involved in the lessons without hovering over their child.
Teaching Textbooks Disadvantages
Teaching Textbooks may not be the best program for parents and children who need to approach math from a higher or more abstract level.
Children who struggle with math are able to excel with this program and parents who need to learn alongside their children can do so with this program. This leads to frustration with some children who already know all of the basics and want to jump right to more advanced lessons at each grade level.
Some parents have taken their children from lower grade levels of Singapore Math and placed them comfortably into the higher levels of Teaching Textbooks. It is important for each student to take the Teaching Textbooks grade placement test so they are placed in the appropriate program. This is the best way for advanced students to find the appropriate level where they will be challenged.
The Parent Perspective
Parents who do not have the time or desire to present math lessons to their children at home tend to love the Teaching Textbooks curriculum.
Since each curriculum was designed with the homeschool child in mind, the textbooks are more thorough and self-explanatory than many other math textbooks. The option to only purchase the CDs allows parents to save money. This is especially helpful for parents using this program as back-up to other mathematics curriculums.
You do not have to worry about the added expense of manipulative tools or instructor's manuals with this program. This is not the cheapest curriculum out there, but it is one of the easiest programs to use.
Where Can You Buy It?
You can buy this curriculum directly from TeachingTextbooks.com. However, the site rarely has discounts or deals on their products there.
Occasionally, you can find it cheaper on ChristianBook.com when they run specials or discounts. So if you keep your eyes open for good deals there, you may be able to save some money | 677.169 | 1 |
3
Grading Policies Three exams and one final (60%) – Each exam is 20%, lowest score dropped – All exams open book & open note Group project (25%) – Model and analyze biological or medical system – Various stages due throughout the semester – Group members receive same grade Homework assignments (15%) – Receive full credit if you gave reasonable effort – I will hand out solutions, wont correct answers – You are encouraged to work together
5
Mathematical Models Sets of equations that approximate the function of a physical system Variables represent physical quantities Equations are applications of physical laws or empirical data that represent transformations between the relevant physical quantities Employ analytical or computational solutions of equations to approximate system behavior You have spent much time in your previous engineering courses learning this process!
6
Why Mathematical Modeling? Low cost – All you need a brain and a computer – Much cheaper than building physical equivalents Accessibility – Can sometimes use models to answer questions that cannot be addressed experimentally – May not be ethical to obtain data from humans or animals Starting point for any new design – Models can indicate whether an idea will work – Can suggest tests/experiments you need to perform once the physical system is constructed
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Course Outline Schematic representations of physical systems Modeling behavior of physical processes with differential equations Simulation and analysis of differential equations Methods to control and improve system performance | 677.169 | 1 |
I have really enjoyed reading this book. I work with a lot of folks on developing their problem solving abilities. In the past, I have used the books by Polya (here is one and another) on problem solving heuristics. While these books are good, I really like the tone of "Solving Mathematical Problems." Tao does a very good job of describing the experimental aspects of problem solving.
Many folks new to problem solving will see a fully formed solution and do not understand that there may have been quite a bit of trial and error involved in coming up with that solution. Thus, the most common question I get from these newbies is "how do I know where to start"? Tao addresses this question by including with his solutions a discussion of alternative approaches and why he chose not to use them. I really appreciate this style of presentation -- all in the voice of a 15-year old. It is conversational and pleasant to read.
The book is focused on the kind of problems encountered in mathematics competitions:
Analytic geometry
Euclidean geometry
Number theory
Polynomials
Diophantine Equations
Sequences and Series
It is a small book, but filled with mathematical gems. If you decide to check it out, I think you will find it worthwhile. | 677.169 | 1 |
Beschreibung
Beschreibung
Fractals and Chaos: An Illustrated Course provides you with a practical, elementary introduction to fractal geometry and chaotic dynamics-subjects that have attracted immense interest throughout the scientific and engineering disciplines. The book may be used in part or as a whole to form an introductory course in either or both subject areas. A prominent feature of the book is the use of many illustrations to convey the concepts required for comprehension of the subject. In addition, plenty of problems are provided to test understanding. Advanced mathematics is avoided in order to provide a concise treatment and speed the reader through the subject areas. The book can be used as a text for undergraduate courses or for self-study.
Inhaltsverzeichnis
INTRODUCTION Introduction A matter of fractals Deterministic chaos Chapter summary and further reading REGULAR FRACTALS AND SELF-SIMILARITY Introduction The Cantor set Non-fractal dimensions: the Euclidean and topological dimension The similarity dimension The Koch curve The quadratic Koch curve The Koch island Curves in the plane with similarity dimension exceeding 2 The Sierpinski gasket and carpet The Menger Sponge Chapter summary and further reading Revision questions and further tasks RANDOM FRACTALS Introduction Randomizing the Cantor set and Koch curve Fractal boundaries The box counting dimension and the Hausdorff dimension The structured walk technique and the divider dimension The perimeter-area relationship Chapter summary and further reading Revision questions and further tasks REGULAR AND FRACTIONAL BROWNIAN MOTION Introduction Regular Brownian motion Fractional Brownian motion: time traces Fractional Brownian surfaces Fractional Brownian motion: spatial trajectories Diffusion limited aggregation The color and power and noise Chapter summary and further reading Revision questions and further tasks ITERATIVE FEEDBACK PROCESSES AND CHAOS Introduction Population growth and the Verhulst model The logistic map The effect of variation in the control parameter General form of the iterated solutions of the logistic map Graphical iteration of the logistic map Bifurcation, stability and the Feigenbaum number A two dimensional map: the Henon model Iterations in the complex plane: Julia sets and the Mandelbrot set Chapter summary and further reading Revision questions and further tasks CHAOTIC OSCILLATIONS Introduction A simple nonlinear mechanical oscillator: the Duffing oscillator Chaos in the weather: the Lorenz model The Rossler systems Phase space, dimension and attractor form Spatially extended systems: coupled oscillators Spatially extended systems: fluids Mathematical routes to chaos and turbulence Chapter summary and further reading Revision questions and further tasks CHARACTERIZING CHAOS Introduction Preliminary characterization: visual inspection Preliminary characterization: frequency spectra Characterizing chaos: Lyapunov exponents Characterizing chaos: dimension estimates Attractor reconstruction The embedding dimension for attractor reconstruction The effect of noise Regions of behavior on the attractor and characterization limitations Chapter summary and further reading Revision questions and further task APPENDIX 1: Computer Program for Lorenz Equations APPENDIX 2: Illustrative Papers APPENDIX 3: Experimental Chaos SOLUTIONS REFERENCES | 677.169 | 1 |
About this product
Description
Description
From signed numbers to story problems eugh mirror teaching methods and classroom protocols Focused, modular content presented in step-by-step lessons * Practice on hundreds of Algebra I problems * Review key concepts and formulas * Get complete answer explanations for all problems
Author Biography
Mary Jane Sterling is the author of Algebra I For Dummies, 2nd Edition, Trigonometry For Dummies, Algebra II For Dummies, Math Word Problems For Dummies, Business Math For Dummies, and Linear Algebra For Dummies. She taught junior high and high school math for many years before beginning her current 30-years-and-counting tenure at Bradley University in Peoria, Illinois. Mary Jane especially enjoys working with future teachers and trying out new technology. | 677.169 | 1 |
Review & Description
Calculus is advanced math for the high school student, but it's the starting point for math in the most selective colleges and universities. Thinkwell's Calculus course covers both Calculus I and Calculus II, each of which is a one-semester course in college. If you plan to take the AP Calculus AB or BC exam, you should consider our Calculus for AP courses, which have assessments targeted to the AP exam.Thinkwell's award-winning math professor, Edward Burger, has a gift for explaining and demonstrating the underlying structure of calculus, so that your students will retain what they've learned. It's a great head start for the college-bound math, science, or engineering student. | 677.169 | 1 |
Practice Makes Perfect Geometry
Overview learning category, Practice Makes Perfect now provides the same clear, concise approach and extensive exercises to key fields within mathematics. The key to the Practice Makes Perfect series is the extensive exercises that provide learners with all the practice they need for mastery.
Not focused on any particular test or exam, but complementary to most geometry curricula
Deliberately all-encompassing approach: international perspective and balance between traditional and newer approaches.
Large trim allows clear presentation of worked problems, exercises, and explained answers. Features
No-nonsense approach: provides clear presentation of content.
Over 500 exercises and answers covering all aspects of geometry
Successful series: "Practice Makes Perfect" has sales of 1,000,000 copies in the language category – now applied to mathematics
Workbook is not exam specific, yet it provides thorough coverage of the geometry skills required in most math tests.
Related Subjects
Meet the Author
Carolyn Wheater teaches middle school and upper school mathematics at the Nightingale-Bamford School in New York City. Educated at Marymount Manhattan College and the University of Massachusetts, Amherst, she has taught math and computer technology for 30 years to students from preschool through college. | 677.169 | 1 |
Cool Math Websites
I have compiled a small list of some great math websites. I constantly update the list, so check back often.
Brightstorm - Over 2000 videos of math topics for courses in Algebra 1, Geometry, Algebra 2 (Adv. Algebra), and Pre-calculus. This is an excellent source of information. Every math student should visit this site on a regular basis. Each course is broken down by topic. Although you do have to sign-up, the math videos are all free, some of the other topics have a cost.
High School ACE - Academic page full of resources for high school students. Many topics are covered, not just math.
Algebra Help - This is a great FREE resource. You can ask questions (many have already been asked and solved), there are lessons and examples to help you in most math concepts. Spend some time looking around the site as there is quite a bit of information.
Math Open Reference - GEOMETRY STUDENTS!!! You must check this site out. It has great interactive online applications to explain many geometry topics. The demonstrations on how to do constructions are excellent. | 677.169 | 1 |
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Unformatted text preview: Linear Algebra Notes Brad Lackey DEPT OF MATHEMATICS, UNIV OF HULL, HULL HU6 7RX, UK E-mail address : [email protected] Contents Preliminaries 1 1. Arithmetic 1 2. Algebra 3 3. Geometry 3 4. Calculus 4 5. Logic and Set Theory 4 Part I. Applicable Linear Algebra 5 Chapter 1. Vectors and Matrices 6 1. Vectors in two dimensions 6 2. Vector in more than two dimensions 8 3. Matrices 8 4. Determinants 13 Chapter 2. Linear Systems 15 1. Substitution 15 2. Elimination 16 3. Row reduction 17 4. Reduced row-echelon form 18 5. Solution spaces of linear systems 20 6. Matrix inverses 21 7. Determinants revisited 23 Chapter 3. Scalar product 24 1. Properties of the scalar product 24 2. Orthogonality 24 3. Projections 25 4. Scalar products over C 26 Chapter 4. Eigenvalues and Eigenvectors 27 1. Eigenvalues and Eigenvectors 27 2. Diagonalization 28 Part II. Abstract Linear Algebra 33 Chapter 5. Vector spaces 34 1. Definitions and Examples 34 2. Linear Independence 35 3. Bases 38 iii CONTENTS iv Chapter 6. Inner product spaces 44 1. Definitions and examples 44 2. Orthonormal Bases 46 3. Projections 49 Chapter 7. Linear transformations 51 1. Definitions and Examples 51 2. Matrix of a Linear Transformation 52 3. Kernels and Images 55 4. Composition 60 Chapter 8. Jordan canonical form 64 1. Similarity (and change of basis) 64 Preliminaries Any University course in mathematics requires experience in secondary school topics. Although many lecturers would say that most skills required can be learned during the first year of University, this is just not the case. A student of mathematics will need a broad knowledge of basic facts and techniques to succeed in University-level courses. This chapter is not designed to enhance, and certainly not to replace, pre-University studies in mathematics. Yet, we must consider the paradoxical situation of knowing what is required for a course without knowing the course itself. It is for this reason we present some general topics which a student will find necessary to succeed at linear algebra. The following five subjects are quite extensive on their own, and we do not pretend to be comprehensive. However, if all the facts presented in the chapter are familiar, then one should be confident to proceed. We should emphasize that the material presented here is background for the following chapters, and it is comprehensive in this sense. Topics within the sections on arithmetic and algebra are naturally needed throughout the course, but the material in the section on geometry, calculus, and logic are not altogether mandatory. To truly master linear algebra, one must borrow from these last three subjects; yet, one can progress far without expertise in them....
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This note was uploaded on 02/24/2012 for the course MATH 513 taught by Professor Igordolgachev during the Winter '09 term at University of Michigan-Dearborn. | 677.169 | 1 |
Algebra and Trigonometry
Author:Cynthia Y. Young
ISBN 13:9780470222737
ISBN 10:470222735
Edition:2
Publisher:Wiley
Publication Date:2009-02-24
Format:Hardcover
Pages:1344
List Price:N/A
 
 
Often, algebra & trigonometry students leave class believing that they understand a concept but are unable to apply that understanding when they get home and attempt their homework problems. This mainstream yet innovative text is written by an experienced professor who has identified this gap as one of the biggest challenges that algebra & trigonometry professors face. She uses a clear, voice that speaks directly to students- similar to how instructors communicate to them in class. Students learning from this text will overcome common barriers to learning algebra & trigonometry and will build confidence in their ability to do mathematics. | 677.169 | 1 |
JEE main 2017 syllabus is well defined by the CBSE (Central Board of Secondary Education). The topics & concept for each subjects can be well understand by the syllabus provided below. Most of the syllabus also carry the weightage of the topics which will be covered by the CBSE class 11th & 12th syllabus. JEE main 2017 syllabus is almost the same as class 10th, 11th & 12th syllabus. A link is provided to download the syllabus.
JEE main Syllabus:-
The JEE main 2017 syllabus covers almost all three important subjects i.e. Physics, Mathematics, Chemistry. How ever the level of difficulty for each topic is different & topics weightage will also be different.
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The JEE Main Syllabus is given in the table below.
Class 11 topics for JEE Main 2017 syllabus
Physics
Chemistry
Mathematics
Unit, dimension, measurement
Periodic table and properties
Logarithms and related expressions
Kinematics
Mole concept
Compound angles
Newtons laws of motion
Nomenclature of organic compounds
Quadratic equations and expressions
Circular motion
Chemical bonding
Sequence and progression
Work, power, energy
Gaseous state
Trignometric functions,equations,inequations
Momentum and collision
Electronic displacement effect
Solutions of triangle
Centre of mass and Inertia
Atomic structure
Straight line and pair of lines
Rotational dynamics
Isomerism
Circles
Elasticity, Calorimetry, Thermal expansion
Redox and equivalent concepts
Permutation and combination
Ktg and thermodynamics
Chemical and ionic equilibrium
Binomial Theorem
Simple Harmonic Motion
Chemical kinetics
Sets, relations, functions
Mechanical Waves
Nuclear chemistry (radioactivity)
Mathematical induction and reasoning
Fluid Mechanics
Class 12 topics for JEE Main 2017 syllabus
Physics
Chemistry
Mathematics
Geometrical Optics
Halogen Derivative
Limit, Continuity, Derivability
Wave Optics
Co-ordination Compounds
Method of differentiation
Electrostatics
Aliphatic Hydrocarbons
Indefinite Integral
Gravitation
Aromatic Hydrocarbons
Definite Integral
Current Electricity
Liquid Solution & Surface Chemistry
Application of Derivatives
Capacitance
Metallurgy
Determinant & Matrices
Magnetic Effect of Current & Magnetism
Alcohol & Ether
Vectors
Electromagnetic Induction & AC
Phenol
3-dimensional geometry
Nuclear physics and radioactivity
Solid State
Probability
Dual nature of matter and radiation
Qualitative Analysis
Differential Equation
Electronic devices
Carbonyl compounds
Area under curve
Communication systems
s, p, d & f block elements
Parabola, Ellipse, Hyperbola
Errors in Measurements & Instruments
Thermodynamics & Thermochemistry
Complex Number
Electrochemistry
Statistics
Biomolecules
Polymers
Carboxylic acids & its derivative
Amines
Practical Organic Chemistry
Importance of JEE main 2017 syllabus:-
The JEE main 2017 syllabus will guide the JEE aspirants to prepare there strategies & to ensure the time management for each topics. The JEE main 2017 syllabus will also guide the candidates as per the topics weightage and other major difficulties. | 677.169 | 1 |
Lial Serieshas helped thousands of students succeed in developmental mathematics by providing the best learning and teaching support to students and instructors.
Author Biography
Marge Lial became interested in math at an early age–it was her favorite subject in the first grade! Marge's intense desire to educate both her students and herself inspired the writing of numerous best-selling textbooks. Marge, who received Bachelor's and Master's degrees from California State University at Sacramento, was affiliated with American River College. An avid reader and traveler, her travel experiences often found their way into her books as applications, exercise sets, and feature sets. She was particularly interested in archeology; trips to various digs and ruin sites produced some fascinating problems for her textbooks involving such topics as the building of Mayan pyramids and the acoustics of ancient ball courts in the Yucatan. We dedicate the new editions of the paperback developmental math series to Marge in honor of her contributions to the field in which she helped thousands of students succeed.
Stan Salzman is a long time resident of Sacramento, California. Stan has taught at American River College for many years, where he was a member of the business department. He is the author of Business Math and Essential Math, published by Pearson Education, Inc., and is coauthor of Basic Math.
Diana Hestwood lives in Minnesota and has taught at Metropolitan Community College in Minneapolis for two decades. She has done research on the student brain and is an expert on study skills. She is the author of Lial/Hestwood's Prealgebra and coauthor of Lial/Salzman/Hestwood's Basic Math and Lial/Hestwood/Hornsby/McGinnis's Prealgebra and Introductory Algebra.
Table of Contents
1. Whole Numbers
1.1 Reading and Writing Whole Numbers
1.2 Adding Whole Numbers
1.3 Subtracting Whole Numbers
1.4 Multiplying Whole Numbers
1.5 Dividing Whole Numbers
1.6 Long Division
1.7 Rounding Whole Numbers
1.8 Exponents, Roots, and Order of Operations
1.9 Reading Pictographs, Bar Graphs, and Line Graphs
1.10 Solving Application Problems
2. Multiplying and Dividing Fractions
2.1 Basics of Fractions
2.2 Mixed Numbers
2.3 Factors
2.4 Writing a Fraction in Lowest Terms
2.5 Multiplying Fractions
2.5 Applications of Multiplication
2.7 Dividing Fractions
2.8 Multiplying and Dividing Mixed Numbers
3. Adding and Subtracting Fractions
3.1 Adding and Subtracting Like Fractions
3.2 Least Common Multiples
3.3 Adding and Subtracting Unlike Fractions
3.4 Adding and Subtracting Mixed Numbers
3.5 Order Relations and the Order of Operations
4. Decimals
4.1 Reading and Writing Decimal Numbers
4.2 Rounding Decimal Numbers
4.3 Adding and Subtracting Decimal Numbers
4.4 Multiplying Decimal Numbers
4.5 Dividing Decimal Numbers
4.6 Fractions and Decimals
5. Ratio and Proportion
5.1 Ratios
5.2 Rates
5.3 Proportions
5.4 Solving Proportions
5.5 Solving Application Problems with Proportions
6. Percent
6.1 Basics of Percent
6.2 Percents and Fractions
6.3 Using the Percent Proportion and Identifying the Components in a Percent Problem
6.4 Using Proportions to Solve Percent Problems
6.5 Using the Percent Equation
6.6 Solving Application Problems with Percent
6.7 Simple Interest
6.8 Compound Interest
7. Geometry
7.1 Lines and Angles
7.2 Rectangles and Squares
7.3 Parallelograms and Trapezoids
7.4 Triangles
7.5 Circles
7.6 Volume and Surface Area
7.7 Pythagorean Theorem
7.7 Congruent and Similar Triangles
8. Statistics
8.1 Circle Graphs
8.2 Bar Graphs and Line Graphs
8.3 Frequency Distributions and Histograms
8.4 Mean, Median, and Mode
9. The Real Number System
9.1 Exponents, Order of Operations, and Inequality
9.2 Variables, Expressions, and Equations
9.3 Real Numbers and the Number Line
9.4 Adding Real Numbers
9.5 Subtracting Real Numbers
9.6 Multiplying and Dividing Real Numbers
9.7 Properties of Real Numbers
9.8 Simplifying Expressions
10. Equations, Inequalities, and Applications
10.1 The Addition Property of Equality
10.2 The Multiplication Property of Equality
10.3 More on Solving Linear Equations
10.4 An Introduction to Applications of Linear Equations
10.5 Formulas and Additional Applications from Geometry
10.6 Solving Linear Inequalities
11. Graphs of Linear Equations and Inequalities in Two Variables
11.1 Linear Equations in Two Variables; The Rectangular Coordinate System | 677.169 | 1 |
Welcome to Secondary Math III!
Use the toolbar above to locate your class and find helpful resources such as select notes and answer keys. Additional resources such as tutorial videos will be located in the "Additional Resources" tab. | 677.169 | 1 |
Showing 1 to 10 of 10
Math 4800 - Elementary Analysis
Exam 2 - Take-home
Directions: This is a take-home exam. You are welcome to use your text, the web, or whatever resource you see
t. You are welcome to work together. You must, however, submit your own written solution (obvi | 677.169 | 1 |
Official Course Information
Mathematics Courses:
Catalog Description:
Topics include solving first degree inequalities, introduction to functions, linear equations in two variables and graphing, solving systems of two or three linear equations and inequalities, brief review of polynomial operations and factoring, algebraic fractions, variation, solving rational equations and proportions, rational exponents and radical expressions, complex numbers, solving radical equations, and four methods for solving quadratic equations, with emphasis on problem solving and applications throughout the course. Not open to students with credit in MAT 136 or higher. Prerequisite: MAT 092 or by placement.
Lecture: 3 hrs.
Course Student Learning Outcomes (CSLOs):
Upon successful completion of this course as documented through writing, objective testing, case studies, laboratory practice, and/or classroom discussion, the student will be able to:
1. Correctly translate and solve first-degree equations and inequalities.
2. Given a relation, identify if it is a function. If so, find its domain and range, and then correctly evaluate the function given a value in its domain.*
3. Given a linear equation, correctly graph it on a rectangular coordinate system.
4. Find the equation of a line given one of the following:
A point on the line and the slope of the line
Two points on the line
A point on the line and the equation of a line parallel or perpendicular to it
5. Given a system of linear equations, correctly solve the system by graphing, substitution or the addition method.
6. Given a system of inequalities, solve the system by the graphing method.
7. Given two or more polynomials, correctly add, subtract, multiply or divide the expressions.
8. Given two or more rational expressions, correctly add, subtract, multiply or divide the expressions.
9. Given an equation involving rational expressions, correctly solve the equation.
10. Given two or more radical expressions, add, subtract, multiply and divide the expressions and write the answer in simplest form.
11. Given two complex numbers, add, subtract, multiply or divide the numbers and write the answer in the form a + bi.
12. Given a quadratic equation, solve it by factoring, taking square roots, completing the square or using the quadratic formula.
* This course objective has been identified as a student learning outcome that must be formally assessed as part of the Comprehensive Assessment Plan of the college. All faculty teaching this course must collect the required data and submit the required analysis and documentation at the conclusion of the semester to the Office of Institutional Research and Assessment. | 677.169 | 1 |
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Unformatted text preview: case, when y = 42.) While these
concepts are more precisely quantified using Calculus, below are two views of the graph of y = N (x),
one on the interval [0, 8], the other on [8, 15]. The former looks strikingly like uninhibited growth;
the latter like limited growth. 84
y = f (x) = 1+2799e−x for
0≤x≤8 6.5.2 84
y = f (x) = 1+2799e−x for
8 ≤ x ≤ 16 Applications of Logarithms Just as many physical phenomena can be modeled by exponential functions, the same is true of
logarithmic functions. In Exercises 6a, 6b and 6c of Section 6.1, we showed that logarithms are
useful in measuring the intensities of earthquakes (the Richter scale), sound (decibels) and acids and
bases (pH). We now present yet a different use of the a basic logarithm function, password strength.
Example 6.5.6. The information entropy H , in bits, of a randomly generated password consisting
of L characters is given by H = L log2 (N ), where N is the number of possible symbols for each
character in the password. In general...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School. | 677.169 | 1 |
This book is an ideal resource for extra classwork, homework and for use in catch-up or Summer classes. Each practice exercise delivers progression through questions which revisit and extend ideas covered in Year 7 Pupil Book 1.
Maths Frameworking offers you the most comprehensive and engaging route to Framework success.Book Description Collins Educational 20/07685
Book Description Collins Educational 20/076851891525
Book Description Collins Educational 20/07685 | 677.169 | 1 |
PreCalculus
Precalculus delves deeper into the functions and trigonometry you studied in algebra II. Additional topics include matrices and determinants, sequences, series, and probability, analytic geometry, limits and an introduction to calculus.
There are student resources available online at Each example in the text book is worked out via a video. Instructional videos, interactive activities, and data downloads are accessible via this site, as are practice tests and a success organizer for taking notes.
Use the calendar below to keep track of assignments and use the included links to access on-line material and resources. Subscribe to the calendar using ical
Ebola 2014 – Use CDC data on the Ebola outbreak and Excel and TI-83+ graphing calculator to create exponential regression curves of best fit.
Exponential and Logarithmic Models – This page contains a video of examples of exponential growth predictions and images of the problems worked out. It also describes various exponential and logarithmic growth models like Gaussian, exponential decay, and logistics growth.
Partial Fractions – The video describes how to write a rational expression as the sum of two or more partial fractions. Five examples are demonstrated including fraction that require linear numerators. | 677.169 | 1 |
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more | 677.169 | 1 |
All of the courses in the junior high, high school, and college mathematics curriculum require a thorough grounding in the fundamentals, principles, and techniques of basic math and pre-algebra, yet many students have difficulty grasping the necessary concepts. Utilizing the author's acclaimed and patented fail-safe methodology for making mathematics easy to understand, Bob Miller's Basic Math and Pre-Algebra for the Clueless enhances students' facility in these techniques and in understanding the basics.
This valuable new addition to Bob Miller's Clueless series provides students with the reassuring help they need to master these fundamental techniques, gives them a solid understanding of how basic mathematics works, and prepares them to perform well in any further mathematics courses they Bob Miller's Basic Math and Pre-Algebra for the Clueless by Bob Miller today - and if you are for any reason not happy, you have 30 days to return it. Please contact us at 1-877-205-6402 if you have any questions.
More About Bob Miller
Bob Miller was a lecturer in mathematics at City College of New York for more than 30 years. He has also taught at Westfield State College and Rutgers. His principal goal is to make the study of mathematics both easier and more enjoyable for students.
Miller is very convinced that he is a great teacher. He spends a good deal of this book telling you that he is. I however have to disagree.
Whenever I tried to learn a concept from this book, I always ended up more confused than when I started. When he derives equations he introduces unnecessary and complicated steps that ultimately contribute nothing to the solution. His descriptions of the functions are confusing and regularly use terms he has not introduced. The values chosen for the examples easily could have been ones that would clearly illustrate the principles being taught, but for some reason Miller usually chose the few values that didn't clearly illustrate anything.
On the plus side he makes notes of which sections will be on the Math SAT. So, if you are practicing for the SATs, you may find that helpful.
Math Made Easy! Apr 5, 2006
Bob Miller takes very difficult concepts and makes them extremely easy to grasp. It is the one math book that will not make you tear your hair out of your head! The author uses a bit of humor and emotion to make math understandable to the average layman. The books are meant to be a supplement to whatever math text you are using. After Bob explains the concepts you will breeze through your textbook's examples. A great book at a great price!
Not quite there May 11, 2003
I can't figure out who this book is for. The author states that it's "written for those who want to get a jump on algebra and for those returning to school, perhaps after a long time". I fall into the latter category, but the book isn't particularly helpful to me. The presentation seems silly (or annoying, as the reviewer below says) and lacking.
The author states, "In order to get maximum benefit from this book, you must practice. Do many exercises until you are very good with each of the skills." There are no exercises in the book, however, for the student to do. Each concept is explained with several problems and solutions as examples, but there are no separate problems for students to solve on their own.
The author states that his "practical dream is to have someone sponsor a math series from prealgebra through calculus so everyone in our country will be able to think well and keep our country number one forever." If this book is an example of thinking well, our country is in deep trouble. Fortunately, I'm not as xenophobic as the author, and would be happy to use a math book from another country if it's written better than this one.
Annoying writing style Aug 7, 2002
Overall this book was helpful to me in reviewing for the GMAT exam. Unfortunately the author had a hideous writing habit that he used throughout the book. As he would explain various topics he would add additional letters to particular words. For example: "..." At first I thought this was a typo, but it occurs on page after page with different words and is terribly distracting. I can't believe an editor would let this get through. I don't know if this was meant to be cute or what, but it is just...annoying.
Additionally, for a book that is obviously designed for those of us who are clueless about algebra, Mr Miller would often start off explaining new topics by stating something to the effect that he once thought this topic was hard, but now he doesn't nor do his students, so the reader should do super well. Excuse me, but if algebra came easily to me I wouldn't have bought this book!
Added to my confusion Jul 23, 2002
I bought this book to help me brush up on the algebra I had in college. The fact that it's for the clueless, I figured I had half a chance to recall what I already thought I knew but needed to brush up on. The writer reminded me of my college professor. The math was easy to him, but he explained it as if it were easy to the reader as well. He seemed to have forgotten I was clueless. The other problem was the writer explained procedures for figuring out a problem but omitt a step until you came across an example where that omitted information would have been helpful. After I got the problem wrong, then the omitted information would be revealed. The writer kept me a step or two behind and therefore confused as I spent most of my time trying to figure out how the correct answer was determined insted of being taught how to arrive at the correct answer. I gave it two stars because it did temporarily motivate me to get started, and I thought price was good compared to some of the other books, but I guess you get what you paid | 677.169 | 1 |
Recommended Books*
*The operation of this site is partly funded by commissions
I make from Amazon.com on books purchased through these links. I
only recommend books I have personally read and found useful. I
only post book recommendations on pages intended for
adults.
Information for Parents
There are lots of free learning activities for children in the Kids section of this site. You can also read
about the process and rationale for my way of teaching in the Teachers section. Details of my teaching
philosophy can be found in the Designers and Researchers sections.
Services
I provide private tutoring for learners K-12. This includes everything from basic arithmetic and introductory Logo programming through AP Calculus and AP Computer Science. Tutoring is provided in person or online in the form of homework help, enrichment activities, or test preparation.
I teach classroom and online courses in math and computer science for K-12 kids. Through CTD and Northwestern's School of Education and Social Policy, I have worked with researchers in learning sciences to deliver cutting edge curriculum as it was being developed and continue to learn about teaching and curriculum development from experts as I develop my own courses.
At Chiaravalle Montessori School, I teach studios (after school courses) in math and computer science to children who attend the school. My Montessori teacher training and three years of experience as a teacher at Chiaravalle have allowed me to extend my understanding and deepen my appreciation of the Montessori method of education. I am informed by Montessori in every effort I make to meet the needs of my students. | 677.169 | 1 |
Introduction to Programming for
Engineers
Lecture 19: Strings
Characters in C: char
In Matlab, we could define a variable to hold a single character
thus:
letter_a = a;
In C, we do the same but must first declare the variable as type
char:
char letter_a =
Question:
0
1
Let A = @ 1
2
Eigenvectors
and eigenvalues
1
0 2
1 1A . Find the eigenvalues and eigenvectors of A.
0 1
Solution: Recall that the eigenvectors of a matrix A are the (nonzero)
vectors x satisfying Ax = x for some number . The number is called
Mathematics IA Tutorial 9 (week 10 )
Semester 1, 2017
1. Consider the plane P in R3 given by the equation x
y + z = 0.
(a) Prove that P is a subspace of R3 and give a basis for P .
(b) Consider the vector (1, 0, 0).
i. Can it be written as a linear combin
Mathematics IA Tutorial 7 (week 8 )
Semester 1, 2017
1. Formulate, but do not solve, the following problem:
The current regulations in Formula 1 require that the power units
each team use are constructed of three main components: the internal
combustion e
THE METHOD OF SECTIONS
The method of sections is often used when forces
in only certain truss members are required.
READING QUIZ
1. In the method of sections, generally a cut passes through no
more than _ members in which the forces are unknown.
A) 1
B) 2
Internal Forces
Objectives:
1. Use the method of sections to determine internal
forces in 2-D frames and beams.
2. Develop relationships to draw internal shear and
moment diagrams throughout a member.
READING QUIZ
1. In a multiforce member, the member is
Reduction of Distributed Loading
Objectives:
Students will be able to
determine an equivalent force
for a distributed load.
READING QUIZ
1. The resultant force (FR) due to a distributed load is
equivalent to the _ under the distributed loading curve,
w =
Chapter 3: EQUILIBRIUM OF A PARTICLE
What is equilibrium?
A particle to which forces are applied is in (static)
equilibrium if the net force is zero:
F 0
which, in 3-D, requires that
F
x
0 and Fy 0 and Fz 0
What is a particle?
Particles have mass but can
Chapter 5: Equilibrium of a Rigid Body 2D
Objectives:
Students will be able to:
a) Identify support
reactions; and
b) Draw a free diagram.
Quiz
1. If a support prevents translation of a body, then the support exerts a
_ on the body.
A) couple moment
B) fo | 677.169 | 1 |
Linear Systems: Eigenvalues and Eigenvectors
In this "Eigenvalues and Eigenvectors" instructional activity, students find the eigenvalue of a matrix and explore the properties of eigenvalues with a matrix. This two page instructional activity contains two multi-step problems. | 677.169 | 1 |
Elementary Classical Analysis
内容简介
· · · · · · calcul... calculus or complex analysis.
目录
· · · · · ·
1. Introduction: Sets and Functions
Supplement on the Axioms of Set Theory
2. The Real Line and Euclidean Space
Ordered Fields and the Number Systems
Completeness and the Real Number System
· · · · · ·
(更多)
1. Introduction: Sets and Functions
Supplement on the Axioms of Set Theory
2. The Real Line and Euclidean Space
Ordered Fields and the Number Systems
Completeness and the Real Number System
Least Upper Bounds
Cauchy Sequences
Cluster Points: lim inf and lim sup
Euclidean Space
Norms, Inner Products, and Metrics
The Complex Numbers
3. Topology of Euclidean Space
Open Sets
Interior of a Set
Closed Sets
Accumulation Points
Closure of a Set
Boundary of a Set
Sequences
Completeness
Series of Real Numbers and Vectors
5. Continuous Mappings
Continuity
Images of Compact and Connected Sets
Operations on Continuous Mappings
The Boundedness of Continuous Functions of Compact Sets
The Intermediate Value Theorem
Uniform Continuity
Differentiation of Functions of One Variable
Integration of Functions of One Variable
6. Uniform Convergence
Pointwise and Uniform Convergence
The Weierstrass M Test
Integration and Differentiation of Series
The Elementary Functions
The Space of Continuous Functions
The Arzela-Ascoli Theorem
The Contraction Mapping Principle and Its Applications
The Stone-Weierstrass Theorem
The Dirichlet and Abel Tests
Power Series and Cesaro and Abel Summability | 677.169 | 1 |
Three million high school students and 172, 000 college students enroll in geometry classes every year. Schaum's Outline of Geometry, Third Edition, is fully updated to reflect the many changes in geometry curriculum, including new terminology and notation and a new chapter on how to use the graphing calculator | 677.169 | 1 |
Mathematics
Secondary Math I (9th grade)
The fundamental purpose of Mathematics I is to formalize and extend the mathematics that students learned in the middle grades. The critical areas, organized into units, deepen and extend understanding of linear relationships, in part by contrasting them with exponential phenomena, and in part by applying linear models to data that exhibit a linear trend. Mathematics I uses properties and theorems involving congruent figures to deepen and extend understanding of geometric knowledge from prior grades. The final unit in the course ties together the algebraic and geometric ideas studied. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.
Secondary Math I Honors (9th grade)
This class will meet with Secondary 1 II (10th grade)
The focus of Secondary Math ll is on quadratic expressions, equations, and functions; comparing their characteristics and behavior to those of linear and exponential relationships from Mathematics l as organized into 6 critical areas or units. The need for extending the set of rational numbers arises and real and complex numbers are introduced so that all quadratic equations can be solved. The link between probability and data is explored through conditional probability and counting methods, including their use in making and evaluating decision. The study of similarity leads to an understanding of right triangle trigonometry and connects to quadratics through Pythagorean relationships. The mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.
Secondary Math II Honors (10th grade)
This class will meet with Secondary 2 III (11th grade)
Students in Secondary Mathematics III will focus on pulling together and applying the accumulation of learning that they have from their previous courses. They will apply methods from probability and statistics, expand their repertoire of functions to include polynomial, rational, and radical functions, they will expand their study of right triangle trigonometry and will bring together all of their experience with functions and geometry to create models and solve contextual problems. The mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. | 677.169 | 1 |
Tuesday, April 29, 2014
New animation: a geometric representation of completing the square. In this post, I present one way to use it as part of an algebra curriculum.
Many secondary school teachers figure that the derivation of the quadratic formula by completing the square can be shown to students, but have little hope of any understanding. They do not worry too much about it, because as they see it, the main point is for students to learn the quadratic formula, and to know how to use it to solve quadratic equations. Understanding it is not a priority.
I disagree.
In fact, I believe the opposite: while I have no objection to students memorizing and being able to use the quadratic formula, this is much less important than it used to be. Nowadays, any quadratic equation can be solved by entering it in Wolfram Alpha, or in a CAS calculator, and pressing the Enter key. On the other hand, completing the square is a nice bit of algebra, with other uses for those students who will pursue more advanced work in mathematics. But even for students who will go no further than Algebra 2, it is interesting and accessible, and helps demystify the quadratic formula.
Well, it's not accessible if taught to eighth or ninth graders strictly through the manipulation of algebraic symbols. I've done that, and to be honest, I was reaching perhaps 10% of my students. I was able to turn this around, and make the whole thing accessible to perhaps 90% of my students, by postponing this topic until Algebra 2 (10th / 11th grade), and by using a geometric approach based on the Lab Gear manipulatives.
Here is a summary of my strategy. The key is to break the long and complicated procedure into understandable chunks.
- Make a rectangle, then make a square:
Doing this several times reveals visually what goes on, and makes it far easier to understand the process of completing the square. The "completing the square" animation is helpful at this point to help students put the process into words, and to generalize to non-whole-number values of b.
- Equal squares
Looking at increasingly complicated examples, I make the point that if we have an "equal squares" equation, we can reduce it to two linear equations. For example, if x2 = 9, then x = 3 or x = -3. If x2 + 6x + 9 = 7, then x + 3 = or x + 3 = - . I avoid the phrase "take the square root of both sides", which only encourages misunderstandings.
- Solving any quadratic
At this point, it becomes possible to combine what was learned into an overall "completing the square" strategy to solve quadratic equations | 677.169 | 1 |
Matrix Calculator This calculator performs all matrix, vector operations. You can add, subtract, find length, find dot and cross product, check if vectors are dependant. For every operation, calculator will generate a detailed explanation
AndroidCalc AndroidCalc is a simple calculator application for the Android PlatformAdvance CALCU Calculator is a simple calculator, with a panel that has more advanced functions | 677.169 | 1 |
• The conceptual model-based problem solving (COMPS) program emphasizes mathematical modeling and algebraic representation of mathematical relations in equations, which are in line with the new Common Core. • "Through building most fundamental concepts pertinent to additive and multiplicative reasoning and making the connection between concrete and abstract modeling, students were prepared to go above and beyond concrete level of operation and be able to use mathematical models to solve more complex real-world problems. As the connection is made between the concrete model (or students' existing knowledge scheme) and the symbolic mathematical algorithm, the abstract mathematical models are no longer "alien" to the students." As Ms. Karen Combs, Director of Elementary Education of Lafayette School Corporation in Indiana, testified: "It really worked with our kids!" • "One hallmark of mathematical understanding is the ability to justify,... why a particular mathematical statement is true or where a mathematical rule comes from" ( Through making connections between mathematical ideas, the COMPS program makes explicit the reasoning behind math, which has the potential to promote a powerful transfer of knowledge by applying the learned conception to solve other problems in new contexts. • Dr. Yan Ping Xin's book contains essential tools for teachers to help students with learning disabilities or difficulties close the gap in mathematics word problem solving. I have witnessed many struggling students use these strategies to solve word problems and gain confidence as learners of mathematics. This book is a valuable resource for general and special education teachers of mathematics. - Casey Hord, PhD, University of Cincinnati
This book collects recent research on posing and solving mathematical problems. Rather than treating these two crucial aspects of school mathematics as separate areas of study, the authors approach them as a unit where both areas are measured on equal grounds in relation to each other. The contributors are from a vast variety of countries and with a wide range of experience; it includes the work from many of the leading researchers in the area and an important number of young researchers. The book is divided in three parts, one directed to new research perspectives and the other two directed to teachers and students, respectively.
A highly practical resource for special educators and classroom teachers, this book provides specific instructional guidance illustrated with vignettes, examples, and sample lesson plans. Every chapter is grounded in research and addresses the nuts and bolts of teaching math to students who are not adequately prepared for the challenging middle school curriculum. Presented are a range of methods for helping struggling learners build their understanding of foundational concepts, master basic skills, and develop self-directed problem-solving strategies. While focusing on classroom instruction, the book also includes guidelines for developing high-quality middle school mathematics programs and evaluating their effectiveness.
ExplicitLearningModel-Based Approaches to Learning provides a new perspective called learning by system modeling. This book explores the learning impact of students when constructing models of complex systems. In this approach students are building their own models and engaging at a much deeper conceptual level of understanding of the content, processes, and problem solving of the domain, which is proven to be successful by research from the area of mindtools. Topics covered include the foundations of knowledge structures and mental model development, modeling for understanding, modeling for assessment, individual versus collaborative modeling, and the use of simulations to support learning and instruction in complex, cognitive domains. The thread tying these chapters together is an emphasis on what the learner is doing when he is engaged in modeling and simulation construction rather than merely interacting with constructed simulations. Model-Based Approaches to Learning is an interesting book for Educators (Instructors, K-12 Teachers), who are looking for forms to use advanced computer technology in classrooms. Also Teachers' educators who are working on the integration of technology into their teacher preparation classrooms can find new concepts and best-practice examples in this book. This also holds true for all Educators and Researchers who are interested in modeling as an activity to successfully work with ill-structured and complex problems. | 677.169 | 1 |
Getting Creative With Options Advice
There are a wide array of calculators. Some are specialized, some quite simple. Some calculators can only do simple arithmetic, and others that can establish a single point on a parabola. Price points for these devices are normally determined by the complexity of the calculator, thus deciding the right calculator can conserve money and annoyance.
Firstly, there are basic calculators. These are used for simple functions. Typically they cost less than ten dollars, and simple functions such as add, multiply, deduct and divide. Some include a memory function, large display, big keys or a transparent design with overhead projectors. Basic calculators are for basic mathematics. Batteries or a solar cell may be used to power them up.
Next, we have the printing calculators. These are made for use in bookkeeping and accounting. They include all of the keypads in a simple calculator but have additional keys for other functions such as finding per centum, subtotal, total, duty rate, as well as several other functions utilized when summing up huge amounts. The print capability enables the consumer to check out for typical entry blunders or skipped numbers, make use of the printout like a receipt or avoid printing it. Most also include an ability to select decimal places' variety and a rounding purpose.
There are also scientific calculators. Scientific calculators are utilized by pupils. They are used for chemistry, physics or math. They are likewise frequently permitted for use in standard assessments. These calculators also come with multifunction keys for a smaller full-featured tool. They are capable of determining root values, with separate keys for square root, cube root and so forth. They also possess logarithmic, trigonometric and exponential function keys.
We also have financial and business calculators. Business and finance calculators are made to be properly used primarily in determining enterprise-related equations, for pupils studying calculus, as well as for planning or monitoring economic info. Financial calculators may build tables for loan obligations, compute APR, evaluate compound interest; total paid, interest and other characteristics of cash overtime.
Business calculators have more outstanding positioning of the most typical functions utilized in business calculus or mathematics. They frequently contain evaluation capabilities and sophisticated mathematical designs. More costly models contain more cash runs greater storage, and much more functions.
Last on the list, we have graphic calculators. These calculators are some of the most expensive ones. They have large memory capability, programmability, the graphing capability and a sizable display itself make graphing calculators different from other types of calculators. They are essential in physics and several college-level mathematics, in addition to in certain finance classes. They have the ability to execute most of the functions though very few have print capabilities within additional specialized calculators. These calculators are designed to chart and plan capabilities, and simplify solving equations | 677.169 | 1 |
Solving Simple Inequalities with Answers
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389 KB|2 pages
Product Description
This one page worksheet includes valuable examples for solving inequalities. Students will be able to understand when to change the inequality sign. Questions of varying formats keeps students challenged. This is an excellent resource for students preparing for any Algebra EOC (End of Course) exam. Everything is included on just one page! | 677.169 | 1 |
Complex Numbers 1
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A 'Teach Further Maths' Resource
37 slides
Lesson Objectives:
To understand what is meant by an 'imaginary number'.
To be able to calculate with powers of i.
To understand what is meant by a 'complex number'.
To be able to solve any quadratic equation.
To know the condition for a quadratic equation to have complex conjugate solutions.
To understand what is meant by an 'Argand Diagram'.
To be able to perform simple arithmetic with complex numbers.
To be able to equate real and imaginary parts to solve some problems involving complex numbers11.00. | 677.169 | 1 |
Comments
Latex is best learn incrementally. The basis of it is really simple, you just need to remember to put your code within \ ( and \ ) (no spaces).
When you see that it fails, then remove the brackets, and just leave it be. E.g. 1 + 2 is as easy to read as \( 1 + 2 \).
When you see someone with an interesting Latex formula that you like you can hover your mouse button over the image to see the actual code. Alternatively, you can hit the "Toggle Latex", which would allow you to copy-and-paste the code so that you can play with it.
–
Calvin Lin
Staff
·
1 year, 12 months ago
Jun, there are some notes on how to use LaTex in this Brilliant site. Just use the search box on the top right corner with the magnifying glass icon. I have copied a link for you.
–
Chew-Seong Cheong
·
1 year, 11 months ago
Well if you are looking forward to using Latex for mathematical equations this might be a bit help : Link rest basic stuff you can learn by the links provided by fellow members..... \(\ddot \smile\)
–
Harshvardhan Mehta
·
1 year, 11 months ago
I am also a beginner in learning latex. All i have learned is from this link and i always see the latex code of others' solutions (you can do that by choosing 'toggle latex' or by hovering over the solution.)
You can read the \(link\) and then wait for more experts to come.Hope it helps you.
@Jun Arro Estrella
–
You can learn from the link that Ashwin has given , or there is an easier way as well . Your profile pic is there on the top-right corner of this page . When you click on it , there'll be an option Toggle Latex .
Click it , now you'll be able to see the \(\LaTeX\) codes present on any page where you perform this small ritual :P
–
Azhaghu Roopesh M
·
1 year, 12 months ago | 677.169 | 1 |
9780321923509
0321923502177.52
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$185.28
More Prices The Sullivan/Struve/Mazzarella Algebraprogram is designed to motivate students to "do the math"— at home or in the lab—and supports a variety of learning environments. The text is known for its two-column example format that provides annotations to the left of the algebra. These annotations explain what the authors are about to do in each step (instead of what was just done), just as an instructor would do.
This package consists of the text, MyMathLab access card, and Video Notebook.
Author Biography
Michael Sullivan, III is a full-time professor of mathematics at Joliet Jr. College. His training is in mathematics, statistics, and economics and he has more than 18 years experience as an instructor. His publications with Pearson cover developmental math, precalculus, and introductory statistics. Most recently, Mike has been highly involved in Course Redesign at Joliet Junior College. His experience in course redesign and writing texts for the college-level math and statistics courses gives him a unique insight into where students are headed after the developmental math track, and what they need to do to be successful. Mike is the father of three children and an avid golfer.
Katherine Struve is a full-time instructor with Columbus State Community College and has more than 30 years experience in the classroom. She is in tune with the challenges of teaching mathematics at a large, urban community college. She has served as the Lead Instructor of Developmental Mathematics and has been involved in the recent course redesign effort at Columbus State.
Janet Mazzarella is a full-time instructor at Southwestern College and has more than 18 years experience teaching a wide range of courses from arithmetic through calculus. She helped develop the self-paced developmental math program and spent two years serving as its director. Along with Mike and Kathy, Janet has also been active in the course redesign initiative at her institution.
Table of Contents
1. Whole Numbers
1.1 Success in Mathematics
1.2 Fundamentals of Whole Numbers
1.3 Adding and Subtracting Whole Numbers
1.4 Multiplying Whole Numbers and Exponents
1.5 Dividing Whole Numbers
1.6 Prime Numbers
1.7 Order of Operations Using Whole Numbers
2. Integers and an Introduction to Algebra
2.1 Fundamentals of Integers
2.2 Adding and Subtracting Integers
2.3 Multiplying and Dividing Integers
2.4 Exponents and Order of Operations
2.5 Simplifying Algebraic Expressions
2.6 Linear Equations: The Addition and Multiplication Properties
2.7 Linear Equations: Using the Properties Together
2.8 Introduction to Problem Solving: Direct Translation Problems
3. Fractions
3.1 Fundamentals of Fractions
3.2 Multiplying and Dividing Fractions
3.3 Adding and Subtracting Like Fractions
3.4 Adding and Subtracting Unlike Fractions
3.5 Order of Operations and Complex Fractions
3.6 Operations with Mixed Numbers
3.7 Solving Equations Containing Fractions
4. Decimals
4.1 Fundamentals of Decimals
4.2 Adding and Subtracting Decimals
4.3 Multiplying Decimals
4.4 Dividing Decimals
4.5 Solving Equations Containing Decimals
5. Ratio, Proportion, and Percent
5.1 Ratios and Unit Rates
5.2 Proportions
5.3 Fundamentals of Percent Notation
5.4 Solving Percent Problems: Proportion Method
5.5 Solving Percent Problems: Equation Method
5.6 Applications Involving Percent
6. Measurement and Geometry
6.1 The U.S. Standard (English) System of Measurement
6.2 The Metric System of Measurement
6.3 Fundamentals of Geometry
6.4 Polygons
6.5 Perimeter and Area of Polygons and Circles
6.6 Volume and Surface Area
7. Introduction to Statistics and the Rectangular Coordinate System
7.1 Tables and Graphs
7.2 Mean, Median, and Mode
7.3 The Rectangular Coordinate System and Equations in Two Variables
7.4 Graphing Equations in Two Variables
8. Equations and Inequalities in One Variable
8.1 Linear Equations: The Addition and Multiplication Properties of Equality | 677.169 | 1 |
TrueShelf
TrueShelf is an AI powered adaptive mathematics learning platform for middle school and high school students. Our platform adaptively generates unlimited number of mathematics problems and helps students learn mathematical concepts rigorously, prepare for competitive exams (like SAT, JEE) and get detailed feedback on their performance. Our AI engine identifies the students' strengths and weaknesses and guides them to improve their performance with step-by-step interactive solutions and bite-sized lessons. Thousands of students and instructors from all over the world use our platform daily. For more details read our CEO's interview. | 677.169 | 1 |
Handbook of Analysis and Its Foundations is a self-contained and unified handbook on mathematical analysis and its foundations. Intended as a self-study guide for advanced undergraduates and beginning graduatestudents in mathematics and a reference for more advanced mathematicians, this highly readable book provides broader coverage than competing texts in the area. Handbook of Analysis and Its Foundations provides an introduction to a wide range of topics, including: algebra; topology; normed spaces; integration theory; topological vector spaces; and differential equations. The author effectively demonstrates the relationships between these topics and includes a few chapters on set theory and logic to explain the lack of examples for classical pathological objects whose existence proofs are not constructive. More complete than any other book on the subject, students will find this to be an invaluable handbook. For more information on this book, see
"synopsis" may belong to another edition of this title.
Review:
This is quite a book! From the table of contents, it would appear to include just about everything one would want to know about the foundations of analysis. It is well-organized and the exposition in the sample chapters is quitegoodclear, concise, and relatively easy to read. It is very good technically; the author knows what he is talking about. --George L. Cain, GEORGIA INSTITUTE OF TECHNOLOGY At the very outset, I would like to say that I am very much impressed by what I have seen. I have read the Preface and understood the authors purpose and his aims. I admire him for his courage in attempting such a daunting task, and I admire him even more for what appears to me to be a very successful completion of this task.....I am very excited over the prospect of this book being made available; it will be a very useful reference not only for beginning graduate students, but also for their teachers. --Robert G. Bartle, EASTERN MICHIGAN UNIVERSITY
From the Author:
For more information see the author's web page. Additional information about this book is available at the web page maintained by the author, at | 677.169 | 1 |
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This course covers mathematical topics in college algebra, with an emphasis on functions. The course is designed to help prepare students to enroll for a first semester course in single variable calculus.
This course covers mathematical topics in trigonometry. Trigonometry is the study of triangle angles and lengths, but trigonometric functions have far reaching applications beyond simple studies of triangles. This course is designed to help prepare students to enroll for a first semester course in single variable calculus.
Prepare for Introductory Calculus courses. Mathematics is the language of Science, Engineering and Technology. Calculus is an elementary Mathematical course in any Science and Engineering Bachelor. Pre-university Calculus will prepare you for the Introductory Calculus courses by revising four important mathematical subjects that are assumed to be mastered by beginning Bachelor students: functions, equations, differentiation and integration | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Word Problems? NO PROBLEM!
Now anyone, even those whose palms begin to sweat at the first sight of math problems that begin "A train left the station going 65 mph..." can overcome anxiety and learn to solve word problems. In Math Word Problems Demystified, experienced math instructor Allan G. Bluman provides an effective, tension-free, approach to conquering the word problems on the SATs and many other standardized tests, in algebra, and in other mathematics and science classes.
With Math Word Problems Demystified, you master the subject one simple step at a time -- at your own speed. This unique self-teaching guide offers practice problems, a quiz at the end of each chapter to pinpoint weaknesses, and a 40 question final exam to reinforce the methods and material presented in the book.
If you want to master math word problems, here's the self-teaching course that will get it done. Get ready to --
Transform word problems into solvable equations
Master a 4-step strategy that empowers you to understand and solve the most common word problems
A fast, effective, and fun way to master word problems, Math Word Problems Demystified is the perfect shortcut to gain the confidence and develop the necessary skills to solve these tough test questions.
Synopsis
Word problems are the most difficult part of any math course –- and the most important to both the SATs and other standardized tests. This book teaches proven methods for analyzing and solving any type of math word problem.
Synopsis
Word problems are the most difficult part of any math course –- and the most important to both the SATs and other standardized tests. This book teaches proven methods for analyzing and solving any type of math word problem. | 677.169 | 1 |
Transcription
2 ABOUT THIS BOOKLET This booklet is intended to help you to prepare for STEP examinations. It should also be useful as preparation for any undergraduate mathematics course, even if you do not plan to take STEP. The questions are all based on recent STEP questions. I chose the questions either because they are nice in the sense that you should get a lot of pleasure from tackling them or because I felt I had something interesting to say about them. In this booklet, I have restricted myself (reluctantly) to the syllabus for Papers I and II, which is the A-level core (i.e. C1 to C4) with a few additions. This material should be familiar to you if you are taking the International Baccalaureate, Scottish Advanced Highers or other similar courses. The first two questions (the sample worked questions) are in a stream of consciousness format. They are intended to give you an idea how a trained mathematician would think when tackling them. This approach is much too long-winded to sustain, but it should help you to see what sort of questions you should be asking yourself as you work through the later questions. I have given each of the subsequent questions a difficulty rating ranging from ( ) to ( ). A question labelled ( ) might be found on STEP I; a question labelled ( ) might be found on STEP II; a question labelled ( ) might be found on STEP III. But difficulty in mathematics is in the eye of the beholder: you might find a question difficult simply because you overlooked some key step, which on another day you would not have hesitated over. You should not therefore be discouraged if you are stuck on a one-star question; though you should probably be encouraged if you get through three-star questions without mishap. Each question is followed by a comment and a full solution. The comment will contain hints and/or direct your attention to key points; or it might put the question in its true mathematical context. The solution should give sufficient working for you to read it through and pick up the gist of the method. It is of course just one way of producing the required result: there may be many other equally good or better ways. Calculators are not required for any of the problems in this booklet; and calculators are not permitted in STEP examinations. In the early days of STEP, calculators were permitted but they were not required for any question. It was found that candidates who tried to use calculators sometimes ended up missing the point of the question or getting a silly answer. My advice is to remove the battery so that you are not tempted. I hope that you will use the comments and solutions as springboards rather than feather beds. You will only really benefit from this booklet if you have a good go at each question before looking at the comment and another good go before looking at the solution. You should gain by reading both even if you have completed the question on your own. If you find errors or have suggestions for ways in which this booklet can be improved, please me: I update it regularly. I hope you enjoy using this booklet as much as I have enjoyed putting it together. STCS
3 STEP MATHEMATICS What is STEP? STEP (Sixth Term Examination Paper) is an examination used by Cambridge colleges as the basis for conditional offers. There were STEPs in many subjects but after 2001 the mathematics papers only remain. They are used not just by Cambridge, but also by Warwick, and many other university mathematics departments recommend that their appicants practise on the past papers even if they do not take the examination. In 2008, 2100 scripts were marked, only about 750 of which were written by Cambridge applicants. There are three STEPs, called papers I, II and III. Papers I and II are based the core A-level syllabus, with some minor additions, and Paper III is based on a typical further mathematics mathematics syllabus (for which there is no recognised core). The questions on Papers II and III are supposed to be of about the same difficulty, and are harder than those on Paper I. Each paper has thirteen questions, including three on mechanics and two on probability/statistics. Candidates are assessed on six questions only. What is the purpose of STEP? From the point of view of admissions to a university mathematics course, STEP has three purposes. First, it acts as a hurdle: success in STEP is thought to be a good indicator of potential to do well on a difficult course. Second, it acts as preparation for the course, because the style of mathematics found in STEP questions is similar to that of undergraduate mathematics. Thirdly, it tests motivation. It is important to prepare for STEP (by working through old papers, for example), which can require considerable dedication. Those who are not willing to make the effort are unlikely to thrive on a difficult mathematics course. STEP vs A-level A-level tests mathematical knowledge and technique by asking you to tackle fairly stereotyped problems. STEP asks you to apply the same knowledge and technique to problems that are, ideally, unfamiliar. Here is a quotation from Roger Porkess 1 which illustrates why A-level examinations do not test satisfactorily the skills required to tackle university mathematics courses at the highest levels: Students following modular syllabuses work much harder and the system rewards them for doing so. This means that there is a new sort of A C grade student: not necessarily a particularly intuitive mathematician but someone who has shown the ability to learn the subject successfully by dint of hard work and, quite possibly, good teaching. No one believes that modular A-levels are uniformly bad; anything that encourages more students to study mathematics must be basically good. However, they are not good preparation for the students who plan to take a degree in mathematics from one of the top universities. 1 From Every cloud has a silver lining: Roger Porkess (1995).
4 Here is an A-level question: By using the substitution u = 2x 1, or otherwise, find 2x (2x 1) 2 dx. Here is a STEP question, which requires both competence in basic mathematical techniques and mathematical intuition. Note that help is given for the first integral, so that everyone starts at the same level. Then, for the second integral, candidates have to show that they understand why the substitution used in the first part worked, and how it can be adapted. Use the substitution x = 2 cosθ to evaluate the integral Show that, for a < b, q where p = (3a + b)/4 and q = (a + b)/2. p 2 3/2 The differences between STEP and A-level are: ( )1 x 1 2 dx. 3 x ( )1 x a 2 (b a)(π ) dx =, b x There is considerable choice on STEP papers (6 out of 13). This is partly to allow for different A-level syllabuses. 2. STEP questions are much longer. Candidates completing four questions in three hours will almost certainly get a grade STEP questions are much less routine. 4. STEP questions may require considerable dexterity in performing mathematical manipulations. 5. Individual STEP questions may require knowledge of different areas of mathematics (especially the mechanics and statistics questions, which will often require advanced pure mathematical techniques). 6. The marks available for each part of the question are not disclosed on the paper. 7. Calculators are not permitted. The Advanced Extension Award Advanced Extension Awards were taken for the first time in The last papers, in all subjects except mathematics, will be sat in Mathematics has a reprieve for at least a few years. The AEA is not a replacement for STEP. It is aimed at the top 10% or so of the A-level cohort; i.e. at the top 7000 or soi whereas STEP is aimed at the top 2000 or so. This difference matters, because in mathematics (though not perhaps in other subjects) it is important to match the difficulty of the
5 question and the ability of the candidates exactly. 2 The AEA contains questions on only the core A-level syllabus: no statistics and (more important for us) no mechanics. Many of my colleagues believe that, apart from the resulting reduction of choice, this is undesirable because it sends out the wrong signals about the importance of these other areas of mathematics. Setting STEP STEP is produced under the auspices of the Cambridge Assessment examining board. The setting procedure starts 18 months before the date of the examination, when the three examiners (one for each paper, from schools or universities) are asked to produce a draft paper. The first drafts are then vetted by the STEP coordinator, who tries to enforce uniformity of difficulty, checks suitability of material and style and tries to reduce overlap between the papers. Examiners then produce a second draft, based on the coordinator s suggestions and taking into account his/her comments. The second drafts are circulated to the three moderators (normally school teachers), and to the other examiners, who produce written comments and discuss the drafts in a two-day meeting. The examiners then produce third drafts, taking into account the consensus at the meetings. These drafts are sent to a vetter, who works through the papers, pointing out mistakes and infelicities. The final draft is checked by a second vetter. At each stage, the drafts are produced camera-ready, using a special mathematical word-processing package called LaTeX so no typesetting errors can arise after the second vetting. STEP Questions STEP questions do not fall into any one category. Typically, there will be a range of types on each papers. An excellent type involves two parts: in the first part, candidates are asked to perform some task and are told how to do it (integration by substitution, or expressing a quadratic as the algebraic sum of two squares, for example); for the second part, candidates are expected to demonstrate that they have understood and learned from the first part by applying the method to a new and perhaps more complicated task. Another good type of question requires candidates to do some preliminary special-case work and then guess and prove a general result. In another type, candidates have to show that they can understand and use new notation or a new theorem to perform perhaps standard tasks in disguise. Multipart questions of the A-level type are generally avoided; but if they occur, there tends to be a sting in the tail involving putting all the parts together in some way. As a general rule, questions on standard material tend to be harder than questions on material that candidates are not expected to have seen before. Sometimes, the first question on the paper is of a basic nature, and on material that will be familiar to all candidates. There is a syllabus for STEP, which is given at the end of this book. Often there is at least one question that does not rely on any part of the syllabus: it might be a common sense question, involving counting or seeing patterns, or it might involve some aspect of GCSE mathematics with an unusual slant. Some questions are devised to check that you do not simply apply routine methods blindly. Remember, for example, that a function can have a maximum value at which the derivative is non-zero 2 You could reasonably ask the question Was Henry VIII a good king on examination papers from GCSE to PhD level. The answers would (or should) differ according to the level. Mathematics does not work like this.
6 (it would have to be at the end of a range), so finding the maximum in such a case is not simply a question of routine differentiation. Some questions take you through a new idea and then ask you to apply it to something else. Some questions try to test your capacity for clear thought without using much mathematical knowledge (like the calendar question mentioned in the next section or questions concerning islands populated by toads who always tell the truth and frogs who always fib ). There are always questions specifically on integration or differentiation, and many others (including mechanics and probability) that use calculus as a means to an end, Graph-sketching is regarded by most mathematicians as a fundamental skill and there are nearly always one or more sketching questions or parts of questions. Some questions simply involve difficult algebraic or trigonometric manipulation. Basic ideas from analysis (such as 0 f (x) k 0 b a f (x)dx (b a)k or f (x) > 0 f (b) > f (a) for b > a or a relationship between an integral and a sum) often come up, though knowledge of such results is never assumed a sketch proof of an obvious result may be asked for. Some areas are difficult to examine and questions tend to come up rarely. Complex numbers fall into this category, and the same applies to (pure mathematical applications of) vectors. The mechanics and probability questions often involve significant ideas from pure mathematics. Questions on pure statistical tests are rare, because questions that require real understanding (rather than cookbook methods) tend to be too difficult. More often, the questions in this section are about probability. The mechanics questions normally require a firm understanding of the basic principles (when to apply conservation of momentum and energy, for example) and may well involve a differential equation. Projectile questions are often set, but are never routine. Advice to candidates First appearances Your first impression on looking at a STEP paper is likely to be that it looks very hard. Don t be discouraged! Its difficult appearance is largely due to it being very different in style from A-levels. If you are considering studying mathematics at Cambridge (or Oxford or Warwick), it is likely that you will manage to work through most or all of an A-level module in the time available: maybe 10 questions in 75 minutes (i.e. 7.5 minutes per question). In STEP, you are only supposed to do six questions in three hours, and you are doing very well if you manage four; that means that each question is designed to take 45 minutes. Thus although the questions are long and arduous, you have much more time to work on them. You may be put off by the number of subjects covered on the paper. You should not be. STEP is supposed to provide sufficient questions for all candidates, no matter which A-level mathematics syllabus they have covered. It would be a very exceptional candidate who had the knowledge required to do all the questions. Once you get used to the idea that STEP is very different from A-level, it becomes much less daunting.
7 Preparation The best preparation for STEP is to work slowly through old papers. 3 Hints and answers for the some years are available, but you should use these with discretion: doing a question with hints and answers in front of you is nothing like doing it yourself, and may well miss the whole point of the question (which is to make you think about mathematics). In general, thinking about the problem is much more important than getting the answer. Should you try to learn up areas of mathematics that are not in your syllabus in preparation for STEP? The important thing to know is that it is much better to be very good at your syllabus than to have have a sketchy knowledge of lots of additional topics: depth rather than breadth is what matters. It may conceivably be worth your while to round out your knowledge of a topic you have already studied to fit in with the STEP syllabus; it is probably not worth your while to learn a new topic for the purposes of the exam. It is worth emphasising that there is no hidden agenda : a candidate who did two complete probability questions and one complete mechanics question will obtain the same grade as one who did four complete pure questions. Just as the examiners have no hidden agenda concerning syllabus, so they have no hidden agenda concerning your method of answering the question. If you can get to the end of a question correctly you will get full marks whatever method you use. Some years ago one of the questions asked candidates to find the day of the week of a given date (say, the 5th of June 1905). A candidate who simply counted backwards day by day from the date of the exam would have received full marks for that question (but would not have had time to do any other questions). You may be worried that the examiners expect some mysterious thing called rigour. Do not worry: STEP is an exam for schools, not university students, and the examiners understand the difference. Nevertheless, it is extremely important that you present ideas clearly and show working at all stages. Presentation You should set out your answer legibly and logically (don t scribble down the first thought that comes into your head) this not only helps you to avoid silly mistakes but also signals to the examiner that you know what you are doing (which can be effective even if you haven t the foggiest idea what you are doing). Examiners are not as concerned with neatness as you might fear. However if you receive complaints from your teachers that your answers are difficult to follow then you should listen. Remember that more space often means greater legibility. Try writing on alternate lines (this leaves a blank line for corrections). Try to read your answers with a hostile eye. Have you made it clear when you have come to the end of a particular argument? Try underlining your conclusions. Have you explained what you are trying to do? (If a question asks Is A true? try beginning your answer by writing A is true if it is true so that the examiner knows which way your argument leads.) If you used some idea (for example, integration by substitution) have you told the examiner that this is what you are doing? 3 These and the other publications mentioned below are obtainable from the STEP web site
8 What to do if you cannot get started on a problem Try the following, in order. Reread the question to check that you understand what is wanted; Reread the question to look for clues the way it is phrased, or the way a formula is written, or other relevant parts of the question. (You may think that the setters are trying to set difficult questions or to catch you out. Usually, nothing could be further from the truth: they are probably doing all in their power to make it easy for you by trying to tell you what to do). Try to work out exactly what it is that you don t understand. Simplify the notation e.g. by writing out sums explicitly. Look at special cases (choose special values which simplify the problem) in order to try to understand why the result is true. Write down your thoughts in particular, try to express the exact reason why you are stuck. Go on to another question and go back later. If you are preparing for the examination (but not in the actual examination!) take a short break. 4 Discuss it with a friend or teacher (again, better not do this in the actual examination) or consult the hints and answers, but make sure you still think it through yourself. REMEMBER: following someone else s solution is not remotely the same thing as doing the problem yourself. Once you have seen someone else s solution to an example, then you are deprived, for ever, of most of the benefit that could have come from working it out yourself. Even if, ultimately, you get stuck on a particular problem, you derive vastly more benefit from seeing a solution to something with which you have already struggled, than by simply following a solution to something to which you ve given very little thought. What if a problem isn t coming out? If you have got started but the answer doesn t seem to be coming out then: Check your algebra. In particular, make sure that what you have written works in special cases. For example: if you have written the series for log(1 + x) as 1 x + x 2 /2 x 3 /3 + then a quick check will reveal that it doesn t work for x = 0; clearly, the 1 should not be there. 5 4 Littlewood (distinguished Cambridge mathematician and author of the highly entertaining Mathematicians Miscellany) used to work seven days a week until an experiment revealed that when he took Sundays off the good ideas had a way of coming on Mondays. 5 Another check will reveal that for very small positive x, log is positive (since its argument is bigger than 1) whereas the series is negative, so there is clearly something else wrong.
9 Make sure that what you have written makes sense. For example, in a problem which is dimensionally consistent, you cannot add x (with dimensions length, say) to x 2 or exp x (which itself does not make sense the argument of exp has to be dimensionless). Even if there are no dimensions in the problem, it is often possible to mentally assign dimensions and hence enable a quick check. Be wary of applying familiar processes to unfamiliar objects (very easy to do when you are feeling at sea): for example, it is all too easy, if you are not sure where your solution is going, to solve the vector equation a.x = 1 by dividing both sides by a. Analyse exactly what you are being asked to do; try to understand the hints (explicit and implicit); remember to distinguish between terms such as explain/prove/define/etc. (There is essentially no difference between prove and show : the former tends to be used in more formal situations, but if you are asked to show something, a proper proof is required.) Remember that different parts of a question are often linked (it is sometimes obvious from the notation and choice of names of variables). If you get irretrievably stuck in the exam, state in words what you are trying to do and move on (at A-level, you don t get credit for merely stating intentions, but STEP examiners are generally grateful for any sign of intelligent life). What to do after completing a question It is a natural instinct to consider that you have finished with a question once you have done got to an answer. However this instinct should be resisted both mathematically and from the much narrower view of preparing for an examination. Instead, when you have completed a question you should stop for a few minutes and think about it. Here is a check list for you to run through. Look back over what you have done, checking that the arguments are correct and making sure that they work for any special cases you can think of. It is surprising how often a chain of completely spurious arguments and gross algebraic blunders leads to the given answer. Check that your answer is reasonable. For example if the answer is a probability p then you should check that 0 p 1. If your answer depends on an integer n does it behave as it should when n? Is it dimensionally correct? If, in the exam, you find that your answer is not reasonable, but you don t have time to do anything about it, then write a brief phrase showing that you understand that you answer is unreasonable (e.g. This is wrong because mass must be positive ). Check that you have used all the information given. In many ways the most artificial aspect of examination questions is that you are given exactly the amount of information required to answer the question. If your answer does not use everything you are given then either it is wrong or you are faced with a very unusual examination question. Check that you have understood the point of the question. It is, of course, true that not all exam questions have a point but many do. What idea did the examiners want you to have? Which techniques did they want you to demonstrate? Is the result of the question interesting in some way? Does it generalise? If you can see the point of the question would your working show the point to someone who did not know it in advance?
10 Make sure that you are not unthinkingly applying mathematical tools which you do not fully understand. 6 In preparation for the examination, try to see how the problem fits into the wider context and see if there is a special point which it is intended to illustrate. You may need help with this. In preparation for the examination, make sure that you actually understand not only what you have done, but also why you have done it that way rather than some other way. This is particularly important if you have had to use a hint or solution. In the examination, check that you have given the detail required. There often comes a point in a question where, if we could show that A implies B, the result follows. If after a lot of thought you suddenly see that A does indeed imply B the natural thing to do is to write triumphantly But A implies B so the result follows 7. Unfortunately unscrupulous individuals who have no idea why A should imply B (apart from the fact that it would complete the question) could and do write the exactly the same thing. Go back through the major points of the question making sure that you have not made any major unexplained leaps. 6 Mathematicians should feel as insulted as Engineers by the following joke. A mathematician, a physicist and an engineer enter a mathematics contest, the first task of which is to prove that all odd number are prime. The mathematician has an elegant argument: 1 s a prime, 3 s a prime, 5 s a prime, 7 s a prime. Therefore, by mathematical induction, all odd numbers are prime. It s the physicist s turn: 1 s a prime, 3 s a prime, 5 s a prime, 7 s a prime, 11 s a prime, 13 s a prime, so, to within experimental error, all odd numbers are prime. The most straightforward proof is provided by the engineer: 1 s a prime, 3 s a prime, 5 s a prime, 7 s a prime, 9 s a prime, 11 s a prime There is a standard anecdote about the distinguished Professor X. In the middle of a lecture she writes It is obvious that A, suddenly falls silent and after a few minutes walks out of the room. The awed students hear her pacing up and down outside. Then after twenty minutes she returns says It is obvious and continues the lecture.
11 Sample worked question Let where a is a constant. Show that, if a 9/8, then for all x. f (x) = ax f (x) 0 x3 1 + x 2, First thoughts When I play tennis and I see a ball that I think I can hit, I rush up to it and smack it into the net. This is a tendency I try to overcome when I am doing mathematics. In this question, for example, even though I m pretty sure that I can find f (x), I m going to pause for a moment before I do so. I am going to use the pause to think about two things. First, I m going to think about what to do when I have found f (x). Second, I m going to try to decide what the question is really about. Of course, I may not be able to decide what to do with f (x) until I actually see what it looks like; and I may not be able to see what the question is really about until I have finished it; maybe not even then. (In fact, some questions are not really about anything in particular.) I am also going to use the pause to think a bit about the best way of performing the differentiation. Should I simplify first? Should I make some sort of substitution? Clearly, if knowing how to tackle the rest of the question might guide me in deciding the best way to do the differentiation. Or it may turn out that the differentiation is fairly straight forward, so that it doesn t matter how I do it. Two more points occur to me as I reread the question. I notice that the inequalities are not strict (they are rather than >). Am I going to have to worry about the difference? I also notice that there is an If... then, and I wonder if this is going to cause me trouble. I will need to be careful to get the implication the right way round. I mustn t try to prove that if f (x) 0 then a 9/8. It will be interesting to see why the implication is only one way why it is not an if and only if question. Don t turn over until you have spent a little time thinking along these lines.
12 Doing the question Looking ahead, it is clear that the real hurdle is going to be showing that f (x) 0. How am I going to do that? Two ways suggest themselves. First, if f (x) turns out to be a quadratic function, or an obviously positive multiple of a quadratic function, I should be able to use some standard method: looking at the discriminant ( b 2 4ac ), etc; or, better, completing the square. But if f (x) is not of this form, I will have to think of something else: maybe I will have to sketch a graph. I m hoping that I won t have to do this, because otherwise I could be here all night. I ve just noticed that f is an odd function, i.e. f (x) = f ( x) (did you notice that?). That is is helpful, because it means that f (x) is an even function. 8 If it had been odd, there could have been a difficulty, because odd functions (at least, those with no vertical asymptotes) always cross the horizontal axis y = 0 at least once (they start in one corner of the graph paper for x large and negative and end in the diagonally opposite corner for x large and positive) and cannot therefore be positive for all x. Now, how should I do the differentiation? I could divide out the fraction, giving x x 2 = x x 1 + x 2 and f (x) = (a 1)x + x 1 + x 2 = bx + x 1 + x 2. (Is this right? I ll just check that it works when x = 2 yes, it does: f (2) = 2a 8/5 = 2(a 1)+2/5.) This might save a bit of writing, but I don t at the moment see this helping me towards a positive function. On balance, I think I ll stick with the original form. Another thought: should I differentiate the fraction using the quotient formula or should I write it as x 3 (1 + x 2 ) 1 and use the product rule? I doubt if there is much in it. I never normally use the quotient rule it s just extra baggage to carry round but on this occasion, since the final form I am looking for is a single fraction, I will. One more thought: since I am trying to prove an inequality, I must be careful throughout not to cancel any quantity which might be negative; or at least if I do cancel a negative quantity, I must remember to reverse the inequality. Here goes (at last): f (x) = a 3x2 (1 + x 2 ) 2x(x 3 ) (1 + x 2 ) 2 = a + (2a 3)x2 + (a 1)x 4 (1 + x 2 ) 2. This is working out as I had hoped: the denominator is certainly positive and the numerator is a quadratic function of x 2. I can finish this off most elegantly by completing the square. There are two ways of doing this, and it is marginally easier to do it the unnatural way: a + (2a 3)x 2 + (a 1)x 4 = a(1 + (2 3a 1 )x 2 ) + (a 1)x 4 = a [ (2 3a 1 )x 2] 2 + [ (a 1) 1 4 a(2 3a 1 ) 2] x 4. We certainly need a 0 to make the first term non-negative. Simplifying the second square bracket gives [2 9a 1 /4], so it is true that if a 9/8 then f (x) 0 for all x. 8 You can see this easily from a by sketching a typical odd function.
13 Post-mortem Now I can look back and analyse what I have done. On the technical side, it seems I was right not to use the simplified form of x 3 /(1 + x 2 ). This would have lead to the quadratic (b + 1) + (2b 1)x 2 + bx 4, which isn t any easier to handle than the quadratic involving a and just gives an extra opportunity to make an algebraic error. Actually, I see on re-reading my solution that I have made a bit of a meal out of the ending. I needn t have completed the square at all; I could used the inequality a 9/8 immediately after finding f (x), since a appears in f (x) with a plus sign always: f (x) = a + (2a 3)x2 + (a 1)x 4 (1 + x 2 ) 2 9/8 + (9/4 3)x2 + (9/8 1)x 4 (1 + x 2 ) 2 = (x 2 3) 2 /8 0 as required. I wonder why the examiner wanted me to investigate the sign of f (x). The obvious reason is to see what the graph looks like. We can now see what this question is about. It is clear that the examiners really wanted to set the question: Sketch the graphs of the function ax x 3 /(1 + x 2 ) in the different cases that arise according to the value of a but it was thought too long or difficult. It is worth looking back over my working to see what can be said about the shape of the graph of f (x) when a < 9/8. My unnecessarily elaborate proof, involving completing the square, makes the role of a a bit clearer than it is in the shorter alternative. (I leave that to you to think about!)
14 Sample worked question 2 The n positive numbers x 1,x 2,...,x n, where n 3, satisfy and also Show that (i) x 1,x 2,...,x n > 1, (ii) x 1 x 2 = x 2 x 3 x 2 x 3, (iii) x 1 = x 2 = = x n. Hence find the value of x 1. First thoughts x 1 = x 2, x 2 = x 3,..., x n 1 = x n, x n = x 1. My first thought is that this question has an unknown number of variables: x 1,..., x n. That makes it seem rather complicated. I might, if necessary, try to understand the result by choosing an easy value for n (maybe n = 3). If I manage to prove some of the results in this special case, I will certainly go back to the general case: doing the special case might help me tackle the general case, but I don t expect to get many marks in an exam if I just prove the result in one special case. Next, I see that the question has three sub-parts, then a final one. The final one begins Hence.... This must mean that the final part will depend on at least one of the previous parts. It is not clear from the structure of the question whether the three subparts are independent; the proof of (ii) and (iii) may require the previous result(s). However, I expect to use the results of parts (i), (ii) and (iii) later in the question. Actually, I think I can see how to do the very last part. If I assume that (iii) holds, so that x 1 = x 2 = = x n = x (say), then each of the equations given in the question is identical and each gives a simple equation for x. I am a bit puzzled about part (i). How can an inequality help to derive the equalities in the later parts? I can think of a couple of ways in which the result x i > 1 could be used. One is that I may need to cancel, say, x 1 from both sides of an equation in which case I would need to know that x 1 0. But looking back at the question, I see I already know that x 1 > 0 (it is given), so this cannot be the right answer. Maybe I need to cancel some other factor, such as (x 1 1). Another possibility is that I get two or more solutions by putting x 1 = x 2 = = x n = x and I need the one with x > 1. This may be the answer: looking back at the question again, I see that it asks for the value of x 1 so I am looking for a single value. I m still puzzled, but I will remember in later parts to keep a sharp look out for ways of using part (i). One more thing strikes me about the question. The equations satisfied by the x i are given on two lines ( and also ). This could be for typographic reasons (the equations would not all fit on one line) but more likely it is to make sure that I have noticed that the last equation is a bit different: all the other equations relate x i to x i+1, whereas the last equation relates x n to x 1. It goes back to the beginning, completing the cycle. I m pleased that I thought of this, because this circularity must be important.
15 Doing the question I think I will do the very last part first, and see what happens. Suppose, assuming the result of part (iii), that x 1 = x 2 = = x n = x. Then substituting into any of the equations given in the question gives x = x i.e. x 2 x 1 = 0. Using the quadratic formula gives x = 1 ± 5 2 which does indeed give two answers (despite the fact that the question asks for just one). However, I see that one is negative and can therefore be eliminated by the condition x i > 0 which was given in the question (not, I note, the condition x 1 > 1 from part (i); I still have to find a use for this). I needed only part (iii) to find x 1, so I expect that either I need both (i) and (ii) directly to prove (iii), or I need (i) to prove (ii), and (ii) to prove (iii). Now that I have remembered that x i > 0 for each i, I see that part (i) is obvious. Since x 2 > 0 then 1/x 2 > 0 and the first equation given in the question, x 1 = 1 + 1/x 2 show immediately that x 1 > 1 and the same applies to x 2, x 3, etc. Now what about part (ii)? The given equation involves x 1, x 2 and x 3, so clearly I must use the first two equations given in the question: x 1 = x 2, x 2 = x 3. Since I want x 1 x 2, I will see what happens if I subtract the two equations: That seems to work!, x 1 x 2 = (1 + 1 x 2 ) (1 + 1 x 3 ) = 1 x 2 1 x 3 = x 3 x 2 x 2 x 3. ( ) One idea that I haven t used so far is what I earlier called the circularity of the equations: the way that x n links back to x 1. I ll see what happens if I extend the above result. Since there is nothing special about x 1 and x 2, the same result must hold if I add 1 to each of the suffices: x 2 x 3 = x 4 x 3 x 3 x 4. I see that I can combine this with the previous result: x 1 x 2 = x 3 x 2 x 2 x 3 = x 3 x 4 x 2 x 2 3 x 4 I now see where this is going. The above step can be repeated to give x 1 x 2 = x 3 x 4 x 2 x 2 3 x 4 and eventually I will get back to x 1 x 2 : x 1 x 2 = x 5 x 4 x 2 x 2 3 x2 4 x 5 = x 5 x 4 x 2 x 2 3 x2 4 x 5. = x 1 x 2 = = ± x 2 x 2 3 x2 4 x2 5...x2 n x2 1 x 2
16 i.e. ( (x 1 x 2 ) 1 ) 1 x 2 = 0. 1 x2 2 x2 3 x2 4 x2 5...x2 n I have put in a ± because each step introduces a minus sign and I m not sure yet whether the final sign should be ( 1) n or ( 1) n 1. I can check this later (for example, by working out one simple case such as n = 3); but I may not need to. I deduce from this last equation that either x 2 1 x2 2 x2 3 x2 4 x2 5...x2 n = ±1 or x 1 = x 2 (which is what I want). At last I see where to use part (i): I know that x 2 1 x2 2 x2 3 x2 4 x2 5...x2 n ±1 because x 1 > 1, x 2 > 1, etc. Thus x 1 = x 2, and since there was nothing special about x 1 and x 2, I deduce further that x 2 = x 3, and so on, as required. Postmortem There were a number of useful points in this question. The first point concerns using the information given in the question. The process of teasing information from what is given is fundamental to the whole of mathematics. It is very important to study what is given (especially seemingly unimportant conditions, such as x i > 0) to see why they have been given. If you find you reach the end of a question without apparently using some condition, then you should look back over your work: it is very unlikely that a condition has been given that is not used in some way. (It may not be a necessary condition and we will see that the condition x i > 0 is not, in a sense, necessary in this question but it should be sufficient.) The other piece of information in the question which you might easily have overlooked is the use of the singular (... find the value of... ), implying that there is just one value, despite the fact that the final equation is quadratic.. The second point concerns using the structure of the question. Here, the position of the word Hence suggested strongly that none of the separate parts were stand-alone results; each had to be used for a later proof. Understanding this point made the question much easier, because I was always on the look out for an opportunity to use the earlier parts. Of course, in some problems (without that hence ) some parts may be stand-alone; though this is rare in STEP questions. You may think that this is like playing a game according to hidden (STEP) rules, but that is not the case. Precision writing and precision reading is vital in mathematics and in many professions (law, for example). Mathematicians have to be good at it, which is the reason that everyone wants to employ people with mathematical training. The third point was the rather inconclusive speculation about the way inequalities might help to derive an equality. It turned out that what was actually required was x 1 x 2... x n ±1. I was a bit puzzled by this possibility in my first thoughts, because it seemed that the result ought to hold under conditions different from those given; for example, x i < 0 for all i (does this condition work??). Come to think of it, why are conditions given on all x i when they are all related by the given equations? This makes me think that there ought to be a better way of proving the result which would reveal exactly the conditions under which it holds. Fourth was the idea (which I didn t actually use) that I might try to prove the result for, say, n = 3. This would not have counted as a proof of the result (or anything like it), but it might have given me ideas for tackling the question. Fifth was the observation that the equations given in the question are circular. It was clear that the circularity was essential to the question and it turned out to be the key the most difficult part.
17 Having identified it early on, I was ready to use it when the opportunity arose. Final thoughts It occurs to me only now, after my post-mortem, that there is another way of obtaining the final result. Suppose I start with the idea of circularity (as indeed I might have, had I not been otherwise directed by the question) and use the given equations to find x 1 in terms of first x 2, then x 3, then x 4 and eventually in terms of x 1 itself. That should give me an equation I can solve, and I should be able to find out what conditions are needed on the x i. Try it. You may need to guess a formula for x 1 in terms of x i from a few special cases, then prove it by induction. You will find it useful to define a sequence of numbers F i such that F 0 = F 1 = 1 and F n+1 = F n + F n 1. (These numbers are called Fibonacci numbers 9.) You should find that if x n = x 1 for any n (greater than 1), then x n = 1± 5 2. No conditions are required for this result to hold, except that all the x i exist, so it was not necessary to set the condition x 1 > 1. (The condition on x 1 for all x i to exist is that x 1 F k /F k 1 for any k n, as you will see if you obtain the general formula for x 1 in terms of x n.) 9 Fibonacci (short for filius Bonacci son of Bonacci) was called the greatest European mathematician of the middle ages. He was born in Pisa (Italy) in about 1175 AD. He introduced the series of numbers named after him in his book of 1202 called Liber Abbaci (Book of the Abacus). It was the solution to the problem of the number of pairs of rabbits produced by an initial pair: A pair of rabbits are put in a field and, if rabbits take a month to become mature and then produce a new pair every month after that, how many pairs will there be in twelve months time?
18 Question 1( ) Find all the solutions of the equation x + 1 x + 3 x 1 2 x 2 = x + 2. Comments This looks more difficult than it is. There is no easy way to deal with the modulus function: you have to look at the different cases individually (for example, for x you have to look at x 0 and x 0). The most straightforward approach would be to solve the equation in each of the different regions determined by the modulus signs: x 1, 1 x 0, 0 x 1, etc. You might find a graphical approach helps you to picture what is going on (I didn t).
20 Question 2( ) (i) Find the coefficient of x 6 in You should set out your working clearly. (1 2x + 3x 2 4x 3 + 5x 4 ) 3. (ii) By considering the binomial expansions of (1 + x) 2 and (1 + x) 6, or otherwise, find the coefficient of x 6 in (1 2x + 3x 2 4x 3 + 5x 4 6x 5 + 7x 6 ) 3. Comments Your first instinct is probably just to multiply out the brackets. A moment s thought will show that it is going to be much quicker to concentrate only on those terms with powers of x that multiply together to give x 6. The difficulty with this would be making sure you have got them all, so a systematic approach is called for. Alternatively, you could work out (1 2x + 3x 2 4x 3 + 5x 4 ) 2 first. Squaring an expression which is the sum of 5 terms leads to an expression with 15 terms after simplification (why?) but in this case, simplification will lead to at most 9 terms (why?). 9 terms shouldn t prove too much of a challenge, but I wouldn t fancy it myself. Obviously, the second part can be done otherwise by multiplying out the brackets as in the first part. However, it is much quicker to follow the hint. If you can t see how considering the expansion of (1 + x) 2 helps, you should write it out.
21 Solution to question 2 (i) Writing out the cube explicitly makes it easier to collect up the terms: (1 2x + 3x 2 4x 3 + 5x 4 )(1 2x + 3x 2 4x 3 + 5x 4 )(1 2x + 3x 2 4x 3 + 5x 4 ) To get the coefficient of x 6, we need to find all ways of choosing one term from each bracket in such a way that, when multiplied together, the result is a multiple of x 6. Start by taking the first term (i.e. 1) from the first bracket. This gives which sums to 46x 6. 1 [(3x 2 ) (5x 4 ) + ( 4x 3 ) ( 4x 3 ) + (5x 4 ) (3x 2 )] The second term from the first bracket gives a contribution of ( 2x) [( 2x) (5x 4 ) + (3x 2 ) ( 4x 3 ) + ( 4x 3 ) (3x 2 ) + (5x 4 ) ( 2x)] = 88x 6. The third term from the first bracket gives a contribution of (3x 2 ) [(1) (5x 4 ) + ( 2x) ( 4x 3 ) + (3x 2 ) (3x 2 ) + ( 4x 3 ) ( 2x) + (5x 4 ) (1)] = 105x 6. The fourth term from the first bracket gives a contribution of ( 4x 3 ) [(1) ( 4x 3 ) + ( 2x) (3x 2 ) + (3x 2 ) ( 2x) + ( 4x 3 ) (1)] = 80x 6. The fifth term from the first bracket gives a contribution of (5x 4 ) [(1) (3x 2 ) + ( 2x) ( 2x) + (3x 2 ) (1)] = 50x 6. The grand total is therefore 369x 6 and 369 is the required coefficient. (ii) The given expression is the beginning of the expansion of ( (1 + x) 2) 3, i.e. of (1 + x) 6. But (1 + x) 6 ( 6) ( 7) = 1 6x + x ! 2! 6! 5! x6 n (n + 6 1)! + + ( 1) x n +... n! 5! so the coefficient of x 6 is 11!/6!5! = 462. Postmortem There was not much to this question. There are two points worth emphasising. The first is the need for a systematic approach (and also for neat presentation and clear explanation) in the first part. Otherwise, there is a real danger of confusing yourself and a certainty of confusing anyone else reading your answer. The second point is the use of the hint in the second part of the question. Although you already knew one sure way of reaching the answer which would have got full marks because of the or otherwise it paid to persevere with the hint, because it led to considerable saving of effort. You might ask yourself why the method of the second part doesn t work for the first part. There were two why? s in the comments section. Why does squaring an expression with five terms give a total of 15 terms when simplified? Of course, multiplying two different 5-term expressions gives 5 5 terms. If the two expressions are the same, then 5 4 terms occur twice, giving a total of Why does the expression we are squaring in this question result in 9 terms? The expression is a polynomial of degree 4, so the result of squaring is a polynomial of degree 8, which can have at most 9 different powers of x and hence (after simplification) at most 9 terms.
22 Question 3( ) Show that you can make up 10 pence in eleven ways using 10p, 5p, 2p and 1p coins. In how many ways can you make up 20 pence using 20p, 10p, 5p, 2p and 1p coins? Comments I don t really approve of this sort of question, but I thought I d better include one in this collection. The one given above seems to me to be a particularly bad example, because there are a number of neat and elegant mathematical ways of approaching it, none of which turn out to be any use. The quickest instrument is the bluntest: just write out all the possibilities. Two things are important: first you must be systematic or you will get hopelessly confused; second, you must lay out your solution, with careful explanations, in a way which allows other people (examiners, for example) to understand exactly what you are doing. Here are a couple of the red herrings. For the first part, what is required is the coefficient of x 10 in the expansion of 1 (1 x 10 )(1 x 5 )(1 x 2 )(1 x). You can see why this is the case by using the binomial expansion (compare question 1). Although this is neat, it doesn t help, because there is no easy way of obtaining the required coefficient. And the second part would be even worse. The second red herring concerns the relationship between the two parts. Since different parts of STEP questions are nearly always related, you might be led to believe that the result of the second part follows from the first: you divide the required twenty pence into two tens and then use the result of the first part to give the number of ways of making up each 10. This would give an answer of 66 (why?) plus one for a single 20p piece. This would also be neat, but the true answer is less than 67 because some arrangements are counted twice by this method and it is not easy to work out which ones.
23 Solution to question 3 Probably the best approach is to start counting with the arrangements which use as many high denomination coins as possible, then work down. We can make up 10p as follows: 10; 5+5 (one way using two 5p coins); , , , (three ways using one 5p coin); , , etc, (six ways using no 5p coins); making a total of 11 ways. We can make up 20p as follows: 20; 10 + any of the 11 arrangements in the first part of the question; ; , etc (3 ways using three 5p coins); , etc (6 ways using two 5p coins); , etc (8 ways using one 5 and making 15 out of 2p and 1p coins); , etc (11 ways of making 20p with 2p and 1p coins). Grand total = 41. Postmortem As in the previous question, the most important lesson to be learnt here is the value of a systematic approach and clear explanations. You should not be happy just to obtain the answer: there is no virtue in that. You should only be satisfied if you displayed your working at least as systematically as I have, above.
24 Question 4( ) Consider the system of equations 2yz + zx 5xy = 2 yz zx + 2xy = 1 yz 2zx + 6xy = 3. Show that and find the possible values of x, y and z. xyz = ±6 Comments At first sight, This looks forbidding. A closer look reveals that the variables x, y and z occur only in pairs yz, zx and xy. The problem therefore boils down to solving three simultaneous equations in these variables, then using the solution to find x, y and z individually. What do you make of the ± in the equation xyz = ±6? Does this give you a clue? There are two ways of tackling simultaneous equations. You could use the first equation to find an expression for one variable (yz say) in terms of the other two variables, then substitute this into the other equations to eliminate the zx from the system. Then use the second equation (in its new form) to find an expression for one of the two remaining variables (zx say), then substitute this into the third equation (in its new form) to obtain a linear equation for the third variable (xy). 10 Having solved this equation, you can then substitute back to find the other variables. This method is called Gauss elimination. Alternatively, you could eliminate one variable (yz say) from the first two of equations by multiplying the first equation by something suitable and the second equation by something suitable and subtracting. You then eliminate yz from the second and third equations similarly. That leaves you with two simultaneous equations in two variables which you can solve by your favorite method. There is another way of solving the simultaneous equations, which is very good in theory by not at all good in practice. You write the equations in matrix form Mx = c, where in this case yz 2 M = 1 1 2, x = zx, c = xy 3 The solution is then x = M 1 c. 10 A linear equation in the variable x is one that does not involve and powers of x except the first power, or any other functions of x. Suppose the equation is f (x) = 0 and the solution is x = a. Then if you replace x by 2x, so that f (2x) = 0, then the solution is x = a/2, and this can be thought of as the defining property of a linearity in this context.
25 Solution to question 4 Start by labelling the equations: We use Gaussian elimination. Rearranging equation (1) gives which we substitute back into equations (2) and (3) : 2yz + zx 5xy = 2, (1) yz zx + 2xy = 1, (2) yz 2zx + 6xy = 3. (3) yz = 1 2 zx + 5 2xy + 1, (4) 3 2 zx + 9 2xy = 0, (5) zx + 2 xy = 2. (6) Thus zx = 3xy (using equation (5)). Substituting into equation (6) gives xy = 2 and zx = 6. Finally, substituting back into equation (1) shows that yz = 3. The question is now plain sailing. Multiplying the three values together gives (xyz) 2 = 36 and taking the square root gives xyz = ±6 as required. Now it remains to solve for x, y and z individually. We know that yz = 3, so if xyz = +6 then x = +2, and if xyz = 6 then x = 2. The solutions are therefore either x = +2, y = 1, and z = 3 or x = 2, y = 1, and z = 3. Postmortem There were two key observations which allowed us to do this question quite easily. Both came from looking carefully at the question. The first was that the given equations, although non-linear in x, y and z (they are quadratic, since they involve products of these variables) could be thought of as three linear equations in yz, zx and xy. That allowed us to make a start on the question. The second observation was that the equation xyz = ±6 is almost certain to come from (xyz) 2 = 36 and that gave us the next step after solving the simultaneous equations. (Recall the next step was to multiply all the variables together.) There was a point of technique in the solution: it is often very helpful in this sort of problem (and many others) to number your equations. This allows you to refer back clearly and quickly, for your benefit as well as for the benefit of your readers. The three simultaneous quadratic equations (1) (3) have a geometric interpretation. First consider the case of linear equations. Each such equation can be interpreted as the equation of a plane, of the form k.x = a (which has normal k and is at distance a/ k from the origin). The solution of the three equations represents a point which is the intersection of all three planes. Clearly, there may not be such a point: for example, the planes may intersect in three pairs of lines forming a Toblerone. In this case, there is no point common to the three planes and therefore no solution to the equations, unless the lines happen to coincide (a Toblerone of zero volume) in which case every point on the line is a solution. The simultaneous equations in the question are quadratic and each is the equation of a hyperboloid (i.e. a surface for which two sets of sections are hyperbolae and the other set of sections are ellipses) like a radar dish. The number of intersections of three infinite radar dishes is not easy to calculate in general. In this case, we know the answer must be at least 2: replacing x, y and z in the equations by x, y and z, respectively, makes no difference to the equations. If x = a, y = b and z = c satisfies the equations, then so also must x = a, y = b and z = c.
26 Question 5( ) Suppose that 3 = 2 x 1 = x x 2 = x x 3 = x x 4 =. Guess an expression, in terms of n, for x n. Then, by induction or otherwise, prove the correctness of your guess. Comments Wording this sort of question is a real headache for the examiners. Suppose you guess wrong; how can you then prove your guess by induction (unless you get that wrong too)? How else can the question be phrased? In the end, we decided to assume that you are all so clever that your guesses will all be correct. To guess the formula, you need to work out x 1, x 2, x 3, etc and look for a pattern. You should not need to go beyond x 4. Proof by induction is not in the core A-level syllabus. I decided to include it in the syllabus for STEP I and II because the idea behind it is not difficult and it is very important both as a method of proof and also as an introduction to more sophisticated mathematical thought.
27 Solution to question 5 First we put the equations into a more manageable form. Each equality can be written in the form 3 = x n + 2 x n+1, i.e. x n+1 = 2 3 x n. We find x 1 = 2/3, x 2 = 6/7, x 3 = 14/15 and x 4 = 30/31. The denominators give the game away. We guess x n = 2n n+1 1. For the induction, we need a starting point: our guess certainly holds for n = 1 (and 2, 3, and 4!). For the inductive step, we suppose our guess also holds for n = k, where k is any integer. If we can show that it then also holds for n = k + 1, we are done. We have, from the equation given in the question, as required. Postmortem x k+1 = 2 3 x k = 2 3 2k k+1 1 = 2(2 k+1 1) 3(2 k+1 1) (2 k+1 2) = 2k k+2 1, There s not much to say about this. By STEP standards, it is fairly easy and short. Nevertheless, you are left to your own devices from the beginning, so you should be pleased if you got it out.
28 Question 6( ) Show that, if tan 2 θ = 2tan θ + 1, then tan 2θ = 1. Find all solutions of the equation tan θ = 2 + tan 3θ which satisfy 0 < θ < 2π, expressing your answers as rational multiples of π. Find all solutions of the equation cot φ = 2 + cot 3φ which satisfy 3π 2 < φ < π, expressing your answers as rational multiples of π. 2 Comments There are three distinct parts. It is pretty certain that they are related, but it is not obvious what the relationship is. The first part must surely help with the second part in some way that will only be apparent once the second part is under way. In the absence of any other good ideas, it looks right to express the double and triple angle tans and cots in terms of single angle tans and cots. You should remember the formula tan(a + B) = tan A + tan B 1 tan Atan B. You can use this for tan 3θ and hence cot 3θ. If you have forgotten the tan(a + B) formula, you can quickly work it out by from the corresponding sin and cos formulae: sin(a + B) sinacos B + cos Asin B = cos(a + B) cos Acos B sin Asin B. You are are not expected to remember the more complicated triple angle formulae. (I certainly don t.) You may well find yourself solving cubic equations at some stage in this question.
29 Solution to question 6 We will write t for tan θ (or tan φ) throughout. First part : tan 2θ = 2t 1 t 2 = 1 (since t2 = 2t + 1 is given). For the second part, we first work out tan 3θ. We have so the equation becomes tan 3θ = tan(θ + 2θ) = tan θ + tan 2θ 1 tan θ tan 2θ = t + 2t/(1 t2 ) 3t t3 1 t(2t/(1 t 2 = )) 1 3t 2, 3t t 3 1 3t 2 = t 2, i.e. t3 3t 2 + t + 1 = 0. One solution (by inspection) is t = 1. Thus one set of roots is given by θ = nπ + π/4. There are no other obvious integer roots, but we can reduce the cubic equation to a quadratic equation by dividing out the known factor (t 1). I would write t 3 3t 2 +t+1 = (t 1)(t 2 +at 1) since the coefficients of t 2 and of t 0 in the quadratic bracket are obvious. Then I would multiply out the brackets to find that a = 2. Now we see the connection with the first part: t 2 + at 1 = 0 tan 2θ = 1, and hence 2θ = nπ π/4. The roots are therefore θ = nπ + π/4 and θ = nπ/2 π/8. The multiples of π/8 in the given range are {2,3,7,10,11,15}. For the last part, we could set cot φ = 1/tan φ and cot 3φ = 1/tan 3φ thereby obtaining 1 t = t2 3t t 3. This simplifies to the cubic equation t 3 + t 2 3t + 1 = 0. There is an integer root t = 1, and the remaining quadratic is t 2 2t + 2t 1 = 0. Learning from the first part, we write this as 1 t 2 = 1, which means that tan 2φ = nπ + π/4. Proceeding as before gives (noting the different range) the following multiples of π/8: {2, 1, 3, 6, 7, 11}. Postmortem There was a small but worthwhile notational point in this question: it is often possible to use the abbreviation t for tan (or s for sin, etc), which can save a great deal of writing. There are two other points worth recalling. First is the way that the first part fed into the second part, but had to be mildly adapted for the third part. This is a typical device used in STEP questions aimed to see how well you learn new ideas. Second is what to do when faced with a cubic equation. There is a formula for the roots of a cubic, but no one knows it nowadays. Instead, you have to find at least one root by inspection. Having found one root, you have a quick look to see if there are any other obvious roots, then divide out the know factor to obtain a quadratic equation. The real detectives among you might have wondered whether there was as deep reason for the peculiar choice of range ( 3π/2 φ π/2) for the third part. You will not be surprised to know that there was. The reciprocal relation between tan and cot used for the first part is not the only relation. We could have instead used cot A = tan(π/2 A). The second equation transforms into the first equation if we set φ = π/2 θ. Furthermore, the given range of φ corresponds exactly to the range of θ given in the second part. We can therefore write down the solution directly from the solution to the second part.
30 Question 7( ) Show, by means of a change of variable or otherwise, that for any given function f. Hence, or otherwise, show that 0 f ( (x 2 + 1) 1/2 + x ) dx = ( (x 2 + 1) 1/2 + x ) 3 dx = (1 + t 2 )f(t)dt, Comments There are two things to worry about when you are trying to find a change of variable to convert one integral to another: you need to make the integrand match up and you need to make the limits match up. Sometimes, the limits give the clue to the change of variable. (For example, if the limits on the original integral were 0 and 1 and the limits on the transformed integral were 0 and π/4, then an obvious possibility would be to make the change t = tan x). Here, the change of variable is determined by the integrand, since it must work for all choices of f. Perhaps you are worried about the infinite upper limit of the integrals. If you are trying to prove some rigorous result about infinite integrals, you might use the definition 0 f (x)dx = lim a a 0 f (x)dx, but for present purposes you just do the integral and put in the limits. The infinite limit will not normally present problems. For example, 1 (x 2 + e x )dx = ( x 1 e x) 1 = 1 e e1 = 1 + e. Don t be afraid of writing things like 1/ = 0. It is perfectly OK to use this as shorthand for lim x 1/x = 0 ; but / and 0/0 are definitely not OK, because of their ambiguity.
31 Solution to question 7 Clearly, to get the argument of f right, we must set t = (x 2 + 1) 1/2 + x. We must check that the new limits are correct (if not, we are completely stuck). When x = 0, t = 1 as required. Also, (x 2 + 1) 1/2 as x, so t. Thus the upper limit is still, again as required. The transformed integral is 1 f (t) dx dt dt so the next task is to find dx. This we can do in two ways: we find x in terms of t and differentiate dt it; or we could find dt and turn it upside down. The snag with the second method is that the dx answer will be in terms of x, so we will have to express x in terms of t anyway in which case, we may as well use the first method. We start by finding x in terms of t: t = (x 2 + 1) 1/2 + x (t x) 2 = (x 2 + 1) x = t2 1 = t 2t 2 1 2t. Then we differentiate: dx dt = t 2 = 1 2 (1 + t 2 ) which agrees exactly the factor that appears in the second integrand. For the last part, we take f (t) = t 3. Thus as required. Postmortem 0 ( (x 2 + 1) 1/2 + x ) 3 1 dx = (t 3 + t 5 )dt 2 1 = 1 ( ) t t = 1 ( ) = This is not a difficult question conceptually once you realise the significance of the fact that the change of variable must work whatever function f is in the integrand. There was a useful point connected with calculating dx. It was a good idea not to plunge into the dt algebra without first thinking about alternative methods; in particular, use of the result dx / dt dt = 1 dx. Finally, there was the infinite limit of the integrand, which I hope you saw was not something to worry about (even though infinite limits are excluded from the A-level core). If you were setting up a formal definition of what an integral is, you would have to use finite limits, but if you are merely calculating the value of an integral, you just go ahead and do it, with whatever limits you are given.
32 Question 8( ) Which of the following statements are true and which are false? Justify your answers. (i) a ln b = b ln a for all positive numbers a and b. (ii) cos(sin θ) = sin(cos θ) for all real θ. (iii) There exists a polynomial P such that P(x) cos x 10 6 for all (real) x. (iv) x x 4 5 for all x > 0. Comments You have to decide first whether the statement is true of false. If true, the justification has to be a proof. If false, you could prove that it is false, though it is often better to find a simple counterexample. Part (iii) might look a bit odd. I suppose it relates to the standard approximation cos x 1 x 2 /2 which holds when x is small. This result can be improved by using a polynomial of higher degree: the next term is x 4 /4!. It can be proved that, given any number k, cos x can be approximated, for k < x < k, by a polynomial of the form N n=0 ( 1) n x2n (2n)!. You have to use more terms of the approximation (i.e. a larger value of N) if you want either greater accuracy or a larger value of k. In Part (iii) of this question, you are being asked if there is a polynomial such that the approximation is good for all values of x.
33 Solution to question 8 (i) True. The easiest way to see this is to log both sides. For the left hand side, we have and for the right hand side we have which agree. ln(a ln b ) = (ln b)(ln a) ln(a ln b ) = (ln a)(ln b), Note that we have to be a bit careful with this sort of argument. The argument used is that A = B because ln A = lnb. This requires the property of the ln function that lna = ln B A = B. You can easily see that this property holds because ln is a strictly increasing function; if A > B, then ln A > ln B. The same would not hold for (say) sin (i.e. sin A = sin B A = B ). (ii) False. θ = π/2 is an easy counterexample. (iii) False. Roughly speaking, any polynomial can be made as large as you like by taking x to be very large (provided it is of degree greater than zero), whereas cos x 1. There is obviously no polynomial of degree zero (i.e. no constant number) for which the statement holds. (iv) True: x x 4 = (x 2 x 2 ) Postmortem The important point here is that if you want to show a statement is true, you have to give a formal proof, whereas if you want to show that it is false, you only need give one counterexample. It does not have to be an elaborate counterexample in fact, the simpler the better. There may still be a question mark in your mind over part (iii). Given a polynomial P(x), how do you actually prove that it is possible to find x such that P(x) > ? This is one of those many mathematical results that seems very obvious but is irritatingly difficult to prove. (Often, the most irritatingly-difficult-to-prove obvious results are false, though this one is true.) You should try to cobble together a proof for yourself. I would start by choosing a general polynomial of degree N. Let N P(x) = a n x n, n=0 where a N 0. Then I would assume that a N = 1 since if I could prove the result for this case, I m sure I could easily modify the proof to deal with the general case. Now what? A simple way forward would be to make use of the fundamental theorem of algebra, which says that P(x) can be written in the form P(x) (x α 1 )(x α 2 ) (x α n ) where α 1, α 2,..., α n are the roots of the equation P(x) = 0. (The theorem says that this equation has n roots.) Then P(x) x α 1 x α 2 x α n which can be made larger than any given number, M say, by choosing x so that x α i > M 1/n for each i. Actually, one one or two things have been brushed under the carpet in this proof (what if some of the roots are complex?), but I m confident that there is nothing wrong that couldn t be easily fixed,
34 Question 9( ) Prove that the rectangle of greatest perimeter which can be inscribed in a given circle is a square. The result changes if, instead of maximising the sum of lengths of sides of the rectangle, we seek to maximise the sum of nth powers of the lengths of those sides for n 2. What happens if n = 2? What happens if n = 3? Justify your answers. Comments Obviously, the perimeter of a general rectangle has no maximum (it can be as long as you like), so the key to this question is to use the constraint that the rectangle lies in the circle. The word greatest in the question immediately suggests differentiating something. The perimeter of a rectangle is expressed in terms of two variables, length x and breadth y, so you must find a way of using the constraint to eliminate one variable. This could be done in two ways: express y in terms of x, or express both x and y in terms of another variable (an angle, say). Finally, there is the matter of deciding whether your solution is the greatest or least value (or neither). This can be done either by considering second derivatives, or by trying to understand the different situations. Often the latter method, or a combination, is preferable. Finding stationary values of functions subject to constraints is very important: it has widespread applications (for example, in theoretical physics, financial mathematics and in fact in almost any area to which mathematics is applied). It forms a whole branch of mathematics called optimisation. Normally, the methods described above (eliminating one variable) are not used: a clever idea, the methods of Lagrange multipliers, is used instead,
35 Solution to question 9 Let the circle have diameter d and let the length of one side of the rectangle be x and the length of the adjacent side be y. Then, by Pythagoras s theorem, and the perimeter P is given by y = d 2 x 2 ( ) P = 2x + 2 d 2 x 2. We can find the largest possible value of P as x varies by calculus. We have dp dx = 2 2 x d 2 x, 2 so for a stationary point, we require (cancelling the factor of 2 and squaring) 1 = x2 d 2 x 2 i.e. 2x 2 = d 2. Thus x = d/ 2 (ignoring the negative root for obvious reasons). Substituting this into ( ) gives y = d/ 2, so the rectangle is indeed a square, with perimeter 2 2d. But is it the maximum perimeter? The easiest way to investigate is to calculate the second derivative, which is easily seen to be negative for all values of x, and in particular when x = d/ 2. The stationary point is certainly a maximum. For the second part, we consider f (x) = x n + (d 2 x 2 ) n/2. The first thing to notice is that f is constant if n = 2, so in this case, the largest (and smallest) value is d 2. For n = 3, we have f (x) = 3x 2 3x(d 2 x 2 ) 1/2, so f (x) is stationary when x 4 = x 2 (d 2 x 2 ), i.e. when 2x 2 = d 2 as before or when x = 0. The corresponding stationary values are 2d 3 and 2d 3, so this time the largest value occurs when x = 0. Postmortem You will almost certainly want to investigate the situation for other values of n yourself, just to see what happens. Were you happy with the proof given above that the square has the largest perimeter? And isn t it a bit odd that we did not discover a stationary point corresponding to the smallest value (which can easily be seen to occur at x = 0 or x = d)? To convince yourself that the maximum point gives the largest value, it is a good idea to sketch a graph: you can check that dp/dx is positive for 0 < x < d/ 2 and negative for d/ 2 < x < d. The smallest values of the perimeter occur at the endpoints of the interval 0 x d and therefore do not have to be turning points. A much better way of tacking the problem is to set x = dcos θ and y = dsin θ and find the perimeter as a function of θ. Because θ can take any value (there are no end points such as x = 0 to consider), the largest and smallest perimeters correspond to stationary points. Try it.
36 Question 10( ) Use the first four terms of the binomial expansion of (1 1/50) 1/2, to derive the approximation Calculate similarly an approximation to the cube root of 2 to six decimal places by considering (1 + N/125) 1/3, where N is a suitable number. [You need not justify the accuracy of your approximations.] Comments Although you do not have to justify your approximations, you do need to think carefully about the number of terms required in the expansions. You first have to decide how many you will need to obtain the given number of decimal places; then you have to think about whether the next term is likely to affect the value of the last decimal. It is not at all obvious where the 2 in the first part comes from until you write (1 1/50) 1/2 = (49/50). In the second part, you have to choose N in such a way that N has something to do with a power of 2. It is a bit easier to do the binomial expansions by making the denominator a power of 10 (so that the expansion in the first part becomes (1 2/100) 1/2 ).
37 Solution to question 10 First we expand binomially: ( 1 2 ) ( 1 2 )( ) ( 1 + 2! = = )( )( 1 1 )( 2 ) 2 ( ! )( )( 1 1 )( 3 )( 2 ) It is clear that the next term in the expansion would introduce the seventh and eighth places of decimals, which it seems we do not need. Of course, after further manipulations we might find that the above calculation does not supply the 6 decimal places we need, in which case we will work out the next term in the expansion. ( ) But = , so / Second part: ( ( 1 3 )1 3 )( = = ) ( 1 + 2! ) ( )( 1 2 )( ) 3 2 ( ! = )( )( 1 2 )( 5 )( 3 ) Successive terms in the expansion decrease by a factor of about 1000, so this should give the right number of decimal places. ( ) But = = so / Postmortem This question required a bit of intuition, and some accurate arithmetic. You don t have to be brilliant at arithmetic to be a good mathematician, but most mathematicians aren t bad at it. There have in the past been children who were able to perform extroardinary feats of arithmetic. For example, Zerah Colburn, a 19th century American, toured Europe at the age of 8. He was able to multiply instantly any two four digit numbers given to him by the audiences. George Parker Bidder (the Calculating Boy) could perform similar feats, though unlike Colburn he became a distinguished mathematician and scientist. One of his brothers knew the bible by heart. The error in the approximation is the weak point of this question. Although it is clear that the next term in the expansion is too small to affect the accuracy, it is not obvious that the sum of all the next hundred (say) terms of the expansion is negligible (though in fact it is). What is needed is an estimate of the truncation error in the binomial expansion. Such an estimate is not hard to obtain (first year university work) and is typically of the same order of magnitude as the first neglected term in the expansion.
38 Question 11( ) How many integers greater than or equal to zero and less than 1000 are not divisible by 2 or 5? What is the average value of these integers? How many integers greater than or equal to zero and less than 9261 are not divisible by 3 or 7? What is the average value of these integers? Comments There are a number of different ways of tackling this problem, but it should be clear that whatever way you choose for the first part will also work for the second part (especially when you realise the significance of the number 9261). A key idea for the first part is to think in terms of blocks of 10 numbers, realising that all blocks of 10 are the same for the purposes of the problem.
39 Solution to question 11 Only integers ending in 1, 3, 7, or 9 are not divisible by 2 or 5. This is 4/10 of the possible integers, so the total number of such integers is 4/10 of 1,000, i.e The integers can be added in pairs: sum = ( ) + ( ) + + ( ). There are 200 such pairs, so the sum is 1, and the average is 500. A simpler argument would be to say that this is obvious by symmetry: there is nothing in the problem that favours an answer greater (or smaller) than 500. Alternatively, we can find the number of integers divisible by both 2 and 5 by adding the number divisible by 2 (i.e. 500) to the number divisible by 5 (i.e. 200), and subtracting the number divisible by both 2 and 5 (i.e. 100) since these have been counted twice. To find the sum, we can sum those divisible by 2 (using the formula for a geometric progression), add the sum of those divisible by 5 and subtract the sum of those divisible by 10. For the second part, either of these methods will work. In the first method you consider integers in blocks of 21 (essentially arithmetic to base 21): there are 12 integers in each such block that are not divisible by 3 or 7 (namely 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20) so the total number is /21 = The average is 9261/2 as can be seen using the pairing argument (1+9260)+(2+9259)+ or the symmetry argument. Postmortem I was in year 7 at school when I had my first encounter with lateral thinking in mathematics. The problem was to work out how many houses a postman delivers to in a street of houses numbered from 1 to 1000, given that he refuses to deliver to houses with the digit 9 in the number. It seemed impossible to do it systematically, until the idea of counting in base 9 occurs; and then it seemed brilliantly simple. I didn t know I was counting in base 9; in those days, school mathematics was very traditional and base 9 would have been thought very advanced. All that was required was the idea of working out how many numbers can be made out of nine digits (i.e. 9 3 ) instead of 10; and of course it doesn t matter which nine. Did you spot the significance of the number 9261? It is 21 3, i.e. the number written as 1000 in base 21. Note how carefully the question is written and laid out to suggest a connection between the two paragraphs (and to highlight the difference); there is not even a part (i) and Part (ii) to break the symmetry. For the examination, the number 1,000,000 was used instead of 1,000 to discourage candidates from spending the first hour of the examination writing down the numbers from 1 to Having realised the that the first part involves (2 5) 3 and the second part involves (3 7) 3 you are probably now wondering what the general result is, i.e: How many integers greater than or equal to zero and less than (pq) 3 are not divisible by p or q? What is the average value of these integers? I leave this to you, with the hint that you need to think about only pq, rather than (pq) 3, to start with.
40 Question 12( ) Sketch the following subsets of the x-y plane: (i) x + y 1 ; (ii) x 1 + y 1 1 ; (iii) x 1 y ; (iv) x y 2 1. Comments Often with modulus signs, it is easiest to consider the cases separately, so for example is part (i), you would first work out the case x > 0 and y > 0, then x > 0 and y < 0, and so on. Here there is a simple geometric understanding of the different cases: once you have worked out the first case, the other three can be deduced by symmetry. Another geometric idea should be in your mind when tackling this question, namely the idea of translations in the plane.
41 Solution to question 12 The way to deal with the modulus signs in this question is to consider first the case when the things inside the modulus signs are positive, and then get the full picture by symmetry, shifting the origin as appropriate. For part (i), consider the the first quadrant x 0 and y 0. In this quadrant, the inequality is x + y 1. Draw the line x + y = 1 and then decide which side of the line is described by the inequality. It is obviously (since x has to be smaller than something) the region to the left of the line; or (since y also has to be smaller than something) the region below the line, which is the same region. Similar arguments could be used in the other quadrants, but it is easier to note that the inequality x + y 1 unchanged when x is replaced by x, or y is replaced by y, so the sketch should have reflection symmetry in both axes. Part (ii) is the same as part (i), except that the origin is translated to (1,1). For part (iii), consider first x y 1, where x > 0 and y > 0, which give an area that is infinite in extent in the positive y direction. Then reflect this in both axes and translate one unit down the y axis and 1 unit along the positive x axis. Part (iv) is the rectangular hyperbola xy = 1 reflected in both axes and the region required is enclosed by the four hyperbolas. The four hyperbolas are then translated 2 units up the y axis.
42 Question 13( ) Show that x 2 y 2 + x + 3y 2 = (x y + 2)(x + y 1) and hence, or otherwise, indicate by means of a sketch the region of the x-y plane for which Sketch also the region of the x-y plane for which x 2 y 2 + x + 3y > 2. x 2 4y 2 + 3x 2y < 2. Give the coordinates of a point for which both inequalities are satisfied or explain why no such point exists. Comments This question gets a two star difficulty rating because inequalities always need to be handled with care. For the very first part, you could either factorise the left hand side to obtain the right hand side, or multiply out the right hand side to get the left hand side. Obviously, it is much easier to do the multiplication than the factorisation. But is that cheating or taking a short cut that might lose marks? The answer to this is no: it doesn t matter if you start from the given answer and work backwards it is still a mathematical proof and any proof will get the marks. (But note that if there is a hence in the question, you will lose marks if you do not do it using the thing you have just proved.) In the second part, you have to do the factorisation yourself, so you should look carefully at where the terms in the (very similar) first part come from when you do the multiplication. It will help to spot the similarities bewteen x 2 y 2 + x + 3y 2 and x 2 4y 2 + 3x 2y + 2 Since no indication is given as to what detail should appear on the sketch, you are have to use your judgement: it is clearly important to know where the regions lie relative to the coordinate axes.
43 Solution to question 13 To do the multiplication, it pays to be systematic and to set out the algebra nicely: as required. (x y + 2)(x + y 1) = x(x + y 1) y(x + y 1) + 2(x + y 1) y = 1 x y = x + 2 ( 1 2, 3 2 ) = x 2 + xy x yx y 2 + y + 2x + 2y 2 = x 2 + x y 2 + 3y 2 For the first inequality, we need x y +2 and x+y 1 both to be either positive or negative. To sort out the inequalities (or inequations as they are sometimes horribly called), the first thing to do is to draw the lines corresponding to the corresponding equalities. These lines divide the plane into four regions and we then have to decide which regions are relevant. The diagonal lines x y = 2 and x + y = 1 intersect at ( 1 2, 3 2 ); this is the important point to mark in on the sketch. The required regions are the left and right quadrants formed by these diagonal lines (since the inequalities mean that the regions are to the right of both lines or to the left of both lines). For the second part, the first thing to do is to factorise x 2 4y 2 +3x 2y+2. The key similarity with the first part is the absence of cross terms of the form xy. This allows a difference-of-two-squares factorisation of the first two terms: x 2 4y 2 = (x 2y)(x + 2y). Following the pattern of the first part, we can then try a factorisation of the form x 2 4y 2 + 3x 2y + 2 = (x 2y + a)(x + 2y + b) where ab = 2. Considering the terms linear in x and y gives a + b = 3 and 2a 2b = 2 which quickly leads to a = 1 and b = 2. Note that altogether there were three equations for a and b, so we had no right to expect a consistent solution (except for the fact that this is a STEP question for which we had every right to believe that the first part would guide us through the second part). Alternatively, we could have completed the square in x and in y and then used difference of two squares: x 2 4y 2 + 3x 2y + 2 = (x )2 9 4 (2y ) = (x )2 (2y )2 = (x y 1 2 )(x y ). 2y = x + 1 ( 3 2, 1 4 ) As in the previous case, the required area is formed by two intersecting lines; this time, they intersect at ( 3 2, 1 4 ) and the upper and lower regions are required, since the inequality is the other way round. It is easy to see from the sketches that there are points that satisfy both inequalities: for example (1,2). 2y = 2 x
44 Question 14( ) Solve the inequalities (i) 1 + 2x x 2 > 2/x (x 0), (ii) (3x + 10) > 2 + (x + 4) (x 10/3). Comments The two parts are unrelated (unusually for STEP questions), except that they deal with inequalities. Both parts need care, having traps for the unwary. In the first part you have to watch out when you multiply an inequality: if the thing you muliply is negative then the inequality reverses. In the second part you have to consider the possibility that your algebraic manipulations have created extra spurious solutions. In the first part, you might find sketching a graph helpful. For the second part, it is worth (after you have finished the question) sketching the graphs of (3x + 10), 2 + (x + 4) and 2 (x + 4). You can get them by translations of x (which you can get by reflection y = x 2 in the line y = x. I would have drawn them for you in the postmortem comment overleaf, but there wasn t room on the page.
45 Solution to question 14 (i) We would like to multiply both sides of the inequality by x in order to obtain a nice cubic expression. However, we have to allow for the possibility that x is negative (which would reverse the inequality). One way is to consider the cases x > 0 and x < 0 separately. For x > 0, we multiply the whole equation by x without changing the inequality: 1 + 2x x 2 > 2/x x + 2x 2 x 3 > 2 x 3 2x 2 x + 2 < 0 (x 1)(x + 1)(x 2) < 0 1 < x < 2, ( ) discarding the possibility x < 1, since we have assumed that x > 0. The easiest way of obtaining the result ( ) from the previous line is to sketch the graph of (x 1)(x + 1)(x 2) < 0. For x < 0, we must reverse the inequality when we multiply by x so in this case, (x 1)(x+1)(x 2) > 0, which gives 1 < x < 0. The smart way to do x > 0 and x < 0 in one step is to multiply by x 2 (which is never negative and hence never changes the direction of the inequality) and analyse x(x 1)(x+1)(x 2) < 0. Again, a sketch is useful. (ii) First square both sides the inequality (3x + 10) > 2 + (x + 4): 3x + 10 > 4 + 4(x + 4) (x + 4) i.e. x + 1 > 2(x + 4) 1 2. Note that the both sides of the original inequality are positive or zero (i.e. non-negative), so the direction of the inequality is not changed by squaring. Now consider the new inequality x + 1 > 2(x+4) 1 2. If both sides are non-negative, that is if x > 1, we can square both sides again without changing the direction of the inequality. But, if x < 1, the inequality cannot be satisfied since the right hand side is always non-negative. Squaring gives x 2 + 2x + 1 > 4(x + 4) i.e. (x 5)(x + 3) > 0. Thus, x > 5 or x < 3. However, we must reject x = 3 because of the condition x > 1. Therefore the inequality holds for x > 5. Postmortem The spurious result x < 3 at the end of the second part arises through loss of information in the process of squaring. (If you square an expression, you lose its sign.) After squaring twice, the resulting inequality is the same as would have resulted from (3x + 10) > 2 (x + 4) and it is to this inequality that x < 3 is the solution.
46 Question 15( ) Sketch, without calculating the stationary points, the graph of the function f (x) given by f (x) = (x p)(x q)(x r), where p < q < r. By considering the quadratic equation f (x) = 0, or otherwise, show that (p + q + r) 2 > 3(qr + rp + pq). By considering (x 2 + gx + h)(x k), or otherwise, show that g 2 > 4h is a sufficient condition but not a necessary condition for the inequality to hold. (g k) 2 > 3(h gk) Comments For the sketch it is not necessary to do more than think about the behaviour for x large and positive, and for x large and negative, and the points at which the graph crosses the x-axis. The argument required for the second part is very similar to that of the first part, except that it also tests understanding of the meaning of the terms necessary and sufficient.
47 Solution to question 15 From the sketch, we see that f (x) has two turning points, so the equation f (x) = 0 has two real roots. p q r Now f (x) = (x p)(x q)(x r) = x 3 (p + q + r)x 2 + (qr + rp + pq)x pqr so f (x) = 3x 2 2(p + q + r)x + (qr + rp + pq) = 0. Using the condition b 2 > 4ac for the quadratic to have two real root gives which is the required result. 4(p + q + r) 2 > 12(qr + rp + pq), For the second part, we can use a similar argument. In this case, the cubic expression is factorised into a linear factor x k and a quadratic factor x 2 + gx + h. If the quadratic equation x 2 + gx + h = 0 has two distinct real roots (i.e. if g 2 > 4h), then the function (x 2 + gx + h)(x k) has three zeros and therefore two turning points. Thus if g 2 > 4h the equation f (x) = 0 has two real roots. Expanding f (x) gives f (x) = (x 2 + gx + h)(x k) = x 3 + (g k)x 2 + (h gk) and differentiating gives f (x) = 3x 2 + 2(g k)x + (h gk) = 0. The condition for two real roots is 4(g k) 2 > 12(h gk). Thus g 2 > 4h is a sufficient condition for this inequality to hold. Why is it not necessary? That is to say, why is it too strong a condition? How could the condition for two turning points hold without f (x) having three zeros. The answer is that both turning points could be above the x axis or both could be below. Saying this would get all the marks. Or you could give a counterexample: the easy counterexample is g 2 = 4h; if this holds, there are still two turning points so (g k) 2 > 3(h gk). (Careful of this counterexample: it will not work if the double root of the quadratic (at x = g/2 coincides with the root of the linear factor (at x = k), so we need the extra condition that g/2 + k 0.) Postmortem I got considerable satisfaction from the way that non-obvious inequalities are derived from understanding simple graphs and the quadratic formula when I set this question. But are the inequalities really obscure? The answer is unfortunately no. The inequality of the first part is equivalent to (q r) 2 + (r p) 2 + (p q) 2 > 0 (the inequality coming from the fact the squares of real numbers are non-negative and in this case the given condition p < q < r means that the expression is strictly positive (not equal to zero). If we rewrite the second inequality as (k + g/2) 2 + 3(g 2 /4 h) > 0 we see that it is certainly holds if (g 2 /4 h) > 0. Clearly, this is not a necessary condition: there is the obvious counterexample (g 2 /4 h) = 0 and (k + g/2) 0. But there are lots of other easy counterexamples for example, g = 0 and k 2 > 3h > 0.
48 Question 16( ) In a cosmological model, the radius R of the universe is a function of the age t of the universe. The function R satisfies the three conditions: R(0) = 0, R (t) > 0 for t > 0, R (t) < 0 for t > 0, ( ) where R denotes the second derivative of R. The function H is defined by H(t) = R (t) R(t). (i) (ii) (iii) Sketch a graph of R(t). By considering a tangent to the graph, show that t < 1/H(t). Observations reveal that H(t) = a/t, where a is constant. Derive an expression for R(t). What range of values of a is consistent with the three conditions ( )? Suppose, instead, that observations reveal that H(t) = bt 2, where b is constant. Show that this is not consistent with conditions ( ) for any value of b. Comments The sketch just means any graph starting at the origin and increasing, but with decreasing gradient. The second part of (i) needs a bit of thought (where does the tangent intersect the x-axis?) so don t despair if you don t see it immediately. Parts (ii) and (iii) are perhaps easier than (i).
49 Solution to question 16 (i) The height of the right-angled triangle in the figure is R(t) and the slope of the hypotenuse is R (t). The length of the base is therefore R(t)/R (t), i.e.1/h(t). The figure shows that the intersection of the tangent to the graph for t > 0 intercepts the negative t axis, so 1/H(t) > t. 1/H(t) t R(t) (ii) One way to proceed is to integrate the differential equation: H(t) = a t R (t) R(t) = a R t (t) a R(t) dt = t dt ln R(t) = aln t + constant R(t) = Ata. The first two conditions ( ) are satisfied if a > 0 and A > 0. For the third condition, we have R (t) = a(a 1)At a 1 which is negative provided a < 1. The range of a is therefore 0 < a < 1, (iii) The obvious way to do this just follows part (ii). This time, we have H(t) = b t 2 R (t) R(t) = b R t 2 (t) b R(t) dt = t 2dt ln R(t) = b t + constant R(t) = Ae b/t. Thus R (t) = H(t)R(t) = Abt 2 e b/t. Clearly A > 0 since R(t) > 0 for t > 0 so R (t) > 0. Furthermore R(0) = 0 (think about this!), so only the condition R (t) < 0 remains to be checked. Differentiating R (t) gives R (t) = Ab( 2t 3 )e b/t + Abt 2 e b/t (bt 2 ) = Abt 4 e b/t ( 2t + b) This is positive when t < b/2, which contradicts the condition R < 0. Instead of solving the differential equation, we could proceed as follows. We have R R = bt 2 R = brt 2 R = br t 2 2bRt 3 = b 2 t 4 R 2bRt 3 = b(b 2t)t 4 R. This is positive when t < b/2, which contradicts the condition R < 0. Postmortem The interpretation of the conditions is as follows. The first condition R(0) = 0 says that the universe started from zero radius in fact, from the big bang. The second condition says that the universe is expanding. This was one of the key discoveries in cosmology in the last century. The quantity called here H(t) is called Hubble s constant (though it varies with time). It measures the rate of expansion of the universe. Its present day value is a matter of great debate (it is of the order of 75 megaparsecs per century). The last condition says that the expansion of the universe is slowing down. This is expected on physical grounds because of the gravitational attraction of the galaxies on each other. Another observational uncertainly is the value of the deceleration. It is not clear yet whether the universe will expand for ever or recollapse.
50 Question 17( ) Show that the equation of any circle passing through the points of intersection of the ellipse (x + 2) 2 + 2y 2 = 18 and the ellipse can be written in the form 9(x 1) y 2 = 25 x 2 2ax + y 2 = 5 4a. Comments When this question was set (as question 1 on STEP I) it seemed too easy. But quite a high proportion of the candidates made no real progress. Of course, is is not easy to keep a cool head under examination conditions. But surely it is obvious that either x or y has to be eliminated from the two equations for the ellipses; and having decided that, it is obvious which one to eliminate. It is worth thinking about how many points of intersection of the ellipses we are expecting: a sketch might help.
51 Solution to question 17 First we have to find the intersections of the two ellipses by solving the simultaneous equations (x + 2) 2 + 2y 2 = 18 9(x 1) y 2 = 25. We can eliminate y by multiplying the first equation by 8 and subtracting the two equations: 8(x + 2) 2 9(x 1) 2 = i.e. x 2 50x + 96 = 0 i.e. (x 2)(x 48) = 0. The two possible values for x at the intersections are therefore 2 and 48. Next we find the values of y at the intersection. Taking x = 2 and substituting into the first simultaneous equation, we have y 2 = 18, so y = ±1. Taking x = 48 gives y 2 = 18 which has no (real) roots. Thus there are two points of intersection, at (2,1) and (2, 1). Suppose a circle through these points has centre (p,q) and radius R. Then the equation of the circle is (x p) 2 + (y q) 2 = R 2. Setting (x,y) = (2,1) and (x,y) = (2, 1) gives two equations:. (2 p) 2 + (1 q) 2 = R 2, (2 p) 2 + ( 1 q) 2 = R 2. Subtracting gives q = 0 (it is obvious anyway that the centre of the circle must lie on the y axis). Thus the equation of any circle passing through the intersections is which simplifies to the given result with p = a. (x p) 2 + y 2 = (2 p) 2 + 1,
52 Question 18( ) Let f (x) = x m (x 1) n, where m and n are both integers greater than 1. Show that ( m f (x) = x n ) f (x). 1 x Show that the curve y = f (x) has a stationary point with 0 < x < 1. By considering f (x), show that this stationary point is a maximum if n is even and a minimum if n is odd. Sketch the graphs of f (x) in the four cases that arise according to the values of m and n. Comments There is quite a lot in this question, but it is one of the best STEP questions on basic material that I came across (I looked through all 15 years of STEP s history). First, you have to find f (x). The very first part (giving a form of f (x)) did not occur in the original STEP question. This particular expression is extremely helpful when if comes to finding the value of f (x) at the stationary point. In the actual exam, a handful of candidates successfully found a general expression for f (x) in terms of x then evaluated it at the stationary point: first class work. You should find that the value of f (x) at the stationary point can be expressed in the form ( )f (x), where the factor in the brackets is quite simple and always negative, so that the nature of the stationary point depends only on the sign of f (x). (Actually, this is intuitively obvious: f (0) = f (1) = 0 so the one stationary point must be, for example, a maximum if f (x) > 0 for 0 < x < 1.) For the sketches, you really have only to think about the behaviour of f (x) when x is large and the sign of f (x) between x = 0 and x = 1. That allows you to piece together the graph, knowing that there is only one stationary point between x = 0 and x = 1. Note that, very close to x = 0, f (x) x m ( 1) n so it is easy to see how the graph there depends on n and m ; you can use this to check that you have got the graphs right. Postmortem [This didn t fit on the next page: read it after you have done the question.] You may have noticed a little carelessness at the top of the next page: what happens in the logarithmic differentiation if any of f (x) or x or x 1 are negative? The answer is that it doesn t matter. One way to deal with the problem of logs with negative arguments is to put modulus signs everywhere using the correct result d dx ln f (x) = f (x) f (x). If you are not sure of this, try it on f (x) = x taking the two cases x > 0 and x < 0 separately. A more sophisticated way of dealing with logs with negative arguments is to note that ln( f (x)) = ln f(x) + ln( 1). We don t have to worry about ln( 1) because it is a constant (of some sort) and so won t affect the differentiation. Actually, a value of ln( 1) can be obtained by taking logs of Euler s famous formula e iπ = 1 giving ln( 1) = iπ.
53 Solution to question 18 The first result can be established by differentiating f (x) directly, but the neat way to do it is to start with ln f (x): as required. ln f (x) = m ln x + n ln(x 1) = f (x) f (x) = m x + n x 1, Thus f (x) has stationary points at x = 0 and x = 1 (since m 1 > 0 and n 1 > 0) and when m x n 1 x = 0, i.e. when m(1 x) nx = 0. Solving this last equation for x gives x = m/(m + n), which lies between 0 and 1 since m and n are positive. Next we calculate f (x). Starting with we obtain f (x) = ( m x 2 f (x) = ( m x n ) f (x), 1 x ) ( n m (1 x) 2 f (x) + x n ) f (x). 1 x At the stationary point, the second of these two terms is zero because f (x) = 0. Thus ( f (x) = m ) x 2 n (x 1) 2 f (x). The bracketed expression is negative so at the stationary point f (x) < 0 if f (x) > 0 and f (x) > 0 if f (x) < 0. The sign of f (x) for 0 < x < 1 is the same as the sign of (x 1) n, since x m > 0 when x > 0. The stationary point is between 0 and 1, so x 1 < 0 and (x 1) n > 0 if n is even and (x 1) n < 0 if n is odd. Thus f (x) < 0 if n is even and f (x) > 0 if n is odd, which is the required result. The four cases to sketch are determined by whether m and n are even or odd. The easiest way to understand what is going on is to consider the graphs of x m and (x 1) n separately, then try to join them up at the stationary point between 0 and 1. If m is odd, then x = 0 is a point of inflection, but if m is even it is a maximum or minimum according to the sign of (x 1) n. You should also think about the behaviour for large x. All the various bits of information (including the nature of the turning point investigated above) should all piece neatly together. m even, n even m odd, n even m even, n odd m odd, n odd
54 Question 19( ) Give a sketch of the curve y = 1, for x x2 Find the equation of the line that intersects the curve at x = 0 and is tangent to the curve at some point with x > 0. Prove that there are no further intersections between the line and the curve. Draw the line on your sketch. By considering the area under the curve for 0 x 1, show that π > 3. Show also, by considering the volume formed by rotating the curve about the y axis, that ln 2 > 2/3. [ Note: x 2 dx = π 4. ] Comments There is quite a lot to this question, so three stars even though none of it is particularly difficult. The most common mistake in finding the equation of the tangent is to muddle the y and x that occur in the equation of the line (y = mx+c) with the coordinates of the point at which the tangent meets the curve, getting constants m and c that depend on x. I m sure you wouldn t make this rather elementary mistake normally, but it is surprising what people do under examination conditions. The reason for sketching the curve lies in the last parts: the shape of the curve relative to the straight line provides the inequality. The note at the end of the question had to be given because integrals of that form (giving inverse trigonometric functions) are not in the core A-level syllabus.
55 Solution to question 19 The sketch should show a graph which has gradient 0 at (0,1) and which asymptotes to the x-axis for large x. The line is the tangent to the graph from the point (0,1) which is required later. y 1 1 x First we find the equation of the tangent to the curve y = (1 + x 2 ) 1 at the point (p,q). The gradient of the curve at the point (p,q) is 2p(1+p 2 ) 2 = 2pq 2, so the equation of the tangent is y = 2pq 2 x + c where c is given by q = 2pq 2 p+c. This line is supposed to pass through the point (0,1), so 1 = c. Thus 1 = p 2 + 2p2 (1 + p 2 ) 2 which simplifies to (1 + p 2 ) 2 = 1 + 3p 2 i.e. p 4 = p 2. The only positive solution is p = 1, so the equation of the line is y = x/ We can check that this line does not meet the curve again by solving the equation Multiplying by (1 + x) 2 gives x = x 2. (1 x/2)(1 + x 2 ) = 1 i.e. x 3 2x 2 + x = 0. Factorising shows that x = 1 and x = 0 satisfy this equation (these are the known roots, x = 1 being a double root corresponding to the tangency) and that there are no more roots. The area under the curve for 0 x 1 is π/4 as given. The sketch shows that this area is greater than the area under the tangent line for 0 x 1, which is (the area of the rectangle plus the area of the triangle above it). Comparing with with π/4 gives the required result. The volume formed by rotating the curve about the y axis is x 2πyxdx = π x2dx = π ln 2. This is greater than the volume formed by rotating the line about the y axis, which is 1 Comparison gives the required result. 0 2πyxdx = 2π 1 0 (1 x/2)xdx = 2π/3.
56 Question 20( ) Let I = a 0 cos x a sin x dx and J = sin x + cos x 0 sin x + cos x dx, where 0 a < 3π/4. By considering I + J and I J, show that 2I = a + ln(sin a + cos a). Find also: (i) π/2 0 cos x dx, where p and q are positive numbers; p sin x + q cos x (ii) π/2 0 cos x + 4 3sin x + 4cos x + 25 dx. Comments I love this question. When I first saw it (one of my colleagues proposed the first part for STEP Paper I a few years ago) I was completely taken by surprise. You might like to think how you would evaluate these integrals without using this method. In order to use it again for STEP in 2002, I added parts (i) and (ii). It took me some time to think of a suitable extension. I was disappointed to find that the basic idea is more or less a one-off: there are very few denominators, besides the ones given, that lead to integrands amenable to this trick. But I was pleased with what I came up with. The question leads you through the opening paragraph and the extra parts depend very much on your having understood why the opening paragraph works. In the examination, candidates who were successful in parts (i) and (ii) nearly always started off with the statement Now let I = and J =. The choice of significant numbers in part (ii) (3, 4 and 25) should be a clue. Finally, you will have noticed that 0 a < 3π/4 was given in the question. Understanding the reason for this restriction does not help to do the question, but you should try to work out its purpose.
58 Question 21( ) In this question, you may assume that if k 1,...,k n are distinct positive real numbers, then 1 n n ( n k r > r=1 n, )1 k r r=1 i.e. their arithmetic mean is greater than their geometric mean. Suppose that a, b, c and d are positive real numbers such that the polynomial has four distinct positive roots. f(x) = x 4 4ax 3 + 6b 2 x 2 4c 3 x + d 4 (i) By considering the relationship between the coefficients of f and its roots, show that c > d. (ii) By differentiating f, show that b > c. (iii) Show that a > b. Comments This result is both surprising and pleasing. I m sorry to say that I don t understand it. I ve looked at it from various points of view, and asked a few selected colleagues who are usually good at this sort of thing, but so far no one has come up with a good explanation. Maybe the penny will drop in time for the next edition. (I haven t even worked out if the converse of this result holds, namely that the above inequalities are sufficient for the roots to lie in the given interval; I don t think this is hard, but I thought I would leave it until I undersood it better.) There are general theorems on the number of roots in any given interval. This problem was studied by Descartes ( ), who came up with his rule of signs. This relates the number of roots to the signs of terms in the polynomial, but it only gives an upper bound for the number of roots. The problem was finally cracked by Sturm in about 1835, but his solution is quite complicated, and does not seem to throw any light on the special case we are looking at here (where the interval is (0, )). It will be clear from the proof that it can be generalised to any equation of the form x N + N 1 k=0 ( ) N ( a k ) N k x k = 0, k where the numbers a k are distinct and positive, which has positive distinct roots. The question looks difficult, but you don t have to go very far before you come across something to substitute into the arithmetic/geometric inequality. Watch out for the condition in the arithmetic/geometric inequality that the numbers are distinct: you will have to show that any numbers (or algebraic expressions) you use for the inequality are distinct. To obtain the relationship between the coefficients and the roots, you need to write the quartic equation in the form (x p)(x q)(x r)(x s) = 0.
59 Solution to question 21 (i) First write f (x) (x p)(x q)(x r)(x s) where p, q, r and s are the four roots of the equation (known to be real, positive and distinct). Multiplying out the brackets and comparing with x 4 4ax 3 +6b 2 x 2 4c 3 x+d 4 shows that pqrs = d 4 and pqr + qrs + rsp + spq = 4c 3. The required result, c > d, follows immediately by applying the arithmetic/geometric mean inequality to the positive real numbers pqr, qrs, rsp and spq : c 3 = pqr + qrs + rsp + spq 4 > [(pqr)(qrs)(rsp)(spq)] 1/4 = [p 3 q 3 r 3 s 3 ] 1/4 = [d 12 ] 1/4. Taking the cube root (c and d are real) preserves the inequality. The arithmetic/geometric mean inequality at the beginning of the question is stated only for the case when the numbers are distinct (though in fact it holds provided at least two of the numbers are distinct). To use the inequality, we have therefore to show that no two of pqr, qrs, rsp and spq are equal. This follows immediately from the fact that the roots are distinct and non-zero. For if, for example, pqr = qrs then qr(p s) = 0 which means that q = 0, r = 0 or p = s any of which is a contradiction. (ii) The polynomial f (x) is cubic so it has three zeros (roots). These are at the turning points of f (x) (which lie between the zeros of f (x)) and are therefore distinct and positive. Now At the turning points, f (x) = 4x 3 12ax b 2 x 4c 3 x 3 3ax 2 + 3b 2 x c 3 = 0. Suppose that the roots of this cubic equation are u, v and w all real, distinct and positive. Then comparing x 3 3ax 2 +3b 2 x c 3 = 0 and (x u)(x v)(x w) shows that c 3 = uvw and vw+wu+uv = 3b 2. The required result, b > c, follows immediately by applying the arithmetic/geometric mean inequality to the positive real numbers vw, wu and uv. (iii) Apply similar arguments to f (x)/12.
60 Question 22( ) The nth Fermat number, F n, is defined by F n = 2 2n + 1, n = 0, 1, 2,..., where 2 2n means 2 raised to the power 2 n. Calculate F 0, F 1, F 2 and F 3. Show that, for k = 1, k = 2 and k = 3, F 0 F 1...F k 1 = F k 2. ( ) Prove, by induction, or otherwise, that ( ) holds for all k 1. Deduce that no two Fermat numbers have a common factor greater than 1. Hence show that there are infinitely many prime numbers. Comments Fermat ( ) conjectured that every number of the form 2 2n + 1 is a prime number. F 0 to F 4 are indeed prime, but Euler showed in 1732 that F 5 ( ) is divisible by 641. As can be seen, Fermat numbers get very big and not many more have been investigated; but those that have been investigated have been found not to be prime numbers. It is now conjectured that in fact only a finite number of Fermat numbers are prime numbers. The Fermat numbers have a geometrical significance as well: Gauss proved that a regular polygon of n sides can be inscribed in a circle using a Euclidean construction (i.e. only a straight edge and a compass) if and only if n is a power of 2 times a product of distinct Fermat primes. This rather nice proof that there are infinitely many prime numbers comes from Proofs from the Book 11, a set of proofs thought by Paul Erdös to be heaven-sent. (Most of them are much less elementary this one.) Erdös was an extraordinarily prolific mathematician. He had almost no personal belongings and no home 12. He spent his life visiting other mathematicians and proving theorems with them. He collaborated with so many people that every mathematician is (jokingly) assigned an Erdös number; for example, if you wrote a paper with someone who wrote a paper with Erdös, you are given the Erdös number 2. It is conjectured that no mathematician has an Erdös number greater than 7. Mine is 8, but that is because I am more a theoretical physicist than a mathematician by M. Aigler and G.M. Ziegler, published by Springer, See The man who loved only numbers by Paul Hoffman published by Little Brown and Company in 1999 a wonderfully readable and interesting book. 13 Since I wrote this, I discovered collaborator distance on which gives the distance between any two researchers. I found that I am not after all counterexample to the conjecture: my Erdös is 5 (via Einstein). Sorry to bore you with all this personal stuff.
61 Solution to question 22 To begin with, we have by direct calculation that F 0 = 3, F 1 = 5, F 2 = 17 and F 3 = 257, so it is easy to verify the displayed result for k = 1,2,3. For the induction, we start by assuming that the result holds for k = m so that F 0 F 1... F m 1 = F m 2. We need to show that this implies that that the result holds for k = m + 1, i.e. that F 0 F 1... F m = F m+1 2. Starting with the left hand side of this equation, we have F 0 F 1... F m 1 F m = (F m 2)F m = (2 2m 1)(2 2m + 1) = ( 2 2m ) 2 1 = 2 2 m+1 1 = F m+1 2, as required. Since we know that the result holds for k = 1, the induction is complete. For the rest of the question: if p divides F l and F m, where l < m, then p divides F 0 F 1... F l...f m 1 and therefore p divides F m 2, by ( ). But all Fermat numbers are odd so no number other than 1 divides both F m 2 and F m, which gives the contradiction. Hence no two Fermat numbers have a common factor greater than 1. For the last part, note that every number can be written uniquely as a product of prime factors and that because the Fermat numbers are co-prime, each prime can appear in at most one Fermat number. Thus, since there are infinitely many Fermat numbers, there must be infinitely many primes. Postmortem This question is largely about proof. The above solution has a proof by induction and a proof by contradiction. The last part could have been written as a proof by contradiction too. The difficulty is in presenting the proof. It is not enough to understand your own proof: you have to be able to set it out so that it is clear to the reader. This is a vital part of mathematics and its most useful transferable skill.
62 Question 23( ) Show that the sum S N of the first N terms of the series is What is the limit of S N as N? The numbers a n are such that n 1 n(n + 1)(n + 2) + ( N ). N + 2 a n (n 1)(2n 1) = a n 1 (n + 2)(2n 3). Find an expression for a n a 1 and hence, or otherwise, evaluate n=1 a n when a 1 = 2 9. Comments If you haven t the faintest idea how to do the sum, then look at the first line of the solution; but don t do this without first having a long think about it, and a long look at the form of the answer given in the question. Limits form an important part of first year university mathematics. The definition of a limit is one of the basic ideas in analysis which is the rigorous study of calculus. Here no definition is needed: you just see what happens when N gets very large (some terms get very small and eventually go away). The second part of the question looks as if it might be some new idea. Since this is STEP, you will probably realise that the series in the second part must be closely related to the series in the first part. The peculiar choice for a 1 (= 2/9) should make you suspect that the sum will come out to some nice round number (not in fact round in this case, but straight and thin). Having decided how to do the first part, please don t use the cover up rule unless you understand why it works: mathematics at this level is not a matter of applying learned recipes.
63 Solution to question 23 The first thing to do is to put the general term of the series into partial fractions in the hope that something good might then happen. (The form of the answer suggests partial fractions have been used; and it is difficult to think of anything else to do.) The general term of the series is 2n 1 n(n + 1)(n + 2) A n + B n C n + 2. (The equivalent sign,, indicates an identity (something that holds for all values of n) rather than an equation to solve for n. The easiest way to deal with this is to multiply both sides of the identity by one of the terms in the denominator and then set this term equal to zero (since the identity must hold for all values of n). For example, multiplying by n gives 2n 1 (n + 1)(n + 2) A + Bn n Cn n + 2. and setting n = 0 shows immediately that A = 1 2. Similarly, multiplying by n + 1 and n + 2 respectively and setting n = 1 and n = 2 respectively gives B = 3 and C = 5 2. Thus The series can now be written ( 1/ /2 ) + 3 ( + + 1/2 N N 1 5/2 N 2n 1 n(n + 1)(n + 2) = 1/2 n + 3 n + 1 5/2 n + 2 ( 1/ /2 4 ) + ) ( + 1/ /2 ) + 5 ) + ( 1/2 N N 5/2 N + 1 ( 1/2 N + 3 N + 1 5/2 N + 2 The first term has 3 as the last number in the denominator so the Nth term has N + 2. All the terms in the series cancel, except those with denominators 1, 2, N + 1 and N +2. The sum of these exceptions is the sum of the series. The limit as N is 3 4 For the second paragraph, we have first part of the question. Thus Alternatively, a n = = = since the other two terms obviously tend to zero. a n a 1 = b n b 1 a n a n 1 = b n b n 1, where b n is the general term of the series in the and 1 a n = a 1 b 1 1 b n = 2/9 1/6 3 4 = 1. (n 1)(2n 1) (n + 2)(2n 3) a (n 1)(2n 1) (n 2)(2n 3) n 1 = (n + 2)(2n 3) (n + 1)(2n 5) a n 2 (n 1)(2n 1) (n 2)(2n 3) (n 3)(2n 5) (n + 2)(2n 3) (n + 1)(2n 5) (n)(2n 7) a 1 2n a 1 = 12 2n 1 n(n + 1)(n + 2) 1 9 n(n + 1)(n + 2), all other terms cancelling. Now using the result of the first part gives a n = 1. 1 ).
64 Question 24( ) (i) Show that, if m is an integer such that (m 3) 3 + m 3 = (m + 3) 3, ( ) then m 2 is a factor of 54. Deduce that there is no integer m which satisfies the equation ( ). (ii) Show that, if n is an integer such that (n 6) 3 + n 3 = (n + 6) 3, ( ) then n is even. By writing n = 2m deduce that there is no integer n which satisfies the equation ( ). (iii) Show that, if n is an integer such that (n a) 3 + n 3 = (n + a) 3, ( ) where a is a non-zero integer, then n is even and a is even. Deduce that there is no integer n which satisfies the equation ( ). Comments I slightly simplified the first two parts of the question, which comprised the whole of the original STEP question, and added the third part. This last part is conceptually tricky and very interesting: hence the three star rating. It is an example of a method of proof by contradiction, called the method of infinite descent. You will understand why it is called infinite descent when you have finished the question (in fact, the name may serve as a hint). It was used by Fermat ( ) to prove special cases of his Last Theorem 14, which of course is exactly what you are doing in this question. The method was probably invented by him and his faith in it sometimes led him astray. It is even possible that he thought he could use it to prove his Last Theorem in full. In fact, the proof of this theorem was only given in 1994 by Andrew Wiles; and it is 150 pages long. 14 The theorem says that the equation x n + y n = z n, where x, y, z and n are positive integers, can only hold if n = 1 or 2
65 Solution to question 24 (i) Simplifying gives m 2 (m 18) = 54. Both m 2 and (m 18) must divide 54, which is impossible since the only squares that divide 54 are 1 and 9, and neither m = 1 nor m = 3 satisfies m 2 (m 18) = 54. You could also argue that m 18 must be positive so m 19 and m which is a contradiction. (ii) Suppose n is odd. The two terms on the left hand side are both odd, which means that that the left hand side is even. But the right hand side is odd so the equation cannot balance. Setting n = 2m gives (2m 6) 3 + (2m) 3 = (2m + 6) 3. Taking a factor of 2 3 out of each term leaves (m 3) 3 + m 3 = (m + 3) 3 which is the same as the equation that was shown in part (i) to have no solutions. (iii) First we show that n is even, dealing with the cases a odd and a even separately. The first case is n odd and a odd. In that case, (n a) 3 is even, n 3 is odd and (n + a) 3 is even, so the equation does not balance. In the second case, n is odd and a is even, the three terms are all odd and again the equation does not balance. Therefore n cannot be odd. Next, we investigate the case n is even and a odd. This time the three terms are odd, even and odd respectively so there is no contradiction. But multiplying out the brackets and simplifying gives n 2 (n 6a) = 2a 3. Since n is even the left hand side is divisible by 4, because of the factor n 2. That means that a 3 is divisible by 2 and hence a is even. Now that we know n and a are both even, we can follow the method used in part (ii) and set n = 2m and a = 2b. This gives (2m 2b) 3 + (2m) 3 = (2m + 2b) 3 from which a factor of 2 3 can be cancelled from each term. Thus m and b satisfy the same equation as n and a. They are therefore both even and we can repeat the process. Repeating the process again and again will eventually result in an integer that is odd which will therefore not satisfy the equation that it is supposed to satisfy: a contradiction. There is therefore no integer n that satisfies ( ).
66 Question 25( ) The function f satisfies 0 f (t) K when 0 t 1. Explain by means of a sketch, or otherwise, why By considering and deduce that Deduce that N n= f (t)dt K. t dt, or otherwise, show that, if n > 1, n(n t) ( ) n 0 ln 1 n 1 n 1 n 1 1 n 1 as N. n 0 ln N N n=2 Noting that 2 10 = 1024, show also that if N < then 1 n 1. N n=1 1 n < 101. Comments Quite a lot of different ideas are required for this question; hence three stars. The first hurdle is to decide what the constant K in the first part has to be to make the second part work. You might try to maximise the integrand using calculus, but that would be the wrong thing to do (you should find if you do this that the integrand has no turning point in the given range: it increases throughout the range. The next hurdle is the third displayed equation, which follows from the preceding result. Then there is still more work to do. N 1 It is not at all obvious at first sight that the series (which is called the harmonic series tends n 1 to infinity as N increases, though there are easier ways of proving this. This way tells us two very interesting things; First it tells us that the sum increases very slowly indeed: the first terms only get to 100. Second it tells us that 0 N 1 1 n ln N 1 for all N. (This is the third displayed equation in the question slightly rewritten.) Not only does the sum diverge, it does so logarithmically. In fact, in the limit N, where γ is Euler s constant. Its value is about 1/2. N 1 1 n ln N γ
67 Solution to question 25 Any sketch showing a squiggly curve all of which lies beneath the line x = K and above the x-axis will do (area under curve is less than the area of the rectangle). Evaluate the integral, noting that t/(n t) = n/(n t) 1: 1 0 t n(n t) dt = 1 0 ( 1 n t 1 ) ( ) n dt = ln 1 n n 1 n. Now note that the largest value of the integrand in the interval [0,1] occurs at t = 1, since the numerator increases as t increases and the denominator decreases as t increases (remember that n > 1) so ( ) n 0 ln 1 n 1 n 1 n(n 1) = 1 n 1 1 n. Summing both sides from 2 to N and cancelling lots of terms in pairs gives 0 ln N N 2 1 n 1 1 N. Note that 1 1/N < 1. Since ln N as N, so also must N 2 1 (it differs from the logarithm n by less than 1). N 1 Finally, rearranging the inequality gives n ln N, i.e. N 1 n ln N + 1. Setting N = 1030, 2 1 and using the identity given and also the inequality e > 2 gives: n ln(1030 ) + 1 < ln( ) + 1 = 100ln < 100ln e + 1 = Postmortem This question seems very daunting at first, because you are asked to prove a sequence of completely unfamiliar results. However, you should learn from this question that if you keep cool and follow the implicit hints, you can achieve some surprisingly sophisticated results. You might think that this situation is very artificial: in real life, you do not receive hints to guide you. But often in mathematical research, hints are buried deep in the problem if only you can recognise them.
68 Question 26( ) Find dy dx if y = ax + b cx + d. ( ) By using changes of variable of the form ( ), or otherwise, show that 1 0 ( ) 1 x + 1 (x + 3) 2 ln dx = 1 x ln ln , and evaluate the integrals (x + 3) 2 ln ( x 2 + 3x + 2 (x + 3) 2 ) dx and 1 0 ( ) 1 x + 1 (x + 3) 2 ln dx. x + 2 By changing to the variable y defined by y = 2x 3 x + 1, evaluate the integral Evaluate the integral x 3 (x + 1) 3 ln ( 2x 3 x + 1 ) dx. ( 2z 3/2 5z 2) ln ( 2 5z 1/2) dz. 9 Comments If you are thinking that this question looks extremely long, you are right. It is two questions in fact, two versions of one question separated by the horizontal line. Beneath the line is the first draft of a STEP question; above the line is the question as it was actually set. I thought you would be interested to see the evolution of a question. Note in particular the way that the ideas in the final draft are closely knit and better structured: the first change of variable is strongly signalled, but the remaining two parts, though based on the same idea, require increasing ingenuity. The first draft just required two unrelated changes of variable. Note also that the integral in the last part of the first draft has an unpleasant contrived appearance, whereas the the integrals of the final draft are rather pleasing: beauty matters to mathematicians. A change of variable of the form ( ) is called a linear fractional or Möbius transformation. It is of great importance in the theory of the geometry of the complex plane.
69 Solution to question 26 Differentiating gives dy a(cx + d) c(ax + b) = dx (cx + d) 2 = For the first integral, set y = x + 1 x + 3 : (x + 3) 2 ln ( x + 1 x + 3 ) dx = 1 2 ad bc (cx + d) 2. 1/2 1/3 ln y dy = 1 2 [ ] 1/2 y ln y y, 1/3 which gives the required answer. (The integral of lny is a standard integral; it can be done by parts, first substituting z = ln y, if you like.) The second integral can be expressed as the sum of two integrals of the same form as the first integral, since ( x 2 ) ( ) ( ) ( ) + 3x + 2 (x + 1)(x + 2) x + 1 x + 2 ln (x + 3) 2 = ln (x + 3) 2 = ln + ln. x + 3 x + 3 We have already done the first of these integrals. Using the substitution y = x + 2 in the second of x + 3 these integrals gives 3/4 ln y dy = ln ln The required integral is therefore 2/3 For the next integral, note that ( ) x + 1 ln = ln x ln ln ( ) x + 1 ln x + 3 ( ) x + 2, x + 3 so the required integral is the difference of the two integrals that we summed in the previous part, i.e ln 3 + ln The change of variable gives 1 1 y ln y dy 5 1/3 which can be integrated by parts. The result is 1 90 ln For the last part, you have to guess the substitution. There are plenty of clues, but the most obvious place to start is the log. The argument of the log can be written as 2(x + 1) 5 x + 1 = 2 5 x + 1 so it looks as if we should take z = (x+1) 2. Making this transformation gives the previous integral almost exactly (note especially the the limits transform as they should). The only difference is a factor of 2 which comes from dz so the answer is twice the previous answer. dx
70 Question 27( ) The curve C has equation y = x (x 2 2x + a), where the square root is positive. Show that, if a > 1, then C has exactly one stationary point. Sketch C when (i) a = 2 and (ii) a = 1. Comments There is something you must know in order to do this question, and that is the definition of (x 2 2x + a). You might think that this is ambiguous, because the square root could be positive of negative, but by convention it means the positive square root. Thus (x 2 ) = x (not x). For the sketches, you just need the position of any stationary points, any other points that the graph passes through, behaviour as x ± and any vertical asymptotes. It should not be necessary in such a simple case to establish the nature of the stationary point(s). You should at some stage think about the condition given on a. In fact, I doubt if many of you will want to leave it there. It is clear that the examiner would really have liked you to do is to sketch C in the different cases that arise according to the value of a but was told that this would be to long and/or difficult for the exam; I am sure that this is what you will do (or maybe what you have already done by the time you read this). Please do not use your calculator for these sketches (except perhaps to check your answers): there is no point the graph will either be wrong, in which case you will be confused, or it will be right and the value that you might have gained from thinking about the question yourself will be lost to you for ever.
71 Solution to question 27 First differentiate to find the stationary points: which is only zero when x = a. 15 dy dx = (x2 2x + a) x(x 1) (x 2 2x + a) 3/2 = a x (x 2 2x + a) 3/2 (i) When a = 2, the stationary point is at (2, 2) (which can be seen to be a maximum by evaluating the second derivative at x = a, though this is not necessary). The curve passes through (0,0). As x, y 1; as x, y 1. x (ii) When a = 1, we can rewrite y as. The modulus x 1 signs arise because the square root is taken to be positive (or zero but that can be discounted here since it is in the denominator). The graph is as before, except that the maximum point has been stretched into a vertical asymptote (like a volcanic plug) at x = 1. Postmortem The first thing to do after finishing the question is to try to understand the condition on a. Part (ii) above is the key. Writing f (x) = x 2 2x + a = (x 1) 2 + a 1 we see that f (x) > 0 for all x if a > 1 but if a < 1 there are values of x for which f (x) < 0. is negative. The borderline value is a = 1, for which f (x) 0 and f (1) = 0. If a < 1, we find f (x) = 0 when x = 1 ± (1 a) and f (x) < 0 between these two values of x. The significance of this is that the square root in the denominator of the function we are trying to sketch is imaginary, so there is a gap in the graph between x = 1 (1 a) and x = 1 + (1 a) and there are vertical asymptotes at these values. The point x = a lies between those values for a < 1. Is that the whole story? Well, no. We now have to decide whether the two significant points of our graph, namely x = 0 where the curve crosses the axis and x = a where the curve has a stationary point, lie inside or outside the forbidden zone 1 (1 a) x 1 + (1 a). Since f (0) < 0 if a < 0 and f (a) < 0 if 0 < a < 1 we see that a has another critical value besides a = 1; if a < 0 the graph is going to look qualitatively different. 15 My drawing package is absolutely useless, so I m not going to attempt to improve these sketches, or label them fully (which you should certainly do). Maybe in the next draft...
72 Question 28( ) Prove that (i) Deduce that, for n 1, n k=0 sinkθ = cos 1 2 θ cos(n )θ 2sin 1 2 θ. ( ) n ( ) kπ ( π sin = cot n 2n) k=0 (ii) By differentiating ( ) with respect to θ, or otherwise, show that, for n 1, n k=0 ( ) kπ k sin 2 (n + 1)2 = + 1 ( π ) 2n 4 4 cot2 2n.. Comments The very first part can be done by multiplying both sides of ( ) by sin 1 2θ and using the identity 2sin Asin B = cos(b A) cos(b + A). It can also be done by induction (worth a try even if you do it by the above method) or by considering the imaginary part of exp(ikθ) (summing this as a geometric progression), if you know about de Moivre s theorem. Before setting pen to paper for part (ii), it pays to think very hard about simplifications of ( ) that might make the differentiation more tractable perhaps bearing in mind the calculations involved with part (i). In the original question, you were asked to show that for large n and n k=0 n ( ) kπ sin 2n n π k=0 ( ) ( kπ 1 k sin 2 2n ) π 2 n 2, using the approximations, valid for small θ, sin θ θ and cos θ θ2. Of course, these follow quickly from the exact results; but if you only want approximate results you can save yourself a bit of work by approximating early to avoid doing some of the trigonometric calculations. I thought that the exact result was nicer than the approximate result (though approximations are an important part of mathematics).
73 Solution to question 28 For the first part, we will show that n 2sin kθ sin 1 2 θ = cos 1 2 θ cos(n )θ. k=0 k=0 Starting with the left hand side, we have n n 2sin kθ sin 1 2 θ = [ cos(k )θ + cos(k 1 2 )θ] k=0 which gives the result immediately, since almost all the terms in the sum cancel. (Write our a few terms if you are not certain of this.) (i) Let θ = π/n. Then n ( ) kπ sin = cos 1 2 (π/n) cos(π (π/n)) n 2sin 1 2 (π n ) k=0 as required. = cos 1 2 (π/n) + cos 1 2 (π/n)) 2sin 1 2 (π/n) = cot 1 2 (π/n) (ii) Differentiating the left hand side of ( ) and using a double angle trig. formula gives n n k cos kθ = k ( n 1 2sin kθ) = 1 2 n(n + 1) 2 k sin 2 1 2kθ. ( ) k=0 k=0 Before attempting to differentiate the right hand side of ( ) it is a good idea to write it in a form that gets rid of some of the fractions. Omitting for the moment the factor 1/2, we have k=0 RHS of ( ) = (1 cos nθ)cot 1 2θ + sin nθ and differentiating gives 1 2 (1 cos nθ)cosec2 1 2 θ + n sin nθ cot 1 2θ + n cos nθ. Now setting θ = π/n gives cosec (π/n) n. ( ) Setting θ = π/n in equation ( ) and comparing with ( ) (remembering that there is a factor of 1/2 missing) gives the required result. Postmortem Now that I have had another go at this question it does not seem terribly interesting. At first, I thought I might ditch it. Then I thought that it perhaps was worthwhile: keeping cool under the pressure of the differentiation for part (ii) it should just be a few careful lines and getting it out is a good confidence booster. Anyway, now that I have slogged 16 through it, I am going to leave it in. (Next day) I recall now that the reason that I included this question in the first place was because in its original form (with approximate answers), the result for part (ii) can be obtained very quickly from part (i) by differentiating with respect to π. The snag with this is that it is not obviously going to work, because the derivatives of terms such as sin π that are omitted from (i) might feature in (ii). It is disappointing but not surprising that this trick cannot be used to obtain the exact formulae. 16 I should come clean at this point: I just differentiated ( ) on autopilot, without considering whether I could simplify the task by rewriting the formula as suggested in my hint on the previous page. It was hard work. Serves me right for not following my own general advice.
74 Question 29( ) Consider the cubic equation where p 0 and r 0. x 3 px 2 + qx r = 0, (i) If the three roots can be written in the form ak 1, a and ak for some constants a and k, show that one root is q/p and that q 3 rp 3 = 0. (ii) If r = q 3 /p 3, show that q/p is a root and that the product of the other two roots is (q/p) 2. Deduce that the roots are in geometric progression. (iii) Find a necessary and sufficient condition involving p, q and r for the roots to be in arithmetic progression. Comments The Fundamental Theorem of Algebra says that a polynomial of degree n can be written as the product of n linear factors, so we can write x 3 px 2 + qx r = (x α)(x β)(x γ), ( ) where α, β and γ are the roots of the equation x 3 px 2 + qx r = 0. The basis of this question is the comparison between the left hand side of ( ) and the right hand side, multiplied out, of ( ). Some of the roots may not be real, but you don t have to worry about that here. The necessary and sufficient in part (iii) looks a bit forbidding, but if you just repeat the steps of parts (i) and (ii), it is straightforward.
75 Solution to question 29 (i) We have i.e. (x ak 1 )(x a)(x ak) x 3 px 2 + qx r x 3 a(k k)x 2 + a 2 (k k)x a 3 = x 3 px 2 + qx r. Thus p = a(k k), q = a 2 (k k), and r = a 3. Dividing gives q/p = a, which is a root, and q 3 /p 3 = a 3 = r as required. (ii) Set r = q 3 /p 3. Substituting x = q/p into the cubic shows that it is a root: (q/p) 3 p(q/p) 2 + q(q/p) (q/p) 3 = 0. Since q/p is one root, and the product of the three roots is q 3 /p 3 (= r in the original equation), the product of the other two roots must be q 2 /p 2. The two roots can therefore be written in the form k 1 (q/p) and k(q/p) for some number k, which shows that they are in geometric progression. (iii) The roots are in arithmetic progression if and only if they are of the form (a d), a and (a+d). If the roots are in this form, then the equation is ( x (a d) )( x a )( x (a + d) ) x 3 px 2 + qx r i.e. x 3 3ax 2 + (3a 2 d 2 )x a(a 2 d 2 ) x 3 px 2 + qx r. Thus p = 3a, q = 3a 2 d 2, r = a(a 2 d 2 ). A necessary condition is therefore r = (p/3)(q 2p 2 /9). Note that one root is p/3. Conversely, if r = (p/3)(q 2p 2 /9), the equation is x 3 px 2 + qx (p/3)(q 2p 2 /9) = 0. We can verify that p/3 is a root by substitution. Since the roots sum to p and one of them is p/3, the others must be of the form p/3 d and p/3 + d for some d. They are therefore in arithmetic progression. A necessary and sufficient condition is therefore r = (p/3)(q 2p 2 /9). Postmortem As usual, it is a good idea to give a bit of thought to the conditions given, namely p 0 and r 0. Clearly, if the roots are in geometric progression, we cannot have a zero root. We therefore need r 0. However, we don t need the condition p 0. If the roots are in geometric progression with p = 0, then q = 0, but there is no contradiction: the roots are be 2πi/3, b and be 2πi/3 where b 3 = r, and these are certainly in geometric progression. So the necessary and sufficient conditions are r 0 and p 3 r = q 3. Neither of these conditions is required if the roots are in arithmetic progression. Therefore, the condition p 0 is only there as a convenience one thing less for you to worry about. The condition r 0 should really have been given only for the first part. It should be said that this sort of question is incredibly difficult to word, which is why the examiner was a bit heavy handed with the conditions.
76 Question 30( ) (i) Let f (x) = (1 + x 2 )e x k, where k is a given constant. Show that f (x) 0 and sketch the graph of f (x) in the cases k < 0, 0 < k < 1 and k > 1. Hence, or otherwise, show that the equation (1 + x 2 )e x k = 0, has exactly one real root if k > 0 and no real roots if k 0. (ii) Determine the number of real roots of the equation (e x 1) k tan 1 x = 0 in the different cases that arise according to the value of the constant k. Note: If y = tan 1 x, then dy dx = x 2. Comments In good STEP style, this question has two related parts. In this case, the first part not only gives you guidance for doing the second part, but also provides a result that is useful for the second part. In the original question you were asked to determine the number of real roots in the cases (a) 0 < k 2/π and (b) 2/π < k < 1. I thought that you would like to work out all the different cases for yourself but it makes the question considerably harder, especially if you think about the special values of k as well as the ranges of k. For part (i) you need to know that xe x 0 as x. This is just a special case of the result that exponentials go to zero faster than any power of x. For part (ii), it is not really necessary to do all the sketches, but I think you should (though you perhaps wouldn t under examination conditions) because it gives you a complete understanding of the way the function depends on the parameter k. For the graphs, you may want to consider the sign of f (0); this will give you an important clue. Remember that, by definition, π/2 < tan 1 x < π/2.
77 Solution to question 30 (i) Differentiating gives f (x) = 2xe x + (1 + x 2 )e x = (1 + x) 2 e x which is non-negative (because the square is non-negative and the exponential is positive. There is one stationary point, at x = 1, which is a point of inflection, since the gradient on either side is positive. Also, f (x) k as x, f (0) = 1 k and f (x) as x +. The graph is essentially exponential, with a hiccup at x = 1. The differences between the three cases are the positions of the horizontal asymptote (x ) and the place where the graph cuts the y axis (above or below the x axis?). Since the graph is increasing and 0 < f (x) <, the given equation has one real root if k > 0 and no real roots if k 0. k < 0 0 < k < 1 k > 1 (ii) First we collect up information required to sketch the graph, as in part (i). We have f (0) = 0, f (x) kπ/2 1 as x, f (x) as x, f (x) = e x k(1 + x 2 ) 1 (which we know from the first part can only be zero only if k > 0), f (0) = 1 k. The critical values of k are 0, 2/π and 1 so we will have to consider four cases. k < 0 0 < k < 2/π k = 2/π 2/π < k < 1 k = 1 k > 1 As you see, the equation (e x 1) k tan 1 x = 0 has one root if k π/2 and two roots otherwise. The case k = 1 can be thought of as having one root at x = 0 or as having two roots, both at x = 0 (it is a double root).
78 Question 31( ) For each positive integer n, let a n = 1 n (n + 1)(n + 2) + 1 (n + 1)(n + 2)(n + 3) + ; b n = 1 n (n + 1) (n + 1) 3 +. (i) Show that b n = 1/n. (ii) Deduce that 0 < a n < 1/n. (iii) Show that a n = n!e [n!e], where [x] is the integer part of x. (iv) Hence show that e is irrational. Comments Each part of this looks horrendously difficult, but it doesn t take much thought to see what is going on. If you haven t come across the concept of integer part some examples should make it clear: [21.25] = 21, [π] = 3, [ ] = 1 and [2] = 2. For part (iii), you need to know the series for the number e, namely n=0 1/n!. If you are stumped by the last part, just remember the definition of the word irrational: x is rational if and only if it can be expressed in the form p/q where p and q are integers; if x is not rational, it is irrational. Then look for a proof by contradiction ( Suppose that x is rational... ). The proof that e is irrational was first given by Euler (1736). He also named the number e, though he didn t invent it; e is the basis for natural logarithms and as such was used implicitly by John Napier in In general, it is not easy to show that numbers are irrational. Johann Lambert showed that π is irrational in 1760 but it is not known if π + e is irrational.
79 Solution to question 31 (i) Although the series for b n does not at first sight look tractable, it is in fact just a geometric progression: the first term is 1/n + 1 and the common ratio is also 1/n + 1. Thus b n = 1 ( ) 1 = 1 n /(n + 1) n. (ii) Each term (after the first) of a n is less than the corresponding term in b n, so a n < b n = 1/n. (iii) Mulitiplying the series for e by n! gives n!e = n! + n! + n!/ a n and the result follows because a n < 1 and all the other terms on the right hand side of the above equation are integers. (iv) We use proof by contradiction. Suppose that there exist integers k and m such that e = k/m. Then m!e is certainly an integer. But if m!e is an integer then [m!e] = m!e, which contadicts the result of part (iii) since we know that a m 0 (it is obvious from the definition that a m > 0).
80 Question 32( ) To nine decimal places, log 10 2 = and log 10 3 = (i) Calculate log 10 5 and log 10 6 to three decimal places. By taking logs, or otherwise, show that < < Hence write down the first digit of (ii) Find the first digit of each of the following numbers: ; ; and Comments This nice little question shows why it is a good idea to ban calculators from some mathematics examinations. When I was at school, before calculators were invented, we had to spend quite a lot of time in year 8 (I think) doing extremely tedious calculations by logarithms. We were provided with a book of tables of four-figure logarithms and the book also had tables of values of trigonometric functions. When it came to antilogging, to get the answer, we had to use the tables backwards, since there were no tables of inverse logarithms (exponentials). A logarithm consists of two parts: the characteristic which is the number before the decimal point and the mantissa which is the number after the decimal point. It is the mantissa that gives the significant figures of the number that has been logged; the characteristic tells you where to put the decimal point. This question is all about calculating mantissas. The characteristic of a number greater than 1 is non-negative but the characteristic of a number less than 1 is negative. The rules for what to do in the case of a negative characteristic were rather complicated: you couldn t do ordinary arithmetic because the logarithm consisted of a negative characteristic and a positive mantissa. In ordinary arithmetic, the number 3.4 means whereas the corresponding situation in logarithms, normally written 3.4, means Instead of explaining this, the teacher gave a complicated set of rules, which just had to be learned not the right way to do mathematics. There are no negative characteristics in this question, I m happy to say.
81 Solution to question 32 (i) For the very first part, we have log 2 + log 5 = log 10 = 1 so log 5 = = to 3 d.p. log 6 = log 2 + log 3 = to 3 d.p. Taking logs preserves the inequalities, so we need to show that i.e. that 47 + log 5 < 100log 3 < 47 + log < < which is true. The first digit of is therefore 5. (ii) To find the first digit of these numbers, we use the method of part (i). We have log = 1000log 2 = = (to 3 d.p.). But log 1 < < log 2, so the first digit of is 1. Similarly, log = 10000log 2 = But log 1 < < log 2, so the first digit of is 1. Finally, log = (4 d.p.) and log 9 = 2log 3 = 0.95 (2 d.p.), so the first digit of is 9. Postmortem Although the ideas in this question are really quite elementary, you needed to understand them deeply. You should feel pleased with yourself it you got this one out.
82 Question 33( ) For any number x, the largest integer less than or equal to x is denoted by [x]. For example, [3.7] = 3 and [4] = 4. (i) Sketch the graph of y = [x] for 0 x < 5 and evaluate 5 0 [x] dx. (ii) Sketch the graph of y = [e x ] for 0 x < lnn, where n is an integer, and show that ln n 0 [e x ]dx = n ln n ln(n!). Hence show that n! n n e 1 n. Comments Again, I have had to add a bit to the original question because it was all dressed up with nowhere to go. The question is clearly about estimating n! so I added in the last line (which makes the question a bit longer but not much more difficult). I could have added another part, but I thought that you would probably add it yourself: you will easily spot, once you have drawn the graphs, that a similar result for n!, with the inequality reversed, can be obtained by considering rectangles the tops of which are above the graph of e x instead of below it. You therefore end up with a nice sandwich inequality for n!.
83 Solution to question 33 (i) y y = x x The graph of [x] is an ascending staircase with 5 stairs, the lowest at height 0, each of width 1 unit and rising 1 unit. The integral is the sum of the areas of the rectangles shown in the figure: area = = 10. (ii) y = e x y x The graph of [e x ] is a also staircase: the height of each stair is 1 unit and the width decreases as x increases (because the gradient of e x increases). It starts at height 1 and ends at height (n 1). The value of [e x ] changes from k to k+1 when e x = k i.e. when x = ln(k + 1). Thus [e x ] = k when ln k x < ln(k + 1) and the area of the corresponding rectangle is k(ln(k + 1) ln k). The total area under the curve is therefore 1(ln 2 ln 1) + 2(ln 3 ln 2) + + (n 1)(ln n ln(n 1)) i.e. n ln n ln n ln(n 1) ln(n 2) ln 1 which sums to n ln n ln(n!) as required. For the last part, note that [e x ] e x (this is clear from the definition) so lnn 0 [e x ]dx lnn Taking exponentials gives the required result. Postmortem 0 e x dx = n 1 i.e. n ln n ln(n!) n 1. Considering [e x + 1] gives rectangles above the graph of y = e x rather than below. The calculations are roughly the same, so you should easily arrive at n! n n+1 e 1 n. We have therefore proved that n n e 1 n n! n n+1 e 1 n. This is a pretty good (given the rather elementary method at our disposal) approximation for n! for large n. Stirling ( ) proved in 1730 that n! 2π n n+1 2e n for large n. This was a brilliant result even reading his book, it is hard to see where he got 2π from (especially as he wrote in Latin) but he went on to obtain the approximation in terms of an infinite series. The expression above is just the first term.
84 Question 34( ) Given that tan π 4 = 1 show that tan π 8 = 2 1. Let I = 1 Show, by using the change of variable x = sin 4t, that 1 1 (1 + x) + (1 x) + 2 dx. I = π/8 0 2cos 4t cos 2 t dt. Hence show that I = 4 2 π 2. Comments This tests trigonometric manipulation and integration skills. You will certainly need tan 2θ in terms of tan θ, cos 2θ in terms of cos θ, and maybe other formulae. A good idea is required in the first part to deal with 1 ± sin4t look away if you want to find it for yourself try writing 1 as cos 2 2t + sin 2 2t. Both parts of the question are what is called multistep: there are half a dozen consecutive steps, each different in nature, with no guidance. This sort of thing is unusual at A-level but normal in university course mathematics. In this question, the most steps require trigonometric manipulation similar to that appearing in earlier parts of the question. I checked the answer on which turned out to be very good indeed at doing this sort of thing. I asked it to do the indefinite integral and, in less than a second, it came up with 1 x [ 1 ( x ) 1 ] 2arcsin (x + 1)/2 + [ x ] 1. Not a pretty sight and not in its neatest form by a long way (the arcsin term reduces, after a bit of algebra, to π/2 + arcsin x(!)). I also asked it to do the same integral with the 2 replaced by a which it did in less than four seconds. The answer was about 20 times longer than the above result. It seemed to enjoy the problem, as far as I could tell. I think that it does it by multiplying top and bottom of the fraction by suitable expressions (maybe (1+x) (1 x)+2 to start with) until the square roots have been removed from the denominator.
85 Solution to question 34 Let t = tan π 8. Then 2t 1 t 2 = 1 t2 + 2t 1 = 0 t = 2 ± 8 2 = 1 ± 2. We take the root with the + sign since we know that t is positive. Now the integral. Substituting x = sin 4t gives: I = π/8 π/8 4cos 4t π/8 dt = sin4t + 1 sin 4t cos 4t (cos 2t + sin2t) + (cos 2t sin 2t) + 2 dt, using 1 ± sin4t = 1 ± 2sin 2t cos 2t = cos 2 2t + sin 2 2t ± 2sin(t/2)cos 2t = (cos 2t ± sin2t) 2 and remembering to take the positive square root (cos 2t sin 2t for values of t in the range of integration). One more trig. manipulation gives the required result. Now so cos 4t = 2cos 2 t 1 = 2(2cos 2 t 1) = 8cos 4 t 8cos 2 t + 1 I = 2 π/8 which gives the required result. Postmortem 0 ( 8cos 2 t 4 + sec 2 t)dt = [ ] π/8 4sin 2t 8t + tan t, 0 Manipulating the integral after the change of variable was really quite demanding. If you didn t think of the idea referred to in the comments for dealing with 1 ± sin 4t you could of course have verified that 1 + sin 4t + 1 ± sin4t + 2 = 4cos 2 t, though even this would require some thought: you would have to subtract 2 from both sides before you squared them, and then you would have to worry about signs, since square-rooting the squares would lead to a ±. It occurs to me that there ought to be a better way of doing the integral, without at any stage spoiling the x x symmetry of the integrand; maybe I ll think again about this one day.
86 Question 35( ) (i) Show that, for 0 x 1, the largest value of What is the smallest value? x 6 (x 2 + 1) 4 is (ii) Find constants A, B, C and D such that, for all x, 1 (x 2 + 1) 4 d dx ( Ax 5 + Bx 3 + Cx (x 2 + 1) 3 ) + Dx6 (x 2 + 1) 4. (iii) Hence, or otherwise, prove that (x 2 dx + 1) Comments You should think about part (i) graphically, though it is not necessary to draw the graph: just set about it as if you were going to (starting point, finishing point, turning points, etc). The equivalence sign in part (ii) indicates an equality that holds for all x you are not being asked to solve the equation for x. After doing the differentiation, you obtain four linear equations for A, B, C and D by equating coefficients of powers of x. You could instead put four carefully chosen values of x into the equation; one good choice would be x = i to eliminate terms with factors of x and setting x = 0 would be useful; after that, it becomes more difficult. Note that there are only odd powers of x inside the derivative. You could include even powers as well, but you would find that their coefficients would be zero: the derivative has to be an even function (even powers of x) so the function being differentiated must be odd. For part (iii), you need to know that inequalities can be integrated: this is obvious if you think about integration in terms of area, though a formal proof requires a formal definition of integration and this is the sort of thing you would do in a first university course in mathematical analysis. This question is about estimating an integral. Although the idea is good, the final result is a bit feeble. It only gives the value of the integral to an accuracy of about (1/16)(11/24) which is about 15%. The actual value of the integral can be found fairly easily using the substitution x = tan t and is 11/48 + 5π/64 so the inequalities give (not particularly good) estimates for π namely (2.933 π 3.733).
88 Question 36( ) Show that 1 xe x dx = 1 and hence evaluate the integral. Evaluate the following integrals: (i) 4 0 x 3 2x 2 x + 2 dx; 0 1 xe x dx xe x dx (ii) π π sin x + cos x dx. Comments The very first part shows you how to do this sort of integral (with mod signs) by splitting up the range of integration at the places where the integrand changes sign. In parts (i) and (ii) you have to use the technique on different examples. This learning process and the nice mixture of ideas make it an ideal STEP question.
90 Question 37( ) A number of the form 1/N, where N is an integer greater than 1, is called a unit fraction. Noting that 1 2 = and 1 3 = , guess a general result of the form 1 N = 1 a + 1 ( ) b and hence prove that any unit fraction can be expressed as the sum of two distinct unit fractions. By writing ( ) in the form (a N)(b N) = N 2 and by considering the factors of N 2, show that if N is prime, then there is only one way of expressing 1/N as the sum of two distinct unit fractions. Prove similarly that any fraction of the form 2/N, where N is prime number greater than 2, can be expressed uniquely as the sum of two distinct unit fractions. Comments Fractions written as the sum of unit fractions are called Egyptian fractions: they were used by Egyptians. The earliest record of such use is 1900BC. The Rhind papyrus in the British Museum gives a table of representations of fractions of the form 2/n as sums of unit fractions for all odd integers n between 5 and 101 a superb achievement when you consider that algebra was years away. It is not clear why Egyptians represented fractions this way; maybe it just seemed a good idea at the time. Certainly the notation they used, in which for example 1/n was denoted by n with an oval on top, does not lend itself to generalisation to fractions that are not unit. One of the rules was that all the unit fractions in an Egyptian fraction should be distinct, so 2 had to be expressed 7 as , which seems pretty daft. 28 It is not obvious that every fraction can be express in Egyptian form; this was proved by Fibonacci in There are however still many unsolved problems relating to Egyptian fractions. Egyptian fractions have been called a wrong turn in the history of mathematics; if so, it was a wrong turn that favoured style over utility which is no bad thing, in my opinion.
91 Solution to question 37 1 A good guess would be that the first term of the decomposition of N is 1. In that case, the N + 1 other term is 1 N 1 N + 1 i.e. 1. That proves the result that every unit fraction can be N(N + 1) expressed as the sum of two unit fractions. The only factors of N 2 (since N is prime) are 1, N and N 2. The possible factorisations of N 2 are therefore N 2 = 1 N 2 or N 2 = N N so, since a b, a N = 1 and b N = N 2 (or the other way round). Thus N + 1 and N 2 + N are the only possible values for a and b and the decomposition is unique. For the second half, set where a b. Proceeding as before, we have ab ( a N )( b N ) = N N = 1 a + 1 b (a + b)n 2 = 0 which we write as i.e. (2a N)(2b N) = N 2. Thus 2a N = N 2 and 2b N = 1 (or the other way round). The decomposition is therefore unique and given by 2 N = 1 (N 2 + N)/2 + 1 (N + 1)/2. The only possible fly in the ointment is the 1/2 in the denominators: a and b are supposed to be integers. However, all prime numbers greater than 2 are odd, so N + 1 and N 2 + N are both even and the denominators are indeed integers. Postmortem Another way of getting the last part (less systematically) would have been to notice that 1 N = 1 (N + 1) + 1 N(N + 1) = 2 N = 1 (N + 1)/2 + 1 N(N + 1)/2 which gives the result immediately. How might you have noticed this? Well, I noticed by trying to work out some examples, starting with what is given at the very beginning of the question. If 1 3 = then 2 3 = which works. 12 Of course, the advantage of the systematic approach is that it allows generalisation: what happens if N is odd but not prime; what happens if the numerator is 3 not 2?
92 Question 38( ) Prove that if (x a) 2 is a factor of the polynomial p(x), then p (a) = 0. Prove a corresponding result if (x a) 4 is a factor of p(x). Given that the polynomial has a factor of the form (x a) 4, find k and m. x 6 + 4x 5 5x 4 40x 3 40x 2 + kx + m Comments The very first part has an obvious feel to it but the difficulty lies in setting out a clear mathematical proof. The key is the factor theorem: if (x a) is a factor of a polynomial, then it can be written as the product of polynomial and (x a). Please take some care over the presentation of next part. There are a few numerical calculations to do. Since calculators are neither permitted nor needed in STEP, it is probably best if you don t use yours. In the original question, k was given (it is 32). But this seemed rather unsatisfactory because it meant that not all the information available is required to solve the problem. Even my new improved version is not perfect in this respect because p (a) = 0 is used only to distinguish between three possibilities for a.
94 Question 39( ) Show that where t = tan(θ/2). Let sin θ = 2t 1 t2, cos θ = 1 + t2 1 + t 2, 1 + cos θ sin θ I = π cos α sin θ dθ. Use the substitution t = tan(θ/2) to show that, for 0 < α < π/2, I = (t + cos α) 2 + sin 2 α dt. By means of the further substitution t + cos α = sinαtan u show that I = α sin α. = tan(π/2 θ/2), Deduce a similar result for π sin α cos φ dφ. Comments The first of the two substitutions is familiarly known as the t substitution. It would have been very standard fare 30 years ago, but seems to have gone out of fashion now. The second is the normal substitution for integrals with quadratic denominators and is only just over the edge of the core A-level syllabus. For the last part, deduce implies that you don t have to do any further integration. This can only really mean a change of variable to make the integrand into the one you want; you have to hope that the limits then transform to the one given. Note that the change of variable is signalled by the use of the variable φ in the last integral; since it is a definite integral, it doesn t matter what the variable is called, and could have been called θ as I ; the use of a different variable was just a kindness on the part of the examiner (me). Of course, you can change α without any such complications. The similar result result should include the conditions under which it is true. It is worth thinking about why the condition 0 < α π/2 is required or indeed, if it is required.
95 Solution to question 39 The three identities just require use of the half-angle formulae cos θ = cos 2 (θ/2) sin 2 (θ/2) and sin θ = 2 sin(θ/2) cos(θ/2). Remember, for the last one, that cot x = tan(π/2 x). For the first change of variable, we have dt = 1 2 sec2 (θ/2)dθ = 1 2 (1 + t2 )dθ and the new limits are 0 and 1, so I = π cos α sin θ dθ = (cos α)2t/(1 + t 2 ) t 2 dt = t cos α + t 2 dt = 2 0 (t + cos α) 2 + sin 2 α dt. For the second change of variable, we have dt = sinαsec 2 udu. When t = 0, tan u = cot α so u = π/2 α. When t = 1, sin αtan u = 1 + cos α so u = π/2 α/2. Thus I = π/2 α/2 π/2 α 2 sin 2 α(1 + tan 2 u) sinαsec2 udu = π/2 α/2 π/2 α 2 sin α du = 2 [ ] π/2 α/2 u = α sin α π/2 α sinα. For the last part, we want to make a change of variable the changes the cosine in the denominator to a sine. One possibility is to set φ = π/2 θ. This will swaps the limits but also and introduces a minus sign since dθ = dφ. Thus α sin α = I = 0 π 2 π cos α cos φ dφ = cos αcos φ dφ. This is almost the integral we want: we still need to replace cos α in the denominator by sin α. Remembering what we did a couple of lines back, we just replace α by π/2 α in the integral and in the answer, giving Postmortem π π/2 α dφ = 1 + cos αcos φ cos α. The restriction on α in the integral might prevent the denominator being zero for some value of θ ; a zero in the denominator usually means that the integral is undefined. However, the only value of α for which cos α sin θ could possibly be as small as 1 (for 0 θ π/2) is π (and of course 3π, etc). From the point of view of the integral, the only restriction should be α π (etc). There is a slight awkwardness in the answer when α = 0 but this can be overcome by taking limits: α lim α 0 sin α = 1 which you can easily verify is the correct answer in this case. You may by now have noticed a curious thing: increasing α by 2π does not change I but does change the answer. If you work through the solution with this in mind you see that the only step where it can make a difference is in the line where you work out tan 1 u, which by definition lies in the range π/2 to π/2. It is this that determines the given restriction. (Note that negative values of α are not required because setting α α changes neither the integral nor the given solution.)
96 Question 40( ) The line l has vector equation r = λs, where s = (cos θ + 3) i + ( 2 sin θ) j + (cos θ 3) k and λ is a scalar parameter. Find an expression for the angle between l and the line r = µ(ai + bj + ck). Show that there is a line m through the origin such that, whatever the value of θ, the acute angle between l and m is π/6. A plane has equation x z = 4 3. The line l meets this plane at P. Show that, as θ varies, P describes a circle, with its centre on m. Find the radius of this circle. Comments It is not easy to set vector questions at this level: they tend to become merely complicated (rather than difficult in an interesting way). Usually, there is some underlying geometry and it pays to try to understand what this is. Here, the question is about the geometrical object traced out by l as θ varies. You will need to know about scalar products of vectors for this question, but otherwise it is really just coordinate geometry. Vectors form an extremely important part of almost every branch of mathematics (maybe every branch of mathematics) and will probably be one of the first topics you tackle on your university course.
97 Solution to question 40 Both l and the line r = µ(ai + bj + ck) pass through the origin, so the angle α between the lines is given by the scalar product of the unit vectors, i.e. the scalar product between the given vectors divided by the product of the lengths of the two vectors: cos α = a(cos θ + 3) + 2bsin θ + (cos θ 3)c [(cos θ + 3) 2 + 2sin 2 θ + (cos θ 3) 2 ] [a 2 + b 2 + c 2 ] = (a + c)cos θ + 2bsin θ + (a c) [a 2 + b 2 + c 2. ] Now we want to show that there is some choice of a, b and c such that cos α does not depend on the value of θ. By inspection, this requires a = c and b = 0. For these values, the terms depending on θ disappear from the expression for cos α leaving cos α = ± according to the whether a is positive or negative. If we choose a = 1 then cos α = and α = π/6 as required. The line m is given by r = µ(i k). Note that as θ varies, the line l generates a cone with axis m; this is what the question is about. The coordinates of a general point on the line l are x = λ(cos θ + 3) y = λ 2sin θ z = λ(cos θ 3). For a point which is also on the plane x z = 4 3 we have λ(cos θ + 3) λ(cos θ 3) = 4 3 so λ = 2. The point P on the intersection between the line and the plane has coordinates (2cos θ + 2 3, 2 2sin θ, 2cos θ 2 3). A general point on the line m has coordinates x = µ, y = 0, z = µ. It meets the plane x z = 4 3 when µ = 2 3 i.e. at the point O with coordinates (2 3, 0, 2 3). We need to check that, as θ varies, P moves round a circle with centre O, i.e. that the length OP is independent of θ. We have OP 2 = (2cos θ) 2 + (2 2sin θ) 2 + (2cos θ) 2 = 8, which is indeed independent of θ as required. The radius of the circle is therefore 2 2.
98 Question 41( ) Given that x 4 + px 2 + qx + r (x 2 ax + b)(x 2 + ax + c), express p, q and r in terms of a, b and c. Show that a 2 is a root of the cubic equation u 3 + 2pu 2 + (p 2 4r)u q 2 = 0. Verify that u = 9 is a root in the case p = 1, q = 6, r = 15. Hence, or otherwise, solve the equation y 4 8y y 2 34y Comments The long-sought solution of the general cubic was found, in 1535 by Niccolò Tartaglia (c ). He was persuaded to divulge his secret (in the form of a poem) by Girolamo Cardano ( ), who promised not to publish it before Cardano did. However, Cardano discovered that it had previously been discovered by del Ferro ( /6) before 1515 so he published it himself in his algebra book The Great Art. There followed an acrimonious dispute between Tartaglia and Cardano, in which the latter was championed by his student Ferrari ( ). The dispute culminated in a public mathematical duel between Farrari and Tartaglia held in the church of Santa Maria in Milan in 1548, in which they attempted to solve each others cubics. The duel ended in a shouting match with Tartaglia storming off. It seems Ferrari was the winner. Tartaglia was sacked from his job as lecturer and Ferrari made his fortune as a tax assessor before becoming a professor of mathematics at Bologna. He was poisoned by his sister, with arsenic, in Ferarri found the a way of reducing quartic equations to cubic equations; his method (roughly) is used in this question to solve a quartic which could probably be solved easier otherwise. But it is the method that is interesting, not the solution. The first step of the Ferrari method is to reduce the general quartic to a quartic equation with the cubic term missing by means of a linear transformation of the form x x a. Then this reduced quartic is factorised (the first displayed equation in this question). The factorisation can be found by solving a cubic equation (the second displayed equation above) that must be satisfied by one of the coefficients in the factorised form. The solutions of the quartic are all complex, but don t worry if you haven t come across complex numbers: you will be able to do everything except perhaps write down the last line. You will no doubt be full of admiration for this clever method of solving quartic equations. One thing you are bound to ask yourself is how the other two roots of the cubic equation fit into the picture. In fact, the cubic equation gives (in general) 6 distinct values of a so there is quite a lot of explaining to do, given that the quartic has at most 4 distinct roots.
99 Solution to question 41 We have so (x 2 ax + b)(x 2 + ax + c) = (x 2 ax)(x 2 + ax) + b(x 2 + ax) + c(x 2 ax) + bc = x 4 + (b + c a 2 )x 2 + a(b c)x + bc p = b + c a 2, q = a(b c), r = bc. ( ) To obtain an equation for a in terms of p, q and r we eliminate b and c from ( ) using the identity (b + c) 2 = (b c) 2 + 4bc. This gives (p + a 2 ) 2 = (q/a) 2 + 4r which simplifies easily to the given cubic with u replaced by a 2. For the purposes of this equation, all that is required is to verify that setting u = a 2 in the given cubic equation works (on use of ( )), but this is probably no easier than constructing the equation by elimination, as above. We can easily verify that u = 9 satisfies the given cubic by direct substitution. To solve the quartic equation, the first task is to reduce it to the form x 4 + px 2 + qx + r, which has no term in x 3. This is done by means of a translation. Noting that (y a) 4 = y 4 4ay 3 +, we set x = y 2. This gives (x + 2) 4 8(x + 2) (x + 2) 2 34(x + 2) which (not surprisingly) boils down to x 4 x 2 6x + 15, so that p = 1, q = 6 and r = 15. We have already shown that one root of the cubic corresponding to these values is u = 9. Thus we can achieve the factorisation of the quartic into two quadratic factors by setting a = 3 (or a = 3 it doesn t matter which we use) and Thus b = 1 ( p + a 2 + q ) = 3 2 a c = 1 ( p + a 2 q ) = 5. 2 a x 4 x 2 6x + 15 = (x 2 3x + 3)(x 2 + 3x + 5). Setting each of the two quadratic factors equal to zero gives so x = 3 ± i 3 2 y = 7 ± i 3 2 and x = 3 ± i 11 2 or y = 1 ± i 11 2., Postmortem Did you work out the relation between the six possible values of a (corresponding to given values of p, q and r) and the roots of the quartic? The point is that the quartic can be written as the product of four linear factors (by the fundamental theorem of algebra) and there are six ways of grouping the linear factors into two quadratic factors. Each way corresponds to a value of a.
100 Question 42( ) Show that tan 3θ = 3tan θ tan3 θ 1 3tan 2 θ Given that θ = cos 1 (2/ 5) and 0 < θ < π/2, show that tan 3θ = Hence, or otherwise, find all solutions of the equations (i) tan(3cos 1 x) = 11 2, (ii) cos ( 1 3 tan 1 y ) = Comments Perhaps three stars is a bit over the top for this question, but I thought it was tricky. Very few candidates got more than half marks in the examination (Paper I, 2001). It is not hard to see what you have to do, but great care is required when writing out the argument. The important thing to remember is that if (say) tan A = tan B then you cannot say that A = B ; rather, you must say A = B + nπ. For the very first part, you might wonder where to start. This did not trouble the candidates in the examination: they all quoted the formula for tan(α + β) and worked from there.
102 Question 43( ) The curve C 1 passes through the origin in the x y plane and its gradient is given by dy dx = x(1 x2 )e x2. Show that C 1 has a minimum point at the origin and a maximum point at ( 1, 1 2 e 1). Find the coordinates of the other stationary point. Give a rough sketch of C 1. The curve C 2 passes through the origin and its gradient is given by dy dx = x(1 x2 )e x3. Show that C 2 has a minimum point at the origin and a maximum point at (1,k), where k > 1 2 e 1. (You need not find k.) Comments No work is required to find the x coordinate of the stationary points, but you have to integrate the differential equation to find the y coordinate. For the second part, you cannot integrate the equation other than numerically, or in terms of rather obscure special functions that you almost certainly haven t come across, such as the incomplete gamma function, defined by Γ(x,a) = a 0 t x 1 e t dt. However, you can obtain an estimate, which is all that is required, by comparing the gradients of C 1 with C 2 and thinking of the graphs for 1 x 1. This is perhaps a bit tricky; an idea that you may well not alight on under examination conditions.
104 Question 44( ) A spherical loaf of bread is cut into parallel slices of equal thickness. Show that, after any number of the slices have been eaten, the area of crust remaining is proportional to the number of slices remaining. A European ruling decrees that a parallel-sliced spherical loaf can only be referred to as crusty if the ratio of volume V (in cubic metres) of bread remaining to area A (in square metres) of crust remaining after any number of slices have been eaten satisfies V/A < 1. Show that the radius of a crusty parallel-sliced spherical loaf must be less than metres. [The area A and volume V formed by rotating a curve in the x y plane round the x-axis from x = a to x = a + t are given by A = 2π a+t a ( y 1 + )1 ( dy ) 2 dx 2 a+t dx, V = π y 2 dx. ] a Comments The first result came as a bit of a surprise to me though no doubt it is well known. I wondered if it was the only surface of revolution with this property. You might like to think about this. For the last part, you will need to minimise a ratio as a function of t (the length of loaf remaining which is of course proportional to the number of slices remaining if the slices are all the same thickness); to find the ratio you have to do a couple of integrals. It is this multi-stepping that makes the problem difficult (and very different from typical A-level questions) rather than any individual step.
105 Solution to question 44 The first thing we need is an equation for the surface of a spherical loaf. The obvious choice, especially given the hint at the bottom of the question, is x 2 + y 2 = a 2 (and z = 0), rotated about the x-axis. If the loaf is cut at a distance t from the end x = a, then the area remaining is 2π a+t a ( y 1 + )1 ( dy ) 2 dx ( 2 a+t ( dx = 2π (a 2 x 2 ) 1 x a (a 2 x 2 ) 1 2 = 2π a+t a = 2πat. adx ) 2 )1 2 dx This is proportional to the length t of remaining loaf, so proportional to the number of slices remaining (if the loaf is evenly sliced). The remaining volume is π a+t a y 2 dx = π a+t a (a 2 x 2 )dx = π(a 2 x x 3 /3) a+t a = π[(a 2 ( a + t) ( a + t) 3 /3] π[(a 2 ( a) ( a) 3 /3] = π(at 2 t 3 /3). As a quick check on the algebra, notice that this is zero when t = 0 and 4πa 3 /3 when t = 2a. Thus V/A = (3at t 2 )/(6a). This is a quadratic curve with zeros at t = 0 and t = 3a, so it has a maximum at t = 3a/2 (by differentiating or otherwise), and where V/A = 3a/8. Since we require this ratio to be less than one, we must have a < 8/3 metres.
106 Question 45( ) A tortoise and a hare have a race to the vegetable patch, a distance X kilometres from the starting post, and back. The tortoise sets off immediately, at a steady v kilometers per hour. The hare goes to sleep for half an hour and then sets off at a steady speed V kilometres per hour. The hare overtakes the tortoise half a kilometre from the starting post, and continues on to the vegetable patch, where she has another half an hour s sleep before setting off for the return journey at her previous pace. One and quarter kilometres from the vegetable patch, she passes the tortoise, still plodding gallantly and steadily towards the vegetable patch. Show that and find v in terms of X. V = 10 4X 9 Find X if the hare arrives back at the starting post one and a half hours after the start of the race. How long does it take the tortoise to reach the vegetable patch? Comments You might find it helpful to draw a distance time diagram. You might also find it useful to let the times at which the two animals meet be T 1 and T 2.
108 Question 46( ) A single stream of cars, each of width a and exactly in line, is passing along a straight road of breadth b with speed V. The distance between successive cars (i.e. the distance between back of one car and the front of the following car) is c. A chicken crosses the road in safety at a constant speed u in a straight line making an angle θ with the direction of traffic. Show that V a u csin θ + acos θ. ( ) Show also that if the chicken chooses θ and u so that it crosses the road at the least possible (constant) speed, it crosses in time b ( c V a + a. c) Comments I like this question because it relates to (an idealized version) of a situation we have probably all thought about. Once you have visualized it, there are no great difficulties. As usual, you have to be careful with the inequalities, though it turns out here that there is no danger of dividing by a negative quantity. When I arrived at ( ), I noticed that there is another inequality that might be relevant (did you?); but when I looked at the rest of the question I realized I had worried unnecessarily. Is the time given in the final part (which corresponds to the least possible speed) the longest time that the chicken can spend crossing the road?
109 Solution to question 46 Let t be the time taken to cross the distance a in which the chicken is at risk. Then a = ut sin θ. For safety, the chicken must choose ut cos θ V t c: this is the case where the chicken starts at the near-side rear of one car and just avoids being hit by the far-side front of the next car. Eliminating t from these two equations gives the required inequality: which is the required result ut cos θ V t c = (ucos θ V )t c a = (ucos θ V ) usin θ c = aucos θ av cusin θ = u(csin θ + acos θ) av For a given value of θ, the minimum speed satisfies u(csin θ + acos θ) = av. The smallest value of this u is therefore obtained when csin θ + acos θ is largest. This can be found by calculus (regard it as a function of θ and differentiate: the maximum occurs when tan θ = c/a) or by trigonometry: csin θ + acos θ = a 2 + c 2 cos (θ arctan(c/a)) so the maximum value is a 2 + c 2 and it occurs when tan θ = c/a. The time of crossing is Postmortem b b(csin θ + acos θ) b(c + acot θ) = = usin θ V asin θ V a There is another inequality besides ( ) that you should have noticed: ucos θt V t + c = b(c + a2 /c). V a otherwise the chicken will run into the back of the car ahead. This leads to (changing the sign of c and reversing the inequality in ( )) u( csin θ + acos θ) av. This looks potentially as if it might also give a lower bound on u (if ( c sin θ + a cos θ) < 0). But it turns out that this is not relevant. Of course, the chicken can take as long as it wants to cross the road just by going at an extremely acute angle at a speed marginally greater than V. (For maximum time, the minimum value of usin θ, not of u) is required.)
110 Question 47( ) 2d G P h d α A lorry of weight W stands on a plane inclined at an angle α to the horizontal. Its wheels are a distance 2d apart, and its centre of gravity G is at a distance h from the plane, and halfway between the sides of the lorry. A horizontal force P acts on the lorry through G, as shown. (i) (ii) If the normal reactions on the lower and higher wheels of the lorry are equal, show that the sum of the frictional forces between the wheels and the ground is zero. If P is such that the lorry does not tip over (but the normal reactions on the lower and higher wheels of the lorry need not be equal), show that where tan β = d/h. W tan(α β) P W tan(α + β), Comments There is not much more to this than just resolving forces and taking moments about a suitable point. You might think it a bit odd to have a force that acts horizontally through the centre of gravity of the lorry: it is supposed to be centrifugal. The first draft of the question was intended as a model of a lorry going round a bend on a cambered road. The idea was to relate the speed of the lorry to the angle of camber: the speed should be chosen so that there is be no tendency to skid. However, the wording was rather difficult, because motion in a circle is not included in the STEP syllabus, and the modelling part of the question was eventually abandoned. You can do the second part algebraically (using equations derived from resolving forces and taking moment), but there is a more direct approach. There was a slight mistake in the wording of the question, which I have not corrected here. Maybe you will spot it in the course of your solution. The diagram is correct.
111 Solution to question 47 Let the normal reactions at the lower and upper wheels be N 1 and N 2, respectively, and let the frictional forces at the lower and upper wheels be F 1 and F 2, respectively, (both up the plane). (i) Taking moments about G, we have so if N 1 = N 2 the sum of the frictional forces is zero. (N 1 N 2 )d = (F 1 + F 2 )h ( ) (ii) The condition for the lorry not tip over down the plane is N 2 0, which is the same as saying that the total moment of W and P about the point of contact between the lower wheel and the plane in the clockwise sense is positive. This gives one of the inequalities directly (after a bit of geometry to find the distance between the lie of action of the forces and the points of contact of the wheels), and the other follows similarly. Alternatively, resolving forces perpendicular to the plane and parallel to the plane gives N 1 + N 2 F 1 + F 2 = W cos α + P sinα = W sin α P cos α Multiplying the first equation by d and the second by h and adding (using ( ) gives 2dN 1 = W(hsin α + dcos α) + P( hcos α + dsin α) 0, i.e. W sin(α + β) P cos(α + β). ( ) Similarly, the first equation by d and the second by h and subtracting (using ( ) gives 2dN 2 = W( hsin α + dcos α) + P(hcos α + dsin α) 0, i.e P cos(α β) W sin(α β). ( ) Postmortem To obtain the required inequalities from ( ) and ( ) above, you have to divide by a cosine. As always with inequalities, you have to worry about the sign of anything you divide by. In this case, cos(α β) > 0, since both angles are acute. But what about cos(α + β)? If this is negative, then ( ) is satisfied with no constraint and it would be wrong to divide through the inequality without changing the direction of the inequality. Can cos(α + β) be negative? The answer, which was overlooked by the setter (me), is that it is negative if α + β > π/2. In this case, the line of action of P passes between the wheels of the lorry (not, as shown in the diagram, higher than the higher wheel). Thus increasing P will not cause to the lorry to tip up the slope and there is, in this case, no upper limit to P. An alternative way of obtaining the inequalities of the second part is to work with the resultant of P and W. The direction of this resultant force (given by tan 1 (W/P)) must be such that the line of action of the force passes between the wheels. The line of action makes an angle of tan 1 (P/W) with the vertical through G. The lines through G to points of contact of the upper wheel and the lower wheel with the plane make angles of β + α and α β, respectively, with the vertical through G. Hence the inequalities in part (ii).
112 Question 48( ) In a game of cricket, a fielder is perfectly placed to catch a ball. She watches the ball in flight takes the catch just in front of her eye. The angle between the horizontal and her line of sight at a time t after the ball is struck is θ. Show that d (tan θ) is constant during the flight. dt The next ball is also struck in the direction of the fielder but at a different velocity. In order to be perfectly placed to catch the ball, the fielder runs at constant speed towards the batsman. Assuming that the ground is horizontal, show that again d (tan θ) is constant during the flight. dt Comments I came across this while looking through some forty year old Entrance Examination papers. Candidates sat these papers in December and were called for interview if they did well enough. This sounds more efficient in terms of everyone s time than the current system in which all applicants are interviewed, but the disadvantage of implementing it now would be that students would have to sit the examination after only four terms in the sixth form. Forty years ago the situation was different because nearly all applicants to Oxford and Cambridge were post A-level. As with all the very best questions, 9/10 of this question is submerged below the surface. It uses the deepest properties of Newtonian dynamics, and a good understanding of the subject makes the question completely transparent. However, it can still be done without too much trouble by a straightforward approach: take the height above the fielder s eye-level at which the ball is struck to be h, and later eliminate h by writing it in terms of the flight time, the speed and angle of elevation at which the ball is hit. The cleverness of this question lies in its use of two fundamental invariances of Newton s law d 2 x = g. ( ) dt2 The first is invariance under what is called time reflection symmetry. The equation ( ) is invariant under the transformation t t, because replacing t by t does not affect the equation. This means that any given solution can be replaced by one where the projectile goes back along the trajectory. The second is invariance under what are called Galilean transformations. Equation ( ) is also invariant under the transformation x x + vt, where v is an arbitrary constant velocity. This means that we can solve the equation in a frame that moves with constant speed.
113 Solution to question 48 We take a straightforward approach. Let the height above the fielder s eye-level at which the ball is struck be h. Let the speed at which the ball is struck be u and the angle which the ball makes with the horizontal be α. Then, taking x to be the horizontal distance of the ball at time t from the point at which the ball was struck and y to be the height at of the ball at time t above the fielder s eye-level, we have x = (ucos α)t, y = h + (usin α)t 1 2 gt2. Let d be the horizontal distance of the fielder from the point at which the ball is struck, and let T be the time of flight of the ball. Then and tan θ = d = (ucos α)t, 0 = h + (usin α)t 1 2 gt 2 ( ) y d x = h + (usin α)t 1 2 gt2 d (ucos α)t = (usin α)t gt 2 + (usin α)t 1 2 gt2 (ucos α)t (ucos α)t = (usin α)(t t) g(t 2 t 2 ) (ucos α)(t t) (using ) = usinα + 1 2g(T + t). (cancelling the factor (T t)) ucos α This last expression is a polynomial of degree one in t, so its derivative is constant, as required. For the second part, let l be the distance from the fielder s original position to the point at which she catches the ball. Then l = vt and tan θ = y (l vt) + d x = y v(t t) + d x = usin α + 1 2g(T + t) v + ucos α cancelling the factor of (T t) as before. This again has constant derivative. Postmortem The invariance mentioned above can be used to answer the question almost without calculation. Using time reflection symmetry to reverse the trajectory shows that the batsman is completely irrelevant: it only matters that the fielder caught a ball. We just think of the ball being projected from the fielder s hands (the time-reverse of a catch). Taking her hands as the origin of coordinates, and using u to denote the final speed of the ball and α to be the final value of θ, we have y = (usin α)t 1 2 gt2, x = (ucos α)t and tan θ = (usin α 1 2gt)/ucos α. The first derivative of this expression is constant, as before. We use the Galilean transformation for the second part of the question. Instead of thinking of the fielder running with constant speed v towards the batsman, we can think of the fielder being stationary and the ball having an additional horizontal speed of v. The situation is therefore not changed from that of the first part of the question, except that ucos θ should be replaced by v + ucos θ.
114 Question 49( ) A rigid straight rod AB has length l and weight W. Its weight per unit length at a distance x from B is αwl 1 (x/l) α 1, where α is a constant greater than 1. Show that the centre of mass of the rod is at a distance αl/(α + 1) from B. The rod is placed with the end A on a rough horizontal floor and the end B resting against a rough vertical wall. The rod is in a vertical plane at right angles to the plane of the wall and makes an angle of θ with the floor. The coefficient of friction between the floor and the rod is µ and the coefficient of friction between the wall and the rod is also µ. Show that, if the equilibrium is limiting at both A and B, then tan θ = 1 αµ2 (1 + α)µ. Given that α = 3/2 and given also that the rod slides for any θ < π/4 find the greatest possible value of µ. Comments This is a pretty standard situation: a rod leaning against a wall, prevented from slipping by friction at both ends. The only slight variation is that the rod is not uniform; the only effect of this is to alter the position of the centre of gravity through which the weight of the rod acts. The question is not general, since the coefficient of friction is the same at both ends of the rod. I think that it is a good idea (at least, if you are not working under examination conditions) to set out the general equations (with µ A and µ B ), and at first no limiting friction (with frictional forces F A and F B ). This will help understand the structure of the equations and highlight the symmetry between the ends of the rod. You might wonder why the weight per unit length is given in such a complicated way; why not simply kx α? The reason is to keep the dimensions honest. The factor of W/l appears so that the dimension is clearly weight divided by length. Then x appears divided by l to make it dimensionless. Finally, the factor of α is required for the total weight of the rod is W Postmortem In the original version, the question spoke about limiting friction without saying that it was limiting at both ends. We then wondered whether the friction could be limiting at one end only. The answer is yes. The three equations that govern the system do not determine uniquely the four unknown forces (normal reaction and frictional forces at each end of the rod); extra information is required. If the angle is also to be determined, two extra pieces of information are required; in this case the information is that the friction is limiting at both ends. The same lack of determinism occurs in many statics problems. For example, the forces on the legs of a four-legged table cannot all be determined from the three available equations (vertical forces, moment about the x-axis and moments about the y-axis) only three legs are required for the table to stand (at least,in the case of a three-legged table with an added fourth leg), so extra information is needed to determine all the forces. This extra information might come from considerations of the internal structure of the table; otherwise, the way the table arrived at its current state might matter; for example, whether all four legs were placed on the floor at the same time.
115 Solution to question 49 The distance, x, of the centre of mass from the end B of the rod is given by x = l 0 αwl α x α dx l 0 αwl α x α 1 dx = l 0 xα dx (α + 1) 1lα+1 l = 0 xα 1 dx (α) 1 l α = αl α + 1. The response to the rest of this question should be preceded by an annotated diagram showing all relevant forces on the rod and with G nearer to A than B. This will help to clarify ideas. F B B R B The equations that determine the equilibrium are F B + R A = W F A R B = 0 G R θ A F A A W (vertical forces) (horizontal forces) R B x sinθ + F B xcos θ = R A (l x)cos θ F A (l x)sin θ (clockwise moments about G) If the rod is about to slip then the frictional and normal forces at A can be specified as F A = µr A and F B = µr B. Substituting in first two of the above questions gives R A = W µ 2 + 1, R B = µw µ 2 + 1, F A = µw µ 2 + 1, F B = µ2 W µ Substituting into the third (moments) equation gives the required result. For the final part, setting α = 3/2 gives tan θ = 2 3µ2 5µ If the rod slips for any angle less than θ = π/4, then the angle at which limiting friction occurs at both ends must be at least π/4. Therefore 2 3µ 2 5µ 1 i.e. 3µ 2 + 5µ 2 0, or (3µ 1)(µ + 2) < 0. The greatest possible value of µ is 1/3.
116 Question 50( ) N particles P 1, P 2, P 3,..., P N with masses m, qm, q 2 m,..., q N 1 m, respectively, are at rest at distinct points along a straight line in gravity-free space. The particle P 1 is set in motion towards P 2 with velocity V and in every subsequent impact the coefficient of restitution is e, where 0 < e < 1. Show that after the first impact the velocities of P 1 and P 2 are respectively. ( 1 eq 1 + q ) V and ( ) 1 + e V, 1 + q Show that if q e, then there are exactly N 1 impacts and that if q = e, then the total loss of kinetic energy after all impacts have occurred is equal to 1 2 me( 1 e N 1) V 2. Comments This situation models the toy called Newton s Cradle which consists of four or more heavy metal balls suspended from a frame so that they can swing. At rest, they are in contact in a line. When the first ball is raised and let swing, there follows a rather pleasing pattern of impacts. In this case, the coefficient of restitution is nearly 1 and the balls all have the same mass, so, as the first displayed formula shows, the impacting ball is reduced to rest by the impact. At the first swing of the ball, nothing happens except that the first ball is reduced to rest and the last ball swings away. Note that this is consistent with the balls being separated by a very small amount; what actually happens is that the ball undergoes a small elastic deformation at the impact, and the impulse takes a small amount of time to be transmitted across the ball to the next ball. You can order a Newton s Cradle (and lots of other interesting toys and games) online from
118 Question 51( ) An automated mobile dummy target for gunnery practice is moving anti-clockwise around the circumference of a large circle of radius R in a horizontal plane at a constant angular speed ω. A shell is fired from O, the centre of this circle, with initial speed V and angle of elevation α. Show that if V 2 < gr, then no matter what the value of α, or what vertical plane the shell is fired in, the shell cannot hit the target. Assume now that V 2 > gr and that the shell hits the target, and let β be the (positive) angle through which the target rotates between the time at which the shell is fired and the time of impact. Show that β satisfies the equation g 2 β 4 4ω 2 V 2 β 2 + 4R 2 ω 4 = 0. Deduce that there are exactly two possible values of β. Let β 1 and β 2 be the possible values of β and let P 1 and P 2 be the corresponding points of impact. By considering the quantities (β1 2 + β2 2 ) and β2 1 β2 2, or otherwise, show that the linear distance between P 1 and P 2 is ( ω ( 2R sin V 2 Rg )). g Comments The rotation of the target is irrelevant for the first part, which contravenes the setters law of not introducing information before it is required. In this case, it seemed better to describe the set-up immediately especially as you are asked for a familiar result. Remember, when you are considering the roots of the quartic (which is really a quadratic in β 2 ) that β > 0. The hint in the last paragraph ( by considering... ) is supposed to direct you towards the sum and product of the roots of the quadratic equation; otherwise, you get into some pretty heavy algebra.
120 Question 52( ) A particle of unit mass is projected vertically upwards with speed u. At any height x, while the particle is moving upwards, it is found to experience a total force F, due to gravity and air resistance, given by F = αe βx, where α and β are positive constants. Calculate the energy expended in reaching a given height z. Show that F = 1 2 βv2 + α 1 2 βu2, where v is the speed of the particle, and explain why α = 1 2 βu2 + g, where g is the acceleration due to gravity. Determine an expression, in terms of y, g and β, for the air resistance experienced by the particle on its downward journey when it is at a distance y below its highest point. Comments This is rather ingenious. The usual approach to this sort of problem is to say at the outset the particle experiences the force due to gravity and also a force due to air resistance of the form kv 2. In this question, the total force is given and you have to deduce the constituent parts of the force, making good use of the basic laws of mechanics (change of energy equals work done, for example). I remember the discussion of the phrasing of the next part of the question explain why α = 1 2 βu2 +g. How best to convey the idea that no real calculation is required, while not suggesting that the answer is trivially obvious? (All that is required, since we are trying to evaluate a constant, is to take a particular value of v.) I m not sure explain why was right; but I can t think of anything better. For the last part, you have to integrate the equation of motion, using the standard form of the acceleration used when there is y dependence instead of t dependence.
121 Solution to question 52 The energy expended in moving a small distance dx is Fdx so the energy expended in reaching height z is z αe βx α(1 e βz ) dx, i.e.. β 0 By conservation of energy, the loss of kinetic energy is equal to the work done against the force in reaching this height, so 1 2 u2 1 2 v2 = α(1 e βz ), β where v is the speed at height y and e βz = 1 β(1 2 u2 1 2 v2 ) α The force experienced at this height is therefore given by as required. F αe βz = α β( 1 2 u2 1 2 v2 ) To obtain an expression for the constant α we just have to evaluate the above equation at a value of v for which F is known. The obvious choice is v = 0: at maximum height the speed is zero so there is no air resistance, the only force on the particle being the force due to gravity. Thus F = g when v = 0, which gives the required result. Substituting for α in F gives F = g βv2, so the air resistance to the particle when it is moving at speed v (up or down) is equal to 1 2 βv2. For the last part, we just have to express 1 2 βv2 in terms of y, β and g. For the downward journey, let v = dy dt. Then v dv dy = g 1 2 βv2 i.e. v g 1 2 βv2dv =. dy. Integrating, and using the initial condition that v = 0 when y = 0, gives ( ) y = 1 β ln(g 1 2 βv2 ) + 1 β ln(g) = 1 g 1 β ln 2 βv2 g ( Thus 1 2 βv2 = g 1 e βy), which is therefore the air resistance..
122 Question 53( ) Hank s Gold Mine has a very long vertical shaft of height l. A light chain of length l passes over a small smooth light fixed pulley at the top of the shaft. To one end of the chain is attached a bucket A of negligible mass and to the other a bucket B of mass m. The system is used to raise ore from the mine as follows. When bucket A is at the top it is filled with mass 2m of water and bucket B is filled with mass λm of ore, where 0 < λ < 1. The buckets are then released, so that bucket A descends and bucket B ascends. When bucket B reaches the top both buckets are emptied and released, so that bucket B descends and bucket A ascends. The time to fill and empty the buckets is negligible. Find the time taken from the moment bucket A is released at the top until the first time it reaches the top again. This process goes on for a very long time. Show that, if the greatest amount of ore is to be raised in that time, then λ must satisfy the condition f (λ) = 0 where f(λ) = λ(1 λ) 1/2 (1 λ) 1/2 + (3 + λ) 1/2. Comments One way of working out the acceleration of a system of two masses connected by a light string passing over a pulley is to write down the equation of motion of each mass, bearing in mind that the force due to tension will be the same for each mass (it cannot vary along the string, because then the acceleration of some portion of the massless string would be infinite). Then you eliminate the tension. Alternatively, you can use the equation of motion of the system of two joined masses. The system has inertial mass equal to the sum of the masses (because both masses must accelerate equally) but gravitational mass equal to the difference of the masses (because the gravitational force on one mass cancels, partially, the gravitational force on the other), so the equation of motion is just (Newton s law of motion) (m 1 + m 2 )a = (m 1 m 2 )g. In the last part, the question says that the process goes on for a very long time. The reason for this is just so that you don t have to worry about whether it is advantageous to stop in mid-cycle: the effect of doing so after completing many cycles is insignificant. It is therefore sufficient to choose λ so as to maximise the rate at which ore is raised in just one cycle.
123 Solution to question 53 For bucket A s downward journey, at acceleration a, the equations of motion for bucket A and bucket B, respectively, are T + 2mg = 2ma, T (1 + λ)mg = (1 + λ)ma, where T is the tension in the rope. Eliminating T gives so a = g(1 λ)/(3 + λ). When bucket A ascends, the acceleration is g. The time of descent (using l = 1 2 at2 ) is 2l/a and the time of ascent is 2l/g. The total time required for one complete cycle is therefore ( 2l 1 + g ) 3 + λ. 1 λ Call this t. The number of round trips in a long time t long is t long /t so the amount of ore lifted in time t long is λmt long /t. To maximise this, we have to maximise λ/t with respect to λ, and λ/t is exactly the f (λ) given. Postmortem If you are feeling exceptionally energetic, you could try to find the value of λ for which f (λ) = 0. This will eventually lead you to the rather discouraging quartic equation λ 4 + 4λ 3 + 2λ 2 8λ + 3 = 0. A quick look at the quartic equation solver reveals, mysteriously, two real solutions: and ; mysteriously because we were expecting, from the wording of the question, only one solution. In any case, there should be an odd number of turning points between λ = 0 and λ = 1, because f (λ) = 0 when λ = 0 and λ = 1 and f (λ) > 0 for 0 < λ < 1. It turns out that in order to remove surds, it was necessary to square an expression and this introduced an extra fictitious solution. The larger of the two solutions quoted above is in fact the correct one, as can be ascertained by substituting both back into the original equation.
124 Question 54( ) A smooth cylinder with circular cross-section of radius a is held with its axis horizontal. A light elastic band of unstretched length 2πa and modulus of elasticity λ is wrapped round the circumference of the cylinder, so that it forms a circle in a plane perpendicular to the axis of the cylinder. A particle of mass m is then attached to the rubber band at its lowest point and released from rest. (i) (ii) Given that the particle falls to a distance 2a below the below the axis of the cylinder, but no further, show that 9πmg λ = (3 3 π) 2. Given instead that the particle reaches its maximum speed at a distance 2a below the axis of the cylinder, find a similar expression for λ. Comments This question uses the most basic ideas in mechanics, such as conservation of energy. I included it in this selection of STEP questions without realising that the properties of stretched strings are not in the syllabus (which is the syllabus for STEP I and II given in the appendix). However, I didn t throw it out: it is a nice question and the only two things you need to know about stretched strings are, for a stretched string of natural (i.e. unstretched) length l and extended length l + x with modulus of elasticity λ: (i) The potential energy stored in the stretched string is λx2 2l (ii) The tension in the stretched string is λx l.
125 Solution to question 54 a a 2a m The length of the extended band is 2πa 2aarccos(1/2) + 2 3a so the extension is 2aarccos(1/2) + 2 3a, i.e. 2a( 3 π/3). (i) By conservation of energy (taking the initial potential energy to be zero), when the particle has fallen a distance d and has speed v, 0 = 1 2 mv2 + λ 2 [2a( 3 π/3)] 2 mgd. 2πa At the lowest point, v = 0 and d = a, which gives the required answer. (ii) The (vertical) acceleration (ÿ) of the particle when it has fallen a distance d is given by mÿ = 2T cos θ mg where sinθ = a/(a + d) and T is the tension. The maximum speed occurs when ÿ = 0 and d = a as before, i.e. when 2 2a( 3 π/3)λ 3 2πa 2 mg = 0 so that in this case λ = 3πmg/(3 3 π).
126 Question 55( ) A wedge of mass M rests on a smooth horizontal surface. The face of the wedge is a smooth plane inclined at an angle α to the horizontal. A particle of mass km slides down the face of the wedge, starting from rest. At a later time t, the speed V of the wedge, the speed v of the particle and the angle β of the velocity of the particle below the horizontal are as shown in the diagram. V v β α Let y be the vertical distance descended by the particle. Derive the following results, stating in (ii) and (iii) the mechanical principles you use: (i) V sinα = v sin(β α); (ii) tan β = (1 + k)tan α; (iii) 2gy = v 2 (1 + k cos 2 β). Write down a differential equation for y and hence show that y = 1 2 g t 2 where g = g sin 2 β (1 + k cos 2 β) = (1 + k)sin2 α k + sin 2 α. Comments I was surprised how difficult this problem is, compared with a particle on a fixed wedge. I don t think that there is an easier method: the numbered parts of the question (i) (iii) seem unavoidable, since they embody the basic principles necessary to solve the problem. For a fixed wedge, the horizontal component of momentum is not conserved, because of the force required to hold the wedge; but (i) and (iii) are just what you would use in the fixed case. Part (ii) is used only (apart from the very last result) to show that the angle β is constant. This is a bit surprising: it means that the particle moves in a straight line. To put it another way, the net force on the particle is parallel to the direction of motion. Why is this? Most of the results of the important intermediate steps are given to you, but it is still very good discipline to check that they hold in special cases: for example, k = 0 corresponding to a massless particle or equivalently a fixed wedge; and α = 0 or α = π/2 corresponding to a horizontal or vertical wedge face. You should check that you understand what should happen in these special cases and that your understanding is consistent with the formulae.
128 Question 56( ) A uniform solid sphere of radius a and mass m is drawn very slowly and without slipping from horizontal ground onto a step of height a/2 by a horizontal force of magnitude F which is always applied to the highest point of the sphere and is always perpendicular to the vertical plane which forms the face of the step. Find the maximum value of F in the motion, and prove that the coefficient of friction between the sphere and the edge of the step must exceed 1/ 3. Comments This is quite straightforward once you have realised that very slowly means so slowly that the sphere can be considered to be static at each position. You just have to solve a statics problem with the usual tools: resolving forces and taking moments.
129 Solution to question 56 F R θ mg f Let the angle between the radius to the point of contact with the step and the downward vertical be θ, as shown. Taking moments about the point of contact with the step gives so Fa(1 + cos θ) = mgasin θ, sin θ F = mg = mg tan(θ/2). 1 + cos θ This takes its maximum value when θ is largest, i.e. when the sphere just touches the hoqizontal ground. At this position, cos θ = 1/2 and sinθ = 3/2 and F max = mg/ 3. At the point of contact, let the frictional force, which is tangent to the sphere, be f and the reaction R (along the radius of the sphere). Taking moments about the centre of the sphere gives F = f, and resolving forces parallel to the radius at the point of contact gives Now R = F sin θ + mg cos θ. F sin θ = mg sin2 θ 1 + cos θ = mg1 cos2 θ 1 + cos θ so R = mg. We therefore need µmg > F max. Postmortem = mg(1 cos θ) To be precise about the meaning of slow in this context, you have to compare the dynamic forces connected with the motion of the sphere with the static forces. The motion of the sphere is rotation about the fixed point of contact with the step. If we assume that the centre moves with constant speed, the only extra force due to the motion is an additional reaction at the point of contact with the step. (No tangential force is required to maintain a constant speed in circular motion.) This reaction is centrifugal in nature and so is of the form kmv 2 /d where d is some number which might be hard to calculate. The force calculated above is roughly mg, so the static approximation ( very slowly ) is valid if v 2 /d g.
130 Question 57( ) Harry the Calculating Horse will do any mathematical problem I set him, providing the answer is 1, 2, 3 or 4. When I set him a problem, he places a hoof on a large grid consisting of unit squares and his answer is the number of squares partly covered by his hoof. Harry has circular hoofs, of radius 1/4 unit. After many years of collaboration, I suspect that Harry no longer bothers to do the calculations, instead merely placing his hoof on the grid completely at random. I often ask him to divide 4 by 4, but only about 1/4 of his answers are right; I often ask him to add 2 and 2, but disappointingly only about π/16 of his answers are right. Is this consistent with my suspicions? I decide to investigate further by setting Harry many problems, the answers to which are 1, 2, 3, or 4 with equal frequency. If Harry is placing his hoof at random, find the expected value of his answers. The average of Harry s answers turns out to be 2. Should I get a new horse? Comments Hans von Osten, a horse, lived in Berlin around the turn of the last century. He was known far and wide for his ability to solve complex arithmetical problems. Distinguished scientists travelled to Berlin to examine Hans and test his marvellous ability. They would write an equation on a chalkboard and Hans would respond by pawing the ground with his hoof. When Hans reached the answer he would stop. Though he sometimes made errors, his success rate was far higher than would be expected if his answers were random. The accepted verdict was that Hans could do arithmetic. Hans s reputation as a calculating horse nosedived when an astute scientist simply made sure that neither the person asking the questions nor the audience knew the answers. Hans became an instant failure. His success was based on his ability to sense any change in the audience: a lifted eyebrow, a sigh, a nodding head or the tensing of muscles was enough to stop him from pawing the ground. Anyone who knew the answer was likely to give almost imperceptible clues to the horse. But we shouldn t overlook Hans s talents: at least he had terrific examination technique. We are investigating the situation when Harry places his hoof at random, so that the probability of the centre of his hoof lands in any given region is proportional to the area of the region. We therefore move swiftly from a question about probability to a question about areas. You just have to divide a given square into regions. each determined by the number of squares that will be partially covered by Harry s hoof if its centre lands in the region under consideration. Remember that the areas must add to one, so the most difficult area calculation can be left until last and deduced from the others. To answer the last part properly, you really need to set out a hypothesis testing argument: you will accept the null hypothesis (random hoof placing) if, using the distribution implied by the null hypothesis, the probability of obtaining the given result is greater than some predecided figure. Obviously, nothing so elaborate was intended here, since it is just the last line of an already long question: just one line would do.
131 Solution to question 57 Harry s hoof will be completely within exactly one square if he places the centre of his hoof in a square of side 1/2 centred on the centre of any square. The area of any such smaller square is 1/4 square units. His hoof will partially cover exactly four squares if he places the centre in a circle of radius 1/4 centred on any intersection of grid lines. The total area of any one such circle is π/16 square units, which may be thought of as four quarter circles, one in each corner of any given square. His hoof will partially cover exactly two squares if he places the centre in any one of four 1/2 by 1/4 rectangles, of total area 4 1 in any given square. 8 Otherwise, his hoof will partially cover three squares; the area of this remaining region is π If Harry placed his hoof at random, the probabilities of the different outcomes would be equal to the corresponding area calculated above (divided by the total area of the square, which is 1). Thus the data given in the question are consistent with random placement. The expected value given random placements is ( π ) + 4 π = 2 + π 16 The expected value (given that Harry gets all questions right) is ( )/4 = 5/2. Harry has a less accurate expected value even than the random expected value. He is clearly hopeless and should go.A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use
An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
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64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the
How to Study Mathematics Written by Paul Dawkins Before I get into the tips for how to study math let me first say that everyone studies differently and there is no one right way to study for a math class.
Club Accounts. 2011 Question 6. Anyone familiar with Farm Accounts or Service Firms (notes for both topics are back on the webpage you found this on), will have no trouble with Club Accounts. Essentially
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86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons andFactoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations
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Induction Problems Tom Davis tomrdavis@earthlin.net November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily | 677.169 | 1 |
Algebra 1 Final Exam
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Algebra 1 Final Exam
This 60 question test works great as a final exam, or practice for a state test. The test covers the following objectives:
-Solve multi-step equations and inequalities.
-Identify properties used while solving an equation or inequality.
-Translate a verbal expression.
-Simplify a numerical expression containing square roots and cube roots.
-Identify the domain and range of a function given ordered pairs, equations, or graphs.
-Determine if a relation represents a function.
-Evaluate a function for a given domain value(s).
-Identify the slope of a graphed line.
-Identify the slope of a line given two ordered points on the line.
-Identify the graph of a line given its equation.
-Identify the equation of a vertical or horizontal line.
-Identify the equation of line after a translation.
-Write the equation of a line given a point and slope.
-Write the equation of a line given two points on the line.
-Solve a system of equations.
-Determine if a system of equations has one solution, infinite solution, or no solution.
-Use a system of equations to solve a real-world application.
-Identify the graph of a system of linear inequalities.
-Simplify an expression using the rules of exponents.
-Simplify a non-perfect square root and non-perfect cube root.
-Simplify a radical expression containing variables.
-Add, subtract, multiply, and divide polynomials.
-Factor polynomials (including GCF, difference of squares, and trinomials)
-Identify the graph of a quadratic equation by identifying its roots.
-Solve a quadratic equation.
-Determine whether a direct or inverse variation exists.
-Identify the correct equation or graph of a direct or inverse variation.
-Identify the line of best fit given a graph.
-Calculate the line of best fit or curve of best fit given a data set.
-Make predictions based upon the equation of a line of best fit or curve of best fit.
-Calculate the mean, median, and mode of a data set.
-Analyze data given in a box-and-whisker plot.
-Calculate the mean absolute deviation, standard deviation, and variance of a data set.
-Calculate the z-score of an element in a data set | 677.169 | 1 |
ESO 208A/ 218
Computational Methods in
Engineering
Summer Semester 2014-15 I
L4(2) Numerical Linear Algebra
Review of basics
What do we mean by Linear Algebra
Matrices and vectors underlie linear algebra, allow us to
represent numbers or functions in an
Data Structures and Algorithms
(CS210A)
Lecture 31
Magical applications of Binary trees -II
1
RECAP OF LAST LECTURE
2
Intervals
S = cfw_[, ], 0 <
Question: Can we have a small set XS of intervals s.t.
every interval in S can be expressed as a union of a
Data Structures and Algorithms
(CS210A)
Lecture 4:
Design of O() time algorithm for Maximum sum subarray
Proof of correctness of an algorithm
A new problem : Local Minima in a grid
1
Max-sum subarray problem
Given an array A storing numbers,
find its suba | 677.169 | 1 |
Library
HOME
About The Department
Teaching of Mathematics started right from the beginning of the college. Mathematics is a sharpened language for expressing science. In 2002 post graduate programme was introduced. Additional section I and II for UG Mathematics was started in the year 2003 and 2009 respectively. The total strength of our Department is around 570.
Our department promotes quality education in Mathematics. It is an upcoming department now engaging itself in various activities like conducting student's seminar, Quiz programme, Mathematics Exhibition etc., It has well experienced and dedicated faculty actively engaged in teaching research activities. They also render services to the student community from rural area, the growing young and aspiring students to achieve their goals. Within a short span of time it created a tradition of academic excellence and universal service.
Vision
To provide quality basic education that is equitably accessible to all and lay the foundation for lifelong learning and service for the common good.
Mission
The aim of education is to cultivate personality integration and to promote quality in higher education and to give overall development to student community.
Objectives
To improve the quality of graduates in order to increase their ability to fit in various places in the job market.
To improve research activities in the department.
To provide training on using mathematical and statistical packages in the real life.
PROGRAMMES OFFERED
S.No.
Name of the Programme
1
B.Sc., Mathematics
2
M.Sc., Mathematics
ACTIVITIES OF THE HYPATIA CLUB
Seminars are conducted on innovative topics by inviting resource persons from various institutions.
Mathematics day is also celebrated on the first Wednesday of March every year.
Mathematics Exhibitions are also conducted to motivate the skills of the students.
NEWSLETTER
The department publishes a Newletter namely Aryabhatta. It contains articles about Mathematicians, Puzzles, Mathematical tricks, drawing, poem etc. The Magazine is edited by the students.
GLOSSARY
Glossary of technical terms of mathematics is prepared by the faculty members and the copies are distributed to the students for better understanding of the subjects.
ENDOWMENT
LIBRARY
The Library extends service beyond the physical walls of the building by providing materials by the assistance of librarian in navigating and analyzing tremendous amounts of information. There are about 975 books of mathematics and 4 journals in our library. In our department we are maintaining a library with 100 books.
9. A.Yogeswari and Dr.T.Chithrakalarani, "Time to reach the break down point of an organization shock model approach" Advance in Mathematics Scientific Development & Application, ISBN-978-81-8487-074-9,2010.
10. A.Yogeswari and Dr.T.Chithrakalarani, "Time To Breakdown Point Of An Organization Cumulative Damage Model", International Journal of Scientific Transaction in Environment and Technovation, Vol 4, Issue 4 ISSN 0973-9157, April to June 2011.
11. A.Yogeswari and Dr.T.Chithrakalarani, " Stochastic model for time to reach the breakdown point of an Organization by two types of losses" Applied Science Periodical Vol – 16, Issue - 2 May – 2014, ISSN – 0972 – 5504. | 677.169 | 1 |
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