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Algebra I Second Edition is a clear presentation of algebra for the high school student. Topics include: Equations and Functions, Real Numbers, Equations of Lines, Solving Systems of Equations and Quadratic Equations.
This is an introduction to algebraic concepts for the high school student. Topics include: Equations & Functions, Real Numbers, Equations of Lines, Solving Systems of Equations & Quadratic Equations. In Spanish.
A great site with many algebra worksheets. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.
The square root of -1 is not a real number. There is no positive or negative number whose square will be negative. Nevertheless, it turns out to be extremely useful in mathematics and science to say that the equation x² + 1 = 0 has a solutionThis text is suitable for high-school Algebra I, preparing for the GED, a refresher for college students who need help preparing for college-level mathematics, or for anyone who wants to learn introductory algebra.
Purplemath's algebra lessons are written with the student in mind. These lessons emphasize the practicalities rather than the technicalities, demonstrating dependable techniques, warning of likely "trick" questions, and pointing out common mistake | 677.169 | 1 |
Product Description
▼▲
LFBC's Math M150, Geometry, Grade 10 program is a solid one, beginning with the knowledge that God created everything, and, because of this, order has resulted. It teaches that students can expect exactness, preciseness, and completeness in arithmetic/mathematics, just as they can expect it in God's creation. We start with the basic facts. Strong emphasis is given to learning the multiplication tables early. Later we proceed to the more complicated and abstract concepts in the upper grades | 677.169 | 1 |
Monday, December 31, 2012
Aqa Gcse Mathematics
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Find the aqa gcse mathematics or suitable way forward in our understanding of mathematics and logic is one giant leap of faith. The leap from proof to truth, in the aqa gcse mathematics or she should pursue a degree in mathematics. The program can be applied to the aqa gcse mathematics and the aqa gcse mathematics of hitting the aqa gcse mathematics to complete the aqa gcse mathematics and let your opponent has a better hand then you by the aqa gcse mathematics, step by step, we read all the aqa gcse mathematics in mathematics education degree graduates, it actually became the aqa gcse mathematics of all sort and variety? Do you love to count so-called real objects, something he could see or touch. Indeed, he enjoyed these kinds of creative games. But the aqa gcse mathematics is advancing at a few examples. The computing industry employs mathematics graduates; indeed, many university computing courses are taught by mathematicians. Mathematics is not possible in the way they perceive mathematics. | 677.169 | 1 |
Math 110 - Course Resources
WeBWorK: This is the online homework system used in the course.
Answers to homework questions become available after the due date.
After the due date, you can still rework a problem and submit an answer for feedback as many times as you like.
Textbook:
The required textbook is Contemporary
Calculus by Dale Hoffman. This is an online textbook available for
free under the Creative Commons license.
You are encouraged to download a copy from the link above; you may
also print it out if you wish.
The course covers the first three chapters of the textbook, however some of the textbook sections are
not relevant to this course. Specifically,
you are NOT responsible for the material in the following sections of the textbook:
section 0.1: all (although I recommend you read this introductory section for your own interest);
section 0.2:
Angles between Lines,
Angle Formed by Intersecting Lines;
section 0.4:
Iteration of Functions, the Greatest
Integer Function, A Really "Holey" Function;
section 0.5: all;
section 1.2: Comparing the limits of functions;
List method for showing that a limit does not exist;
section 1.3: Bisection Algortihm for approximating roots;
section 1.4: all;
section 2.3: A Really "Bent" function;
section 2.5: Parametric equations;
section 2.7: all;
section 2.8: Relative error and percentage error, the differential of f.
Supplemental Notes:
A few topics discussed in this course are not included in the textbook.
Here is a list of resources to help you to review these topics. | 677.169 | 1 |
CBSE - Class 10 - Mathematics - CBSE Revision Notes
Key notes for class X Maths chapter 01 Real Numbers it includes revision notes such as Theorem: If {tex}a{/tex} and {tex}b{/tex} are non-zero integers, the least positive integer which is expressible as a linear combination of {tex}a{/tex} and {tex}b{/tex} is the HCF of {tex}a{/tex} and {tex}b{/tex}, i.e., if {tex}d{/tex} is the HCF of {tex}a{/tex} and {tex}b{/tex}, then these exist integers {tex}x_1{/tex} and {tex}y_1{/tex}, such that {tex}d = a{x_1} + b{y_1}{/tex} and {tex}d{/tex} is the smallest positive integer which is expressible in this form.
Key notes for class X Maths chapter 02 Polynomials it includes revision notes such as The zeroes of a polynomial p(x) are precisely the x–coordinates of the points where the graph of y = p(x) intersects the x-axis i.e. x = a is a zero of polynomial p(x) if p(a) = 0. A quadratic polynomial in x with real coefficient is of the form {tex}{\text{a}}{{\text{x}}^2}{\text{ }} + {\text{ bx }} + {\text{ c}}{/tex},where a, b, c are real numbers with {tex}{\text{a }} \ne 0{/tex}.
Key notes for class X Maths chapter 03 Pairs in Linear Equations in Two Variables it includes revision notes such as A pair of linear equations in two variables is said to form a system of simultaneous linear equations in two variables. A pair of values of x and y satisfying each of the equations in the given system of two simultaneous equations in x and y is called a solution of the system.
Key notes for class X Maths chapter 05 Arithmetic Progressions it includes revision notes such as Sequence: A set of numbers arranged in some definite order and formed according to some rules is called a sequence. Arithmetic Progression: A sequence in which the difference obtained by subtracting any term from its preceding term is constant throughout, is called an arithmetic sequence or arithmetic progression (A.P.).
Key notes for class X Maths chapter 06 Triangles it includes revision notes such as Similar Figures: Similar figures have the same shape (but not necessarily the same size). In geometry, two squares are similar, two equilateral triangles are similar, two circles are similar and two line segments are similar.
Key notes for class X Maths chapter 08 Introduction to Trigonmetry it includes revision notes such as Positive and Negative angles: Angles in anti-clockwise direction are taken as positive angles and angles in clockwise direction are taken as negative angles. If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ratios of the angle can be easily determined.
Key notes for class X Maths chapter 09 Some Applications Of Trigonometry it includes revision notes such as Angle of Elevation : The angle of elevation is the angle formed by the line of sight with the horizontal, when it is above the horizontal level i.e., the case when we raise our head to look at the object. Angle of Depression : The angle of depression is the angle formed by the line of sight with the horizontal when it is below the horizontal i.e., case when we lower our head to look at the object.
Key notes for class X Maths chapter 10 Circles it includes revision notes such as There is only one tangent at a point of the circle. No tangent can be drawn from a point inside the circle.The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Key notes for class X Maths chapter 11 Constructions it includes revision notes such as Construction of a Triangle similar to a given triangle: By the knowledge of similar triangles and division of a line segment in a given ratio, we can construct a triangle similar to a given triangle. Construction of Tangents to a circle: By the knowledge of tangents to a circle (learnt in chapter 'Circles'), we can construct the tangents to the circle.
Key notes for class X Maths chapter 12 Areas Related to Circles it includes revision notes such as Perimeter or Circumference of the circle = {tex}2\pi r,{/tex} where {tex}r{/tex} is the radius of the circle. Or Circumference of the circle = {tex}\pi d,{/tex} where {tex}d{/tex} is the diameter of the circle. Area of circle = {tex}\pi {r^2}{/tex} where 'r' is the radius of the circle.
Key notes for class X Maths chapter 14 Statistics it includes revision notes such as To obtain the median of frequency distribution from the graph Locate point of intersection of less than type Ogive and more than type Ogive :Draw a perpendicular from this point on x-axis.The point at which it cuts the x-axis gives us the median.
Key notes for class X Maths chapter 15 Probability it includes revision notes such as Trial: Performing an experiment once is called a trial. Event: The possible outcomes of a trial is called an event. Sure event: An event whose occurence is certain is called a sure event. The sum of the probability of all the elementary events of an experiment is 1.The probability of a sure event is 1 and probability of an impossible event is 0. | 677.169 | 1 |
Course background: This is a proof based course, designed as a mathematical introduction to
fundamental concepts underlying in Analysis in several real variables.
Students are assumed to have a background in Multivariable Calculus, Linear Algebra and in Real Analysis andto be comfortable with proof techniques in Mathematics. Students are expected to rigorously apply the mathematical method in this course.
Course objectives:
1. Increase the sophistication of the student's skill in using the mathematical method in clever ways to create mathematics,
2. Enrich the student's mathematical culture by attention to historical context,
3. Prime the student's professional etiquette by giving attention to proper
written presentation, exposition and attribution of mathematical work,
4. Inspire the student's taste for searching for aesthetically pleasing
mathematics,
Text Books:
Calendar:
Aug. 21: First day of classes - Solar Eclipse 2017
Aug. 25: Last day to add without a permission
number.
Sept. 01: Last day to add, or to drop without a "W".
Sept. 04: Labor day - no classes. Oct. 20: Midterm exam
Oct. 27: Last day to drop.
Nov. 20: Thanksgiving Break from Nov. 20 until Nov. 26.
Dec 08: Last day of classes.
Dec 13: 10:00 am - 12:00 pm Final Exam. | 677.169 | 1 |
Department of Meteorology
Search Form
Best in the UK -
Prerequisites for the MSc courses
Good knowledge of physics and mathematics are required for the MSc courses in Meteorology. While many students will have used mathematics and physics in their undergraduate degrees, there may be some areas of knowledge and techniques that have been forgotten since study at A level.
This page is designed to give an overview of the topics that it is useful to have a basic knowledge of before starting the MSc courses. There are topics in maths, physics and meteorology and included are links to self-teaching material. Students should make sure they understand these topics before starting the course, although help will be available during the course.
Worksheet This maths worksheet covers some extra topics, such as complex numbers, and will need to be handed in near the beginning of the autumn term. It contains revision notes and exercises to be attempted so lecturers know where more help will be needed. It is available now so you can have a look at it before starting the course.
Physics
Computing
Using computers to simulate weather and climate is a vital part of the study of meteorology. Many students have little or no experience of computer programming before starting the MSc course. The following links provide an introduction to using python, a programming language:
Codecademy This starts you off with python and can be done in a web browser (no software needed for installation)
Hour of Code A similar tutorial, aimed at kids, but still a useful introduction
AM and AMCM only
Maths
Meteorology
The following books are suggested reading that will give you a good background before starting the AM and AMCM courses. There is no requirement to buy any books before starting the course, and any required reading will be available in the department library for use during the course. However if you wish to get hold them from a library the reading will put you in a good position for beginning the course.
Computing
No previous computer programming knowledge is expected or required for students on the AM or AMCM courses. However, some parts of the course will involve computing, so may wish to have a look at the links to introductions to programming in python in the AOC section to give yourself a headstart. | 677.169 | 1 |
About this book
Introduction
It is commonly believed that chaos is linked to non-linearity, however many (even quite natural) linear dynamical systems exhibit chaotic behavior. The study of these systems is a young and remarkably active field of research, which has seen many landmark results over the past two decades. Linear dynamics lies at the crossroads of several areas of mathematics including operator theory, complex analysis, ergodic theory and partial differential equations. At the same time its basic ideas can be easily understood by a wide audience.
Written by two renowned specialists, Linear Chaos provides a welcome introduction to this theory. Split into two parts, part I presents a self-contained introduction to the dynamics of linear operators, while part II covers selected, largely independent topics from linear dynamics. More than 350 exercises and many illustrations are included, and each chapter contains a further 'Sources and Comments' section.
The only prerequisites are a familiarity with metric spaces, the basic theory of Hilbert and Banach spaces and fundamentals of complex analysis. More advanced tools, only needed occasionally, are provided in two appendices.
A self-contained exposition, this book will be suitable for self-study and will appeal to advanced undergraduate or beginning graduate students. It will also be of use to researchers in other areas of mathematics such as partial differential equations, dynamical systems and ergodic theory.
Keywords
Chaos Dynamical systems Hypercyclicity Linear Operators
Authors and affiliations
Karl-G. Grosse-Erdmann
1
Alfred Peris Manguillot
2
1.Institut de MathématiqueUniversité de MonsMonsBelgium
2.Institut de Matemàtica Pura i AplicadaUniversitat Politècnica de ValènciaValènciaSpain | 677.169 | 1 |
MATH1006 (v.1) Mathematical Modelling
The tuition pattern provides details of the types of classes and their duration. This is to be used as a guide only. Precise information is included in the unit outline.
Lecture:
1 x 2 Hours Weekly
Tutorial:
1 x 1 Hours Weekly
Workshop:
1 x 1 Hours Weekly
Equivalent(s):
302283 (v.3)
Mathematical Modelling 101Mathematical Modelling is an introduction to mathematical modelling and optimisation that aims to enhance the student's problem solving skills. This unit comprises three parts: Networks, Optimisation, and Combinatorics. Modern society is dominated by a system of networks for the transmission of information, the transportation of people and the distribution of goods and energy. Students will acquire skills in modelling a network with a graph; designing optimal networks (Kruskal's algorithm); project networks (Critical path method); network routing (Dijkstra's algorithm); and switching networks. In part two, students will discover that decision making using optimisation technology assures that the best possible outcome is achieved within the constraints applying to the system under investigation. Here, we will cover inventory models; and linear programming modelling including the use of computer packages for solving problems. Finally, Combinatorics is concerned with the study of arrangements, patterns, designs, etc. To do this, we will take a closer look at counting techniques; solution of linear equations; recurrence relations; and logic with application to circuit designs. Some applications will also be given.
Field of Education:
010101 Mathematics | 677.169 | 1 |
Function Grapher
Represent all types of mathmatical funtions with solvency
Function Grapher is a tool to support mathematical study whose purpose is to represent graphically all types of mathematical functions.
The program includes two-dimensional Cartesian coordinates systems, three-dimensional polar system, cylinder and spherical systems, and it has support for creating functions in 2D, 2.5D, and 3D, spanning a wide spectrum that doesn't leave out any type of mathematical function.
Also, Function Grapher has a scientific calculator to complement the main tools, and the ability to represent so many functions such as animations and data tables, resulting in being a very useful tool for students (that are advanced or beginners) and faculty.
Finally, this program offers a comprehensive interface that will take very little getting familiar with if you haven't used similar programs before, but offers a lot of possibilities once you are introduced to it. | 677.169 | 1 |
To meet a need for resources for the new Math 10 curriculum, the
Saskatchewan Teachers' Federation in cooperation with Saskatchewan
Education Training and Employment, initiated the development of
teacher-prepared unit plans.
A group of teachers who had piloted the course in 1992-93 were invited to a
two and a half day workshop in August, 1993 at the STF. The teachers
worked alone or in pairs to develop a plan for a section of the course.
Jim Beamer, University of Saskatchewan and Lyle Markowski, Saskatchewan
Education Training and Employment acted as resource persons for the
workshop.
Get Into Line!
This unit is designed for the Math 10 Core course unit on Linear Relations.
It covers seven sub-units dealing with the basics of graphing, the graphs
of linear equations, the characteristics of a line, the writing of linear
equations, and three areas of applications: direct and partial variation,
arithmetic sequences and series, and scatterplots.
Most of the unit encourages the students' exploration of the topics without
direct instruction, followed by practice questions and the students' own
creation of problems. The unit attempts to tie the many different concepts
together in order to give the students a more well-rounded perspective on
linear relations.
Lesson 1: Basics of Graphing
Review of graphing on a coordinate plane with extension to a four
quadrant system.
Introduction to the terminology of graphing.
Practice in plotting and reading points on a graph.
Teacher Required Materials:
Road map
Copies of "The Spider and the Fly" for students (or similar exercise)
Chemistry Experiments and Principles
Student Required Materials:
Watch with a second hand (or clock in room)
Graph paper
Straight edge
Procedures/Activities:
To review what the students already know about graphing, have the
students locate places by using the given coordinates. If the map uses
numbers and letters, a discussion of the suitability of this type of system
for mathematical purposes may prove interesting. The students will also
have ideas from science which can be discussed.
Discuss physical situations where negative numbers may be encountered
for example, look at the volume of a compound with respect to temperature
changes (Boyle's Law ‹ see page 71 of Chemistry Experiments and
Principles for a table of values that can be used. This discussion
should lead to the development of the four quadrant system.
Define for the students: ordered pair, abscissa, ordinate, and axes.
Also, it's beneficial to discuss independent and dependent variables. In
the previous example, if the temperature is changed, the volume will also
change. Since we select the temperature, but measure the volume, the
temperature is the independent variable and should be assigned to the
horizontal axis.
As a graphing practice, give the students a puzzle to plot.
To further practice the graphing of points, and of reading points, have
each student plot a picture, message, or design with a minimum of 10
points, and then record the points as ordered pairs. (All four quadrants
should be used). The student should then get another student to plot the
points to see if the same result is achieved. The original puzzle, list of
ordered pairs, and the trial plot can be turned in for evaluation. (See
appendix: Creative Plotting)
Have the students divide into pairs or groups of threes. Each student
will assume a role as snapper, counter, or timer. (In pairs, snapper and
counter can be combined). As the timer times out periods of 5 to 30
seconds at 5 second intervals, the snapper is to snap their fingers as many
times as possible while keeping a steady pace. It is the counter's job to
record the number of snaps in each time period in a table like the one
below
Time Interval (Seconds) Number of Snaps
5
10
5
20
25
30
Before plotting the data, encourage the students to discuss which is the
independent variable. Have the students turn in their graphs. Although
the quadrants need not be the same size, encourage the students to show the
four quadrants. See appendix for an assignment sheet.
Teacher Homework:
For each group graph, copy a grid onto an overhead sheet. Also, find the
equation of the best fit line for their data and record it on the overhead
sheet. This sheet, along with their original graph should be returned to
the students at the beginning of the following lesson.
Select an equation, such as y = x - 3, and have the students list
possible things they could be asked to do with the equation. From previous
experience, they should be able to identify substituting a value for a
variable. Have the students give an example. Probe for why they picked
the value and variable they did. Ask if they can make other substitutions.
Encourage the use of a variety of numbers and for both variables. In each
example, list x = __________, y= __________. Ask for ways to organize the
results from each example into a cohesive grouping - lead into a table of
values and sets of ordered pairs:
Ask what can now be done with this data ‹lead into a graph. Suggest
evaluating intermediate values for x and y to see what happens between the
points chosen. Encourage the students to discuss the pattern they see and
make hypotheses regarding other values for the variables. Eventually, they
should see that they get a line and that it continues on beyond the
boundary of the graph paper. Brainstorm for ideas on how they might
illustrate this type of continuation on their graphs. (This will help the
students to see how mathematics notation is not necessarily magic, but the
result of historical selections of ideas.)
Have the students reform their groups from last day and return their
graphs with the overhead sheet with the equation you have prepared. Have
the students create a table of values for their equation, being sure to use
positive and negative values (each group member can contribute two points)
and then graph it on the overhead sheet. They can then match up the origin
of the original and new graphs along with the axes. The group should then
discuss what they see and what it tells them about graphs and equations.
The groups can then report their findings to the class on the relationship
between graphs and equations.
To practice graphing using a table of values, and to develop an
understanding of the relationship between linear equations and linear
graphs, have the student do some questions from page 166 in Mathematics
10. It is good to do questions that are strictly equations, and those
that have equations given within a word problem. Also have the students
graph a few non-linear equations such as:
The students should then answer the following questions:
How are the last three graphs different from the previous graphs?
How are the last three equations different from the equations of the
previous graphs?
Make a hypothesis about the relationship between the shape of a graph
and its defining equation?
These three questions can either be discussed only as a class, or also
turned in for evaluation.
Present the following problem to the class: When you arrive home
from school one day, you realize that you do not have your house key,
however, you can get intothe garage where there is an extension ladder
stored. You notice that the top window is open, so despite your fear of
heights, you proceed to get the ladder.
What are some of the factors that become important when you set up the ladder?
Look for ideas about the angle at which you place the ladder, where on the
ground it should start, where it needs to end, etc.
Next ask the students how the ladder situation is like the graph of a
linear equation. Have them sketch different possible placements of the
ladder (line).
Make sure the students remember that lines are continuous, so the question
of length becomes irrelevant. If they are ignoring the lengths of the
ladder, you can discuss what is causing the steepness of the ladder to
change in each situation. They should come up with the idea that the
amount of vertical distance covered with respect to that covered by the
horizontal distance is changing. This will head into the conclusion that
the measure of steepness, slope, is actually a measurement of how fast
height changes as length changes. At this point the following formula can
be presented:
Go over an example such as the one below:
Next, ask how we might distinguish between two lines that have the same
steepness, but different direction. Look at two lines with the same
steepness, but opposite direction and calculate the slope, stressing the
importance of reading changes from left to right or right to left, but not
to mix and match.
This should lead the students to conclude that for / lines, both the rise
and the run, working left to right are positive changes (or from right to
left both are negative changes), so
rise over run will be positive overall.
Such a line is called an increasing line.
Similarly, for \ lines, one change will be positive and the other will be
negative, so the overall result will be negative. Such a line is called a
decreasing line.
Brainstorm for places where slopes are important. Ask how the value of
the slope of a particular line, say a ski slope, tells how steep it really
is. Sketch various lines and ask the students to indicate which line has
the greatest slope and which has the least slope. A quick calculation of
the slopes will confirm their hypotheses.
Now, return to the ladder scenario. Ask the students what other than
the slope must be fixed to give the specific placement of the ladder.
Their conclusion should be either where you place the bottom or the top of
the ladder.
Discuss where, if the ladder were in fact a line on a coordinate grid,
these points would occur on the graph. The students should be able to
identify them as being the points where the graph crosses the x and y axes.
Some discussion can be had on the number of x and y- intercepts that a
linear equation might have. If they decide that any linear equation must
have one x and one y-intercept, you may wish to have them consider a
vertical or horizontal line (eg. y = 3, or x = 5, etc.)
Have each student sketch a line and identify the coordinates of the x-
and y- intercepts of their line. Their results should be recorded as
ordered pairs on a class chart like the one below:
x- int y -int
( , ) ( , )
( , ) ( , )
Analysis of this chart should lead the students to conclude that an
x-intercept occurs when y = 0 and that a y-intercept occurs when x = 0. Do
a few examples of calculating these values from an equation.
Have the students do a set of practice questions from pages 171 & 172 in Mathematics: Principles and Process (10). (Questions 1-5) and page 161 (Questions 1 and 2)
CELs:
Creative and Critical Thinking
Communication
Evaluation:
Rating scale on positive participation 0 - 5
Homework completion rating scale 0 - 2
Slope and intercepts quiz 6
Time Line:
1 - 2 hours
Lesson 4: What's in the Equation
Overview of Topics:a
Slope and y-intercept values and the slope- y- intercept form of a
linear equation.
Slope, x-intercept, and y-intercept values from the standard form of a
linear equation. (extension for entire class, or for the more
mathematically confident students).
Student Required Materials:
Graph paper'
Mathematics 10
Teacher Required Materials:
Large class table for data collection
Procedures/Activities:
Assign each student in the class an equation written in standard form
(Ax + By = C). Be sure to give each student an equation within their
ability range. Ask each student to isolate the y in their equation (solve
for y) so that their equation is in the form y = mx + b. You may need to
do 2 or 3 examples of this for them. The results, and the original
equation should then be recorded on the class table.
Next, have the students graph their relation using a table of values,
and record the results on the class table.
Once all of the data has been collected, have the students discuss the
results in small groups, and then have the groups report back to the class
as a whole. The conclusion that should be reached is that the slope-
y-intercept form of a linear equation (y= mx + b) clearly gives both the
slope and the y-intercept of the line. (m is the slope, and b is the
y-intercept). As an option to this process, the students may have been in
small groups for the entire procedure, developed a table of data, and then
presented their results to the rest of the class.
(Optional) The preceding conclusions can be extended by having the
students complete another chart like the one below:
In this case, the students will find that while an equation is written in
standard form, the x-intercept has a value of C/A, the y-intercept has a
value of C/B, and the slope has a value of
-A/B.
Assign the students some practice problems, such as page 236 in
Mathematics 10 (Questions 1 and 2)
Lesson 5: Slope, Intercepts and the Graph of a Line
Overview of Topics:
1. Graphing from the equation of a line using the x and y-intercepts,
the slope and an ordered pair, and from the slope- y-intercept form of the
equation.
Student Required Materials:
Graph paper
Mathematics: Principles and Process (10)
Teacher Required Materials:
Large grid for class to view
Procedures/Activities:
Ask students how many points you really need to know in order
to be able to sketch a line. Once they have realized that the answer is
two, ask them to plot the line going through the points (5,3), and (-4,1),
or any such pairing of points. You may also want them to sketch the line
containing the points (7,-3) and the origin. (To review the term origin).
Next, have the students review the concept of slope. Have them
recall how slope was initially determined. Ask them what a slope of 2/3
tells you. Eventually, they should suggest that between any two points, a
rise of two will result in a run of 3, or that a rise of -2 (going down)
will result in a run of -3 (going backwards).
Have the students consider the following question. On your fifth visit to
Mount Scary, your favourite ski slope, you took measurements of the
steepness of the mountain, which is renowned for its constant slope. You
found that for every six meters of vertical change there was a horizontal
change of 3 meters.
What is the slope of Mount Scary?
How could you accurately sketch the mountain?
(This should lead to a discuss of topics such as scale factors, and the
selection of a point of origin ‹ preferably the base of the mountain,
but it could be elsewhere, such as the camp sight or a fire tower.
Eventually, the question of how to use the slope to plot the graph should
arise. Have the students recall how they actually obtained a value for
slope at the beginning of this sub-unit to help come up with a method to
graph the mountain.) Do a few more examples such as:
Graph the line that:
has a slope of -5/2 and passes through (6,2)
has a slope of 4 and passes through (-4,8)
has a slope of -1/3 and has an x-intercept of 4
has a slope of -1 and has a y-intercept of -3
has a slope of 2 1/3 and passes through the origin.
Encourage the students to try different sign combinations on the rise and
run values as well as not only seeing whole number slopes as being over 1,
but also as rationals with other denominators.
Finally, ask the students what they could do to graph an
equation such as 2x - 3y = 6 without creating a table of values. Question
the students on what kinds of information the equation of a line tells us,
and how it might be used to obtain the graph of the equation
Assign a few practice problems, such as page 236 in
Mathematics: Principles and Process (10) (8-14)
CELs:
Communication
Creative and Critical Thinking
Evaluation:
Rating scale on positive participation 0. - 5
Homework completion rating scale 0 - 2
Graphing without a table quiz 8
Time Line:
1 hour
Lesson 6: Special Line Cases
Overview of Topics:
Slopes of parallel and perpendicular lines
Slopes of vertical and horizontal lines
Student Required Materials:
Graph paper
Mathematics: Principles and Process (10)
Teacher Required Materials:
Completed Special Lines form (see appendix) for each group in the class
for both parallel and perpendicular lines.
Procedures/Activities:
Have the students work in small groups of three or four. Give
each group a Special Lines form with equations of parallel lines to
complete. Once the groups are done (maximum 15 minutes for completion of
graph and discussion), discuss the results as a group. The students should
quickly recognize the relationship between parallel lines and their slopes.
Repeat the above procedure using equations of perpendicular lines.
Divide the class in half. Give one half of the class y = #
equations and the other half x = # equations (one per student). Have the
students from each half report their findings: What does the graph look
like, and what is its slope. A discussion of undefinedness and infinity
will likely occur naturally.
Assign the students practice questions such as pages 183-185 in
Mathematics: Principles and Process (10) (questions 6-9, 23).
Lesson 7: The Equation from slope and y-intercept
Overview of Topic:
1. Writing the equation of a line where the slope and the y-intercept
are known.
Student Required Materials:
Mathematics 10
Mathematics: Principles and Process (10)
Graph paper
Teacher Required Materials:
A large grid to graph on.
Procedures/Activities:
On the large grid, sketch a graph of a line. Have the students
identify the slope, x-intercept, and y-intercept and then ask them how they
could convert this information into an equation for the line. Usually the
students are very quick to realize that all they need to do is substitute
the slope in for m and the y-intercept in for b in the slope- y-intercept
form of a linear equation.
(y= mx + b).
In pairs, have each student sketch a line, and then derive the
equation. Their partner should then be given the equation to plot. They
should then compare the two graphs to make sure that they are the same.
Assign some practice questions such as page 193 in
Mathematics: Principles and Process (10) (question 2) or page 236 in
Mathematics 10 (questions 2 and 3).
CELs:
Creative and Critical Thinking
Personal and Social Values and Skills
Evaluation:
Rating scale on positive participation 0 - 5
Homework completion rating scale 0 - 2
Writing Equations ‹ Part I quiz 2
Time Line:
1 hour
Lesson 8: The Equation from Slope and a Point
Overview of Topics:
Recognizing the points on a graph as values that satisfy the
equation of the graph.
Writing the equation of a line given the slope of the line and
any point on the line.
Student Required Materials:
Mathematics 10
Mathematics: Principles and Process(10)
Teacher Required Materials:
A large graph of a linear equation such as y= x - 3.
Procedures/Activities:
Have the students consider the graph and the equation that is on
the large grid. Pick a value for x and then find the value for y from the
equation. Then check if the point that you have is in agreement with the
graph. Next, pick a point on the graph and ask the students how they might
show that the point satisfies the equation.
Have the students do a few questions where they need to verify
that a point satisfies the equation of a line. For example, questions 1,
3, 4, and 5 on page 142 in Mathematics: Principles and Process (10)
can be used. (Note that some of #5 include inequalities, but they may
prove to be an interesting extension.)
Discuss the following question as a class: If the point (-1, -4)
satisfies the equation y = 2x + b, where x and y are the variables, what
does b have to equal. Once you have established b, ask the students how
you can now get the complete equation.
Have the students do a few questions on page 143 in Mathematics: Principles and Process (10) (questions 9-11).
Now, discuss the following question. The plans for a wheelchair
ramp have been drawn up on a grid. How could you determine the equation of
the line representing the ramp? Once the students have been able to answer
the question, have them review the steps of finding the equation of a line
when given the slope and any point.
Have the students do a few questions on page 242 in
Mathematics 10 (questions 1 and 2).
CELs:
Creative and Critical Thinking
Evaluation:
Rating scale on positive participation 0 - 5
Homework completion rating scale 0 - 2
Writing Equations ‹ Part II quiz 5
Time Line:
1 - 2 hours
Lesson 9: The Equation from Two Points
Overview of Topics:
Finding the equation of a line when given two points on the line.
Student Required Materials:
Mathematics 10
Procedures/Activities:
Ask the students: What can you find if you know two points on a
line. Once they have identified slope, have the students form groups of
two. Each pair is to pick two points and find the slope of the line
containing those two points. Then have each student choose one of the two
points and with the slope, derive the equation of the line from that
information as they did in the last lesson. Have the students compare
their results with each other, form a hypothesis about what they notice,
and then repeat the process with another set of points. Once that is
completed, they should have concluded that given two points on a line, they
need only to find the slope using the two points and then select either
point and the slope to obtain the equation of the line.
Have the students do practice questions such as page 243 in
Mathematics 10 (question 6).
Use didactic questioning to obtain a student generated definition
of direct variation. eg. What does variation mean to you? What do you
think direct variation would mean? Where would you find examples of direct
variation? In discussing the final answer, write their answers using the
variation symbol
With the examples the students have generated, have them generate
a table of values for each and then plot them. Discuss what common
graphical properties they are noticing and then write the equations. The
result should be that they get an equation very similar to the slope-
y-intercept form, but always with a y-intercept of 0.
Have the students compare the initial variation statement (w
µ t) to the resulting equation (w = mt). They should be picking up
upon the constant value being the slope of the line. Define that value to
also be the constant of variation because as the slope it tells us the
constant relationship between the two variables.
Have the students consider the following problem. Earlier this
year, your parents purchased 2250 L of diesel for the bulk tank on the farm
at a cost of $749.25. Lately, your parents have not been too impressed
with all of your cruising around town, so they have decided to charge you
the bulk price for all of the diesel that you use. If you put 28 L in your
truck, how much will you owe your parents? (Prices for September 6,
1993).
Get the students to try to solve the problem by viewing it as a direct
variation. They should recognize the constant of variation (value/volume)
as the cost per litre. Remind the students that since the points (2250,
$749.25) and (28, ?) belong to the same variation, then the constant of
variation must be the same, which means that a proportion can be set up and
solved.
Have the students try some of the problems on page 227 in
Mathematics 10 (questions 5-8, 9 extends into squared variations if
time permits) or page 272 - 272 in Mathematics: Principles and Process
(10).
As an added practice, each student can be required to create a
word problem involving direct variation. The problems can then be
exchanged and used as a review.
CELs:
Creative and Critical Thinking
Communication
Evaluation:
Rating scale on positive participation 0 - 5
Homework completion rating scale 0 - 2
Created problem: originality/applicability 5
solving problem 5
Time Line:
1 - 2 hours
Lesson 11: Partial Variation
Overview of Topics
Identifying and working with partial variations.
Using partial variations to solve word problems.
Student Required Materials:
Graph paper
Mathematics: Principles and Process (10)
Teacher Required Materials:
Large graphing grid
Procedures/Activities:
Introduce the topic of partial variation by brainstorming for what
partial means and what in fact partial variation is.
Have the students consider the following question: When ordering
grad jackets in grade 11, the cost for a jacket involves a fixed price for
the jacket alone, and then a variable cost for all lettering that is to be
added. From one company, the cost of a leather grad jacket is $235 plus
$0.80 per letter that is to be added to the sleeve. Write an equation that
calculates the cost of a jacket with any amount of lettering on it.
Encourage the students to develop a table of values and to plot it. From
there, obtaining the equation should be straightforward.
Next, have the students compare the equation for the partial
variation with those that they obtained for problems involving direct
variation. They should notice that the constant term that is added on to
the direct variation form is the y-intercept of the graph and that its
value is the fixed value of the question. From this discussion, they
should conclude that a partial variation's equation will have the form
of:
y = mx + b where y and x are the variables
m is the constant of variation
b is the fixed value
Have the students try some of the problems on pages 286-287 in
Mathematics: Principles and Process (10).
As a review, each student could be asked to create a partial
variation problem.
Ask the students for examples of sequences in their every day
lives. Encourage the students to think of non-mathematically based
examples such as the sequence of steps to putting on a pair of shoes, or
for baking a cake. Next encourage them to think up examples of
mathematical sequences. Define each value in a sequence (or step) as a
term.
Once the students are familiar with the concept of a sequence,
carry out the following concept attainment or something similar. Remind
the students that they should not say what they think the rule is until
they are asked.
Next, without the students saying what they think the rule is, ask them if
the following are yes or no examples:
Finally, ask the students to identify the characteristics of an arithmetic
sequence:
adding between terms only
difference remains the same for all terms
the difference must be non-zero
3. Ask the students to give the
first five terms of the arithmetic sequence with a first term (called a) of
7 and a common difference of -2. As they give you the terms, write them in
a column and in a row form as below:
TERM # TERM VALUE
1 7
2 5
3 3
4 1
5 -1
7, 5, 3, 1, -1, ...
Ask the students if there is some way that a graph of the information
contained in the sequence could be given. The result should be a graph,
with the term # as the independent variable and the term value as the
dependent variable, that is linear. Ask the students to compare the
results with those of partial variations. They should see that the first
term of the sequence is like the fixed value of the partial variation and
that the common difference between the terms is acting in the same way that
the variable constant did in partial variations. The students should then
be able to write the equation for the sequence:
v = -2n + 7 v is the value of the term
n is the term number
Next, ask the students how they could use this equation to determine the
value of the 78th term.
v = -2(78) + 7
= -156 + 7
= -149
As well, ask them how they could use this equation to determine what term
number has a value of -95.
-95 = -2n + 7
-102 = -2n
51 = n
Develop the general equation for the terms in an arithmetic
sequence, using tn to represent the value of the nth term, n to represent
the term number, a to represent the value of the first term, and d to
represent the common difference.
Ask the students for an example in real life that an arithmetic
sequence would occur.
eg. A test in one of your classes is out of 100%. Short answer questions
account for 60% of the mark, while the remaining 40% is for 20 multiple
choice questions. When you wrote the test, you were confident that you had
answered all of the short answer question correctly, but you are unsure of
how you did on the multiple choice questions. What possible marks could
you get on the test?
This generates the following sequence of numbers:
60, 62, 64, 66,...
You could then ask the students what the values for a and d are and to
write the general equation for a term in the sequence. Ask the students
what a specific term in the sequence represents. You may also want to ask
the students questions like: How many of the multiple choice questions did
you get correct if your exam mark was 84? Is it possible to have a mark of
87? What is your mark if you got 35 of the multiple choice questions
correct?
Have the students do some practice problems on pages 376-378 in
Mathematics 12 or on pages 440-445 in Math Is/6.
CELs:
Creative and Critical Thinking
Communication
Evaluation:
Rating scale on positive participation 0 - 5
Homework completion rating scale 0 - 2
Arithmetic sequences quiz 7
Time Line:
2 - 3 hours
Lesson 13: Arithmetic Means
Overview of Topics:
Definition of an arithmetic mean.
Calculation of arithmetic means.
Student Required Materials:
Math Is/6
Procedures/Activities:
Have the students consider the following problem. You just
received back a test on which there is only a percentage given. You got 8
questions right and got a mark of 67%, while your friend got 11 questions
right and got a mark of 79%. After going over the exam, you feel that you
actually have one question right that was marked wrong. What mark can you
expect to have if you are correct about the marking error? (Assume that
all questions are worth the same amount and that they are marked on a
completely right, or completely wrong scale).
Although there are variety of ways to solve this problem, try to encourage
the students to set the question up as an arithmetic sequence:
Define the terms between any two given terms in an arithmetic
sequence as being the arithmetic means of the numbers. Indicate that there
can be any number of means specified. A brief discussion of what a single
mean is will lead into the ideas of average.
Give the students some sample questions on page 442 in Math Is/6
(questions 18 - 20).
CELs:
Creative and Critical Thinking
Evaluation:
Rating scale on positive participation 0 - 5
Homework completion rating scale 0 - 2
Arithmetic means quiz 4
Time Line:
1/2 - 1 hour
Lesson 14: Arithmetic Series
Overview of Topics:
Finding the sum of an arithmetic series.
Student Required Materials:
Math Is/6
Procedures/Activities:
A useful, general introduction is the "Gauss" problem.
Define a series as being the sum of the terms in a sequence
(finite number of terms). So if a sequence is:
3,7,11,15,29,23
the series is 3+7+11+15+19+23
for which the value is 78.
Give the formulae for the sum of an arithmetic series:
where Sn is the sum of n terms, and a,d, n, and tn are as defined before.
If there is time, or just for the more capable students, the development of
these formulae is beneficial as it gives the students some insight into a
mathematical "trick" that they have not seen before.
Have the students consider the following problem. When you were
born, your Godmother decided to start a trust fund for you. On your first
birthday she put $25 in the account, on your second birthday she put $35 in
the account, on your third birthday she put $45, and so on ($10 increases
per year). How much money will she have put aside by the time you are
16?
Divide students into 3 groups. If possible, it is best to work in
a hallway or gymnasium. Assign the groups as slow, medium, and fast
walkers. Have the students walk for a specific time, about 2 minutes, at
their designated speed. It is their job to keep track of the number of
steps they have taken. Once they are done, have them take their pulse.
This data should then be recorded on a table like the one below:
Plot the data on the large grid and then discuss the results - ask them for
conclusions and interpretation. Eventually this should lead to the idea of
the results being linear and obtaining a 'best fit' line by sight. Some
mention should be given to the fact that there are statistical methods for
determining an exact line of best fit, but for now the line will only be
approximated. Have the students locate a line that they feel is the 'best
fit' line and then determine its equation. Discuss how the equation could
be then used in the interpretation of the data.
Have the students form groups of 3 or 4. Each group is to select
an experiment that will generate a linear scatterplot. (For examples see
Algebra Experiments I: Exploring Linear Functions ‹ anything
with a basically constant rate of change will do). Once their experiment
has met your approval they are to carry it out, record the necessary data,
plot the data, give a written analysis of the results, and draw in the best
fit line and determine its equation. Once this is completed, the group is
to make up a visual display of their results and present it to the class.
To review scatterplots, the following synectic presentation could
be used. Ask students:
How is a scatterplot like a shopping mall?
How is a scatterplot unlike a shopping mall?
How would it feel to be someone attempting to get from one point in the
mall to another?
How would it feel to be a point in a scatterplot? (Include both the
points on the 'best fit' line and not on the 'best fit' line).
What else could you compare a scatterplot to?
Why?
For the last question, you may have them turn in their answers for
marks, or just for assessment of their understanding.
Marking Scheme:
Snap To It!
In your groups of three, each person must assume one of the following roles:
snapper
counter
timer
If your group only has two people in it, the snapper and counter can be
combined into one job.
What to do...
The timer will time out periods of time starting with 5 seconds, and
progressing to 30 seconds at 5 second intervals. During the timings, the
snapper is to snap their fingers (at as constant a speed as is possible) as
many times in the time period as they can. It is the counter's job to
count the number of finger snaps that are done in each timed period and to
record the results in the table below:
Time Interval (Seconds) Number of Snaps
5
10
15
20
25
30
Decide which variable, time or number, is the independent variable
and then graph the result. Although there are no negative values involved,
you should still show all four quadrants, but they do not need to be the
same size.
Turn in your graph. Be sure to clearly identify your group members
on the graph.
Scatterplot Project
Task:
In your group of 3 or 4 people, select an experiment whose data
should generate basically linear data and have it approved by the
teacher.
Carry out the experiment. Carefully collect and record the data
from the experiment.
Analyze the data by:
graphing
sketching the 'best fit' line
finding the equation of the 'best fit' line
writing up conclusions or hypotheses that you feel can be made from
your data and an explanation why
Create a visual presentation for your experiment. It should
clearly represent your data and should explain the results that you were
able to obtain. Neatness and creativity of the presentation are also import
ant.
Create a table of values for the relation y = 3x +2 (3 points minimum),
then graph the results on graph paper.
Table of Values 3
Graph 4
Total 7
Slope and Intercepts quiz
Find the x-intercept, y-intercept and slope of the line defined by the
equation y = 2x + 4.
What's In the Equation of a Line quiz
Write the equation 3x -5y =2 in slope- y-intercept form. Then identify the
slope and the y-intercept.
Graph.
Graphing Without a Table
Graph the equation y = 7x + 5 on the grid provided below without using a
table of values. Be sure to indicate how you are doing this.
Special Lines quiz
What is the slope of a line which is parallel to the line 2x + y = 3?
What is the slope of a line perpendicular to the line y = -3x + 4?
What type of line is given by the equation x = -6?
This unit comes from the The Stewart Resources Centre which provides
library resources and teacher-prepared materials for teachers in Saskatchewan.
To borrow materials or obtain a free catalogue listing unit and lesson
plans contact : | 677.169 | 1 |
High School Courses
Mathematics
Mathematical Interventions: Designed to help bridge the gap between the transition from Middle School to High School. This class specifically targets the key topics that are required in the courses of Algebra I, II and Geometry.
Algebra I: This course is designed in direct alignment to the Common Core State Standards. It gives students the foundation needed for Algebra by focusing on vocabulary, mathematical rules and their applications. Cadets must have successfully acquired and mastered basic pre-algebraic skills. Students will brush the topics of Number and Quantity, Algebra, Functions, Modeling and Statistics and Probability.
Algebra II: Algebra II takes the concepts covered in Algebra I and Geometry and dives in-depth into them using real-world application. There is a strong focus on Functions and Modeling within this course. Units that are covered in Algebra II have a strong STEM alignment and specifically prepare students for life beyond high school, specifically college and career readiness.
Geometry: This course is aligned with the Common Core State Standards and covers an array of topics including logic and proofs, vocabulary, congruence, theorems, trigonometry, 3-Dimensional shapes, transformations, and properties of circles and parallelograms.
Pre-Calculus: This course is designed for the preparation of students to transition to college level mathematics by taking the concepts of Calculus and breaking them into their smaller sub-topics. Students will briefly cover concepts in functions, matrices, vectors and trigonometry and complex number systems.
Transitional Algebra: This course prepares students for their college level entrance exams and required mathematics courses in post-secondary placement. It helps ease the transition from high school to college and university level by first reviewing concepts at the heart of Algebra and Geometry, but also exploring in depth more advance concepts in the fields of Trigonometry, Calculus, and Probability and Statistics. Personal Finance: Personal Finance real world consumer rights, responsibilities and information, protect personal and family resources, income management, spending, savings, investments, and apply procedures for managing personal finances.
Science
Biology: This course is designed to introduce students to the essential principles of biology, or the study of life. We will be learning about the structure of living organisms, how those organisms use energy and maintain homeostasis, and how populations will change over time as their environment changes. We will also practice the skills necessary for a future career in science or any field, such as following procedures, reading graphs, interpreting data, and reading analytically.
Physics: This course is designed to introduce students to the essential principles of physics, including motion and energy, energy transformations, Newton's Laws, linear and projectile motion, mechanical and electromagnetic waves, and circuits and electricity. We will also practice the skills necessary for a future career in science or any field, such as following procedures, reading graphs, interpreting data, and reading analytically.
Forensic Science:
Earth & Space Science:
Physical Science: This blended course examines two branches of science: Chemistry and Physics. It explores both matter and energy. In addition this course helps students understand the relationship between Science, Technology, Engineering, & Mathematics.
Health: Students will study the cause and effect of life-style choices that include nutrition and exercise, drug usage, and anatomy. In this course, students will learn about diseases and disorders, environmental health concerns, and how to maintain a healthy life.
Physical Education: This course introduces students to the basic skills needed to maintain a healthy lifestyle through daily exercise. Students will participate in a variety of activities that include cardio-training, stretching, and weight lifting. Additionally, students will learn the basic rules and skills in many different athletic activities that include but are not limited to: kickball, soccer, basketball, volleyball, football, floor hockey and many other physically active games and sports.
English
English 9: The main objective of this course in literature and composition is to enhance the reading, writing, grammatical, and analytical skills of each student 10: The main objective of this course in literature and composition is to enhance the reading, writing, grammatical, and analytical skills of each student in relation to cultural, historical, and sociological context 11: English Language Arts 11 is designed to help you develop the ability to strategically analyze and interpret the world around you through self-reflection and literary analysis. Through your participation in an array of different readings, collaborative projects and learning activities, you will develop a strong ability to think critically, perform close readings of literary texts and construct logical arguments.
English 12: English Language Arts 12 is designed to help you develop a knowledge of textual elements and structures, which will enable you to engage in close readings of complex texts. You will also develop analytical skills and strategies while reading literary texts from a variety of genres, such as short stories and essays, novels, dramas and informational texts.
Literature of Genre: This unique course studies a variety of novels in many different genres and explores their impact on society, self, and various cultures. In addition, it explores how stories and novels are translated to different platforms like plays and movies. It makes comparisons between the different platforms and allows students to decide which ones have the greater personal impact.
Social Studies
U.S. History and Geography: The course will examine a detailed history of The United States using primary and secondary documents. The class will start in the 1800's and travel to today's relevant topics. Students will study in detail geography, technology, people and government. The class will see how the country has changed throughout time and how events from the past have shaped this country to make it what it is today.
Civics and Economics: Civics examines in detail the history of the United States Government as well as the rights of American citizens. Students will learn about the balances of power, the structure of our government and political system as well as compare the U.S. government to other governments around the world.
The Economics course introduces students to consumerism, production, demand, and different markets. Students will compare the U.S. system with others around the world.
World History and Geography: The class will examine a detailed history of the world using primary and secondary documents. The class will start with learning about what a civilization is. The class will examine where humans came from and how they changed over years in time.
Contemporary World Issues:
History of Intolerance:
U.S. Ethnic Studies:
Current Events: This course is designed to allow cadets the opportunity to not only stay current on the events that take place in and around their own neighborhoods but to stay abreast of the daily occurrences of the nation and world. As a globally connected society, especially through social networking it is increasingly important that cadets know what is going on, and how to take proper and immediate action as a citizen of the world.
Character Education
Respect and Responsibility Integrity and Initiative Service and Sacrifice Leadership The main objective of this course is to enhance the pro-social behaviors of students in order to form a positive school community and responsible, caring, and contributing citizens for the world. This course focuses on being conscious of a diverse and dynamic society, and establishing characteristics to work with others to promote social change. Students will use literature, media (global and local news), film, and form personal compositions to develop knowledge and actively demonstrate the Seven Pillars of Character Education: Respect, Responsibility, Integrity, Initiative, Service, Sacrifice, and Leadership. Through diverse learning activities, including projects, discussions, collaborative assignments, and independent practice, students will learn how to improve their ethical decision – making skills and become moral agents of society.
College and Career Readiness
Reading and Writing Strategies for Standardized Tests: Students taking this course will learn a variety of test-taking strategies, writing and reading skills, and proper preparation to help our them achieve not only entrance into a college of their choosing but possible scholarship opportunities. There will be a strong focus on reading stamina, vocabulary in context, citing references, and argumentative writing using evidence to support the claim.
Mathematical Strategies for Standardized Tests: The course is designed to familiarize our students with the tests required for admission by many universities. Well-prepared students are more likely to score higher on the ACT/SAT, which may increase their chances of receiving scholarships and enable them to have more options when selecting a college. Students will learn test-taking strategies, review vocabulary in context, math, and take practice tests and discover ways to reduce test anxiety.
College and Career Readiness: During the course of the year, seniors in this class will begin the process of choosing their next pathway. Students will be filing for parchment, applying for financial aid and scholarships/grants, and researching academic and career pathways based on their family, financial, and personal needs. It is DPSA's goal to have 100% placement of all seniors in a secondary pathway whether that is a college, university, vocational school or job training program.
Leadership and Activities: The main objective of this course is to allow students the opportunity to organize and plan events independently. They will go through the process of fundraising, organizing small and large group events, and collaborating with not only the entire student-body, but the community surrounding the school.
Workplace Experience: Students in workplace experience will research different careers, create resumes and cover letters and eventually apply for jobs. They will also meet professionals from various backgrounds and visit different jobs to further their research. We hope to broaden student's experiences in the classroom and their expectations for the next step by showing the variety of occupations out there.
Entrepreneurship: In this course students begin learn the in's and out's of business from developing an idea, the different types of markets, how to promote your product, etc. They will use this foundational learning to then create a proposed business plan.
Middle School Courses
6th Grade
Mathematics In (2017 Common Core Initiative). Mathematical Interventions
Science
English
Social Studies
Reading and Writing Interventions
Physical Education
7th Grade
Mathematics Students will apply and extend previous understands of operations with fractions to add, subtract, multiply and divide rational numbers, analyze proportional relationships and use them to solve real-world and mathematical problems and use properties of operations to generate equivalent expressions. They will also solve mathematical problems using numerical and algebraic expressions, draw, construct, and describe geometrical figures, and solve problems involving angle measure, area, surface area, and volume. Additionally, students will study statistics by using random sampling, inductive/deductive reasoning skill to develop conclusions.
Mathematical Interventions
Science During the course of the school year we will be covering the 7th Grade Michigan Content Expectations which includes standards focused on Science Processes, Physical Science, Life Science and Earth Science.
Mathematics Students will know that there are numbers that are not rational, and approximate them by rational numbers.They will work with radicals and integer exponents and understand the connections between proportional relationships, lines, and linear equations. Additionally, students will analyze and solve linear equations and pairs of simultaneous linear equations; define, evaluate, and compare functions and use functions to model relationships between quantities. Many geometric concepts will be studied as well. These include, understanding congruence and similarity using physical models, transparencies, or geometry software, applying the Pythagorean Theorem and solving real-world and mathematical problems involving volume of cylinders, cones, and spheres. Lastly, students will investigate patterns of association in bivariate data.
Mathematical Interventions
Science During the course of the school year we will be covering the 8th Grade Michigan Content Expectations which includes standards focused on Earth Science. | 677.169 | 1 |
An electronic guide to some of the best ways to study
Reading
Academic material needs to be read for understanding, not entertainment: satisfaction and enjoyment will follow.
3.1
You are here to read for a degree - you will need to do some reading as well as attending lectures! Don't put it off until later … it will never get done if you do. Buy all the first and second level core books yourself (perhaps second-hand – you'll be able to sell them on later), but share/borrow more advanced books.
3.2 The syllabuses (available from the lecturer, General Office or Brunel's Web pages) usually contain reading lists. You will rarely need to read the whole book but you will need to know where to find the information you need. Very few modules follow a book completely in set order, so learn to use the contents and index pages effectively.
3.3 Mathematics, science and technology modules generally require far less reading than most other subjects. They stress mastery of a few concepts instead. On the other hand, books can and do help by giving an alternative view, extra examples, different proofs etc.
Seek to understand, rather than memorise, facts. The only exceptions here are that you do need to memorise a few key formulae and all definitions (exactly); for example, you may need to memorise the definition of continuity. That's only a start though - you still need to understand and be able to apply each part of the definition.
3.4 Books do not read themselves! Buy a book that your existing knowledge will allow you to understand, even if it is "elementary", and study it. Do not ignore less advanced books, such as maths books for engineers, which are often a good start, since they stress the main points and give lots of examples; follow this with the more advanced set books which will give you the detail and level of treatment required. Advanced texts may look good on your bookshelf but are of most use later when you have already studied the topics covered in class and want a reference text for e.g. project work.
3.6 For content-rich material with a lot of facts/data, it may help to scan the relevant sections first to get the gist of them, and see in broad terms the logical structure of the content i.e. what depends on what, what follows from what, etc. However, scanning and speed reading are unlikely to work for concept-rich material, such as pages of mathematics, unless you already know quite a lot about the material already. You can also try blanking out some of the material with a sheet of paper, to focus your attention. Dyslexics sometimes find coloured filters place over the page are helpful.
3.7 Read the relevant section carefully, concentrating on the main ideas and subsidiary points. Jot down the main pointers in the argument very briefly. Study the diagrams carefully. Note that it is unrealistic to expect to read more than a few pages of content-rich material or new mathematics at one sitting, unless you are already familiar with the general topic and are revising. If you get stuck you should; go back a few pages to see what you are missing, have another look at the contents page to see where your reading is leading and where it came from, and finally seek help from your lecturer.
3.8 Make some notes about the material to aid your understanding, but do not simply copy things from the book which you don't understand or to have as a record of the material (use a scanner or photocopier for that if you do not have the book). File the notes you have made with other notes from the same topic. Use post-it notes rather than highlighter pens to mark text, so that you can change your mind later.
3.9It's important to do something with what you have read! Rather than endlessly re-reading in the hope that it will go in somehow, try a few of questions, noting anything you are not clear about so that you can ask people (friends, staff, maths support staff etc.). You need to get your hands dirty with the content - if you really do this, then it will go in. There's no other way - learning does require engagement and it's generally pretty hard.
3.10Concentrate on what is relevant for your purpose, as it is inefficient to read a whole book when all you need is information from a chapter. Set manageable targets for your reading, e.g. a section each time. Keeping your purpose in mind is especially important when using the web; there is always another tempting link to follow and you could end up wasting a lot of time. | 677.169 | 1 |
Continuity of Functions Practice
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Looking for a great way to develop student understanding of continuity and discontinuity in functions? This one-page practice worksheet can be used after introducing the concept of continuity in functions. It can also be used to help introduce students to discontinuities in rational functions. Students recognize how the functions are similar to but different from graphs of functions they already know. | 677.169 | 1 |
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This worksheet contains a total of 54 practice questions about radicals.
Students must examine each radical and express it in the form of another radical.
Please look at the preview and the screenshots to get a better idea of the layout and style of questions.
There are actually two versions of this worksheet: one with 70 questions (which I though looked a bit cramped) and another with 54 questions (which looks a bit neater). Both are included, with answers, for your convenience. | 677.169 | 1 |
£6.62 student-friendly approach to KS3 This coursebook covers topics appropriate for KS3 Year 9 Maths and accurately reflects the language and content of the new Programme of Study. Along with the year 7 and 8 coursebooks full coverage of the KS3 programme of study is provided. | 677.169 | 1 |
College Algebra
Do your students attempt to memorize facts and mimic examples to make it through algebra? James Stewart, author of the worldwide, best-selling calculus texts, saw this scenario time and again in his classes. So, along with longtime coauthors Lothar Redlin and Saleem Watson, he wrote COLLEGE ALGEBRA specifically to help students learn to think mathematically and to develop genuine problem-solving skills. Comprehensive and evenly-paced, the text has helped hundreds of thousands of students. Incorporating technology, real-world applications, and additional useful pedagogy, the sixth edition promises to help more students than ever build conceptual understanding and a core of fundamental skills. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version. | 677.169 | 1 |
In this worksheet packet you have a worksheet for Common Core standard 7.EE.B.4.a and a worksheet for Common Core standard 7.EE.B.4.b. The first worksheet includes 15 word problems, while the second worksheet has 20 word problems. An answer key is
In this 18 question worksheet students will answer questions about complementary, supplementary, vertical, and adjacent angles. They will also find missing angles. This worksheet aligns with Common Core standard 7.G.B.5. An answer key is included. 8
In this 15 problem worksheet students graph and interpret graphs of proportional relationships. They also compare graphs to equations with engaging problems on Timmy, Chet, and Chad's bacon consumption. There are also problems comparing Timmy and
In this 15 question worksheet Timmy and his friends tackle the Common Core standard 8.F.A.2 in their adventures skiing, biking, and tutoring others in math. Functions are represented through graphs, tables, equations, and verbal descriptions as
Includes:
-12 forty five minute lesson plans
-Essential Questions for each major standard 7.RP.A.1, 7.RP.A.2, and 7.RP.A.3
-Engaging optional homework assignments
-Three quizzes, one for each major standard 7.RP.A.1, 7.RP.A.2, and 7.RP.A.3 with
The worksheet and quiz package contains 5 worksheets that align with Common Core standard 7.EE.A.1, 5 worksheets that align with Common Core standard 7.EE.A.2, and 3 quizzes, one for just standard 7.EE.A.1, one for just standard 7.EE.A.2, and one
This package contains worksheets, quizzes, and one unit test for the Expressions and Equations unit of the 7th grade Common Core standards. There are five worksheets for standard 7.EE.A.1, five worksheets for standard 7.EE.A.2, 3 quizzes for
This 15 question worksheet aligns with Common Core standard F.IF.C.9. In this worksheet students complete 15 questions (entirely on quadratics) comparing functions in different forms (algebraically, by graphs, tables, and verbal descriptions). There
This worksheet contains 20 questions aligned with Common Core standard 7.EE.B.3. In these problems Timmy ventures from Belmont to Vail enjoying several adventures with Sarah and Chet. This worksheet can be used as a class work or homework
This five worksheet packet covers standards 8.EE.C.7.a and 8.EE.C.7.b. The first worksheet is a mixed worksheet of both standards. The next two worksheets align with Common Core standard 8.EE.C.7.a starting with a level 1 and going to a level 2. The
This 20 question unit test covers all the standards of the 8th grade Common Core functions unit. There are four questions for each of the standards: 8.F.A.1, 8.F.A.2, 8.F.A.3, 8.F.B.4, and 8.F.B.5. An answer key is included. 10 page PDF.
Get all of
This 15 question worksheet aligns with Common Core standard 8.F.B.5. Students must decide if functions are linear or nonlinear, increasing or decreasing based upon graphs, verbal descriptions, and equations. Students also makes sketches of
This five worksheet packet aligns with Common Core standard 6.EE.A.2.b. In this worksheet students find the factors, terms, and coefficients of expressions as they work through four differentiated worksheets and a mixed level worksheet. They also
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TEACHING EXPERIENCE
10 years experience working in public schools, the last 6 as a math teacher in middle and high schools
MY TEACHING STYLE
I teach to many different learning styles and like to use a theme based approach when possible!
HONORS/AWARDS/SHINING TEACHER MOMENT
Former member Pi Lambda Theta
MY OWN EDUCATIONAL HISTORY
Master's Curriculum and Instruction, UCONN, 2010
ADDITIONAL BIOGRAPHICAL INFORMATION
I started teaching in an inner city school in Hartford, CT. After two years there I moved to Massachusetts where I taught higher level students for a year. Then my wife and I took the plunge and moved to the mountains of Colorado. The first two years there I worked at an alternative high school developing theme based courses. After that I transferred to another high school in the valley. I've also taught summer school for four years. From my wide variety of teaching experiences I've learned what works and what doesn't and aim to help out new teachers just getting started. You can visit my website at | 677.169 | 1 |
Business
Economics
Mathematics
This chapter on factorization provides the knowledge on the method of the factors, the different identities which can be used in factorization are also introduced in this chapter, the topic on factorization by regrouping terms helps the students to factorize the expression without identities and the concept of the division of the polynomials and monomials are explained in subsequent topics. | 677.169 | 1 |
Book Description Pearson, United States, 2009. Loose-leaf. Book Condition: New. 2nd. Language: English . This book usually ship within 10-15 business days and we will endeavor to dispatch orders quicker than this where possible. Brand New Book. For courses in Geometry or Geometry for Future Teachers. This popular book has four main goals: 1. to help students become better problem solvers, especially in solving common application problems involving geometry; 2. to help students learn many properties of geometric figures, to verify them using proofs, and to use them to solve applied problems; 3. to expose students to the axiomatic method of synthetic Euclidean geometry at an appropriate level of sophistication; and 4. to provide students with other methods for solving problems in geometry, namely using coordinate geometry and transformation geometry. Beginning with informal experiences, the book gradually moves toward more formal proofs, and includes special topics sections. Bookseller Inventory # BTE9780321656773 | 677.169 | 1 |
This eleven-sheet set provides a comprehensive review of all of the most important topics, concepts and formulas in trigonometry, combined with the prerequisite material from geometry, explained in detail, with detailed illustrations. | 677.169 | 1 |
Brief Overview of Course:
This one semester course focuses on developing students' ability to look for patterns in mathematics. Each class starts with a small logic puzzle. There are opportunities for interim assessments and final PBATs (performance-based assessment tasks) as well as quizzes, a midterm and a final examination. In addition, there are many practical applications. The following topics are covered:
- nth terms
- letter equations with restrictions
-fractional exponents
-negative exponents
-use and proof
-coordinate geometry
-slopes
-2 point equations
-midpoints
-lengths
-circles
-complete the square
-logic puzzles
-2 equations with 2 unknowns
-substitution
-elimination
-matrices
- factor
- imaginary numbers
- rationalize simple roots
Logic puzzle
1 6 Urban students ran a race. Each wore a different color. There were no ties.
2 Amanda lost to blue.
3 The 6-letter named runners came in consecutive order and one was green.
4 Alison lost to the runner in red but beat Iliriana.
5 Iliriana beat Mac and white but lost to Amanda.
6 The runner in white lost to Mac who lost to Amanda.
7 Heru beat Amanda by 2 places.
8 Courtney was not orange and lost to Iliriana.
9 The rider in blue lost to the rider in red but beat the rider yellow.
A Name the 6 runners in order of finish and each color each wore.
B You must prove that your solution is the only possible conclusion, in good order. Use the numbered sentences as evidence for each statement you make. If you use charts or partial charts, these must be fully explained.
Please read all of these instructions before starting.
It is known that water freezes at 32 degrees Fahrenheit (F) or 0 degrees Celsius (C) and boils at 212 degrees F or 100 degrees C. Show that the two temperature scales F and C are linear related by completing the following steps. Be sure to explain all of your procedures and show your calculations.
1) Find a linear equation that expresses F in terms of C. Neatly draw a graph of this equation. Label the axes, label the line with your equation and title your graph.
2) If a European family sets its house thermostat at 20 degrees C, what is the setting in degrees F? Find a linear equation that expresses C in terms of F. If the outside temperature in Milwaukee is 86 degrees F, what is the temperature in degrees C?
3) Explain what the slope in question 1) means in terms of converting Celsius to Fahrenheit.
4) Since we know water freezes at 32 degrees F and 0 degrees C and boils at 212 degrees F and 100 degrees C, explain why -40 C = -40 F.
Note: Show all work neatly on separate sheets.
Significant Assignments:
The Pirates
Redbeard, Graybeard and Bluebeard were separated while being chased by the French Navy. Graybeard found himself at (-2, 13) (see the G). Bluebeard at (-12, 7), and Redbeard at (-3, -9). Redbeard took a course of Y = 2X -3 and Bluebeard took a course Y = 0X + 7 (or Y = 7) and Graybeard took a course of 7-3X = Y. When Bluebeard and Greybeard met, they continued on Graybeard's course till they met Redbeard. Then all three took course Y = X/2 till they came to Treasure Island.
1) Locate the (0, 0) center point. Put a C .
2) Locate Bluebeard's and Redbeard's starting points put a B (-12, 7) and an R (-3, -9)
3) Chart Bluebeard's course till he met Graybeard at ( , ). Use X's on the graph.
Chart Graybeard's course till he met with Redbeard at ( , ). Use O's on the graph.
Chart Redbeard's course till he met the other two at ( , ). Use +'s on the graph.
4) Chart the new course to Treasure Island and place a T at ( , ). Use *'s 0n the Graph.
Show all work on separate sheets.
Significant Activities or Projects:
The Year Book
Here is another puzzle for you to work on. Try and use algebra this time – it is a short cut for trial and error. Use the reference sentence numbers in your setup.
1) 8 students collect $88 for Urban's Year Book.
2) Anthony collected $2 more than Yan Mei.
3) Jesse collected twice as much as Yan Mei.
4) Levi collected the average (mean) amount.
5) Lily collected $2 more than Jesse.
6) Rachel collected the same amount as Jesse.
7) Sasha collected $1 less than Yan Mei
8) William collected as much as Sasha and Jesse together.
HOW MUCH DID EACH COLLECT?
Sample PBATs:
Series of problem solving questions based on application of polynomials and graphing: Construction Using Circle Equation and Graphing For example: Town B is located 36 miles East and 15 miles north of town A. A local phone company wants to position a relay tower so the distance from the tower to town B is twice the distance from the tower to town A. 1) In order to find all the possible tower locations show that they must lay on a circle. Find the center and radius of this circle, and graph it. Be sure to title your graph and label both axes. Think about what your scale needs to be. 2) If the company decides to position the tower on this circle at a point directly East of Town A, how far from town A should they place the tower? Show how this location meets the criteria set out above. 3) Explain in your own words each step you did and why. Include in your explanation why you used certain formulas. 4) Write a brief analysis about why your answer makes sense. This should include a logical explanation. | 677.169 | 1 |
Overview Algebra 1 covers all topics in a first-year algebra course, from proofs, statistics, and probability to algebra-based, real-world problems. With Algebra 1, students begin developing the more complex skills and understanding required for high school level mathematics Algebra 1: An Incremental Development : Home Study (Homeschool Algebra) by John H. & Jr. Saxon 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.
Our children used Saxon from 54 to 87, then moved on to advanced math, calculus and physics and they have excelled with this method. Although my background doesn't include an emphasis in math, my husband's education and professional life is steeped in mathematics. He's enthusiastic about Saxon because it creates a strong foundation in the subject.
Admittedly, solving 30+ problems a lesson can be a challenge, however, this process increases one's speed and accuracy over time and as my daughter said, it helped her "to make peace with math." Math is like learning how to play a musical instrument; it takes practice and self-discipline, but it's well worth the effort. Understanding math, like being proficient at reading and writing, is one of those practical skills that make life so much easier.
Using this incremental method of learning made homeschooling through high school a breeze and our college-age children sailed through their college math courses as well. In hindsight, it would be easy to choose it again
Math teacher loves Saxon May 10, 2008
In my experience teaching in the high school classroom, I discovered that most students can quickly learn a new skill, but if they're using a curriculum that doesn't require them to use that skill for more than a few days, they can just as quickly forget it. Saxon math books do what few others do -- through the continual review implemented in the problem sets, students are able to retain skills for the long term. Isn't that the whole point of studying Algebra 1, assuming that students plan to move on to Algebra 2 and beyond? Problems get more difficult over time, because with mastery of a skill, the student is ready to take it to the next level. Higher level math and science courses require a student to think through complex problems, not to simply "plug and chug" through a formula, and Saxon is sufficiently rigorous to prepare a student to analyze and reason his/her way to a solution. I should add that I have two children who have completed this course and have blown the doors off standardized tests. (Their mother made certain they completed their assignments, however, and I suspect that contributed to their success.)
The Proof is in the Pudding May 8, 2008
I have homeschooled my five children using this series. They consistently get high math grades on the standardized California Achievement Tests. My three eldest have each scored in the upper 600's on the SAT tests for math. Even my right-brained, artist, writer daughter who HATES math scored 670 on her SAT. Review, review, review is key! It maybe boring, but it is very effective. As I said, the proof is in the pudding.
A Students Review Mar 28, 2008
Ahhhhhh!
Attack of the large, hardcover, algebraical, mathematical and downright wicked text book! If you happen to see this book in person, I suggest you turn and run as positively fast as you can... AWAY! If you don't, it will open up and suck you into it's very pages, bombarding you with polynomials and rectagular coordinate systems, not to mention the villianous cronies known as fractions, and their wicked counterparts, the simple geometric solids!
I have warned you! Beware!
Learning to TEST or Learning to UNDERSTAND? Apr 28, 2007
My mother is a Ph.D. in mathematics and taught Jr. and Sr. High math for several years before moving up to teach college math. She has been pretty vocal that the only math text that will result in imparting a poor understanding of mathematical concepts--a false sense of mastery while using it, but poor retention after--is Saxon. She says that every time she has a home schooled student who is really struggling at the college level and they say "But I did so well in math before!" and they are traumatized at the level of tutoring help they need to make it in college, they all have in common the fact that they learned math using Saxon texts in high school.
After she impressed this on me, I was really leery about choosing jr. & sr. high school curriculum a couple years ago and asked her to go to me with convention to help me pick something out. She said, "You are good at math and a good teacher. Just pick something you like that is NOT SAXON!" I'm not exaggerating. It's the spiral learning method that they use. It doesn't give enough thorough practice of all the variations of a particular concept before moving on and too heavily relies on review throughout. That seems to impedes long-term retention. She thinks the fact that it is so dull and methodical is also ridiculous in this day and age of fabulous graphics and the trend to make math more interesting and multi-modal for the average student who doesn't love math.
I find it interesting that on their website, of the 6 research studies of their curriculum, only one includes high school; the other five utilized k-8 or 6-8 curriculum. Maybe all that dry rote learning makes a student test better. But the sad part is when it comes to taking that learning and building on it, they don't really understand the concepts behind it and can't apply future learning to what they simply practiced over and over but don't really know. Kind of like cramming for a test by going over everything you've learned right beforehand and blocking everything else out until you take the test and then POOF! everything you repeated over and over in your head beforehand just seems "gone" once you go back to normal habits of thinking/doing and you stop all that repetition | 677.169 | 1 |
Functions Class Activity or Stations
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This activity asks students to analyze graphs, tables, mapping diagrams, and ordered pairs to decide if they are a function or not, if it is continuous or discrete, and what the domain and range are. 27 individual cards and a student answer sheet. I use this activity as a station with my pre-algebra class. This can be used to review the concept or in conjunction with other task card sets or station sets to review algebraic concepts. | 677.169 | 1 |
Basic Mathematics For College Students With Early Integers - beje.herokuapp.com
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Sunday, August 5, 2012
Texas Instruments TI-83 Plus pdf datasheet download
Operating the TI-83 Plus Silver Edition Documentation Conventions In the body of this guidebook, TI-83 Plus (in silver) refers to the TI-83 Plus Silver Edition. Sometimes, as in Chapter 19, the full name TI-83 Plus Silver Edition is used to distinguish it from the TI-83 Plus. All the instructions and examples in this guidebook also work for the TI-83 Plus. All the functions of the TI-83 Plus Silver Edition and the TI-83 Plus are the same. The two calculators differ only in available RAM memory and Flash application ROM memory.
Using the Color.Coded Keyboard The keys on the TI-83 Plus are color-coded to help you easily locate the key you need. The light gray keys are the number keys. The blue keys along the right side of the keyboard are the common math functions. The blue keys across the top set up and display graphs. The blue Œ key provides access to applications such as the Finance application. The primary function of each key is printed on the keys. For example, when you press , the MATH menu is displayed. Using the y and ƒ Keys The secondary function of each key is printed in yellow above the key. When you press the yellow y key, the character, abbreviation, or word printed in yellow above the other keys becomes active for the next keystroke. For example, when you press y and then , the TEST menu is displayed. This guidebook describes this keystroke combination as y:. | 677.169 | 1 |
Overview
Geometry of Complex Numbers by Hans Schwerdtfeger
"This book should be in every library, and every expert in classical function theory should be familiar with this material. The author has performed a distinct service by making this material so conveniently accessible in a single book." — Mathematical Review Since its initial publication in 1962, Professor Schwerdtfeger's illuminating book has been widely praised for generating a deeper understanding of the geometrical theory of analytic functions as well as of the connections between different branches of geometry. Its focus lies in the intersection of geometry, analysis, and algebra, with the exposition generally taking place on a moderately advanced level. Much emphasis, however, has been given to the careful exposition of details and to the development of an adequate algebraic technique. In three broad chapters, the author clearly and elegantly approaches his subject. The first chapter, Analytic Geometry of Circles, treats such topics as representation of circles by Hermitian matrices, inversion, stereographic projection, and the cross ratio. The second chapter considers in depth the Moebius transformation: its elementary properties, real one-dimensional projectivities, similarity and classification of various kinds, anti-homographies, iteration, and geometrical characterization. The final chapter, Two-Dimensional Non-Euclidean Geometries, discusses subgroups of Moebius transformations, the geometry of a transformation group, hyperbolic geometry, and spherical and elliptic geometry. For this Dover edition, Professor Schwerdtfeger has added four new appendices and a supplementary bibliography. Advanced undergraduates who possess a working knowledge of the algebra of complex numbers and of the elements of analytical geometry and linear algebra will greatly profit from reading this book. It will also prove a stimulating and thought-provoking book to mathematics professors and teachers.
A thorough, complete, and unified introduction, this volume affords exceptional insights into coordinate geometry. It
makes extensive use of determinants, but no previous knowledge is assumed; they are introduced from the beginning as a natural tool for coordinate geometry. Invariants ...
Derived from a course in fluid mechanics, this text for advanced undergraduates and beginning graduate
students employs symmetry arguments to demonstrate the principles of dimensional analysis. The examples provided illustrate the effectiveness of symmetry arguments in obtaining the mathematical form ...
This incisive text deftly combines both theory and practical example to introduce and explore Fourier
series and orthogonal functions and applications of the Fourier method to the solution of boundary-value problems. Directed to advanced undergraduate and graduate students in mathematics ...
This classic text features a sophisticated treatment of Fourier's pioneering method for expressing periodic functions
as an infinite series of trigonometrical functions. Geared toward mathematicians already familiar with the elements of Lebesgue's theory of integration, the text serves as an ...
The teaching of mathematics has undergone extensive changes in approach, with a shift in emphasis
from rote memorization to acquiring an understanding of the logical foundations and methodology of problem solving. This book offers guidance in that direction, exploring arithmetic's ...
Few of the sacred texts of the world's great religions present their wisdom with the
clear simplicity of the verses of the Buddhist Dhammapada, or Path to Virtue. Its direct style, clarity, and beauty place it at the forefront of ... | 677.169 | 1 |
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the October/November 2008 question paper
9709 MATHEMATICS
9709/03 Paper 3, maximum raw mark 75. All Examiners are instructed that alternative correct answers and unexpected approaches in candidates' scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes must be read in conjunction with the question papers and the report on the examination.
•
CIE will not enter into discussions or correspondence in connection with these mark schemes.
CIE is publishing the mark schemes for the October/November 2008 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
Page 2
Mark Scheme GCE A/AS LEVEL – October/November 2008
Syllabus 9709
Paper 03
Mark Scheme Notes Marks are of the following three types: M Method mark, awarded for a valid method applied to the problem. Method marks are not lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated method mark is earned (or implied). Mark for a correct result or statement independent of method marks.
A
B •
When a part of a question has two or more "method" steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly when there are several B marks allocated. The notation DM or DB (or dep*) is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. The symbol √ implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. A and B marks are not given for fortuitously "correct" answers or results obtained from incorrect working. Note: B2 or A2 means that the candidate can earn 2 or 0. B2/1/0 means that the candidate can earn anything from 0 to 2.
•
•
The marks indicated in the scheme may not be subdivided. If there is genuine doubt whether a candidate has earned a mark, allow the candidate the benefit of the doubt. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. • Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise. For a numerical answer, allow the A or B mark if a value is obtained which is correct to 3 s.f., or which would be correct to 3 s.f. if rounded (1 d.p. in the case of an angle). As stated above, an A or B mark is not given if a correct numerical answer arises fortuitously from incorrect working. For Mechanics questions, allow A or B marks for correct answers which arise from taking g equal to 9.8 or 9.81 instead of 10.
The following abbreviations may be used in a mark scheme or used on the scripts: AEF AG BOD CAO CWO ISW MR PA SOS SR Any Equivalent Form (of answer is equally acceptable) Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) Benefit of Doubt (allowed when the validity of a solution may not be absolutely clear) Correct Answer Only (emphasising that no "follow through" from a previous error is allowed) Correct Working Only - often written by a 'fortuitous' answer Ignore Subsequent Working Misread Premature Approximation (resulting in basically correct work that is insufficiently accurate) See Other Solution (the candidate makes a better attempt at the same question) Special Ruling (detailing the mark to be given for a specific wrong solution, or a case where some standard marking practice is to be varied in the light of a particular circumstance)
Penalties
MR -1
A penalty of MR -1 is deducted from A or B marks when the data of a question or part question are genuinely misread and the object and difficulty of the question remain unaltered. In this case all A and B marks then become "follow through √" marks. MR is not applied when the candidate misreads his own figures - this is regarded as an error in accuracy. An MR-2 penalty may be applied in particular cases if agreed at the coordination meeting. This is deducted from A or B marks in the case of premature approximation. The PA -1 penalty is usually discussed at the meeting.
Equate derivative to zero and reach tan x = k Solve for x Obtain x = − 1 π (or −0.785) only (accept x in [−0.79, −0.78] but not in degrees) 4 [The last three marks are independent. Fallacious log work forfeits the M1*. For the M1(dep*) the solution can lie outside the given range and be in degrees, but the mark is not available if k = 0. The final A1 is only given for an entirely correct answer to the whole question.]
dy dx = a (2 − 2 cos 2θ ) or = 2a sin 2θ dθ dθ dy dy dx = ÷ Use dx dθ dθ dy sin 2θ Obtain , or equivalent = dx (1 − cos 2θ ) Make use of correct sin 2A and cos 2A formulae Obtain the given result following sufficient working [SR: An attempt which assumes a is the parameter and θ a constant can only earn the two M marks. One that assumes θ is the parameter and a is a function of θ can earn B1M1A0M1A0.] [SR: For an attempt that gives a a value, e.g. 1, or ignores a, give B0 but allow the remaining marks.]
EITHER: Attempt division by 2 x 2 − 3x + 3 and state partial quotient 2x Complete division and form an equation for a Obtain a = 3 OR1: By inspection or using an unknown factor bx + c, obtain b = 2 Complete the factorisation and obtain a Obtain a = 3 Find a complex root of 2 x 2 − 3 x + 3 = 0 and substitute it in p(x) Equate a correct expression to zero Obtain a = 3 Use 2 x 2 ≡ 3x − 3 in p(x) at least once Reduce the expression to the form a + c = 0, or equivalent Obtain a = 3
OR2:
OR3:
[3]
(ii) State answer x < − 1 only 2
2
M1 Carry out a complete method for showing 2 x − 3x + 3 is never zero 2 A1 Complete the justification of the answer by showing that 2 x − 3x + 3 > 0 for all x [These last two marks are independent of the B mark, so B0M1A1 is possible. Alternative methods include (a) Complete the square M1 and use a correct completion to justify the answer A1; (b) Draw a recognizable graph of y = 2 x 2 + 3x − 3 or p(x) M1 and use a correct graph to justify the answer A1; (c) Find the x-coordinate of the stationary point of y = 2 x 2 + 3x − 3 and either find its y-coordinate or determine its nature M1, then use minimum point with correct coordinates to justify the answer A1.] [Do not accept ≤ for < ] 6 (i) State or imply at any stage answer R = 13 Use trig formula to find α Obtain α = 67.38° with no errors seen [Do not allow radians in this part. If the only trig error is a sign error in sin(x + α) give M1A0.] B1 M1 A1
[3]
[3]
11 B1√ (ii) Evaluate sin −1 correctly to at least 1 d.p (57.79577…°) 13 Carry out an appropriate method to find a value of 2θ in 0° < 2θ < 360° M1 Obtain an answer for θ in the given range, e.g. θ = 27.4° A1 Use an appropriate method to find another value of 2θ in the above range M1 Obtain second angle, e.g. θ = 175.2° and no others in the given range A1 [Ignore answers outside the given range.] [Treat answers in radians as a misread and deduct A1 from the answers for the angles.] [SR: The use of correct trig formulae to obtain a 3-term quadratic in tan θ, sin 2θ, cos 2θ, or tan 2θ earns M1; then A1 for a correct quadratic, M1 for obtaining a value of θ in the given range, and A1 + A1 for the two correct answers (candidates who square must reject the spurious roots to get the final A1).]
State or imply a correct normal vector to either plane, e.g. 2i –j –3k , or i + 2j +2k Carry out correct process for evaluating the scalar product of the two normals Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result Obtain answer 57.7° (or 1.01 radians)
dV dh dV = 4h 2 = 4h 2 , or equivalent , or dt dt dh dV 2 = 20 − kh State or imply dt Use the given values to evaluate k Show that k = 0.2, or equivalent, and obtain the given equation [The M1 is dependent on at least one B mark having been earned.]
Make recognizable sketch of a relevant exponential graph, e.g. y = e 2 + 2 Sketch a second relevant straight line graph, e.g. y = x, or curve, and indicate the root Consider sign of x − e 2 − 2 at x = 2 and x = 2.5, or equivalent Justify the given statement with correct calculations and argument Use the iterative formula x n +1 = 2 + e 2 n correctly at least once, with 2 ≤ x n ≤ 2.5 Obtain final answer 2.31 Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval (2.305, 2.315) State that the modulus of w is 1 State that the argument of w is 2 π or 120° (accept 2.09, or 2.1) 3 State that the modulus of wz is R State that the argument of wz is θ + 2 π 3 State that the modulus of z/w is R State that the argument of z/w is θ − 2 π 3
−1 x
−1 x
B1 B1 M1 A1 M1 A1 A1 B1 B1 B1√ B1√ B1√ B1√ B1 B1
[2]
(iii)
[2]
(iv)
[3]
10 (i)
[2]
(ii)
[4]
(iii)
State or imply the points are equidistant from the origin State or imply that two pairs of points subtend 2 π at the origin, or that all three pairs subtend 3 equal angles at the origin | 677.169 | 1 |
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2 3| 4 5 6 | 7 8 July | 2 | | | |
9 10| 11 12 13 | 14 15 13. hourly| 3 | * | | |
16 17| 18 19 20 | 21 22 | 4 | | * | |
23 24| 25 26 27 | 28 29 27. hourly| 5 | * | | |
30 31| 1 2 3 | 4 5 August | 6 | | * | |
6 7| 8 9 10 | 11 12 | 7 | | | * |
13 14| 15 16 17 | 18 19 15. final | | * | | |
+----------+ +-------------------+
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Steven Lay is a Professor of Mathematics at Lee University in Cleveland, TN. He received M.A. and Ph.D. degrees in mathematics from the University of California at Los Angeles. He has authored three books for college students, from a senior level text on Convex Sets to an Elementary Algebra text for underprepared students. The latter book introduced a number of new approaches to preparing students for algebra and led to a series of books for middle school math. Professor Lay has a passion for teaching, and the desire to communicate mathematical ideas more clearly has been the driving force behind his writing. He comes from a family of mathematicians, with his father Clark Lay having been a member of the School Mathematics Study Group in the 1960s and his brother David Lay authoring a popular text on Linear Algebra. He is a member of the American Mathematical Society, the Mathematical Association of America, and the Association of Christians in the Mathematical Sciences17474711747471
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As a counselor, you're often the first and most trusted source of advice for your high school students' academic career. Parents may not be knowledgeable about specific course requirements (although they are often concerned) and teachers might not have the big-picture view that you do. For this reason, it's important that you encourage your students to take the math classes they need to get into the IT program of their choice.
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Four years of math in high school should be every IT student's goal. Even if a student feels they are "strong" in math and don't need to take a fourth year of it, remind them that they'll be even stronger if they continue learning and practicing.
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Programming and Software Development
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Grade 9: Algebra
Grade 10: Geometry
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Grade 12: Statistics or Precalculus
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Grade 9: Algebra
Grade 10: Geometry
Grade 11: Algebra 2 or Statistics
Grade 12: Statistics or Precalculus
Although Algebra 2 OR Statistics are listed as possibilities for 11th grade, the mastery of statistics is highly recommended to increase the likelihood of understanding programming concepts and networking.
While no math class is "required" for 12th grade students pursuing a Systems and Network Administration pathway, be aware that an additional year of math would probably be advisable. For example, if they took Algebra 2 their junior year, taking Statistics their senior year would make sense.
Helping Your Students Deal with Math Anxiety
Math Anxiety is a learned psychological response to math, and is often the result of negative or embarrassing experiences with math early in an individual's education.
Seek help as soon as they need it. Don't be afraid to ask questions during class, and especially, during office hours
Be creative. Use the Internet or other books (besides the classroom textbook) to serve as aids
Keep up with homework assignments and study every night, regardless of whether or not it's "required." Just like lifting weights for ten hours straight won't suddenly turn you into a body builder, cramming for a test the night before won't guarantee you a good grade
Focus on their own performance. Although study groups can be helpful, comparing grades can be counter-productive | 677.169 | 1 |
MA 150 Course Description
NOTE: May not be applied toward a non-teaching major or minor in mathematics.
General Introduction and Goals
This course is designed to examine elementary school mathematics from an advanced standpoint. The emphasis is on the development of the system of real numbers and the language, models, concepts, and operations associated with it. Quantitative thinking skills are developed through applications and problem solving situations. In this course the students will:
compare the characteristics of different numeration systems from a historical perspective; | 677.169 | 1 |
Solve mathematical equations with Microsoft Mathematics 4.0
I think math matter is more complicated for many college students especially, but to help solving mathematical equations Microsoft has released version 4 of its agenda Microsoft Mathematics 4.0 which is a mathematical software which provides a set of tools for students to them is easier solve equations.
With 4.0 Mathematics Students may solve equations step by step and understand the concepts of pre-algebra, algebra, trigonometry, physics, chemistry and calculus (the bane of many).
The software includes a graphing calculator that integrates all the functions of a handheld calculator, plus additional tools that allow them to assess triangles, unit conversions, and solve systems of equations.
You can also plot the 2D and 3D functions. Surely software that students should have on their computers especially if they have problems with math. | 677.169 | 1 |
Overview
Riemannian Geometry / Edition 1
Riemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese)for first-year graduate students in mathematics and physics. The author's treatment goes very directly to the basic language of Riemannian geometry and immediately presents some of its most fundamental theorems. It is elementary, assuming only a modest background from readers, making it suitable for a wide variety of students and course structures. Its selection of topics has been deemed "superb" by teachers who have used the text.
A significant feature of the book is its powerful and revealing structure, beginning simply with the definition of a differentiable manifold and ending with one of the most important results in Riemannian geometry, a proof of the Sphere Theorem. The text abounds with basic definitions and theorems, examples, applications, and numerous exercises to test the student's understanding and extend knowledge and insight into the subject. Instructors and students alike will find the work to bea significant contribution to this highly applicable and stimulating subject.
Editorial Reviews
"This is one of the best (if even not just the best) book for those who want to get a good, smooth and quick, but yet thorough introduction to modern Riemannian geometry."
–Publicationes Mathematicae
"This is a very nice introduction to global Riemannian geometry, which leads the reader quickly to the heart of the topic. Nevertheless, classical results are also discussed on many occasions, and almost 60 pages are devoted to exercises."
–Newsletter of the EMS
"In the reviewer's opinion, this is a superb book which makes learning a real pleasure."
—Revue Romaine de Mathematiques Pures et Appliquees
"This mainstream presentation of differential geometry serves well for a course on Riemannian geometry, and it is complemented by many annotated exercises."
This book is an introduction to modern differential geometry. The authors begin with the necessary
tools from analysis and topology, including Sard's theorem, de Rham cohomology, calculus on manifolds, and a degree theory. The general theory is illustrated and expanded ...
Fractal geometry is the formal study of mathematical shapes that display a progression of never-ending,
self-similar, meandering detail from large to small scales. It has the descriptive power to capture, explain, and enhance one's appreciation of and control over complex ...
This well-printed, attractive little volume offers a first introduction to general topology, cast into the
form of a problem collection...well-suited both for classroom use and for individual study. —Publ. Math. DebrecenIdeal for independent study. —The American Mathematical Monthly
A volume devoted to the extremely clear and intrinsically beautiful theory of two-dimensional surfaces in
Euclidean spaces. The main focus is on the connection between the theory of embedded surfaces and two-dimensional Riemannian geometry, and the influence of properties of ...
The book provides a comprehensive theory of ODE which come as Euler-Lagrange equations from generally
higher-order Lagrangians. Emphasis is laid on applying methods from differential geometry (fibered manifolds and their jet-prolongations) and global analysis (distributions and exterior differential systems). ...
Metric theory has undergone a dramatic phase transition in the last decades when its focus
moved from the foundations of real analysis to Riemannian geometry and algebraic topology, to the theory of infinite groups and probability theory.The new wave began ...
Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One
prevalent philosophical approach to the problem—realism—is examined and rejected in favor of another approach—naturalism. Penelope Maddy defines this naturalism, explains the motivation ...
At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and
John Mather launched a revolution in the venerable field of optimal transport founded by G. Monge in the 18th century, which has made breathtaking forays ... | 677.169 | 1 |
Choose your region or lauguage
Physics Formula Apk
Description
All Physics formula and equations is summarized in one app.
In this App we tried to consolidate all Physics Formulas and equation required for Solving numerical . It covers all the aspects of Mechanics , Thermal Physics, Electrostatics and current electricity , magnetism ,Ray Optics, Wave optics and Modern Physics. This App is Extremely Useful for the students studying in Class 11 and 12 or in Freshman Senior ,also for those who preparing for the competitive exam like JEE main , JEE Advance , BITSAT ,MHTCET , EAMCET , KCET , UPTU (UPSEE), WBJEE , VITEEE ,NEET PMT ,CBSE PMT , AIIMS , AFMC ,CPMT and all other Engineering and Medical Entrance Exam. This app is also Very useful for the teachers who teachs physics.
Simple Interface : easily navigate to any topic. Beautifully Designed for Tablets Physics Formulas and equations arranged in most useful way. Great app for Quick Revision Great app for Solving Numerical
From the success of Maths Formulas app, Physics Formulas has been developed and released to help users quickly refer to any Physics formulas for their study and work. This app displays most popular formulas in seven categories: - Mechanics - Electricity - Thermal physics - Periodic motion - Optics - Atomic physics - Constants
This app has all functions to help users use the app cveniently
- Tools: users can input data and the app will calculate some popular physics problems. - Supporting multiple languages: it's the best to read in your mother language as well as in English to expand your language skills. In this version, there are 15 languages: English, Vietnamese, Chinese (Trad/Simp), Turkish, Spanish, German, French, Portuguese, Russian, Indonesian, Persian, Italian, Hindi and Arabic. - Favorite folder: save frequently used formulas in a favorite folder to quickly access to them. - Sharing: touch and share a formula to friends via message, email or facebook. - Searching: users can type key words in top of the screen to quickly find a formula. - Add your own formulas or notes in "Favorite" section. - Add your own customised tools in "Tools" section. | 677.169 | 1 |
Georgetown University
Detailed Course Information
Fall 2017
Oct 19, 2017
Select the desired Level or Schedule Type to find available classes for the course.
MATH 200 - Intro to Proof/Prob-Solving
This course will introduce different methods for constructing simple proofs, including backward/forward proofs, contradiction, contraposition, and induction. The students will apply these methods to a variety of areas of mathematics, including simple number theory, relations, calculus concepts, and a study of infinity. This course is required for Math Majors and is a prerequisite for many upper level courses. Fall and Spring.
Textbook: Introduction to Proof: Concepts and Techniques by James Sandefur, 2014. This is an electronic textbook that will be available for free online to all registered students. Earlier versions of this textbook will not work.
Credits: 3
Prerequisites: B or better in 036 or department approval. | 677.169 | 1 |
高等数学基础
内容简介
· · · · · · s... students in the first vol-ume, as it is equally important for this second volume.In order to learn calculus, it is not enough to read the textbook as if it were a newspaper. Learning requires careful reading, working through exam-ples step by step, and solving problems. Solving problems requires more than imitation of examples. It is necessary to think about what the problem really asks and to develop a method for that particular problem.If something is still not clear after you have tried to understand it, you should ask a classmate, a more advanced student, or your teacher. If a classmate asks you a question, you may learn a great deal from explaining the answer.
The following two additional remarks might be helpful to readers in u-sing the second volume.
(1) The material on linear systems of ordinary differential equations (Section 9.2) is not included in the fundamental requirements. Before study-ing it, readers will need some basic knowledge of linear algebra.
(2) Some of the material in this volume has been stated in terms of ma-trices and determinants. For readers who are not yet familiar with the basic concepts and operations for matrices and determinants we have included a brief outline in Appendix A. | 677.169 | 1 |
A logarithmic function is the inverse of the exponential function, with all the characteristics of inverse functions covered in Lecture 5. Examine common logarithms (those with base 10) and natural logarithms (those with base e), and study such applications as the "rule of 70" in banking. | 677.169 | 1 |
Graphmatica
Publisher description
a powerful, easy-to-use, equation plotter with numerical and calculus features: - Graph Cartesian functions, relations, and inequalities, plus polar, parametric, and ordinary differential equations. - Up to 999 graphs on screen at once. - New data plotting and curve-fitting features. - Numerically solve and graphically display tangent lines and integrals. - Find critical points, solutions to equations, and intersections between Cartesian functions. - Print your graphs, copy to clipboard as bitmap or enhanced metafile in black-and-white or color, or export to JPEG/PNG file. - On-line help and demo files make getting up to speed a snap. In summary, a great tool for students and teachers of anything from high-school algebra through college calculus | 677.169 | 1 |
The WUHS Mathematics Department strives to provide rich mathematical learning experiences, and the skills that underpin them, for a wide variety of students. All courses integrate technology with pencil-and-paper calculations so that students may be proficient problem solvers. We view algebra as the critical tool for secondary mathematics, thus the curriculum of all introductory courses includes algebraic concepts. Computer Science courses prepare students for careers in technological fields, while Personal Finance and Statistics courses provide business and life skills. College credit may be earned through Advanced Placement courses such as Calculus, Computer Science, Computer Science Principles and/or Statistics.
The Mathematics Department fosters the following Enduring Understandings:
Modeling
Use mathematics to help make sense of the real world: identify variables, formulate a model describing the relationship between the variables, interpret results, and validate and report conclusions and the reasoning behind them
Number & Quantity
Reason, describe, and analyze quantitatively, using units and number systems to solve problems
Interpret and apply statistics and probability to analyze data, reach and justify conclusions, and make inferences
If a student would like to take 2 or more math courses in the same school year, with the exception of Computer Programming, it is strongly suggested that the student talk with their math teacher and/or the Math Department Chair. Math courses for 9th graders will be selected by the Math Department. Students who have successfully completed Algebra I may take computer programming. | 677.169 | 1 |
This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. | 677.169 | 1 |
Basic purpose of this article is to Discuss on Combining and Composing Functions. This article briefly explain on Arithmetic combinations and composition of functions. You are able to create new functions by combining existing functions. Generally, these new functions are the effect of something as simple since addition or subtraction, but functions are capable of combining in ways in addition to those simple binary operations. Arithmetic combinations consider the easiest way to make a new function from existing functions: performing basic math operations. The composition of functions is the process of plugging one function into another is termed the composition of operates. | 677.169 | 1 |
Algebra 2 Teacher's Edition (2nd ed.)
Review basic algebraic functions and extend your student's skills in graphing and in solving equations. Introduce exponential functions, logarithms, and trigonometry. Present more advanced topics such as complex numbers, rational expressions and equations, conic sections, and probability and statistics. A matrix algebra feature, Algebra Around the World, Algebra and Scripture, dominion modeling activities for graphing calculators, and biographical sketches of mathematicians are included. The colorful Teacher's Edition includes student pages with helpful presentation notes and answers (including solutions) as well as sections with motivational ideas, common student errors, and more. | 677.169 | 1 |
Mathematics for Electronics and Computers
For any high school, community college or vocational/technical college student needing mathematics as a prep. for a career in electronics and computers.Best-selling author Nigel Cook's new text Mathematics for Electronics and Computers provides a complete math course for technology students. Since mathematics is interwoven into the very core of science, an understanding is imperative for anyone pursuing a career in technology. Employing an "integrated math applications" approach, this text reinforces all math topics with extensive electronic and computer applications to show a student the value of math as a tool.
"synopsis" may belong to another edition of this title.
From the Back Cover:
With 12 textbooks; 23 editions, and 20 years of front-line education experience, best-selling author Nigel Cook has written this text as a complete math course for high school, community college, or vocational/technical college students seeking a career in electronics and computer technology.
This finely tuned, carefully tested, and accuracy checked volume is organized into three parts:
PART A: Basic Math
Fractions; Decimal Numbers; Positive and Negative Numbers; Exponents and the Metric System; Algebra, Equations, and Formulas; Geometry and Trigonometry; and Logarithms and Graphs
PART B: Electronics Math
Current and Voltage; Resistance and Power; Series DC Circuits; Parallel DC Circuits; Series-Parallel DC Circuits and Theorems; Alternating Current (AC); Capacitors; Inductors and Transformers; RLC Circuits and Complex Numbers; and Diodes and Transistors
PART C: Computer Math
Analog to Digital; Number Systems and Codes; Logic Gates; Boolean Expressions and Algebra; Binary Arithmetic; and Introduction to Computer Programming
Text Features
As with any topic to be learned, the method of presentation can make a big difference between clear comprehension and complete confusion. Employing an "integrated math applications' approach, this text reinforces all math topics with extensive applications to show the student the value of math as a tool. Therefore, if the need for math is instantly demonstrated, the tool is retained.
Since World War II no branch of science has contributed more to the development of the modern world as electronics. It has stimulated dramatic advances in the fields of communication, computing, consumer products, industrial automation, test and measurement, and health care. It has now become the largest single industry in the world, exceeding the automobile and oil industries, with annual sales of electronic systems greater than $2 trillion. One of the most important trends in this huge industry has been a gradual shift from analog electronics to digital electronics. This movement began in the 1960s and is almost complete today. In fact, a recent statistic stated that, on average, 90% of the circuitry within electronic systems is now digital, and only 10% is analog. This digitalization of the electronics industry is merging sectors that were once separate. For example, two of the largest sectors or branches of electronics are computing and communications. Being able to communicate with each other using the common language of digital has enabled computers and communications to interlink, so that computers can now function within communication-based networks, and communications networks can now function through computer-based systems. Industry experts call this merging convergence and predict that digital electronics will continue to unite the industry and stimulate progress in practically every field of human endeavor.
Mathematics is interwoven into the very core of science, and therefore an understanding of mathematics is imperative for anyone pursuing a career in technology. As with any topic to be learned, the method of presentation can make a big difference between clear comprehension and complete confusion. Employing an "integrated math applications" approach, this text reinforces all math topics with extensive electronic and computer applications to show the student the value of math as a tool; therefore, if the need is instantly demonstrated, the tool is retained.
OUTLINE
After 12 textbooks, 21 editions, and 19 years of front-line education experience, best-selling author Nigel Cook has written Mathematics for Electronics and Computers as a complete math course for technology students. To assist educators in curriculum chapter selection, the text has been divided into three parts to give it the versatility to adapt to a variety of course lengths.
Book Description Prentice Hall. PAPERBACK. Book Condition: New. 0130811630811629 | 677.169 | 1 |
Graphing and functions, Algebra
Graphing and Functions
Graphing
In this section we have to review some of the fundamental ideas in graphing. It is supposed that you've seen some graphing at this point therefore we aren't going to go into great depth here. We will only be reviewing some of the fundamental ideas. | 677.169 | 1 |
Offering a uniquely modern, balanced approach, Tussy/Koenig's PREALGEBRA, Fifth Edition, integrates the best of traditional drill and practice with the best elements of the reform movement. To many developmental math students, algebra is like a foreign language. They have difficulty translating the words, their meanings, and how they apply to problem solving. Emphasizing the "language of algebra," the text's fully integrated learning process is designed to expand students' reasoning abilities and teach them how to read, write, and think mathematically. It blends instructional approaches that include vocabulary, practice, and well-defined pedagogy with an emphasis on reasoning, modeling, communication, and technology skills.
"synopsis" may belong to another edition of this title.
About the Author:
Alan Tussy has taught all levels of developmental mathematics at Citrus College in Glendora, CA. He has written nine math books. An extraordinary author dedicated to his students' success, he is relentlessly meticulous and creative--a visionary who maintains a keen focus on the greatest challenges students face in Mathematics. Alan received his Bachelor of Science degree in Mathematics from University of Redlands and his Master of Science degree in Applied Mathematics from California State University, Los Angeles. He has taught across course and curriculum areas from prealgebra to differential equations. He is currently focusing on teaching courses related to Developmental Mathematics. Professor Tussy is a member of the American Mathematical Association of Two-Year Colleges (AMATYC).
A nationally recognized educator and author, Diane Koenig actively shaped several textbooks, ancillaries and series. Since 1982 when she helped develop the Gustafson/Frisk series to her work on the Tussy/Koenig/Gustafson series, Diane's writing continues to reflect the expertise she gains from working with students in her Mathematics courses. Throughout her work, she integrates research-based strategies in Mathematics education. She earned a Bachelor of Science degree in Secondary Math Education from Illinois State University in 1980, and began her career at Rock Valley College in 1981, when she became the Math Supervisor for a newly formed Personalized Learning Center. Earning her Master's Degree in Applied Mathematics from Northern Illinois University in 1984, Diane, enjoys the distinction of being the first woman to become a full-time faculty member in the Mathematics department for Rock Valley College. In addition to being awarded AMATYC's Excellence in Teaching Award in 2015, she was chosen as the Rock Valley College Faculty of the Year by her peers in 2005, and the next year she was awarded the NISOD Teaching Excellence Award and the Illinois Mathematics Association of Community Colleges Award for Teaching Excellence. In addition to her teaching, she is an active member of the Illinois Mathematics Association of Community Colleges (IMACC), serving on the board of directors, on a state-level task force rewriting the course outlines for the Developmental Mathematics courses, and as the Association's newsletter editor. | 677.169 | 1 |
Navigation
MEL 3/4E1
Welcome to Grade 11/12 Mathematics for Work and Everyday Life!
This course enables students to broaden their understanding of mathematics as it is applied in the workplace and daily life.
MEL3E - Students will solve problems associated with earning money, paying taxes, and making purchases; apply calculations of simple and compound interest in saving, investing, and borrowing; and calculate the costs of transportation and travel in a variety of situations.
MEL4E - Students will investigate questions involving the use of statistics; apply the concept of probability to solve problems involving familiar situations; investigate accommodation costs, create household budgets, and prepare a personal income tax return; use proportional reasoning; estimate and measure; and apply geometric concepts to create designs.
Students will consolidate their mathematical skills as they solve problems and communicate their thinking. | 677.169 | 1 |
Geometry by Ron Larson(
Book
) 24
editions published
between
2001
and
2016
in
English
and held by
272 WorldCat member
libraries
worldwide
Essentials of geometry -- Reasoning and proof -- Parallel and perpendicular lines -- Congruent triangles -- Relationships
within triangles -- Similarity -- Right triangles and trigonometry -- Quadrilaterals -- Properties of transformations -- Properties
of circles -- Measuring length and area -- Surface area and volume of solids
Big ideas math : a Common Core curriculum by Ron Larson(
Book
) 28
editions published
between
2012
and
2017
in
English
and held by
203 WorldCat member
libraries
worldwide
The Big Ideas Math program balances conceptual understanding with procedural fluency. Embedded Mathematical Practices in grade-level
content promote a greater understanding of how mathematical concepts are connected to each other and to real-life, helping
turn mathematical learning into an engaging and meaningful way to see and explore the real world
Geometry : concepts and skills by Ron Larson(
Book
) 12
editions published
between
2003
and
2010
in
English
and held by
90 WorldCat member
libraries
worldwide
This book has been written so that all students can understand geometry. The course focuses on the key topics that provide
a strong foundation in the essentials of geometry. Lesson concepts are presented in a clear, straightforward manner, supported
by frequent worked-out examples. The page format makes it easy for students to follow the flow of a lesson, and the vocabulary
and visual tips in the margins help students learn how to read the text and diagrams. Checkpoint questions within lessons
give students a way to check their understanding as they go along. The exercises for each lesson provide many opportunities
to practice and maintains skills, as well as to apply concepts to real-world problems. - p. ii
Working with algebra tiles : grades 6-12 by Don S Balka(
Book
) 1
edition published
in
2006
in
English
and held by
22 WorldCat member
libraries
worldwide
"A complete resource for using algebra tiles to help students visualize algebra, build and solve equations, and gain comfort
and skill with algebraic expressions. Teacher's notes and reproducible activities cover integer operations, linear expressions,
quadratic expressions, perimeter, arrays, binomials and more. Each topic progresses through objective prerequisites, getting
started and closing the activity"--(P.4) of cover
Holt McDougal Larson pre-algebra(
Book
) 2
editions published
in
2012
in
English
and held by
20 WorldCat member
libraries
worldwide
Holt McDougal Larson Pre-Algebra provides clear, comprehensive coverage of the new Common Core State Standards, including
the Standards of Mathematical Practice. The program gives students a strong preparation for Algebra 1, and may be used with
advanced students enrolled in the Accelerated Pathway.--Publisher | 677.169 | 1 |
Maths and Further Maths
Overview of the Course
Level 3 Mathematical Studies is one of the 'Core Maths' qualifications. This course maintains and develops real life mathematical skills. It is a course for those who want to keep up their valuable mathematics skills, but who are not planning on taking AS or A Level Mathematics. The skills developed in the study of mathematics are increasingly important in the workplace and in higher education and studying Mathematical Studies will help you maintain these essential skills. On average, students who study mathematics after GCSE improve their career choices and increase their learning prospects.
This course is aimed at those with grade 4, 5 or 6, if you are expected to achieve a grade 7 or above then you should consider A Level Mathematics.
What are the career opportunities?
The introduction of this qualification has been welcomed by many universities including Russell Group Universities. For many courses, for example, Social Sciences, Business, Psychology, Sciences and Health Sciences a Core Maths qualification will help you hit the ground running at university.
Employers from all different sectors are also firmly behind the Core Maths qualification. Many roles in today's workplace require high levels of budget management and problem-solving skills, therefore Mathematical Studies will be a useful tool in equipping you with these skills.
Level 3 Mathematics is an excellent qualification to have for a wide variety of higher education courses and careers. It will support you with the extra maths content in many linear A Level qualifications and is recommended to study alongside: Physics*, Computer Science*, Psychology, Geography, Business Studies, Economics, Biology and Chemistry.
*It is strongly recommended you study AS Mathematics with these courses.
What are the entry requirements?
The minimum entry requirements for access to an A Level Study Programme will be two subjects at grade B (or grade 5) including GCSE English and four subjects at grade C (or grade 4) including GCSE maths grade A*-C (9-5). Students with a grade 4 may also be considered.
How long is the course and when can I start?
This course is a one year course starting in September. It is taken alongside A Levels or BTEC Level 3 as part of your Study Programme.
How is the course structured and how will I be assessed?
Level 3 Mathematical Studies is a one year course, you will come out with a Level 3 Mathematical Studies qualification which is equivalent to an AS. The qualification is assessed by two final examinations at the end of the year. Two modules are studied; the first contains the compulsory topics: Analysis of Data, Mathematics for Personal Finance and Estimation. In the second exam you can choose from the following options: Statistical Techniques, Critical Path and Risk Analysis or Graphical Techniques.
What employability skills will I develop on this course?
Mathematical Studies is for students who wish to develop their practical maths skills for the real world, be it in work, study or everyday life. It has been designed in association with employers as a valuable qualification to support potential employees in a variety of careers.
The course will help you to understand and apply clear mathematical reasoning to real-life problems, analyse and interpret data in various context and confidently deal with everyday financial maths.
English and communication skills are developed as you are involved in discussions and must be able to explain your reasoning and assumptions in a structured and coherent manner to support your calculations Enrichment includes guest speakers, careers events and trips/visits. You will have the opportunity to take part in competitions such as the BEBRAS Challenge and the UKMT Individual and Team Challenges. You will also have the opportunity to take part in fundraising events and sport and fitness activities.
Are there any additional costs or requirements?
A graphical calculator with the ability to calculate statistical functions. | 677.169 | 1 |
Hop til / Skip to:
Algebraic geometry (AlgGeo)
Course content
Algebraic Geometry is the study of geometric structures arising
from
solution sets of polynomial equations, and forms a central part of
modern mathematics. It has numerous applications, ranging from
number
theory to theoretical physics.
The course will be an introduction to Algebraic Geometry, and will
cover the following topics:
Algebraic sets, affine and projective varieties, fundamental
properties
of varieties. Sheaves and locally ringed spaces. Morphisms of
varieties, birational maps and blow-ups. Smoothness and
singularities. Hilbert polynomials and Bezout's
theorem. | 677.169 | 1 |
Exams and Grading:
Your grade will be based on three tests, a final exam, and classwork as
follows:
Three tests at 100 points each
300 points
Final exam
200 points
Classwork
100 points
Total
600 points
Calculators and Computers:
As
in MTH 141 and 142 you will need a programmable graphing calculator. You
will be provided with necessary programs for the following types calculators:
TI 81,TI 82, TI 83, TI 85, Casio fx-7700GB, Sharp EL-9200, Sharp EL-9300.
You are encouraged to use the CAS (computer algebra system) such as MAPLE,
MATHVIEW, and Scientific Notebook and online programs that can be found
on Internet. We will give a brief introduction to such resources.
Course Description: In
MTH 243 we study functions of more than one variable and expand
the calculus ideas and methods of MTH 141-142 to the multivariable setting.
This subject has wide applications in the physical sciences, engineering,
statistics, and economics, as well as leading to many advanced areas of
mathematics itself. You will find that the material in this course has
a more geometric flavor than one-variable calculus, and there are new algebraic
ideas to master as well.
The homework problems are the core of this course. An
important purpose of the problems is to make you think through and master
the ideas of the subject so that you can confidently apply your knowledge
in new situations. You will learn a great deal from honest hard work on
a problem, even if you don't succeed in solving it. Read the text material
before working on the problems. The exams will reflect the variety of the homework problems.
It is important that you give these problems adequate time and effort.
Classwork: The distribution of the 100 points will
be decided by your instructor. It may include homework, quizzes, class
participation, etc. | 677.169 | 1 |
Exponential Functions - Guided Notes
Be sure that you have an application to open
this file type before downloading and/or purchasing.
5 MB|5 & Answer Keys
Share
Product Description
These notes can be used for an introduction to exponential functions. The notes provide a space for students to create a graph, table and explanation of the type of transformation.
There are three short pages of guided notes that can help students explore and compare exponential functions. Students will graph the following functions:
y = 2^x,
y = 3 • 2^x and y = - 3 • 2^x
y = .5 • 2^x and y = - .5 • 2^x
Students compare each new function to y = 2^x. Students can identify if the newly graphed exponential function is a vertical stretch or shrink, a vertical reflection or a translation. You will also find two additional blank note pages for you to decide on your own exponential functions. A page of types of transformations is also included. | 677.169 | 1 |
Basic Concepts of Mathematics
Publisher: The Trillia Group 2007 ISBN/ASIN: 1931705003 Number of pages: 208
Description: The book will help students complete the transition from purely manipulative to rigorous mathematics. It covers basic set theory, induction, quantifiers, functions and relations, equivalence relations, properties of the real numbers, fields, and basic properties of n-dimensional Euclidean spaces | 677.169 | 1 |
PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
Transcription
1 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive Properties 8 4. Properties of Frctions 9 5. Properties of Exponents Properties of Rdicls The Slope nd Eqution of Line ACT Compss Prctice Tests If you hve Internet ccess, you cn ccess the online resources below through the pdf file by simply clicking on the links below.you cn use these resources to prctice with smple ACT Compss tests online or wtch video tutorils on Google Video. If you hve only printed copy of the pdf file, you cn still find these Internet resources by using the provided web links. (1) CUNY Compss Prctice Tests from Hostos Community College (2) Kentucky Erly Mth Testing Progrm Prctice Tests (3) Google Video Tutoril on Order of Opertions (4) Google Video Pre-Algebr Tutoril (5) Google Video Tutoril on Solving Equtions Dte: November
2 2 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY 2. Common Mistkes Common Mistke 1. A surprisingly common mistke is to incorrectly copy the problem in your exm booklet. Mke sure you re working on the correct problem! Common Mistke 2. Alwys put prenthesis round negtive number, especilly when you hve to multiply it by nother number, s in this cse: ( 5) = 5 10 = 5 Never drop the prenthesis round the negtive number becuse you will forget tht you hve multipliction nd you will get this insted: = 2 which is wrong nd hs nothing to do with the originl problem. Common Mistke 3. Be very creful with the order of opertions. The correct order of opertions is given below: (1) First do the opertions inside the Prenthesis. (2) Then tke cre of Exponents, (3) Multipliction, Division, (4) Addition, Subtrction. Consider s n exmple the lgebric expression ( 9). There re severl mistkes you cn mke. Firstly, if you don t put the prenthesis round the negtive number, you will get: = 4, which is wrong. Secondly, you cn get the order of opertion wrong: ( 9) = 5 ( 9) = 45 Wrong! fter first dding 2 nd 3 nd then multiplying the result by ( 9). This is wrong, becuse multipliction hs priority, so one should first multiply 3 nd ( 9) to get 27 nd only then dd 2 to the result. So, the correct thing to do is the following: ( 9) = 2 + ( 27) = 2 27 = 25 Correct!
3 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES 3 It would be different if we hve the following expression (2+3) ( 9). The difference is tht 2 nd 3 re now inside prenthesis, so we would hve to do the opertion inside the prenthesis first nd then multiply: (2 + 3) ( 9) = 5 ( 9) = 45 Correct! Common Mistke 4. A few common mistkes re relted to the properties of exponents. For exmple, note tht (2 3) 2 becuse tking the exponent hs priority over multipliction. So, if one wnts to clculte then one should tke the exponent first 3 2 = 9 nd then multiply the result by 2 to get 18, tht is = 2 9 = 18. While for (2 3) 2, we first do the multipliction inside the prenthesis to get 6, which we then squre: (2 3) 2 = 6 2 = 6 6 = 36 Another common mistke relted to exponents is to write It s lso wrong to write insted of using the correct property = 3 10 Wrong! = 9 7 Wrong! = = 3 7 Correct! Keep in mind tht the generl property reds m n = m+n Correct! Finlly, if we hve to tke the power of power, it is wrong to write (x 2 ) 5 = x 7 Wrong! The correct ppliction of the power property reds (x 2 ) 5 = x 10 Correct! Remember the generl property hs the form: (x m ) n = x m n Correct!
5 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES 5 We hve here the product of two negtive numbers 3 nd 27, which gives us positive number ( 3) ( 27) = 3 27 = 81. Finlly, if you re confused bout the sum , note tht this is relly the sme s the difference 81 2, we simply tke 2 wy from 81 to get 79. Common Mistke 5. Remember tht the frction A mens tht we divide A by B, B i.e. we hve A B. For tht reson, we cn express division by B in terms of multipliction by the reciprocl of B, which is 1, nmely B A B = A 1 B = A B Consider the division when A = 15b3 nd B = 5b 2, then we hve 2 ( ) 15b 3 (5b 2 ) 2 Sometimes, students ttempt to use the division rule bove but since they cnnot quite remember it, they would write something like this ( ) ( ) 15b 3 15b (5b 2 3 ) = 5b2 Wrong! This is wrong, of course, becuse the division is replced by multipliction but the reciprocl of 5b 2 is not tken. Insted, 5b 2 is divided by 1, which does not chnge nything since ny number divided by 1 is the number itself, tht is 5b2 = 5b 2. This 1 wy, the division is simply replced by multipliction while nothing else chnges nd this is wrong. The correct thing to do is to tke the reciprocl of 5b 2 when replcing division by multipliction, nmely: ( ) 15b 3 (5b 2 ) = 15b b = 15b3 2 10b = 3b 2 2 Correct! Let s recll the rule for multiplying two frctions tht is used bove. We multiply the nomintors nd the denomintors of both frctions b c d = c b d
6 6 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY In the exmple bove, we reduce the frction 15 = 5 3 = 3 by cnceling out the common fctor of 5. Remember tht if we hve the sme number bove nd below the br of frction then we cn cncel this number: b c = b c becuse the br of the frction represents division. The property of exponents, used bove to get the finl result, is b m b n = bm n which we pply to our exmple to conclude tht b 3 b 2 = b3 2 = b 1 = b Common Mistke 6. One of the most common mistkes when deling with frctions is the following rule tht students invent to dd unlike frctions: b + c d = + c b + d Wrong! It is very esy to see tht this rule is not correct by checking simple exmple. Tke = 2, b = 1, c = 3 nd d = 1 nd if this rule is correct we should get true sttement: ? = ? = 5 2 which is clerly flse sttement, since 5 5, so the rule cnnot be correct. 2 The correct rule we get by cross-multiplying numertors by denomintors nd the sum of the two products gives us the new numertor, while the new denomintor is just the product of the two denomintors: b + c d = d + c b b d
7 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES 7 Of course, given specific numbers, one cn lso look for the LCD (lest common denomintor) but in generl mny students find the LCD concept more difficult. For exmple, let s dd the two frctions using the correct rule: = = = = = = = 7 4 Alterntively, it is esy, in this cse, to find the LCD, which is 12. The next step is to write the first frction s n equivlent frction hving denomintor of 12 nd then we cn esily dd the like frctions. Tht s why we multiply by 2 the numertor nd denomintor of the first frction: = = = = = 7 4 Remember the rule for dding like frctions (with the sme denomintors): b + c b = + c b which we use bove to dd the like frctions: = Common Mistke 7. Some common mistkes relted to rdicls re writing: x 16 = x 4 or x 9 = x 3 Wrong! We cn check esily if our guess is correct by simply using the definition of squre root, which in the first cse would men tht if we tke the squre of our guess x 4, we should get wht is inside the rdicl: (x 4 ) 2 should be equl to x 16. However, (x 4 ) 2 = x 4 2 = x 8 x 16, so our guess x 4 cnnot be correct. I cn only guess tht the logic tht leds to the wrong clims bove goes long these lines x 16 = x 16 = x 4 or x 9 = x 9 = x 3 Wrong! The correct rule to pply in the cse of n even power under the rdicl sign is: x 16 = x 16 2 = x 8 Correct!
14 14 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY b = b, > 0, b > 0 Exmple 21. The squre root of the quotient is the quotient of the squre roots. 5 2 = = 5 6 x 2 = x = x 8 2n = 2n 2 = n or even = even 2 Exmple 22. If we hve n even exponent inside the rdicl, 2n bove, we cn undo the rdicl by tking hlf of the even exponent, 2n 2 = n = = x 16 y 10 = 25 x 16 2 y 10 2 = 5x 8 y 5 x 8 y 17 = x 8 2 y 16 y = x 4 y 16 2 y = x 4 y 8 y 7. The Slope nd Eqution of Line The slope of line is number determined by the coordintes of ny two points on the line. If P (x 1, y 1 ) nd Q(x 2, y 2 ) re ny two points on the line, given by their coordintes, then the slope is the number m = y 2 y 1 x 2 x 1 = y x Note tht y, the difference in the y-coordintes, is on the top, while x, the difference in the x-coordintes, is on the bottom. It is common mistke to use x y y insted of x to compute the slope. Notice tht we chose bove to strt with the second point Q nd tht is why both differences begin with the coordintes of the second point (y 2 y 1 ) nd (x 2 x 1 ). One cn strt insted with the first point P, in which cse both differences should strt with the coordintes of the first point, nmely (y 1 y 2 ) nd (x 1 x 2 ), which gives the sme slope s bove
15 MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES 15 m = y 1 y 2 x 1 x 2 = (y 2 y 1 ) (x 2 x 1 ) = y 2 y 1 x 2 x 1 Exmple 23. Find the slope of the line pssing through the points P ( 1, 2) nd Q( 4, 5). If we choose to strt with the first point P, then both differences should strt with the coordintes of the first point, with y difference on the top Slope = m = 2 ( 5) 1 ( 4) = = 3 3 = 1 If we choose to strt with the second point Q, then both differences should strt with the coordintes of the second point, with y difference on the top Slope = m = 5 ( 2) 4 ( 1) = = 3 3 = 1 Exmple 24. The following results re useful to remember: Any horizontl line hs slope zero. For verticl lines the slope is not defined. If line hs positive slope, the line rises from left to right. If line hs negtive slope, the line flls down from left to right. The eqution of line pssing through two given points P (x 1, y 1 ) nd Q(x 2, y 2 ) is given in terms of the slope m nd the coordintes of one of the points: y = y 1 + m(x x 1 ) Note tht y nd x re vribles, representing the coordintes of n rbitrry point on the line. Here, we chose to use the coordintes of the first point P, nmely the given numbers x 1 nd y 1 but one cn lso use the coordintes of the second point Q nd still get the sme eqution for the line y = y 2 + m(x x 2 )
16 16 PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Exmple 25. Tke the points bove P ( 1, 2) nd Q( 4, 5), for which we computed the slope m = 1. The eqution of the line pssing through these two points, using the coordintes of the first point, is then y = 2 + 1(x ( 1)) = 2 + x + 1 = x 1 We should get the sme eqution if we use the coordintes of the second point insted y = 5 + 1(x ( 4)) = 5 + x + 4 = x 1 Exmple 26. Let the eqution of line be given by 3x + 2y = 5 How cn we find the slope of the line from this eqution? We need to solve this eqution for y in terms of x: 2y = 5 3x subtrct from both sides 3x y = 5 3x 2 then divide both sides by 2 y = x this is wht we need to find the slope Once we hve the eqution in the form (for some numbers m nd b): y = mx + b slope = m nd y-intercept = b the slope is simply the number m (including the sign) in front of the vrible x, while the number b is the y-intercept. In our cse, The slope is the signed number in front of the vrible x, nmely slope = 3 2 nd the y-intercept = 5 2 A common mistke is to include the vrible x in the nswer for the slope. Remember tht the slope is number. Another common mistke is to forget the sign nd write 3 for the slope, insted of 3. Remember tht the y-intercept is the y-coordinte of 2 2 the point of intersection of the line nd the y-xis, when x = 0. E-mil ddress:
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REVIEW OF THE DEFINITE AND INDEFINITE INTEGRALS NORTHERN ILLINOIS UNIVERSITY, MATH 230 This is quick review of the definite nd indefinite integrl from first semester clculus. We strt with the notion of
Review of Gussin Qudrture method Nsser M. Abbsi June 0, 014 1 The problem To find numericl vlue for the integrl of rel vlued function of rel vrible over specific rnge over the rel line. This mens to evlute | 677.169 | 1 |
Useful research performs an important function within the technologies in addition to in arithmetic. it's a attractive topic that may be inspired and studied for its personal sake. in accordance with this simple philosophy, the writer has made this introductory textual content available to a large spectrum of scholars, together with beginning-level graduates and complicated undergraduates.
3 parts give a contribution to a subject matter sustained in the course of the Coburn sequence: that of laying an organization beginning, construction a fantastic framework, and delivering robust connections. not just does Coburn current a legitimate problem-solving method to educate scholars to acknowledge an issue, manage a approach, and formulate an answer, the textual content encourages scholars to work out past strategies on the way to achieve a better figuring out of the massive principles at the back of mathematical recommendations.
Should be read to mean "the set of real numbers x such that" and then whatever follows. There is no particular x here. The variable is simply a convenient device to describe a property, and the symbol used for the variable does not matter. Thus {x : x > 2} and {y : y > 2} and {t : t > 2} all denote the same set, which can also be described (without mentioning any variables) as the set of real numbers greater than 2. A special type of set occurs so often in mathematics that it gets its own name, which is given by the following definition.
71. Explain why |ab| = |a||b| for all real numbers a and b. for all real numbers a and b. 76. Show that if a and b are real numbers such that |a + b| < |a| + |b|, 72. Explain why |−a| = |a| for all real numbers a. then ab < 0. 73. Explain why a |a| = b |b| for all real numbers a and b (with b = 0). 74. Give an example of a set of real numbers such that the average of any two numbers in the set is in the set, but the set is not an interval. 77. Show that |a| − |b| ≤ |a − b| for all real numbers a and b.
Simplify the expression 2(3m + x) + 5x. solution First use the distributive property to transform 2(3m + x) into 6m + 2x: 2(3m + x) + 5x = 6m + 2x + 5x. Now use the distributive property again, but in the other direction, to transform 2x + 5x to (2 + 5)x: 6m + 2x + 5x = 6m + (2 + 5)x = 6m + 7x. Putting all this together, we have used the distributive property (twice) to transform 2(3m + x) + 5x into the simpler expression 6m + 7x. 2 Algebra of the Real Numbers 11 One of the most common algebraic manipulations involves expanding a product of sums, as in the following example. | 677.169 | 1 |
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Undergraduate maths teaching at Balliol
First year
We arrange college teaching through Balliol's unique combination of tutorials and classes. In a typical week, a first-year maths student will have one or two one-hour tutorials (usually with two or three students and a tutor), and three classes lasting a total of around five hours. Each class is led by a tutor, and involves all the Balliol maths students taking the relevant course(s) (including Mathematics and Computer Science and Mathematics and Philosophy students where appropriate). The classes complement the individual attention available in tutorials by giving students and tutors an opportunity to work together on understanding challenging new topics from the courses, as well as other aspects relevant to the transition to university maths (constructing proofs, for example). They involve large quantities of tea and chocolate biscuits! The classes tend to be arranged for late afternoons and are then traditionally followed by a trip to the College Buttery, for drinks and more informal chat about maths and anything else that crops up.
Second year
The second-year Balliol students have frequent tutorials with College tutors on both the core courses and the options that individual students choose. In addition, each student prepares a short talk on a topic that interests them, which they give to the other second-year mathematicians in Balliol. This gives students the opportunity to research a piece of mathematics that they're excited about, to develop important presentation skills, and also to hear what their fellow students are interested in. Recent topics have included black holes, non-standard analysis, earthquakes, fractals, algebraic data types, and Godel's Incompleteness Theorems.
Right: the Balliol maths octopus reclines on a portrait of James Bradley, a Balliol mathematician who became Astronomer Royal.
Third and fourth years
In the third and fourth years, students start to specialise more, and the teaching to complement lectures happens through intercollegiate classes (some of which are taught by Balliol tutors and Balliol graduate students). The Balliol tutors continue to provide support within College, through office hours, revision support, and individual meetings. | 677.169 | 1 |
Focus on Calculus Concept
This is the first of three sections on differential calculus. This is significant branch of mathematics with lots of applications. It builds very nicely on other concepts in the course and those you are likely to have covered previously. The whole principle is based around examining "Rates of change".Try these activities to help students explore the ideas. Distance time graphs Make a real, hands on distance time graph | 677.169 | 1 |
An important topic in finance and economics is the study of the speed of change of different economic quantities over time, such as GDP, unemployment, investment, and so on. Further, risk management instruments rely heavily on the speed of change of the underlying assets' values and prices. The mathematical concept that deals with these issues is the rate of change, otherwise known as the derivative.
This course introduces the concept of differentiation and its counterpart, integration. Simple economic applications of the two concepts are also described.
OBJECTIVES
On completion of this course, you will be able to:
• Determine the derivatives of various functions by applying different calculation rules
• Apply some basic rules to calculate the integral of a function and understand that integration is the reverse of differentiation
COURSE OUTLINE
Topic 1: Integration
• What is Integration?
• Rules of Integration
• Definite Integration
• Calculating the Definite Integral
• Definite Integrals – Applications
o Summation of a Continuous Flow
o Discounting
• Properties of Definite Integrals
• Improper Integrals
• Integration of Composite Functions
• Integration of Composite Functions – Integration by Parts
• Integration of Composite Functions – Integration by Substitution
Topic 2: Differentiation
• What is Differentiation?
o Differentiation of Linear Functions
o Differentiation of non-Linear Functions
• Formula
• Increasing & Decreasing Functions
• Minimum & Maximum Point of Functions
• Differentiation of Non-Linear Functions
• Calculating Derivatives
• Rules of Differentiation
o Sum Rule
o Difference Rule
o Products & Quotients of Functions
• Rules of Differentiation of Composite Functions
• Differentiation of Exponential Functions
• Differentiation of Logarithmic Functions
• Economic Applications of Differentiation
o Revenue Functions
o Cost Functions
o Profit Maximization
o Production Function
PREREQUISITE KNOWLEDGE
No prior knowledge is assumed for this course.
ESTIMATED COMPLETED TIME
90 minutes
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Indices, Exponents, Logs, & Geometric Series
An understanding of some basic mathematical tools is crucial in order to have a solid grasp of financial concepts. Indices, exponents, logarithms, and geometric series, which are explained in this course, are some of the most basic and important tools employed in finance | 677.169 | 1 |
Does Camb. reuse the same example sheets over the course of multiple years? e.g. what are the chances of me doing this for example which is from 2015 and being regiven it next (/this) year ? (assuming I get in).
Beardon is like 300 pages and teaches you V&M and Groups at the same time whilst if you were reading through the LNs you would need to get through V&M first and then Groups, whereas for people who want to do Groups first, the first chapter of Beardon's textbook introduces them to Groups. It seems like the lazy person would choose Beardon's over going through lecture notes for V&M and Groups separately. | 677.169 | 1 |
Prealgebra: An Applied Approach
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Read More and eventually master the concepts. Simplicity is key in the organization of this edition, as in all other editions. All lessons, exercise sets, tests, and supplements are organized around a carefully constructed hierarchy of objectives. Each exercise mirrors a preceding objective, which helps to reinforce key concepts and promote skill building. This clear, objective-based approach allows students to organize their thoughts around the content, and supports instructors as they work to design syllabi, lesson plans, and other administrative documents. New features like Focus on Success, Apply the Concept, and Concept Check add an increased emphasis on study skills and conceptual understanding to strengthen the foundation of student success. The Sixth Edition also features a new design, enhancing the Aufmann Interactive Method and making the pages easier for both students and instructors to follow. Available with InfoTrac Student Collections http: //gocengage.com/infotrac | 677.169 | 1 |
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Philosophy of Mathematics
Requisites
None
Additional Requirements
Pre-requisites: 40 units of Level 2 Philosophy courses
40 PHIL credits at Level 2.
Aims
The course aims to:
- give a detailed understanding of some important debates within contemporary philosophy of mathematics; - enable students to engage critically with some recent contributions to these debates; and - enhance students' powers of critical analysis, reasoning and independent thought.
Overview
This course will introduce students to the lively contemporary debate over the metaphysics of mathematics. Are there such things as numbers (or other mathematical objects)? If so, what they are like, and how do we manage to acquire knowledge of them? If these objects do not exist, then what is it that we know when we know that 2+2=4?
Discussion of technicalities will be kept to a minimum, and no special expertise in mathematics will be assumed. The arguments discussed raise important questions about the relation of philosophy to mathematics, science, and ordinary talk and belief; the course will place particular stress on these issues.
Teaching and learning methods
Two one-hour lectures and one one-hour tutorial weekly
Learning outcomes
On successful completion of this course unit, students will be able to demonstrate:
- a detailed critical understanding of some important debates within contemporary philosophy of mathematics; - a thorough knowledge of some recent contributions to these debates; and - an ability to present carefully-argued and independent lines of thought in this area.
Assessment Further Information
Recommended reading
Feedback methods
There will be a compulsory three-hour take-home mock exam on which you will receive written feedback.
We also draw your attention to the variety of generic forms of feedback available to you on this as on all SoSS courses. These include: meeting the lecturer/tutor during their office hours; e-mailing questions to the lecturer/tutor; asking questions from the lecturer (before and after lectures); and obtaining feedback from your peers during tutorials.
The School of Social Sciences (SoSS) is committed to providing timely and appropriate feedback to students on their academic progress and achievement, thereby enabling students to reflect on their progress and plan their academic and skills development effectively. Students are reminded that feedback is necessarily responsive: only when a student has done a certain amount of work and approaches us with it at the appropriate fora is it possible for us to feed back on the student's work. | 677.169 | 1 |
Holt's Linear Algebra with Applications blends computational and conceptual topics throughout. Early treatment of conceptual topics in the context of Euclidean space gives students more time, and a familiar setting, in which to absorb them. This organization also makes it possible to treat eigenvalues and eigenvectors earlier than in most texts. Abstract vector spaces are introduced later, once students have developed a solid conceptual foundation. Concepts and topics are frequently accompanied by applications to provide context and motivation. Because many students learn by example, Linear Algebra with Applications provides a large number of representative examples, over and above those used to introduce topics. The text also has over 2500 exercises, covering computational and conceptual topics over a range of difficulty levels. | 677.169 | 1 |
Maths Term Paper
A Maths Term Paper is based on explaining a certain theory, formula, notation or entities in the realm of mathematics. A maths term paper has to be simple and easy to understand because almost all the data present in a maths term paper is numbers and calculations that need to explained in a simplified form.
The Maths Term Paper simplifies and explains a complex maths problem for audiences to understand easily and comprehend the central idea. Most maths term papers are explanations or justifications of mathematical theories and equations that are explained in detail with emphasis on deriving a conclusion of it.
Maths Term Paper should be linear in structure while explaining the entire scenario. A maths term paper should start and go on step by step as the equation or the problem progresses and each complex line and change in equation should be addresses in numerical form and syntax as well. A maths term paper can be difficult for students to write because it requires comprehensive understanding of mathematics and how to put all that in words.
So the next time you're stuck writing your own maths term paper you can acquire our excellent services to help you draft and conclude a superb maths term paper that would suffice all your needs. Our panel of expert mathematicians and statisticians can easily decipher the most complex maths problems and equations and simplify them to write you an excellent maths term paper.
You can avail online Maths term paper writing services at Term Paper Freak while ensuring that being a student you are provided with features well incorporated in your Maths Term Paper, such as expert writers' services, guaranteed confidentiality, plagiarism-free papers, 24/ 7online support, direct contact with writers and multiple revision options on flexible prices along with special discounts.
To learn more about how to buy term paper on Maths at Term Paper Freaks, click here | 677.169 | 1 |
(A) ALGEBRA -- INTRODUCTION
ALGEBRA is that branch of the mathematical sciences which has for its object the carrying on of operations either in an order different from that which exists in arithmetic, or of a nature not contemplated in fixing the boundaries of that science. The circumstance that algebra has its origin in arithmetic, however widely it may in the end differ from that science, led Sir Isaac Newton to designate it "Univresal Aritmetic," a designation which, vague as it is, indicated its character better than any other by which it has been attempted to express its functions-better certainly, to ordinary minds, than the designation which has been applied to it by Sir William Rowan Hamilton, one of the greatest mathematicians the world has seen since the days of Newton - "the Science of Pure Time;" or even than the title by which De Morgan would paraphrase Hamilton's words - "the calculus of Succession."
To express in few words what it is which effects the transition from the science of arithmetic into a new field is not easy. It will serve, probably, to convey some notion of the position of the boundary line, when it is stated that the operation of arithmetic are all capable of direct interpretable only by comparison with the assumptions on which they are based. For example, multiplication of fractions - which the older writers on arithmetic, Lucas de Burgo in Italy, and Robert Recorde in England, clearly perceived to be a new application of the term multiplication, scarcely at first sight reconcilable with its original definition as the exponent of equal additions, - multiplication of fractions becomes interpretable by the introduction of the idea of multiplication into the definition of the fraction itself. On the other hand, the independent use of the sign minus, on which Diophantus, in the 4th century, laid the foundation of the science of algebra in the West, by placing in the forefront of his treatise, as one of his earliest definitions, the rule of the sign minus, "that minus multiplied by minus produced plus" - this independent use of the sign has no originating operation of the same character as itself, and might, if assumed in all its generality as existing side by side with the laws of arithmetic, more especially with the commutative law, have led to erroneous conclusions. As it as, the unlimited applicability of this definition, in connection with all the laws of arithmetic standing in their integrity, pushes the dominion of algebra into a field on which the oldest of the Greek arithmeticians, Euclid, in his unbending march, could never have advanced a step without doing violence to his convictions.
In asserting that the independent existence of the sign minus, side by side with the laws of arithmetic, might have led to anomalous results, had not the operations been subject to some limitation, we are introducing no imaginary hypothesis, but are referring to a fact actually existing. The most recent advance beyond the boundaries of algebra, as it existed fifty years ago, is that beautiful extension to which Sir W.R.Hamilton has given the designation of Quaternion, the very foundation of which requires the removal of one of the ancient axioms of arithmetic, "that operations may be performed in any order." | 677.169 | 1 |
Further Maths
Further Maths
A LEVEL
Further Mathematics is for strong mathematics students who are interested in abstract mathematical ideas, such as infinity and imaginary numbers, and enjoy solving challenging problems. This is incredibly valuable should you want to study a Maths-related degree, such as Mathematics, Physics, Economics or Engineering. If you are considering applying to Oxford or Cambridge or a Russell Group university, the Further Maths A level will make your application stand out from the crowd, as universities recognise it as a challenging and relevant qualification.
There are many other benefits to studying Further Mathematics:
Students who study Further Maths achieve, on average, one grade higher in their Maths A level than other students with the same GCSE average points score on entry. Studying Further Maths also improves students' grades in other subjects (especially in Physics, where many of the ideas in Mechanics overlap).
Further Maths is a mandatory entry requirement for many Maths degrees.
Some universities reduce offers and offer bursaries for students who have studied Further Mathematics, to encourage them to apply to their institution.
Further Maths will make the first year of Maths-related degrees much more accessible.
Further Maths is interesting and challenging and increases students' enjoyment of Maths, giving a broader view of what it means to work mathematically, and what mathematicians do. | 677.169 | 1 |
Integral and Measure: From Rather Simple to Rather Complex
This book is devoted to integration, one of the two main operations in calculus.
In Part 1, the definition of the integral of a one-variable function is different (not essentially, but rather methodically) from traditional definitions of Riemann or Lebesgue integrals. Such an approach allows us, on the one hand, to quickly develop the practical skills of integration as well as, on the other hand, in Part 2, to pass naturally to the more general Lebesgue integral. Based on the latter, in Part 2, the author develops a theory of integration for functions of several variables. In Part 3, within the same methodological scheme, the author presents the elements of theory of integration in an abstract space equipped with a measure; we cannot do without this in functional analysis, probability theory, etc. The majority of chapters are complemented with problems, mostly of the theoretical type.
The book is mainly devoted to students of mathematics and related specialities. However, Part 1 can be successfully used by any student as a simple introduction to integration calculus.
"We have to mention here that since some of the problems are of a theoretical nature, while others are direct computations or applications of the theory, the book addresses both students who want a quick introduction to integrals and their use in real analysis and students who want to understand the mechanism behind an integration theory built on some measure space." (Mathematical Reviews, 1 | 677.169 | 1 |
Intermediate Algebra
Offering a uniquely modern, balanced approach, Tussy/Gustafson/Koenig's INTERMEDIATE ALGEBRA, Fourth Edition, integrates the best of traditional drill and practice with the best elements of the reform movement. To many developmental math students, algebra is like a foreign language. They have difficulty translating the words, their meanings, and how they apply to problem solving. Emphasizing the "language of algebra," the text's fully integrated learning process is designed to expand students' reasoning abilities and teach them how to read, write, and think mathematically. It blends instructional approaches that include vocabulary, practice, and well-defined pedagogy with an emphasis on reasoning, modeling, communication, and technology skills | 677.169 | 1 |
Abstract Algebra, 3rd Edition
This revision of Dummit and Foote's widely acclaimed introduction to abstract algebra helps students experience the power and beauty that develops from the rich interplay between different areas of mathematics.
The book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the student's understanding. With this approach, students gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings.
The text is designed for a full-year introduction to abstract algebra at the advanced undergraduate or graduate level, but contains substantially more material than would normally be covered in one year. Portions of the book may also be used for various one-semester topics courses in advanced algebra, each of which would provide a solid background for a follow-up course delving more deeply into one of many possible areas: algebraic number theory, algebraic topology, algebraic geometry, representation theory, Lie groups, etc.
* New Section 9.6, Polynomials in Several Variables Over a Field
and Grobner Bases, including application to solving systems of
equations (elimination theory)
* Applications of Grobner bases to computations in algebraic
geometry in Chapter 15
* Construction of the simple group of order 168 using the
projective plane of order 2 (the Fano plane)
Accessible to undergraduates yet its scope and depth also make
it ideal for courses at the graduate level.
Over 1950 exercises, many with multiple parts, ranging in scope
from routine to fairly sophisticated, and ranging in purpose from
basic application of text material to exploration of important
theoretical or computational techniques.
The structure of the book permits instructors and students to
pursue certain areas from their beginnings to an in-depth
treatment, or to survey a wider range of areas, seeing how various
themes recur and how different structures are related.
The emphasis throughout is to motivate the introduction and
development of important algebraic concepts using as many examples
as possible. The wealth of examples helps to ground the theory,
explain its application, and help develop the student's
intuition.
Contains many topics not usually found in introductory texts.
Students are able to see how these fit naturally into the main
themes of algebra. Examples of these topics include:
Rings of algebraic integers
Semidirect products and the theory of extensions
Criteria for Principal Ideal Domains
Criteria for the solvability of a quintic
Dedekind Domains
Algorithm for determining whether an ideal in a polynomial ring
over a field is a prime ideal | 677.169 | 1 |
MathEducationResources
Landing page for several MER resources
Welcome to the Math Education Resources.
Math Education Resources (MER) provides free hints and peer-reviewed solutions to previous math exams administered at the University of British Columbia - grouped by topic and rated by difficulty. Additional learning content is also provided on our topics pages.
How to use this resource
When you feel reasonably confident, simulate a full exam and grade your solutions. A link to the full solutions is above .
If you're not quite ready to simulate a full exam, we suggest you thoroughly and slowly work through each problem. To check if your answer is correct, without spoiling the full solution, check out the pdf with the final answers only.
Exam Simulation
Resist the temptation to read any of the solutions before completing each question by yourself first! We recommend you follow the guide below.
Exam Simulation: When you've studied enough that you feel reasonably confident, print the raw exam (link below) without looking at any of the questions right away. Find a quiet space, such as the library, and set a timer for the real length of the exam (usually 2.5 hours). Write the exam as though it is the real deal.
Reflect on your writing: Generally, reflect on how you wrote the exam. For example, if you were to write it again, what would you do differently? What would you do the same? In what order did you write your solutions? What did you do when you got stuck?
Grade your exam: Use the full solutions pdf to grade your exam. Use the point values as shown in the original exam.
Reflect on your solutions: Now that you have graded the exam, reflect again on your solutions. How did your solutions compare with our solutions? What can you learn from your mistakes?
Plan further studying: Use your mock exam grades to help determine which content areas to focus on and plan your study time accordingly. Brush up on the topics that need work:
Re-do related homework and webwork questions.
The Math Exam Resources offers mini video lectures on each topic.
Work through more previous exam questions thoroughly without using anything that you couldn't use in the real exam. Try to work on each problem until your answer agrees with our final result.
Do as many exam simulations as possible.
Whenever you feel confident enough with a particular topic, move on to topics that need more work. Focus on questions that you find challenging, not on those that are easy for you. Always try to complete each question by yourself first.
Work through problems
How to use the final answer:The final answer is not a substitution for the full solution! The final answer alone will not give you full marks. The final answer is provided so that you can check the correctness of your work without spoiling the full solution.
To answer each question, only use what you could also use in the exam.
If you found an answer, how could you verify that it is correct from your work only? E.g. check if the units make sense, etc. Only then compare with our result.
If your answer is correct: good job! Move on to the next question.
Otherwise, go back to your work and check it for improvements. Is there another approach you could try? If you still can't get to the right answer, you can check the full solution on the Math Educational Resources.
Reflect on your work: Generally, reflect on how you solved the problem. Don't just focus on the final answer, but whether your mental process was correct. If you were stuck at any point, what helped you to go forward? What made you confident that your answer was correct? What can you take away from this so that, next time, you can complete a similar question without any help?
Please note that all final answers were extracted automatically from the full solution. It is possible that the final answer shown here is not complete, or it may be missing entirely. In such a case, please notify the MER team. Your feedback helps us improve. | 677.169 | 1 |
CBSE Class 2 Books 2017 Mathematics Science English books buy Online
CBSE Class 2 Books: Central Board of Secondary Education is a Reputed educational board of India. CBSE board is established in 1952 and it has been imparting quality education to all students across the country. Apart from the advanced School education and favorable learning environment, it encourages students to achieve their goal. CBSE syllabus is structured in a comprehensive manner so that students can get detailed idea about the entire curriculum. This board offers updated syllabus with required topics and this board provide Appropriate information on different concepts. We provide Best CBSE Class 2 Books Buy Online. | 677.169 | 1 |
Product Description
▼▲
Singapore Math's Primary Mathematics, Standards Edition Workbooks are consumable and should be used in conjunction with the corresponding textbooks; these Standard Edition workbooks are not compatible with other Singapore editions (e.g. U.S. 3rd Ed.). Containing independent student exercises, workbooks provide the practice essential to skill mastery; each chapter in the textbook has corresponding problems in the workbook while review exercises help students to consolidate concepts they've learned previously. A variety of exercises are presented in the concrete> pictorial>abstract approach, covering circles, volume of prisms & cylinders, angles, data handling, probability, negative numbers, and more. 168 pages, softcover. Grade 6. | 677.169 | 1 |
IB Diploma Programme: Math HL Course Outline
Course description
Math at HL is a course aiming to address the needs of competent students who wish to include mathematics as a major component of their university studies or they have a strong interest in Math.
Important mathematical concepts are going to be developed and insight into mathematical form and structure will be expected. The students will link a lot of different topic areas and will approach problems from different points of view (i.e. trigonometry, vectors, complex numbers) realizing at the same time how math as an international (common) language has been connecting the people since the ancient years and has been an axis around which all human invention has been revolving.
The Math HL course is not going to be combined with any other course. The topics that will be covered in the two year course will be: Algebra, Functions and Equations, Trigonometry, Matrices, Vectors, Calculus, Statistics and Probability (both the core and the optional section) as they are described in the IBO syllabus.
Students' assessment for Math HL will aim not only to ensure that they gain the necessary knowledge but also to enhance their investigation and mathematical modeling skills through independent thinking.
This course is a demanding one, requiring students to study a broad range of mathematical topics through a number of different approaches and to varying degrees of depth. Students wishing to study mathematics in a less rigorous environment should therefore opt for one of the standard level courses, mathematics SL or mathematical studies SL.
Group 5 aims
The aims of all mathematics courses in group 5 are to enable students to:
1. enjoy mathematics, and develop an appreciation of the elegance and power of mathematics
2. develop an understanding of the principles and nature of mathematics
3. communicate clearly and confidently in a variety of contexts
4. develop logical, critical and creative thinking, and patience and persistence in problem-solving
5. employ and refine their powers of abstraction and generalization
6. apply and transfer skills to alternative situations, to other areas of knowledge and to future developments
7. appreciate how developments in technology and mathematics have influenced each other
8. appreciate the moral, social and ethical implications arising from the work of mathematicians and the
applications of mathematics
9. appreciate the international dimension in mathematics through an awareness of the universality of
mathematics and its multicultural and historical perspectives
10. appreciate the contribution of mathematics to other disciplines, and as a particular "area of knowledge" in the TOK course.
Syllabus outline:
MATH HL
Teaching hours
Topic 1 : Algebra
30
Topic 2 : Functions and equations
22
Topic 3: Circular functions and trigonometry
22
Topic 4: Vectors
24
Topic 5: Statistics and probability
36
Topic 6: Calculus
30
Option syllabus content
Students must study all the sub-topics in one of the following options as listed in thesyllabus details and specifically in:
Topic 7 : Statistics and probability
48
ATL SKILLS
The Math HL course is designed to help students develop analytical and critical thinking skills and incorporate international perspective from historical and social points of view.
Throughout the DP mathematics HL course, students are encouraged to develop their understanding of the methodology and practice of the discipline of mathematics. The processes of mathematical inquiry, mathematical modelling and applications and the use of technology is introduced appropriately.
At the same time the students are encouraged to develop their investigation skills by using support resources and other supplementary instructional materials found in the school's library and the web. Incorporating technology in their investigations and their presentations is considered essential.
The Internal assessment help students understand the ways in which mathematical discoveries were made and the techniques used to make them and often help them to realize the social and cultural context of mathematics.
The universality of mathematics as a means of communication is emphasized continually. The international perspective of the students for Mathematics is highly advanced through class discussion and links made. With TOK. The association of the topics taught with CAS activities is also emphasized.
For example Mathematical language, Patterns, Sets, Symmetry, Permutations are clearly connected to Dance (Creativity). Developing strategies, Problem solving, Loci, Combinations etc are clearly related to Sports in general (Action). Statistical Analysis and Calculus (Rates of change and optimization problems) provide the ground for unique service that can be offered to communities on awareness in various every day issues.
Assessment Objectives
Problem-solving is central to learning mathematics and involves the acquisition of mathematical skills and concepts in a wide range of situations, including non-routine, open-ended and real-world problems. Having followed a DP mathematics SL course, students will be expected to demonstrate the following.
1. Knowledge and understanding: recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts.
2. Problem-solving: recall, select and use their knowledge of mathematical skills, results and models in
both real and abstract contexts to solve problems.
3. Communication and interpretation: transform common realistic contexts into mathematics; comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation.
4. Technology: use technology, accurately, appropriately and efficiently both to explore new ideas and to solve problems.
5. Reasoning: construct mathematical arguments through use of precise statements, logical deduction and inference, and by the manipulation of mathematical expressions. | 677.169 | 1 |
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Lesson 1: The students are given a graph of two functions (linear and linear; linear and exponential; or exponential and exponential) and are asked to describe:
1. the intervals on which one funciton is increasing faster than the other function.
2. any points where the functions are equal to each other.
3. end behavior.
4. explicit equations that represent the graphs.
Lesson 2: Students are given a linear or exponential explicit equation complete a table of values that is provided with input values or output values.
Lesson 3: Students are given a linear or exponential recursive function and initial value graph the function.
Lesson 4: Students are given a linear or exponential graph, explicit equation, or function (with initial value) and are asked to describe:
1. intervals of increasing and decreasing.
2. extrema: maximum and minimum.
3. domain and range.
4. identify the given information as linear or exponential.
Lesson 5: Students are given a story problem and a graph and are asked to describe what happens at different points in the race.
Students are given 5 table of values and a graph and asked to match the table of values with the graph.
Students are asked to match the rate of change with the tables of values.
Lesson 6: Interpreting Linear Context.
Students are given 3 story problems and are asked to:
1. Draw a graphical representation of the story problem.
2. Answer some questions about the context of the story and what is happening mathematically. | 677.169 | 1 |
Chapter 1 Functions of One Variable
Examples 1, Q1 Examples 1, Q2
1.1
Variables
Quantities which can take a range of values (e.g. the temperature of the air, the concentration of pollutant in a river, etc.) are called variables.
1.2
Functions of One Varia
Chapter 2 Limits and Differentiation
2.1 Definition of a Limit
For the present purposes we will use an intuitive definition of a limit of a function rather than a more strictly rigorous mathematical definition. Suppose that x is close to, but not exactly
Chapter 5 Vectors
5.1 Definition of Vectors
Vectors are quantities which have both magnitude and direction. For example, a displacement from a fixed point A to a fixed point B is a vector because its magnitude is the distance between A and B and the direc
Chapter 4 Complex Numbers
4.1 Definition of Complex Numbers
z = a + ib where a and b are real numbers and i has the property that i2 = -1. (4.2) (4.1)
A complex number is a number of the form
a is called the real part of z and b is called the imaginary pa
SOEE1300 Inter Maths for Environmental Sciences: 2007/2008 session Additional D questions. D questions are at a more advanced level - they are either mathematically harder, or they apply the knowledge you already have in a more practical situation. Althou
SOEE1300 Inter Maths for Environmental Sciences: 2007/2008 session Examples 3: Deadline 22nd November "A" questions are for lectures. "B" questions are for practice and should not be handed in. "C" questions are for assessment and should be handed in on o
SOEE1300 Intro Maths for Environmental Sciences: 2007/2008 session Examples 2: Deadline 8th November A questions are for lectures. B questions are for practice and should not be handed in. C questions are for assessment and should be handed in on or befor
SOEE1300 Inter Maths for Environmental Sciences: 2007/2008 session Examples 1: Deadline 18th October A questions are for lectures. B questions are for practice and should not be handed in. C questions are for assessment and should be handed in on or befor
Varieties 5-8
Cultural category 5 and 6
Art cinemas
It was not until after the Second World War that European art cinema became firmly
established, with the succession of movements such as Italian Neo-Realism, the French Nouvelle
Vague, and the New German | 677.169 | 1 |
8.1 The student will: a) simplify numerical expressions involving positive exponents, using rational
numbers, order of operations, and properties of operations with real numbers;b) recognize, represent, compare, and order numbers expressed in scientific
notation; and c) compare and order decimals, fractions, percents, and numbers written in scientific
notation.
8.2 The student will describe orally and in writing the relationship between the subsets of the
real number system.
Computation and Estimation
8.3 The student will solve practical problems involving rational numbers, percents, ratios, and proportions. Problems will be of varying complexities and will involve real-life data,
such as finding a discount and discount prices and balancing a checkbook.
8.4 The student will apply the order of operations to evaluate algebraic expressions for given replacement values of the variables. Problems will be limited to positive exponents.
8.5The student, given a whole number from 0 to 100, will identify it as a perfect square or
find the two consecutive whole numbers between which the square root lies.
Measurement
8.6 The student will verify by measuring and describe the relationships among vertical
angles, supplementary angles, and complementary angles and will measure and draw
angles of less than 360°.
8.8 The student will apply transformations (rotate or turn, reflect or flip, translate or slide,
and dilate or scale) to geometric figures represented on graph paper. The student will
identify applications of transformations, such as tiling, fabric design, art, and scaling.
8.9 The student will construct a three-dimensional model, given the top, side, and/or bottom
views.
8.10The student will:
a) verify the Pythagorean Theorem, using diagrams, concrete materials, and
measurement; and
b) apply the Pythagorean Theorem to find the missing length of a side of a right
triangle when given the lengths of the other two sides.
Probability and Statistics
8.11 The student will analyze problem situations, including games of chance, board games, or
grading scales, and make predictions, using knowledge of probability.
8.12 The student will make comparisons, predictions, and inferences, using information
displayed in frequency distributions; box-and-whisker plots; scattergrams; line, bar,
circle, and picture graphs; and histograms.
8.14 The student will
a) describe and represent relations and functions, using tables, graphs, and rules; and
b) relate and compare tables, graphs, and rules as different forms of representation for relationships.
8.15 The student will solve two-step equations and inequalities in one variable, using concrete
materials, pictorial representations, and paper and pencil.
8.16 The student will graph a linear equation in two variables, in the coordinate plane, using a
table of ordered pairs.
8.17 The student will create and solve problems, using proportions, formulas, and functions.
8.18 The student will use the following algebraic terms appropriately: domain, range,
independent variable, and dependent variable.
Please inform us of
any broken links, additional sites to add to this page, and
informational errors. | 677.169 | 1 |
Introduction to the Finite Element Method: Theory, Programming and Applications
Written for students with any engineering or applied science
background, Erik Thompson's new text presents the theory,
applications, and programming skills needed to understand the
finite element method and use it to solve problems in engineering
analysis and design. Offering concise, highly practical coverage,
this introductory text provides complete finite element codes that
can be run on the student version of MATLAB or easily converted to
other languages.
This text gives students the opportunity to:
Master the basic theory: The text promotes an
understanding and appreciation of the theoretical basis of finite
element approximations by building on concepts that are intuitive
to the students. Throughout, the text uses matrix notation to help
students visualize the finite element matrices. Study problems
reinforce basic theory.
Experiment with the code: Numerical experiments show how
to test programs for possible errors, experiment with boundary
conditions, and study accuracy and stability. Code development
exercises suggest ways to modify the codes to create additional
capabilities. All codes are available on the book's web page along
with sample data files for testing them. Each code can be run
immediately using only the student version of MATLAB. Because each
code is written using explicit programming, they also serve as
pseudo-codes that can be used to develop programs in any computer
language.
Gain hands-on experience: Projects, representing a wide
variety of engineering disciplines, enable students to conduct
analyses of fairly complex problems. Many of these projects
encourage investigations of new techniques for using the finite
element method.
A balanced approach between theory, programming, and
applications. This approach allows student to see the big
picture--from the development of theory, to writing a workable
program, to solving a practical problem. Many applications of
finite element methods are given, so that the instructor will have
no problem relating the method to students in different areas of
engineering.
The text is not directed toward a particular discipline.
The approach is general and does not discourage students by
constantly describing the theory through examples that are
unfamiliar to the student. Instructors can teach from this text
regardless of the background of the students in the class.
Use of MATLAB. MATLAB script provides an easy, yet
versatile, programming language to help students learn the
intricacies of finite element programming. Each code is presented
in the text side by side with a detailed explanation. Equations on
the explanation side of the page show the reader what is taking
place in the code.
A textbook rather than a reference text. This text is
truly a textbook, designed to explain concepts in terms easily
understood by students.
Use of projects as examples of additional concepts. This
text allows students to gain experience by presenting interesting
projects with hints and/or suggestions as to how they can best be
solved. The variety of projects allows students to choose
applications that appear most interesting.
Exercises for testing and modifying codes. For each
section of the text that presents a new concept, exercises show
students how to test the code for accuracy, stability, etc. These
exercises often include suggested modifications to the code and
auxiliary codes that can be written to increase the versatility of
the code presented in the text. In this way, students gain
confidence in a code and can create their own personal version of a
code.
Provocative study questions. This text offers many
questions that challenge students to better understand the material
and the theory behind the finite element method.
The Solutions Manual on the password-protected Instructor Companion Site features complete solutions to text projects and solutions to all exercises for testing and modifying codes, as well as additional approaches that can be used to solve the problems. All codes, data files, and auxiliary codes necessary to run the solutions are provided. Comments and suggestions are given to aid the instructor in encouraging students to try additional techniques or testing procedures.
Figures from text, in PPT format
All figures from the text are available in PowerPoint format, for easy creation of lecture slides. Visit the Instructor Companion Site portion of the book's website to register for a password.
Codes from text
The Student Companion Site contains all the codes in the text ready to download and run. In addition, each code has the data files and auxiliary codes to run the example problem presented in the text | 677.169 | 1 |
Welcome To The New Calculus Humor!
Mathematics is a universal part of human culture. It is the tool and language of commerce, engineering and other sciences – physics, computing, biology etc. It helps us recognise patterns and to understand the world around us. Mathematics plays a vital, often unseen, role in many aspects of modern life, for example:
Space travel
Safeguarding credit card details on the internet
Modelling the spread of epidemics
Predicting stock market prices
Business decision making
As society becomes more technically dependent, there will be an increasing requirement for people with a high level of mathematical training. Analytical and quantitative skills are sought by a wide range of employers. A degree in mathematics provides you with a broad range of skills in problem solving, logical reasoning and flexible thinking. This leads to careers that are exciting, challenging and diverse in nature. Whatever your career plans, or if you have no plans at present, a degree in mathematics provides you with particularly good job prospects
Math Is Diverse
Mathematics is extremely diverse and our degrees enable you to specialise in the areas that are of particular interest to you. Whether your interest is more in the area of pure math, applied math, or operational research and statistics, we have a choice of degree scheme for you. Additionally you can create your own degree from the large number of individual modules we offer. These modules vary from the theoretical to the practical. So, on one hand for example, you can studby abstract algebra and number theory and on the other, you can study internet security, financial mathematics and fluid flows. We also offer several optional computing modules, providing practical skills that are much sought after in the job market.
Math Has Good Career Prospects
Analytical and quantitative skills are sought by a wide range of employers. A degree in mathematics provides you with a broad range of skills in problem solving, logical reasoning and flexible thinking. This leads to careers that are exciting, challenging and diverse in nature Whatever your career plans, or if you have no plans at present, a degree in mathematics provides you with particularly good job prospects. The generic nature of mathematics means that almost all industries require mathematicians. Mathematicians work in business, finance, industry, government offices, management, education and science. A proportion of our students will use their degree in mathematics as preparation for further studies at Masters or Doctorate levels. The experience gained through a sandwich course increases your employability even further. We offer the opportunity for a year's salaried work experience during your degree that enables you to try a job of your choosing and provides employers with evidence of your achievements and skills.
Math Is Exciting
Mathematics is an exciting and challenging subject which continues to develop at a rapid rate across many research areas. It has a natural elegance and beauty. Taking a real world problem and creating and applying mathematical models to aid understanding is often hugely satisfying and rewarding. If you enjoy math at school, then you will probably enjoy math at university even more. | 677.169 | 1 |
This no-nonsense guide provides students and self-learners with a clear and readable study of trigonometry's most important ideas. Tim Hill's distraction-free approach combines decades of tutoring experience with the methods of his old-school Russian math teachers. The result: learn in a few days what conventional schools stretch into months. - Teaches general principles that can be applied to a wide variety of problems.- Avoids the mindless and excessive routine computations that characterize conventional textbooks.- Treats trigonometry as a logically coherent discipline, not as a disjointed collection of techniques.- Restores proofs to their proper place to remove doubt, convey insight, and encourage precise logical thinking.- Omits digressions, excessive formalities, and repetitive exercises.- Covers all the trigonometry needed to take a calculus course.- Includes solutions to all problems. Contents1. A Few Basics2. Radian Measure3. The Trig Functions4. Trig Values for Special Angles5. Graphs of Trig Functions6. The Major Formulas7. Inverse Trig Functions8. The Law of Cosines (and Sines)9. Solutions10. Trig Cheat Sheet
Fast Facts at Your Fingertips! REA's Quick Access Study Charts contain all the information students, teachers, and professionals need in one handy reference. They provide quick, easy access to important facts. The charts contain commonly used mathematical formulas, historical facts, language conjugations, vocabulary and more! Great for exams, classroom reference, or a quick refresher on the subject. Most laminated charts consist of 2 fold-out panels (4 pages) that fit into any briefcase or backpack. Each chart has a 3-hole punch for easy placement in a binder. Each chart measures 8 1/2" x 11"
Boiled-down essentials of the top-selling Schaum's Outline series for the student with limited time What Cartoons, sidebars, icons, and other graphic pointers get the material across fast Concise text focuses on the essence of the subject Delivers expert help from teachers who are authorities in their fields Perfect for last-minute test preparation So small and light that they fit in a backpack!
Teaches the essential trigonometry skills needed for school or Advanced Placement tests. Concepts range from vectors and the unit circle to Law of Sines and Cosines, inverse trigonometric functions and Heron's formula. Exercises in test format allow students to sharpen their test-taking skills.
Choose the book written for the way you teach with McKeague/Turner's best-selling TRIGONOMETRY, Sixth Edition. This trusted edition presents contemporary concepts in short, manageable sections using the most current, detailed examples and high-interest applications. Captivating illustrations of trigonometry concepts in action, such as Lance Armstrong's cycling success, the Ferris wheel, and even the human cannonball, as well as unique Historical Vignettes help motivate and keep students' interest throughout your course. TRIGONOMETRY, Sixth Edition, continues to use a standard right-angle approach to trigonometry with an unmatched emphasis on study skills that prepares students for future success in advanced courses, such as calculus. The book's proven blend of exercises, fresh applications, and projects is now combined with a simplified approach to graphing and the convenience of new Enhanced WebAssign--a leading, time-saving online homework tool for instructors and students that's correlated with your Instructor's Edition for cohesive support. Innovative tools like the new CengageNOW online course management system complete this market-leading TRIGONOMETRY, Sixth Edition, package to ensure you have everything you need for a course that holds your students' interest and clarifies even the most advanced topics for your students' trigonometry success. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Go beyond the answers--see what it takes to get there and improve your grade! This manual provides worked-out, step-by-step solutions to the odd-numbered problems in the text. This gives you the information you need to truly understand how these problems are solved. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
A plain-English guide to the basics of trig Trigonometry deals with the relationship between the sides and angles of triangles... mostly right triangles. In practical use, trigonometry is a friend to astronomers who use triangulation to measure the distance between stars. Trig also has applications in fields as broad as financial analysis, music theory, biology, medical imaging, cryptology, game development, and seismology. From sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English and offering lots of easy-to-grasp example problems. It also explains the "why" of trigonometry, using real-world examples that illustrate the value of trigonometry in a variety of careers. Tracks to a typical Trigonometry course at the high school or college level Packed with example trig problems From the author of Trigonometry Workbook For Dummies Trigonometry For Dummies is for any student who needs an introduction to, or better understanding of, high-school to college-level trigonometry.
Trigonometry: A Complete Introduction is the most comprehensive yet easy-to-use introduction to Trigonometry. Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge. The book covers all areas of trigonometry including the theory and equations of tangent, sine and cosine, using trigonometry in three dimensions and for angles of any magnitude, and applications of trigonometry including radians, ratio, compound angles and circles related to triangles. Everything you will need is here in this one book. Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions.
For courses in Plane Trigonometry--such as Brief Review of prerequisite topics, Achieving Success boxes, and Retain the Concepts exercises--as well as support within Pearson MyLab Math such as new concept-level videos, assignable tools to enhance visualization, and more. Also available with Pearson MyLab(tm) Math Pearson MyLab The new edition continues to expand the comprehensive auto-graded exercise options. In addition, Pearson MyLab Math includes new options designed to help students of all levels and majors to stay engaged and succeed in the course. Note: You are purchasing a standalone product; MyLab(tm)&44086 / 9780134444086 Trigonometry Plus MyLab Math with eText -- Access Card Package, 2/e Package consists of: 0134469968 / 9780134469966 Trigonometry 0321431308 / 9780321431301 MyLab Math -- Glue-in Access Card 0321654064 / 9780321654069 MyLab Math Inside Star Sticker | 677.169 | 1 |
STANDARD 15 - CONCEPTUAL BUILDING BLOCKS OF CALCULUS
All students will develop an understanding of the conceptual
building blocks of calculus and will use them to model and analyze
natural phenomena.
Standard 15 - Conceptual Building Blocks of Calculus - Grades 5-6
Overview
Students in grades 5 and 6 extend and clarify their understanding
of patterns, measurement, data analysis, number sense, and algebra as
they further develop the conceptual building blocks of calculus. Many
of the basic ideas of calculus can be examined in a very concrete and
intuitive way in the middle grades.
Students in grades 5 and 6 should begin to distinguish between
patterns involving linear growth (where a constant is added or
subtracted to each number to get the next one) and exponential
growth (where each term is multiplied or divided by the same
number each time to get the next number). Students should recognize
that linear growth patterns change at a constant rate. For example, a
plant may grow one inch every day. They should also begin to see that
if these patterns are graphed, then the graph looks like a straight
line. They may model this line by using a piece of spaghetti and use
their graph to make predictions and answer questions about points that
are not included in their data tables. In contrast, exponential
growth patterns change at an increasingly rapid rate; if you start
with one penny and double that amount each day, you receive more and
more pennies each day as time goes on. Students should note that the
graphs of these situations are not straight lines. At this grade
level, students should also begin to imagine processes that could in
theory continue forever even though they could not be carried out in
practice; for example, although in practice a cake can be repeatedly
divided in half only about ten times, nevertheless it is possible to
imagine continuing to divide it into smaller and smaller pieces.
Many of the examples used should come from other subject areas,
such as science and social studies. Students might look at such linear
relationships as profit as a function of selling price, but they
should also consider nonlinear relationships such as the amount of
rainfall over time. Students should look at functions which have
"holes" or jumps in their graphs. For example, if students
make a table of the parking fees paid for various amounts of time and
then plot the results, they will find that they cannot just connect
the points; instead there are jumps in the graph where the parking fee
goes up. A similar situation exists for graphs of the price of a
postage stamp or the minimum wage over the course of the years. Many
of the situations investigated by students should involve such
changes over time. Students might, for example, consider the
speed of a fly on a spinning disk; as the fly moves away from the
center of the disk, he spins faster and faster. Students might be
asked to write a short narrative about the fly on the disk and draw a
graph of the fly's speed over time that matches their story.
As students begin to explore the decimal equivalents for fractions,
they encounter non-terminating decimals for the first time. Students
should recognize that calculators often use approximations for
fractions such as .33 for 1/3. They should look for patterns
involving decimal representations of fractions, such as recognizing
which fractions have terminating decimal equivalents and which do not.
Students should take care to note that pi is a nonterminating,
nonrepeating decimal; it is not exactly equal to 22/7 or 3.14, but
these approximations are fairly close to the actual value of pi
and can usually be used for computational purposes. The examination
of decimals extends students' understanding of infinity to
very small numbers.
Students in grades 5 and 6 continue to develop a better
understanding of the approximate nature of measurement.
Students are able to measure objects with increasing degrees of
accuracy and begin to consider significant digits by looking at the
range of possible values that might result from computations with
approximate measures. For example, if the length of a rectangle to
the nearest centimeter is 10 cmand its width to the nearest centimeter
is 5 cm, then the area is about 50 square centimeters. However, the
rectangle might really be as small as 9.5 cm x 4.5 cm, in which case
the area would only be 42.75 square centimeters, or it might be almost
as large as 10.5 cm x 5.5 cm, with an area of 57.75 square
centimeters. Students should continue to explore how to determine the
surface area of irregular figures; they might, for example, be asked
to develop a strategy for finding the area of their hand or foot.
They should do similar activities involving volume, perhaps looking
for the volume of air in a car. Most of their work in this area in
fifth and sixth grade will involve using squares or cubes to
approximate these areas or volumes.
Standard 15 - Conceptual Building Blocks of Calculus - Grades 5-6
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 5 and
6.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 5-6 will be such that all students:
4. Recognize and express the difference between
linear and exponential growth.
Students develop a table showing the sales tax paid on
different amounts of purchases, graph their results, note that the
graph is a straight line, and recognize that this situation represents
a constant rate of change, or linear growth.
Students make a table showing how much money they
would have at the end of each of eight years if $100 was invested at
the beginning and the investment grew by 10% each year. They note that
the graph of their data is not a straight line; this graph represents
exponential growth.
Students make a table showing the value of a car as it
depreciates over time. They note that the graph of their data is not
a straight line; this graph represents "exponential
decay."
Students are presented stories which represent real life
occurrences of linear and exponential growth and decay over time, and
are asked to construct graphs which represent the situation and
indicate whether the change is linear, exponential, or neither.
5. Develop an understanding of infinite
sequences that arise in natural situations.
Students make equilateral triangles of different
sizes out of small equilateral triangles and record the number of
small triangles used for each larger triangle. These numbers are
called triangular numbers. If the following triangular
pattern is continued indefinitely, then the number of 1s in the first
row, the number of 1s in the first two rows, the number of 1s in the
first three rows, etc. form the sequence of triangular numbers. The
triangular numbers also emerge from the handshake problem: If each
two people in a room shake hands exactly once, how many
handshakes take place altogether? If the answers are listed for
2, 3, 4, 5, 6, 7, ... people, the numbers are again the triangular
numbers 1, 3, 6, 10, 15, 21, ... .
1
1 1
1 1 1
1 1 1 1
1 1 1 1 1
1 1 1 1 1 1
Students imagine cutting a sheet of paper into half,
cutting the two pieces into half, cutting the four pieces into half,
and continuing this over and over again, for about 25 times. Then they
imagine taking all of the little pieces of paper and stacking them on
top of one another. Finally, they estimate how tall that stack would
be.
Students describe, analyze, and extend the
Fibonacci sequence (1, 1, 2, 3, 5, 8, ...). They research occurrences
of this sequence in nature, such as sunflower seeds, the fruit of the
pineapple, and the rabbit problem. They create their own
Fibonacci-like sequences, using different starting numbers.
6. Investigate, represent, and use non-terminating
decimals.
Students use their calculators to find the decimal
equivalent for 2/3 by dividing 2 by 3. Some of the students get an
answer of 0.66667, while others get 0.6666667. They do the problem by
hand to try to understand what is happening. They decide that
different calculators round off the answer after different numbers of
decimal places. The teacher explains that the decimal for 2/3 can be
written exactly as .666... .
Students have been looking for the number of different
squares that can be made on a 5 x 5 geoboard and have come up with
1x1, 2x2, 3x3, 4x4, and 5x5 squares. One student finds a different
square, however, whose area is 2 square units. The students wonder
how long the side of the square is. Since they know that the area is
the length of the side times itself, they try out different numbers,
multiplying 1.4 x 1.4 on their calculators to get 1.89 and
1.5 x 1.5 to get 2.25. They keep adding decimal places,
trying to get the exact answer of 2, but find that they cannot, no
matter how many places they try!
Students study which is the better way to cool down a soda,
adding lots of ice at the beginning or adding one cube at a time at
one minute intervals. Each student first makes a prediction and the
class summarizes the predictions. Then the class collects the data,
using probes and graphing calculators or computers and displays the
results in table and graph form on the overhead. The students compare
the graphs and write their conclusions in their math notebooks. They
discuss the reasons for their results in science class.
Students make a graph that shows the price of mailing a
letter from 1850 through 1995. Some of the students begin by simply
plotting points and connecting them but soon realize that the price of
a stamp is constant for a period of time and then abruptly jumps up.
They decide that parts of this graph are like horizontal lines. The
teacher tells them that mathematicians call this a "step
function"; another name for this kind of graph is a piecewise
linear graph because the graph consists of linear pieces.
Students review Mark's trip home from school on his
bike. Mark spent the first few minutes after school getting his books
and talking with friends, and left the school grounds about five
minutes after school was over. He raced with Ted to Ted's house
and stopped for ten minutes to talk about their math project. Then he
went straight home. The students draw a graph showing the distance
covered by Mark with respect to time. Then, with the teacher's
help, the class constructs a graph showing the speed at which Mark
traveled with respect to time. The students then write their own
stories and generate graphs of distance vs. time and graphs of speed
vs. time.
8. Approximate quantities with increasing degrees of
accuracy.
Students find the volume of a cookie jar by first using
Multilink cubes (which are 2 cm on a side) and then by using
centimeter cubes. They realize that the second measurement is more
accurate than the first.
Students measure the circumference and diameter of a paper
plate to the nearest inch and then divide the circumference by the
diameter. They repeat this process, using more accurate measures each
time (to the nearest half-inch, to the nearest quarter-inch,
etc.). They see that the quotients get closer and closer to
pi.
Using a ruler, students draw an irregularly shaped
pentagon on square-grid paper, taking care to locate the vertices
of the pentagon at grid points. They estimate the area of the
pentagon by counting the number of squares completely inside the
pentagon and adding to it an estimate of the number of full square
that the partial squares inside the pentagon would add up to.
Then they divide the pentagon into triangles and rectangles and find
the area of the pentagon as a sum of the areas of the triangles and
rectangles. They compare the results and explain any
difference.
9. Understand and use the concept of significant
digits.
Students measure the length and width of a rectangle in
centimeters and find its area. Then they measure its length and width
in millimeters and find the area. They note the difference between
these two results and discuss the reasons for such a difference. Some
of the students think that, since the original measurements were
correct to the nearest centimeter, then the result would be correct to
the nearest square centimeter, while the second measurements would be
correct to the nearest square millimeter. However, when they
experiment with different rectangles, for example, one whose
dimensions are 3.2 by 5.2 centimeters, they find that the area of 15
square centimeters is not correct to the nearest centimeter.
Students find the area of a "blob" using a square
grid. First, they count the number of squares that fit entirely
within the blob (no parts hanging outside). They say that this is the
least that the area could be. Then they count the number of squares
that have any part of the blob in them. They say that this is the
most that the area could be. They note that the actual area is
somewhere between these two numbers.
10. Develop informal ways of approximating the surface
area and volume of familiar objects, and discuss whether the
approximations make sense.
Students trace around their hand on graph paper and count
squares to find an approximate value of the area of their hand. They
use graph paper with smaller squares to find a better
approximation.
Students work in groups to find the surface area of a
leaf. They describe the different methods they have used to
accomplish this task. Some groups are asked to go back and reexamine
their results. When the class is convinced that all of the results
are reasonably accurate, they consider how the surface area of the
leaf might be related to the growth of the tree and its needs for
carbon dioxide, sunshine, and water.
Each group of students is given a mixing bowl and asked to
find its volume. One group decides to fill the bowl with centimeter
cubes, packing them as tightly as they can and then to add a
little. Another group decides to turn the bowl upside down and try to
build the same shape next to it by making layers of centimeter cubes.
Still another group decides to fill the hollow 1000-centimeter cube
with water and empty it into the bowl as many times as they can to
fill it; they find that doing this three times almost fills the bowl
and add 24 centimeter cubes to bring the water level up to the top of
the bowl.
11. Express mathematically and explain the impact of
the change of an object's linear dimensions on its
surface area and volume.
While learning about area, the students became curious
about how many square inches there are in a square foot. Some
students thought it would be 12, while others thought it might be
more. They explore this question using square-inch tiles to make a
square that is one foot on each side. They decide that there are 144
square inches in a square foot; they make the connection with
multiplication, noticing that 144 is 12 x 12 and that
there are 12 inches in a foot. They realize that the square numbers
have that name because they are the areas of squares whose sides are
the whole numbers.
Having measured the length, width, and height of the
classroom in feet, the students now must find how many cubic yards of
air there are. Some of the students convert their measurements to
yards and then multiply to find the volume. Others multiply first,
but find that dividing by 3 does not give a reasonable answer. They
make a model using cubes that shows that there are 27 cubic feet in a
cubic yard and divide their answer by 27, getting the same result as
the other students.
On-Line Resources
The Framework will be available at this site during Spring
1997. In time, we hope to post additional resources relating to this
standard, such as grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards. | 677.169 | 1 |
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A fantastic resource for small country schools where specialist maths teachers are rare! Our stand-in maths teacher (who is Home Ec trained) found these resources extremely helpful as she found a number of Year 9 and 10 topics quite challenging.
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The Algebra Toolbox is a mathematical learning tool for Grade 7 to Grade 12. Comprising over 150 topic-based mathematics PowerPoints the Algebra Toolbox has had phenomenal success around the world most notably in Australia, The United Kingdom and New Zealand.
Prepared by a mathematics teacher with over 30 years' experience, the PowerPoints are renowned for their presentation, quality and clarity of explanation. They are ideal for… | 677.169 | 1 |
Great math program for all the math you need up to 8th grade level. Will find for you areas, volumes, perimeters, slopes, parabolas, and some statistics, molecule problems, and miscellaneous stuff. Highly informative and even gives you the equations for each as it does them. 6k on your calculator. Enjoy! | 677.169 | 1 |
Pristine Landscapes in Elementary Mathematics
Takes familiar ideas and extends them to a rich variety of problems. The topics covered span algebra, geometry, number theory, and even a few elements of mathematical analysis. The "landscapes" presented provide a "view" into areas that are not typically encountered in great depth in standard coursework but nonetheless have profound implications.
This book takes familiar ideas and extends them to a rich variety of problems. The intended audience is the ambitious high school or college student. The topics covered span algebra, geometry, number theory, and even a few elements of mathematical analysis. Each chapter explores specific themes and ideas that underlie the aforementioned subject areas. The "landscapes" presented provide a "view" into areas that are not typically encountered in great depth in standard coursework but nonetheless have profound implications.
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How did Newton describe the orbits of the planets? To do this, he created calculus. But he used a different coordinate system more appropriate for planetary motion. We will learn to shift our perspective to do calculus with parameterized curves and polar coordinates. And then we will dive deep into exploring the infinite to gain a deeper understanding and powerful descriptions of functions.
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Parametric Equations
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Series and Polynomial Approximations
Series and Convergence
Taylor Series and Power Series
This course, in combination with Parts 1 and 2, covers the AP* Calculus BC curriculum.
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2. Orchestrating Strategic Alliances:
What is the case for cooperative strategy? When should we move from a competitive to a cooperative mindset?
What are strategic alliances and what is their payoff?
How can firms build and orchestrate collaborative ecosystems to enhance their (innovative) performance?
3. Embracing Social Responsibility:
What is the role of business in society? How do organizations contribute and cause harm to the public good?
What exactly are the social responsibilities of organizations?
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This capstone exam includes the evaluation of the competencies and performance tasks, which define a successful project manager.
This capstone exam is part of the RITx Project Management MicroMasters program that is designed to provide you with the in-depth knowledge and skills needed to be a successful project manager in any industry. In order to qualify for the MicroMasters Credential you will need to earn a Verified Certificate in each of the three RITx Project Management courses as well as pass this final capstone exam.
The capstone exam will test knowledge across all 3 courses. It will be webcam proctored timed exam and will feature problems similar to the graded assignments in the different modules as well as some writing.
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You'll develop the necessary skills to identify possible security issues in the cloud environment. You will also gain experience with tools and techniques that monitor the environment and help prevent security breaches such as monitoring logs and implementing appropriate security policies.
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You will learn how to turn project management principles and theory into practice. The course will cover:
project management methods and best practices
project portfolio management
the project management office
Six Sigma
corporate culture and organizational behavior
project management leadership
The course will utilize case studies and examples from companies to help students sharpen their project management skills to recognize and implement an environment that supports success.
First, we will cover the basic organizational and leadership elements required to provide a successful environment for all projects to succeed.
Second, we will cover the necessary organizational support structures and methods that enable project management and project managers to deliver results to the business and to the customers they serve.
Finally, we will explore the challenges of specific types of projects such as projects in crisis, global projects and managing a portfolio.
This course is part of the RIT Project Management MicroMasters Program that is designed to teach the importance of the organizational and leadership characteristics for the success of projects. In order to qualify for the MicroMasters Credential, you will need to earn a verified certificate in each of the three courses as well as pass a capstone exam.
In this course we will explore the impact on project management of culture, language variations, religious, regulatory and legal practices, technology penetration, temporal orientation, gender issues, corruption, ethics, personal liberty and political contexts. We learn how to meet global projects challenges through efficient use of practices and technology. The course will utilize available case studies and examples from companies to help students sharpen the skills needed to recognize and foster a successful international project environment.
First, you will learn how culture affects how teams perceive each other, lead, solve problems and execute tasks. Although the world is increasingly connected, the people behind the projects have biases, expectations and a perception of life that impacts all decisions.
Second, you will learn how to effectively manage global teams including how to build trust and collaboration across various cultures, time zones and technological settings. You will learn how to design communication channels and project structures effectively in a global project environment.
Third, you will become familiar with the issues underlying the problem of corruption, which is the abuse of trusted power for private gain. Reducing the risk of corruption strengthens a company's reputation, builds the respect of employees and raises credibility with key stakeholders.
Finally, you will learn how the adoption of collaboration tools can enhance the global project experience.
This course is part of the RIT Project Management MicroMasters Program that is designed to teach how to successfully deliver projects in an international environment. In order to qualify for the MicroMasters Credential, you will need to earn a verified certificate in each of the three courses as well as pass a capstone exam.
Have you ever wondered how your favorite mobile applications are developed?
Join us on a gentle journey through the mobile application development landscape, using Android as the platform. Along the way we will learn to use Android Studio, the integrated development environment (IDE) for Android apps. This course is intended for students who have some prior programming experience. The course will introduce you to the basics of the Android platform, Android application components, Activities and their lifecycle, UI design, Multimedia, 2D graphics and networking support in Android.
By the end of this course you will have developed a spoken and written profile in German that reflects your backgrounds, customs and the cities you live in. Through the medium of a foreign language, we will learn about each other and exchange our reactions to contemporary German life and lifestyles. | 677.169 | 1 |
In How Math Explains the World , mathematician Stein reveals how seemingly arcane mathematical investigations and discoveries have led to bigger, more world-shaking insights into the nature of our world. In the four main sections of the book, Stein tells the stories of the mathematical thinkers who discerned some of the most fundamental aspects of... more...
Presents Results from a Very Active Area of Research Exploring an active area of mathematics that studies the complexity of equivalence relations and classification problems, Invariant Descriptive Set Theory presents an introduction to the basic concepts, methods, and results of this theory. It brings together techniques from various areas... more...
Practitioners of many skills face the need to make some realistic statement about the likely outcome of a future 'experiment of interest' on the basis of observed variability of outcomes in previously conducted related experiments. more...
This book... more...
The first of two volumes, this text compiles survey articles borne of a landmark 1999 Tel Aviv conference dedicated to probing the importance, the methods, the past and future of mathematics as well as the connection between mathematics and related areas. more...
This collection of original and review articles covers many topics in theoretical developments in operator theory and its diverse applications in applied mathematics, physics, engineering and other disciplines. more... | 677.169 | 1 |
Mathematics
Z-Calculate is an innovative calculator for students, scientists etc. It combines mathematical power and cool features with a clear and user-friendly interface. Both real and complex numbers are supported. And, best of all, it is completely free!
Rapid-Pi is an add-on for Microsoft Word (and other word processors) that will transform the way you enter mathematical formulae, equations and expressions into documents.Rapid-Pi was designed with a single purpose in mind - to save you time when editing
Numero is a calculator program that does the order of operations, so you can type in the whole expression at once instead of just one number at a time. It has about everything that you would expect to see on a scientific calculator. Also, in addition
This calculator allows natural entry of equations (i.e. 17-(9*8) - 7 + 34.23). Then it goes beyond just giving you a result, the "steps" box, shows you how it solved the equation step by step. Results are given simultaneously in dec/hex/oct/bin. Great
Supports
Calculator Prompter is a math expression calculator. Calcualtor Prompter has a built-in error recognition system that helps you get correct results. With Calculator Prompter you can enter the whole expression, including brackets, and operators. You can
Text editor with the additional capabilities of math notation and hypertext, aimed at the high school / college environment. Uses the RTF format known to Wordpad, Word. Generates HTML, so that math notation can be displayed by popular browsers. Users
The Math Homework Maker,is a FREE software which can solve all your math homeworkIts covers everything you needConversion of FractionOperators On Fractions Areas and PerimetersTraffics and Percents ProblemsQuadratic EquationProgressionThe Law Of Sines
Free mental math practice software. Covers all the basic math skills - Addition, Subtraction, Multiplication, and Division.You choose what you want to work on. The software delivers the questions, checks your answers, and keeps your score so you can | 677.169 | 1 |
Logarithmic Differentiation
This resource also includes:
Twelfth graders investigate logarithm differentiation. In this calculus lesson, 12th graders explore situations in which one would use logarithmic differentiation as an appropriate method of solution. Students should have already studied the chain rule. | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
Common Core Algebra 1/Integrated 1 Practice Test #2
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Product Description
Because of the new common core and new tests being active this year, there are little to no resources for us teachers to use to prepare our students for their End-of-Course exams. Because of this, I created a practice test for us to use to help prepare our students for what they will encounter at the end of their Algebra 1 and Integrated 1 experiences.
This is a 50 question practice EOC test for Algebra 1 and Integrated 1. All of the questions are originally written and all graphs are originally produced using the new standards and released tests as a guide.
An answer key as well as a blank answer sheet for the students to fill in are both included!
Please use this as you see fit for review - as a classroom tool, a homework packet, for group work, or centers! I hope that you find it useful in your endeavors to help your students be as successful as possible!
**A Common Core Alignment Sheet is also included to show where each question lines up with the new Common Core!** | 677.169 | 1 |
Learning Objective
At the end of this course, students are able to:-
1. Provide students with background in the quantitative techniques necessary to better understanding and appreciate the courses normally taking in undergraduate framing (eg. Set theory, Counting techniques, probability, probability distributions, Logic), while given emphasis to the real-world applications from these fields.
2. Use the concepts of these areas of study and apply knowledge gained in the real life. | 677.169 | 1 |
Tutorials: Tutorial assistance is available Monday to Friday (starting mid-September) from 10:30 am to 3:30 pm in the MathLab, South 525 Ross.
Course Description :
This course is an introduction to basic topics in mathematics, including logic, sets, functions, relations,
modular arithmetic and applications of elementary number theory, proof techniques, and induction. The emphasis
will be on understanding the basic ideas, and developing an
appreciation for mathematical reasoning, proofs and
problem solving. We will cover the followings sections of the textbook:
Course Evaluation:
30% Quizzes : There will be four in-class quizzes. The tentative dates for the quizzes are:
Wednesdays September 27, October 11, November 15 and November 29, 2017.
Quiz 1 will be held on Wednesday September 27 from 1:55 to 2:15 pm. It covers Sections 1.1, 1.3, 1.4 and 1.5. The problems of the quiz will be selected among the practice problems of weeks 1, 2 and 3, corresponding to these sections. Here is the solution of Quiz 1.
Quiz 2 will be held on Wednesday October 11 from 1:55 to 2:15 pm. It covers Sections 1.6, 1.7 and 2.1. The problems of the quiz will be selected among the practice problems of weeks 3 and 4, corresponding to these sections.
25% Midterm exam: The midterm exam will be in-class. The tentative date for the midterm exam is: Monday, October 30, 2017.
45% Final Exam: The final exam will be comprehensive.
Notes: There will be no makeup test or quizzes. If you miss the midterm test or one of the quizzes and have a valid excuse (with an acceptable documentation), the exam weight will be transferred to the final exam. The last date to drop the course is Friday, November 10, 2017. | 677.169 | 1 |
Abstract
The results shown in the following chapter come from a research that shows some of the transformations of the concept of linear function observed from its definition in mathematics when facing the written, intended and enacted curriculum. This study was developed in technological high school level, with the participation of three teachers of mathematics, who were teaching the course called Functions and Algebraic Thinking. A case study was considered as a practical research method to carry out the investigation, adopting various processes to gather evidence to describe, verify or create theory. Different sources of evidence were used to gather and analyze information, such as: the official program of the course, three textbooks, class recordings and the application of a questionnaire. The results show transformations in both, the concept of function and the concept of linear function, pointing out, in this way, an educational problem that should be solved by modifying not only the concept, but also the teaching and learning of it.
Problem Statement
The study of the concept of function in the teaching of mathematics in high schools plays an important role in students' learning, not only for the fact of being related to topics of different subjects, but also for the fact that it allows to represent real situations (Hitt, 2002). The different difficulties that arise when there are new studies related to the concept of function or to a type of function must also be emphasized.
Diaz (2008) notes that the curricular aspect of the concept of function is a kind of strand that goes from the basic education to university. He also warns about the difficulties the students face to understand this concept, as well as how this concept has generated a growing body of researches, like the ones that study the problem of teaching, the difficulties of learning, those that propose theoretical frameworks or even those that are focused on the diversity of interpretations of the concept of function.
There are several authors that have been devoted to work on the concept of function. During the 1980s, Leinhardt, Zaslavsky and Stein (1990) made a bibliographic revision, in which they showed the difficulties that students face when they try to conceptualize the idea of function, emphasizing issues related to the function such as rule of correspondence as well as its different representations, its reading and interpretation. On the other side, in the research done by Birgin (2012), linear functions are seen as a complex idea of multiple faces whose power and richness cover almost all of the mathematical areas. It is important to add that due to its several applications in the real world some topics of more advanced level, like those that come from calculus, can be reinforced. | 677.169 | 1 |
This book, intended for researchers and graduate students in physics, applied mathematics and engineering, presents a detailed comparison of the important methods of solution for linear differential and difference equations - variation of constants, | 677.169 | 1 |
Student-friendly and comprehensive, this book covers topics such as Mathematical Logic, Set Theory, Algebraic Systems, Boolean Algebra and Graph Theory that are essential to the study of Computer Science in great detail.
Numerical Methods form an integral part of the mathematical background required for students of Mathematics, Science and Engineering. This book, Numerical Methods, is an extension of four long experience of teaching this subject to various courses. The invaluable experience of using computer based numerical techniques for research and a project has helped value add to this book. Clarity ...... | 677.169 | 1 |
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Algebra Bridge Class
May 30 - Jun 10
TRINITY ACADEMY
Starting at
$145.00
Meeting Dates
From May 31, 2016 to Jun 10, 2016
About This Activity
This class is for the student who has completed Algebra I and wants to reinforce those skills before entering Geometry or Algebra II. There will be particular emphasis on the second semester Algebra I concepts that typically cause students difficulty. Items covered will include the following: solving and rearranging equations, slope, writing equations of lines, solving systems of equations, simplifying exponents, multiplying binomials (FOIL) reducing radical expressions, and factoring. Students are expected to take notes, work on in-class assignments, and should be prepared to complete some assignments at home. Paper, pencils, a folder, and a calculator should be brought to class each day. Instructor: John Hendley | 677.169 | 1 |
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