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Sunday, February 28, 2010
Innovative Ideas That Make Sense for Those Hungry for Math Instruction
It is a mistake to suppose that requiring the nonmathematical to take more advanced math courses will enhance their understanding and not merely exacerbate their sense of inadequacy. -- William Raspberry
I have a dream. A recurring dream and I don't think I'm alone. In fact, I just had this dream AGAIN last night. My dream is that I am contacted by someone from my undergrad school who tells me that they discovered there was a mistake in their records and I didn't actually pass algebra (sometimes this is geometry, but usually algebra) so I will need to go back to school for a semester and take the class or all my degrees will be canceled. I'm shocked to learn this but confirm it is true. So I enroll in the class and once there I struggle with the class and hate it as much as I did when I was actually in high school and college. What does this mean? When will I ever use this?
That was my math mantra and to my frustration these questions were left unanswered by my instructors and met with rolled eyes from other students who wanted to move on because they had accepted that math had a god-like quality and was not meant to be questioned.
Recently, the PBS News Hour featured a series of YouTube math videos that cover everything from basic addition to calculus. In them, the narrator poses a problem, and walks through the steps to solve it. After just four years, these videos have attracted tens of thousands of students a day, and are used by schools and students around the world.
Yet as popular as the videos are, and, as powerful as they certainly have been for those having difficulty with a concept, it still doesn't answer those two questions that resulted in my distaste for the subject. What does this mean? When will I ever use this?
The classes are ideal for the student who has accepted math should not be questioned, but for a student like me where the real learning lies beyond the "how" and into the "why," such videos are of little value without the answers. Academic studies and anecdotal evidence alike support a simple (and perhaps obvious) fact: students learn best when the instruction is meaningful and relevant. This is particularly true in mathematics where, starting in the middle grades, content becomes increasingly abstract. During my high school and math classes, while I was often the only one to actually speak up, I don't think I was alone. In hindsight I probably had silent supporters in the shadows too embarrassed to share their frustration.
To take this to a real-world example let's take the algebraic concept of "slope" which is defined as "rise over run," "∆y over ∆x," or "y2 – y1 over x2 – x1."
HUH???
What does any of this mean to me? Does any of this make sense? Isn't this just another procedure/algorithm to memorize? And, "Why should I care?" My answer to the equation: Nothing; No/Yes; I don't.
While the aforementioned videos are a valuable tool for differentiating instruction, for me reformatting traditional content for YouTube and the iPhone helps students learn the algorithm better, but the fundamental questions are left unanswered and my annoying mantra still exists. What does this mean? When will I use it?
These are not just good questions, but critical ones. Like many students, I was led to believe Algebra was an isolated subjected created for the sole purpose of teaching critical and higher order thinking skills out of context. But the reality of what math actually is, that they never taught me in class is that math is:
1) A set of logical tools that we [humans] created to 2) Explore the world around us.
Math skills such as slope were not, as many students might assume, codified in the Big Bang. Divide thy riseth by thy runeth was not the Eleventh Commandment. Instead, at some point in our human development we had a question, we needed a tool, and this is what we came up with. To illustrate this point, here's another real-world example.
Question: what are percents, and why did we invent them? Answer: because they allow us to compare things that are otherwise difficult to compare.
In one store, we save $4 for every $10 that you spend. In another, we save $9 for every $25. Where should you shop?
There are any number of ways to approach this. One method would be to compare how much we'd save if we spent the same amount: we could spend $50 in both stores, and save either $20 or $18, respectively.
But what if the numbers weren't so clean? What if, instead of $10 and $25, the amounts were $12 and $25? Here, the "multiples" approach is a bit more cumbersome, yet the underlying question remains: Where should we shop?
At some point in our history, mankind faced a situation like this and said, Lets just pick some number and compare everything to that. For whatever reason, we picked 100. So now instead of finding a common multiple, we simply ask, How much would we save if we spent $100 at each store.
So when could we use this?
According to the Wheel of Fortune wheel, bankrupt should come up once out of every 24 spins. If in an actual episode it comes up three times in 60 spins, can we conclude that the show is rigged? For every 100 spins…
That's the logic of the percent. It's not magic. It's not ordained. It's simply a useful tool.
What about slope? A traditional source might ask us to calculate the slope between (16 , 629) & (32 , 729). But if a student is only taught the procedure, what does he actually know, and how long will he remember it?
But what if we instead approached the problem as: The 16GB iPad costs $629. The 32GB costs $729. How much is each additional gigabyte of hard drive space?
With this simple question, a student might reason:
Slope If an additional 16GB costs an additional $100, then Apple is charging $6.25/GB.
Y-Intercept If Apple charges $6.25/GB, then 32GB would cost $200. But since the actual cost of the 32GB iPad is $729, the base-cost must be $529.
Equation The cost, C = 529 + 6.25g
Evaluation Based on the equation, the 64GB model should cost 529 + 6.25(64), or $929, but it actually costs $829. Therefore, iPad pricing isn't linear.
Of course, this emphasis on context does not mean that math classes should revolve entirely around real-world problems such as Wheel of Fortune, batting averages and the iPad. At its extreme, this would be just as limiting as rote-procedure, albeit in the opposite direction.
Instead, effective math instruction involves a three-step process:
1) contextualize a problem to explore a skill ($/gig) 2) generalize the skill (change in y due when x increases by 1) 3) apply the skill to a wide-range of real-world topics (effect of music tempo on running time, marginal benefit of another piece of Halloween candy).
Unfortunately, teaching too often addresses only the second step. It's understandable, then, why so many students construe mathematics as an arbitrary collection of meaningless steps; why so many ask why they have to learn it; and why so many absolutely hate it.
Fine. This all sounds good. But don't teachers already feel overwhelmed by the demands of teaching? Won't this approach take three times as long? Doesn't the author get that I have to cover this material before the end-of-year test?
These are legitimate questions. Fortunately, addressing the meaning behind and applications of mathematics has a strange effect: it actually saves time, and allows teachers to cover more material in more depth, and with better results.
The earlier question about saving money at a store? In the "spend $50" approach, we implicitly addressed months of instruction: common multiples; the lowest common multiple; equivalent fractions; simplifying fractions; and ratios & proportions. We then extended this to percents with the "out of 100" step, and could have easily included decimals by asking, How much do we save for every one dollar that we spend?
Likewise, the iPad example addressed most of the topics surrounding linear functions. Yet were any of the steps arbitrary? Was there any place where students would have asked, What does this mean?, or When will I use this?
Of course, this is not to say that rote practice does not have its place in math education. But for students like me, the practice comes after the fundamental questions are answered. The practice is not a substitute for learning procedure or a replacement for understanding. But, practice, after-school tutoring programs and drills-based YouTube videos play a more effective role after, "What does this mean? When will I use it?"has been answered.
In the end, true innovation and lasting progress in math education will come not by repackaging or rebranding methodologies, but by emphasizing meaningful and intentional instruction. And this requires math teach-ers, not simply math do-ers. Once a context is set for the videos in the Frontline special their on-demand nature has a more valuable place.
At its heart math is simple. We would do well to pull back the curtain and remember that.
Math is a tool. It's a tool that we created—that we continue to create—to make sense of the world. And in our efforts to guide students through math, we can't ignore the world. We can't ignore the sense. To do so is to ignore mathematics itself.
I imagine some math teachers, many who grew up just accepting the idea that it was okay to teach math without answering these questions, may agree with this philosophy but feel it would be unrealistic to expect them to be able to answer these questions for all math concepts. Furthermore, they already have a curriculum to follow, standards to meet, and a textbook they use. How could one begin to teach this way??? There is help in a site featured in the New York Times' Freakonomics blog this month in a post called, "Making Math More Appetizing" The blog explains the site as follows: Mathalicious provides free math lessons, including supporting materials, for teachers and parents. The organization hopes to "transform the way math is taught and learned by focusing not only on skills but on the real-world applications of math, from sports to politics to video games to exercise." So far, they've used the Pythagorean Theorem to determine how big a 42-inch TV really is; used percentages to examine environmental issues; and asked whether music can kill you.
The lessons are written in an ease-to-use, teacher-friendly format which makes sense since the site founder was a public school math teacher and later a math coach who worked with teachers to improve instruction by teaching for conceptual understanding and relevance. This site helps provide the answer for students like me who were hungry for the answer to the questions "What does this mean? When will I use it?" before being able to consume an out-of-context lesson. With a foundation like this, kids may find they no longer need tutors, and the question of "when will I use this?" will be a thing of the past.
This post was written jointly by Lisa Nielsen, The Innovative Educator and Karim Kai Logue, the founder and CEO of Mathalicious, which creates meaningful and real-world math content for parents, students and teachers. | 677.169 | 1 |
Algebra I in the 8 th Grade: Considerations and Consequences.
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Algebra I in the 8 th Grade: Considerations and Consequences
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Do you have middle school students who should have the option of taking Advanced Placement (AP) Mathematics, or two advanced mathematics courses as part of their high school experience? Some Pathways for Students Who Complete Algebra I in Grade 8 Geometry Geometry Geometry Algebra II W/Trig Algebra II W/Trig Algebra II W/Trig Precalculus Discrete MathematicsPrecalculus Analytical Mathematics Precalculus Advanced Placement (AP) Mathematics Course (ACOS: Mathematics, 2010, p. 127)
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"Systems offering Algebra I in the eighth grade have the responsibility of ensuring that all Algebra I course content standards and Grade 8 course content standards be included in instruction." (ACOS: Mathematics, 2010, p. 81) The State Department of Education will provide further guidance and training (Phase II) in the fall of 2011 relative to issues local education agencies may encounter in providing an Algebra I course in Grade 8.
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Decisions to accelerate students into a high school Algebra I course before Grade 9 should not be rushed. Placing students into an Algebra I course too early should be avoided at all costs. Local education agency's decision should: Be Advertised Be Equitable Provide Written Policy Decisions to accelerate students into a high school Algebra I course before Grade 9 should be based on solid evidence of student learning.
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Not all students are ready for Algebra I in Grade 8. The 2010 COS Algebra I content is not the same as the Algebra I content in earlier Alabama Courses of Study. Much of what was previously included in Algebra I will now be taught in Grades 6-8 in the 2010 COS. A Southern Regional Education Board (SREB) study found a difference in readiness for higher-level mathematics between students scoring in the bottom quartile and those scoring in the top 3 quartiles. Graduates who completed Geometry or Algebra II as ninth graders earned an average Grade 12 NAEP mathematics at the Proficient level.
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"Mathematics leaders need to ensure equitable access to courses by carefully monitoring barriers to participation." (A Guide to Mathematics Leadership, 2010, p. 4) There are valid reasons for placing students in certain courses, but mathematics leaders are expected to ensure that there are no systematic barriers that discriminate against specific populations. (A Guide to Mathematics Leadership, 2010, pp ) An accelerated Grade 7 course and a Grade 8 Algebra I course differ from the Grade 7, Grade 8, and Algebra I courses found in the 2010 COS in that they contain additional content by comparison and demand a faster pace for instruction and learning.
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Ensure that the Grade 8 Algebra I course is not watered down. Ensure that there is equity in support materials for all courses, that is, that more and better materials and supports are not just in place for the accelerated and advanced courses. Ensure that interventions are in place so that all students are prepared for high school Algebra I coursework.
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Allow students to take two math electives simultaneously, provided the course progression chart is followed. Use block scheduling to take a math course both semesters. Offer Credit Advancement. Offer Dual Enrollment. Offer summer courses that are designed to provide an equivalent experience of a full course. | 677.169 | 1 |
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Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space.Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.
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11.1: Matrices
Introduction to Matrices
The following matrix, stolen from a rusted lockbox in the back of a large, dark lecture hall in a school called Hogwart's, is the gradebook for Professor Severus Snape's class in potions.When I say this is a "matrix" I'm referring to the numbers in boxes. The labels (such as "Granger, \begin{align*}H\end{align*}" or "Poison") are labels that help you understand the numbers in the matrix, but they are not the matrix itself.
Each student is designated by a row. A row is a horizontal list of numbers.
1. Below, copy the row that represents all the grades for "Malfoy, \begin{align*}D\end{align*}."
Each assignment is designated by a column, which is a vertical list of numbers. (This is easy to remember if you picture columns in Greek architecture, which are big and tall and...well, you know...vertical.)
2. Below, copy the column that represents all the grades on the "Love philter" assignment.
I know what you're thinking, this is so easy it seems pointless. Well, it's going to stay easy until tomorrow. So bear with me.
The dimensions of a matrix are just the number of rows, and the number of columns...in that order. So a "\begin{align*}10 \times 20\end{align*}" matrix means \begin{align*}10\end{align*} rows and \begin{align*}20\end{align*} columns.
3. What are the dimensions of Dr. Snape's gradebook matrix?
For two matrices to be equal, they must be exactly the same in every way: same dimensions, and every cell the same. If everything is not precisely the same, the two matrices are not equal.
4. What must \begin{align*}x\end{align*} and \begin{align*}y\end{align*} be, in order to make the following matrix equal to Dr. Snape's gradebook matrix?
Finally, it is possible to add or subtract matrices. But you can only do this when the matrices have the same dimensions!!! If two matrices do not have exactly the same dimensions, you cannot add or subtract them. If they do have the same dimensions, you add and subtract them just by adding or subtracting each individual cell.
As an example: Dr. Snape has decided that his grades are too high, and he needs to curve them downward. So he plans to subtract the following grade-curving matrix from his original grade matrix.
563. Rewrite problem \begin{align*}^\#1\end{align*} as a multiplication problem. (Remember what multiplication means—adding something to itself a bunch of times!)
This brings us to the world of multiplying a matrix by a number. It's very straightforward. You end up with a matrix that has the same dimensions as the original, but all the individual cells have been multiplied by that number.
Now, we saw how Professor Snape could lower his grades (which he loves to do) by subtracting a curve matrix. But there is another way he can lower his grades, which is by multiplying the entire matrix by a number. In this case, he is going to multiply his grade matrix by \begin{align*}\frac{9}{10}\end{align*}. If we designate his grade matrix as \begin{align*}[S]\end{align*} then the resulting matrix could be written as \begin{align*}\frac{9}{10}[S]\end{align*}. (\begin{align*}^*\end{align*}Remember that the cells in a matrix are numbers! So \begin{align*}[S]\end{align*} is just the grades, not the names.)
4. Below, write the matrix \begin{align*}\frac{9}{10}[S]\end{align*}.
Finally, it's time for Professor Snape to calculate final grades. He does this according to the following formula For instance, to calculate the final grade for "Granger, \begin{align*}H\end{align*}" he does the following calculation: \begin{align*}(30\%)(100)+(20\%)(105)+(15\%)(99)+(35\%)(100)=100.85\end{align*}.
To make the calculations easier to keep track of, the Professor represents the various weights in his grading matrix which looks like the following:
A "row matrix" means a matrix that is just one row. A "column matrix" means...well, you get the idea. When a row matrix and a column matrix have the same number of items, you can multiply the two matrices. What you do is, you multiply both of the first numbers, and you multiply both of the second numbers, and so on...and you add all those numbers to get one big number. The final answer is not just a number—it is a \begin{align*}1 \times 1\end{align*} matrix, with that one big number inside it.
5. Below, write the matrix multiplication that Professor Snape would do to find the grade for "Potter, \begin{align*}H\end{align*}." Show both the problem (the two matrices being multiplied) and the answer (the \begin{align*}1 \times 1\end{align*} matrix that contains the final grade).
Just for a change, we're going to start with...Professor Snape's grade matrix!As you doubtless recall, the good Professor calculated final grades by the following computation He was able to represent each student's final grade as the product of a row matrix (for the student) times a column matrix (for weighting).
1. Just to make sure you remember, write the matrix multiplication that Dr. Snape would use to find the grade for "Malfoy, \begin{align*}D\end{align*}." Make sure to include both the two matrices being multiplied, and the final result!
I'm sure you can see the problem with this, which is that you have to write a separate matrix multiplication problem for every student. To get around that problem, we're going to extend our definition of matrix multiplication so that the first matrix no longer has to be a row—it may be many rows. Each row of the first matrix becomes a new row in the answer. So, Professor Snape can now multiply his entire student matrix by his weighting matrix, and out will come a matrix with all his grades! Let's try it. Do the following matrix multiplication. The answer will be a \begin{align*}3 \times 1\end{align*} matrix with the final grades for "Malfoy, \begin{align*}D\end{align*}," "Potter, \begin{align*}H\end{align*}," and "Weasley, \begin{align*}R\end{align*}."
OK, let's step back and review where we are. Yesterday, we learned how to multiply a row matrix times a column matrix. Now we have learned that you can add more rows to the first matrix, and they just become extra rows in the answer.
For full generality of matrix multiplication, you just need to know this: if you add more columns to the second matrix, they become additional columns in the answer! As an example, suppose Dr. Snape wants to try out a different weighting scheme, to see if he likes the new grades better. So he adds the new column to his weighting matrix. The first column represents the original weighting scheme, and the second column represents the new weighting scheme. The result will be a \begin{align*}3 \times 2\end{align*} matrix where each row is a different student and each column is a different weighting scheme. Got all that? Give it a try now!
b. Test your answer by doing the multiplication to make sure it works out.
Name: __________________
The "Identity" and "Inverse" Matrices
This assignment is brought to you by one of my favorite numbers, and I'm sure it's one of yours...the number\begin{align*}1\end{align*}. Some people say that\begin{align*}1\end{align*}is the loneliest number that you'll ever do. (Bonus: who said that?) But I say,\begin{align*}1\end{align*}is the multiplicative identity.
Allow me to demonstrate.
1. \begin{align*}5 \times 1 =\end{align*}
2. \begin{align*}1 \times \frac{2}{3} =\end{align*}
3. \begin{align*}-\pi \times 1 =\end{align*}
4. \begin{align*}1 \times x =\end{align*}
You get the idea?\begin{align*}1\end{align*}is called the multiplicative identity because it has this lovely property that whenever you multiply it by anything, you get that same thing back. But that's not all! Observe...
5. \begin{align*}2 \times \frac{1}{2} =\end{align*}
6. \begin{align*}\frac{-2}{3} \times \frac{-3}{2}\end{align*}
The fun never ends! The point of all that was that every number has an inverse. The inverse is defined by the fact that, when you multiply a number by its inverse, you get\begin{align*}1\end{align*}.
7. Write the equation that defines two numbers \begin{align*}a\end{align*} and \begin{align*}b\end{align*} as inverses of each other.
8. Find the inverse of \begin{align*}\frac{4}{5}\end{align*}.
9. Find the inverse of \begin{align*}-3\end{align*}.
10. Find the inverse of \begin{align*}x\end{align*}.
11. Is there any number that does not have an inverse, according to your definition in \begin{align*}^\#7?\end{align*}
So, what does all that have to do with matrices? (I hear you crying.) Well, we've already seen a matrix which acts as a multiplicative identity! Do these problems.
Pretty nifty, huh? When you multiply \begin{align*}\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}\end{align*} by another \begin{align*}2 \times 2\end{align*} matrix, you get that other matrix back. That's what makes this matrix (referred to as\begin{align*}[I]\end{align*}) the multiplicative identity.
Remember that matrix multiplication does not, in general, commute: that is, for any two matrices\begin{align*}[A]\end{align*}and\begin{align*}[B]\end{align*}, the product\begin{align*}AB\end{align*}is not necessarily the same as the product\begin{align*}BA\end{align*}. But in this case, it is: \begin{align*}[I]\end{align*}times another matrix gives you that other matrix back no matter which order you do the multiplication in. This is a key part of the definition of\begin{align*}I\end{align*}, which is...
Definition: The matrix \begin{align*}I\end{align*} is defined as the multiplicative identity if it satisfies the equation:
\begin{align*}AI = IA = A\end{align*}
Which, of course, is just a fancy way of saying what I said before. If you multiply I by any matrix, in either order, you get that other matrix back.
14. We have just seen that \begin{align*}\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}\end{align*} acts as the multiplicative identity for a \begin{align*}2 \times 2\end{align*} matrix.
a. What is the multiplicative identity for a \begin{align*}3 \times 3\end{align*} matrix?
b. Test this identity to make sure it works.
c. What is the multiplicative identity for a \begin{align*}5 \times 5\end{align*} matrix? (I won't make you test this one...)
d. What is the multiplicative identity for a \begin{align*}2 \times 3\end{align*} matrix?
e. Trick question! There isn't one. You could write a matrix that satisfies \begin{align*}AI=A\end{align*}, but it would not also satisfy \begin{align*}IA=A\end{align*}—that is, it would not commute, which we said was a requirement. Don't take my word for it, try it! The point is that only square matrices (*same number of rows as columns) have an identity matrix.
So what about those inverses? Well, remember that two numbers\begin{align*}a\end{align*}and\begin{align*}b\end{align*}are inverses if\begin{align*}ab=1\end{align*}. As you might guess, we're going to define two matrices\begin{align*}A\end{align*}and\begin{align*}B\end{align*}as inverses if\begin{align*}AB=[I]\end{align*}. Let's try a few.
You see? These two matrices are inverses: no matter which order you multiply them in, you get\begin{align*}[I]\end{align*}. We will designate the inverse of a matrix as\begin{align*}A^{-1}\end{align*}, which looks like an exponent, but isn't really, it just means inverse matrix—just as we used\begin{align*}f^{-1}\end{align*}to designate an inverse function. Which leads us to...
Definition
The matrix \begin{align*}A^{-1}\end{align*} is defined as the multiplicative inverse of \begin{align*}A\end{align*} if it satisfies the equation:
a. Since we don't know the inverse yet, we will designate it as a bunch of unknowns: \begin{align*}\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}\end{align*} will be our inverse matrix. Write down the equation that defines this unknown matrix as our inverse matrix.
b. Now, in your equation, you had a matrix multiplication. Go ahead and do that multiplication, and write a new equation which just sets two matrices equal to each other.
c. Now, remember that when we set two matrices equal to each other, every cell must be equal. So, when we set two different \begin{align*}2 \times 2\end{align*} matrices equal, we actually end up with four different equations. Write these four equations.
Today you are going to find the inverse of the generic\begin{align*}2 \times 2\end{align*}matrix. Once you have done that, you will have a formula that can be used to quickly find the inverse of any\begin{align*}2 \times 2\end{align*} matrix.
Our goal is to find a formula for \begin{align*}w\end{align*} in terms of our original variables \begin{align*}a, b, c\end{align*}, and \begin{align*}d\end{align*}. That formula must not have any \begin{align*}w, x, y\end{align*}, or \begin{align*}z\end{align*} in it, since those are unknowns! Just the original four variables in our original matrix \begin{align*}[A]\end{align*}. Then we will find similar formulae for \begin{align*}x, y\end{align*}, and \begin{align*}z\end{align*} and we will be done.
Our approach will be the same approach we have been using to find an inverse matrix. I will walk you through the steps—after each step, you may want to check to make sure you've gotten it right before proceeding to the next.
1. Write the matrix equation that defines \begin{align*}A^{-1}\end{align*} as an inverse of \begin{align*}A\end{align*}.
2. Now, do the multiplication, so you are setting two matrices equal to each other.
3. Now, we have two \begin{align*}2 \times 2\end{align*} matrices set equal to each other. That means every cell must be identical, so we get four different equations. Write down the four equations.
4. Solve. Remember that your goal is to find four equations—one for \begin{align*}w\end{align*}, one for \begin{align*}x\end{align*}, one for \begin{align*}y\end{align*}, and one for \begin{align*}z-\end{align*}where each equation has only the four original constants \begin{align*}a, b, c\end{align*}, and \begin{align*}d!\end{align*}
5. Now that you have solved for all four variables, write the inverse matrix \begin{align*}A^{-1}\end{align*}.
\begin{align*}A^{-1}=\end{align*}
6. As the final step, to put this in the form that it is most commonly seen in, note that all four terms have an \begin{align*}ad-bc\end{align*} in the denominator. (\begin{align*}^*\end{align*}Do you have a \begin{align*}bc-ad\end{align*} instead? Multiply the top and bottom by \begin{align*}-1!\end{align*}) we can write our answer much more simply if we pull out the common factor of \begin{align*}\frac{1}{ad-bc}\end{align*}. (This is similar to "pulling out" a common term from a polynomial. Remember how we multiply a matrix by a constant? This is the same thing in reverse.) So rewrite the answer with that term pulled out.
\begin{align*}A^{-1}=\end{align*}
You're done! You have found the generic formula for the inverse of any \begin{align*}2 \times 2\end{align*} matrix. Once you get the hang of it, you can use this formula to find the inverse of any \begin{align*}2 \times 2\end{align*} matrix very quickly. Let's try a few!
a. Find the inverse—not the long way, but just by plugging into the formula you found above.
b. Test the inverse to make sure it works.
9. Can you write a \begin{align*}2 \times 2\end{align*} matrix that has no inverse?
Name: ______________________
Using Matrices for Transformation
You are an animator for the famous company Copycat Studios. Your job is to take the diagram of the "fish" below (whose name is Harpoona) and animate a particular scene for your soon-to-be-released movie.
In this particular scene, the audience is looking down from above on Harpoona who begins the scene happily floating on the surface of the water. Here is a picture of Harpoona as she is happily floating on the surface.
1. Explain, in words, how this matrix represents her position. That is, how can this matrix give instructions to a computer on exactly how to draw Harpoona?
2. The transformation \begin{align*}\frac{1}{2}[H]\end{align*} is applied to Harpoona.
a. Write the resulting matrix below.
b. In the space below, draw Harpoona after this transformation.
c. In the space below, answer this question in words: in general, what does the transformation \begin{align*}\frac{1}{2}[H]\end{align*} do to a picture?
3. Now, Harpoona is going to swim three units to the left. Write below a general transformation that can be applied to any \begin{align*}2 \times 4\end{align*} matrix to move a drawing three units to the left.
4. Harpoona—in her original configuration before she was transformed in either way—now undergoes the transformation \begin{align*}\begin{bmatrix}
0 & -1\\
1 & 0
\end{bmatrix}[H]\end{align*}.
a. Write the new matrix that represents Harpoona below.
b. In the space below, draw Harpoona after this transformation.
c. In the space below, answer this question in words: in general, what does the transformation \begin{align*}\begin{bmatrix}
0 & -1\\
1 & 0
\end{bmatrix}[H]\end{align*} do to a picture?
5. Now: in the movie's key scene, the audience is looking down from above on Harpoona who begins the scene happily floating on the surface of the water. As the scene progresses, our heroine spins around and around in a whirlpool as she is slowly being sucked down to the bottom of the sea. "Being sucked down" is represented visually, of course, by shrinking.
a. Write a single transformation that will rotate Harpoona by \begin{align*}90^\circ\end{align*} and shrink her.
b. Apply this transformation four times to Harpoona's original state, and compute the resulting matrices that represent her next four states.
c. Now draw all four states—preferably in different colors or something.
Name: ______________________
Homework: Using Matrices for Transformation
1. Harpoona's best friend is a fish named Sam, whose initial position is represented by the matrix:
is multiplied by any matrix, it effects a powerful transformation on that matrix. Below, write the matrix \begin{align*}S_2=T \ S_1\end{align*}. (You may use \begin{align*}1.7\end{align*} as an approximation for \begin{align*}\sqrt{3}\end{align*}.)
3. Draw Sam's resulting condition, \begin{align*}S_2\end{align*}.
4. The matrix \begin{align*}T^{-1}\end{align*} will, of course, do the opposite of \begin{align*}T\end{align*}. Find \begin{align*}T^{-1}\end{align*}. (You can use the formula for the inverse matrix that we derived in class, instead of starting from first principles. But make sure to first multiply the \begin{align*}\frac{1}{2}\end{align*} into \begin{align*}T\end{align*}, so you know what the four elements are!)
5. Sam now undergoes this transformation, so his new state is given by \begin{align*}S_3=T^{-1} \ S_2\end{align*}. Find \begin{align*}S_3\end{align*} and graph his new position.
6. Finally, Sam goes through \begin{align*}T^{-1}\end{align*} again, so his final position is \begin{align*}S_4=T^{-1} \ S_3\end{align*}. Find and graph his final position.
7. Describe in words: what do the transformations \begin{align*}T\end{align*} and \begin{align*}T^{-1}\end{align*} do, in general, to any shape?
Solving Linear Equations
I'm sure you remember our whole unit on solving linear equations...by graphing, by substitution, and by elimination. Well, now we're going to find a new way of solving those equations...by using matrices!
Oh, come on...why do we need another way when we've already got three?
Glad you asked! There are two reasons. First, this new method can be done entirely on a calculator. Cool! We like calculators.
Yeah, I know. But here's the even better reason. Suppose I gave you three equations with three unknowns, and asked you to do that on a calculator. Think you could do it? Um...it would take a while. How about four equations with four unknowns? Please don't do that. With matrices and your calculator, all of these are just as easy as two. Wow! Do those really come up in real life? Yes, all the time. Actually, this is just about the only "real-life" application I can give you for matrices, although there are also a lot of other ones. But solving many simultaneous equations is incredibly useful. Do you have an example? Oh, look at the time! I have to explain how to do this method.
So, here we go. Let's start with a problem from an earlier homework assignment. I gave you this matrix equation:
1. The first thing you had to do was to rewrite this as two equations with two unknowns. Do that now. (Don't bother solving for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, just set up the equations.)
The point is that one matrix equation is the same, in this case, as two simultaneous equations. What we're interested in doing is that process in reverse: I'll give you simultaneous equations, and you'll turn them into a matrix equation that represents the same thing. Let's try a few.
a. Write a single matrix equation that represents these three equations.
b. Just to make sure it worked, multiply it out and see what three equations you end up with.
OK, by now you are convinced that we can take simultaneous linear equations and rewrite them as a single matrix equation. In each case, the matrix equation looks like this:
\begin{align*}AX = B\end{align*}
where \begin{align*}A\end{align*} is a big square matrix, and \begin{align*}X\end{align*} and \begin{align*}B\end{align*} are column matrices. \begin{align*}X\end{align*} is the matrix that we want to solve for—that is, it has all our variables in it, so if we find what \begin{align*}X\end{align*} is, we find what our variables are. (For instance, in that last example, \begin{align*}X\end{align*} was \begin{align*}\begin{bmatrix}
a\\
b\\
c
\end{bmatrix}\end{align*}.) So how do you solve something like this for \begin{align*}X\end{align*}? Time for some matrix algebra! We can't divide both sides by \begin{align*}A\end{align*}, because we have not defined matrix division. But we can do the next best thing.
4. Take the equation \begin{align*}AX=B\end{align*}, where \begin{align*}A, X,\end{align*} and \begin{align*}B\end{align*} are all matrices. Multiply both sides by \begin{align*}A^{-1}\end{align*} (the inverse of \begin{align*}A\end{align*}) in front. (Why did I say, "in front?" Remember that order matters when multiplying matrices. If we put \begin{align*}A^{-1}\end{align*} in front of both sides, we have done the same thing to both sides.)
5. Now, we have \begin{align*}A^{-1}A\end{align*}___gee, didn't that equal something? Oh, yeah...rewrite the equation simplifying that part.
6. Now, we're multiplying I by something...what does that do again? Oh, yeah...rewrite the equation again a bit simpler.
We're done! We have now solved for the matrix \begin{align*}X\end{align*}.
So, what good is all that again?
Oh, yeah...let's go back to the beginning. Let's say I gave you these two equations:
\begin{align*}3x + y & = -2 \\
6x - 2y & = 12\end{align*}
You showed in \begin{align*}^\#2\end{align*} how to rewrite this as one matrix equation \begin{align*}AX=B\end{align*}. And you just found in \begin{align*}^\#6\end{align*} how to solve such an equation for \begin{align*}X\end{align*}. So go ahead and plug \begin{align*}A\end{align*} and \begin{align*}B\end{align*} into your calculator, and then use the formula to ask your calculator directly for the answer!
7. Solve those two equations for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} by using matrices on your calculator.
Did it work? We find out the same way we always have—plug our \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values into the original equations and make sure they work.
8. Check your answer to \begin{align*}^\#7\end{align*}.
9. Now, solve the three simultaneous equations from \begin{align*}^\#3\end{align*} on your calculator, and check the answers.
Name: __________________
Homework—Solving Linear Equations
\begin{align*}4x + 2y = 3\\
3y - 8x = 8\end{align*}
Solve these two equations by either substitution or elimination.
Now, rewrite those two equations as a matrix equation.
Solve the matrix equation. Your answer should be in the form of a matrix equation: \begin{align*}[X]=\end{align*}
Now, using your calculator, find the numbers for your equation in part \begin{align*}(c)\end{align*}. Do they agree with the answers you found in part \begin{align*}(a)\end{align*}?
\begin{align*}6x - 8y = 2\\
9x - 12y = 5\end{align*}
Solve by using matrices on your calculator.
Hey, what happened? Why did it happen, and what does it tell you about these two equations?
8. a. Find the inverse of the matrix \begin{align*}\begin{bmatrix}
4 & x \\
1 & -2 \end{bmatrix}\end{align*} by using the definition of an inverse matrix. Note: if you are absolutely flat stuck on part\begin{align*}(a)\end{align*}, ask for the answer. You will receive no credit for part \begin{align*}(a)\end{align*} but you may then be able to go on to parts\begin{align*}(b)\end{align*}and\begin{align*}(c)\end{align*}.
b. Test it, by showing that it fulfills the definition of an inverse matrix. | 677.169 | 1 |
A TRANSITION TO ADVANCED MATHEMATICS helps students make the transition from calculus to more proofs-oriented mathematical study. The most successful book of its kind, the 5th edition continues to provide a firm foundation in major concepts needed for continued study and guides users to think and express themselves mathematicallyato analyze a situation, extract pertinent facts, and draw appropriate conclusions.
The authors place continuous emphasis throughout on improving users' ability to read and write proofs, and on developing their critical awareness for spotting common errors in proofs. Concepts are clearly explained and supported with detailed examples, while abundant and diverse exercises provide thorough practice on both routine and more challenging problems.
Students will come away with a solid intuition for the types of mathematical reasoning they'll need to apply in later courses and a better understanding of how mathematicians of all kinds approach and solve problems | 677.169 | 1 |
GETTING STARTED
Store Profits
A company is hiring staff for its new
mega-bookstore. If there are too few
staff members, they will not be able
to run the store effectively. If there are
too many staff members, their salary
costs will be too high.
A consult
A Tour of Your Textbook
To understand the textbooks structure, begin by taking a brief tour.
CHAPTER INTRODUCTION
SPECIFIC EXPECTATIONS
The specific expectations listed on the first page of each chapter describe the
concepts and skills that you are expec
REVIEW OF KEY CONCEPTS
4.1 Reviewing the Trigonometry of Right Triangles
Refer to the Key Concepts on page 271.
Solve each triangle. Round each side length to the nearest tenth of a
unit, and each angle to the nearest tenth of a degree.
1.
a)
b)
A
9.7 cm
CHALLENGE PROBLEMS
1. Buy or LeaseA
company is considering the possibility of acquiring
new computer equipment for $400 000 cash. The salvage value is
estimated to be $50 000 at the end of the 6-year life of the equipment.
Maintenance costs will be $4000
8.3
Technology: Loci and Conics
The diagram shows a double cone. The two cones have one point
in common.
The intersection of a double cone and a plane is called a conic section
or a conic.
The circle, ellipse, parabola, and hyperbola are the cross section
6.6
Geometric Series
The following is a geometric sequence.
3, 6, 12, 24, , where
3 aand r 2.
A geometric series is the sum of the terms
of a geometric sequence.
The geometric series that corresponds to
the geometric sequence above is
3 6 12 24 , where a
7.2
Compound Interest
For compound interest, the interest is reinvested
at regular intervals. The interest is added to the
principal to earn interest for the next interval of
time, or compounding period. If $4000 is
invested at 6.25% per annum, compounded
CUMULATIVE REVIEW: CHAPTERS 7 AND 8
Chapter 7
Chapter 8
1. a) Graph the locus of points that are
Leila is borrowing $65 000 for 4 years.
She is deciding between a loan at 6.95%
equidistant
per
from the lines
3xy7 and
annum, compounded monthly, and a loan
8.5
The Ellipse
Kidney stones are crystal-like objects
that can form in the kidneys.
Traditionally, people have undergone
surgery to remove them. In a process
called lithotripsy, kidney stones can
now be removed without surgery. To
remove the stones, doct
8.9
Intersection of
Lines and Conics
The centre circle
of a hockey rink
has a radius of
4.5 m. A diameter
of the centre circle
lies on the centre
red line.
centre (red)
centre circle
line
I NVESTIGATE & I
NQUIRE
A coordinate grid is superimposed on a hock
7.5
Amount of an Ordinary Annuity
Nigel is saving $700 each year for a trip.
Rashid is saving $200 at the end of each
month for university. Jeanine is depositing
$875 at the end of each 3 months for 3
years. Marcel is saving for a home
entertainment centr
7.4
Present Value
Often, when people invest money, they have a goal
for which they want a specific amount of money at
a future date. Ramona hopes to buy a house in
3 years and estimates that a down payment of
$70 000 should be sufficient. Ravis grandparen
8.4
The Circle
The epicentre of an earthquake is where the earthquake
originates on the Earths surface. Seismologists find an
epicentre by taking seismic readings from three
recording stations in different locations. Each reading
indicates the distance of
GETTING STARTED
Comparing Costs
1. Max can buy
a)What factors
a stereo system, paying $599 now or paying $750 in a year.
would you suggest Max consider when deciding which plan to
accept? How might these vary from factors you would suggest to someone else
8.6
The Hyperbola
Some ships navigate using a radio
navigation system called LORAN,
which is an acronym for LOng RAnge
Navigation. A ship receives radio
signals from pairs of transmitting
stations that send signals at the same
time. The LORAN equipment de
C
APPENDIX
COMPUTER SOFTWARE
THE G EOMETER S SKETCHPAD
Preferences
Before you begin using The Geometers Sketchpad, you may
need to change some of the default settings in the program.
Click on the Display menu and choose Preferences.
A window like the one
8.8
Conics With Equations in the Form
2
2
ax + by + 2gx + 2fy + c = 0
The CF-18 Hornet is a supersonic jet flown in Canada. It has a
maximum speed of Mach 1.8. The speed of sound is Mach 1.
When a plane like the Hornet breaks the sound barrier, it
produce
7.8
Mortgages
A knowledge of mortgages enables
you to make decisions about financial
arrangements when buying property.
The variety in amortization periods,
terms, amounts of mortgages, and
even interest rates, allows you to make
or combine choices. These
Investigate & Apply
Frieze Patterns
A frieze pattern is a pattern that repeats in one
direction. The patterns depend on the use of
transformations. Many cultures have used frieze
patterns to make decorative designs on
buildings, textiles, pottery, and so
6.5
Arithmetic Series
The following is an arithmetic sequence.
1, 3, 5, 7, 9,
A series is the sum of the terms of a sequence.
An arithmetic series is the sum of the terms of an
arithmetic sequence.
The series that corresponds to the sequence above is
1 3
7.7
Technology: Amortization Tables and
Spreadsheets
Generally, people must borrow money
when they purchase a car, house, or
condominium, so they arrange a loan
or mortgage. Loans and mortgages are
agreements between a money lender and
a borrower to finan
8.7
The Parabola
The Hubble Space Telescope orbits
the Earth at an altitude of
approximately 600 km. The
telescope takes about ninety minutes
to complete one orbit.
Since it orbits above the Earths
atmosphere, the telescope can
perform its scientific work
7.1
Investigation: Simple Interest, Arithmetic
Sequences, and Linear Growth
When you invest money in
a bank or other financial
institution, you are paid
interest for the use of your
money. The financial
institution uses your money
to earn money. When you
GETTING STARTED
Daylight Hours
A photoperiod is the number of daylight hours in a given day. Photoperiodism
is the response to changes in the photoperiod.
Photoperiodism is important in biology, because different species react to
changes in the photoperio | 677.169 | 1 |
Ticker Tape Calculator Simple calculator, but keeps a ticker tape that saves all your calculations, like an accountant's calculator that spits out ticker tape
Matrix Calculator This calculator performs all matrix, vector operations. You can add, subtract, find length, find dot and cross product, check if vectors are dependant. For every operation, calculator will generate a detailed explanation
Pearson Square Pearson Square software will provide an easy way to calculate mixtures of two ingredients | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
This concise text offers undergraduates in mathematics and science a thorough and systematic first course in elementary differential equations. Presuming a knowledge of basic calculus, the book first reviews the mathematical essentials required to master the materials to be presented.
The next four chapters take up linear equations, those of the first order and those with constant coefficients, variable coefficients, and regular singular points. The last two chapters address the existence and uniqueness of solutions to both first order equations and to systems and n-th order equations.
Throughout the book, the author carries the theory far enough to include the statements and proofs of the simpler existence and uniqueness theorems. Dr. Coddington, who has taught at MIT, Princeton, and UCLA, has included many exercises designed to develop the student's technique in solving equations. He has also included problems (with answers) selected to sharpen understanding of the mathematical structure of the subject, and to introduce a variety of relevant topics not covered in the text, e.g. stability, equations with periodic coefficients, and boundary value problems.
Synopsis
A thorough, systematic first course in elementary differential equations for undergraduates in mathematics and science, requiring only basic calculus for a background. Includes many exercises and problems, with answers. Index.
Synopsis
A thorough and systematic first course in elementary differential equations for undergraduates in mathematics and science, with many exercises and problems (with answers). Index.
Description
Includes bibliographical references and index.
Table of Contents
Chapter 0. Preliminaries
1. Introduction
2 Complex numbers
3 Functions
4 Polynomials
5. Complex series and the exponential function
6. Determinants
7. Remarks on methods of discovery and proof
Chapter 1. Introduction--Linear Equations of the First Order
1. Introduction
2. Differential equations
3. Problems associated with differential equations
4. Linear equations of the first order
5. The equation y'+ay=0
6. The equation y'+ay=b(x)
7. The general linear equation of the first order
Chapter 2. Linear Equations with Constant Coefficients
1. Introduction
2. The second order homogeneous equation
3. Initial value problems for second order equations
4. Linear dependence and independence
5. A formula for the Wronskian
6. The non-homogeneous equation of order two
7. The homogeneous equation of order n
8. Initial value problems for n-th order equations
9. Equations with real constants
10. The non-homogeneous equation of order n
11. A special method for solving the non-homogeneous equation
12. Algebra of constant coefficient operators
Chapter 3. Linear Equations with Variable Coefficients
1. Introduction
2. Initial value problems for the homogeneous equation
3. Solutions of the homogeneous equation
4. The Wronskian and linear independence
5. Reduction of the order of a homogeneous equation
6. The non-homogeneous equation
7. Homogeneous equations with analytic coefficients
8. The Legendre equation
9. Justification of the power series method
Chapter 4. Linear Equations with Regular Singular Points
1. Introduction
2. The Euler equation
3. Second order equations with regular singular points--an example
4. Second order equations with regular singular points--the general case
5. A convergence proof
6. The exceptional cases
7. The Bessel equation
8. The Bessel equation (continued)
9. Regular singular points at infinity
Chapter 5. Existence and Uniqueness of Solutions to First Order Equations
1. Introduction
2. Equations with variables separated
3. Exact equations
4. The method of successive approximations
5. The Lipschitz condition
6. Convergence of the successive approximations
7. Non-local existence of solutions
8. Approximations to, and uniqueness of, solutions
9. Equations with complex-valued functions
Chapter 6. Existence and Uniqueness of Solutions to Systems and n-th Order Equations | 677.169 | 1 |
sábado, 1 de março de 2014
Descrição:Reports on seven specific programs for improving mathematics education in the schools of Hungary, Indonesia, Japan, the Philippines, the Union of Soviet Socialist Republics, the United Kingdom, and the United Republic of Tanzania are presented. The report from the United Kingdom deals with the Continuing Mathematics Project that focuses on innumeracy in the 16 to 19 age group and new materials designed to deal with this problem. The remaining six articles describe developments in secondary mathematics as a part of the general secondary program. Biographical notes about the contributing authors are appended.
Descrição:This volume was geared to answering the question, does the teaching of mathematics correspond to the needs of the majority of pupils and the society. There are three types of chapters: (1) descriptions of goals reflecting some need of society; (2) case studies of national goal setting; and (3) a summary of the May 1980 meeting of the United Nations Educational, Scientific, and Cultural Organization (UNESCO), which undertook a review of the goals of mathematics teaching. Individual chapters are: (1) Goals as a Reflection of the Needs of Society; (2) Goals as a Reflection of the Needs of the Learner; (3) Goals of Mathematics for Rural Development; (4) School Mathematics-Links with Commerce and Industry; (5) Goals of Mathematics as a Reflection of the Requirements of Production; (6) Educational Objectives for Mathematics Compatible with its Development as a Discipline; (7) New Goals for Old: An Analysis of Reactions to Recent Reforms in Several Countries; (8) The NCTM PRISM Project: An Attempt to Make Curriculum Change More Rational and Systematic; (9) The Evolution of Mathematics Curricula in the Arab States; (10) Goals of the Mathematics Curriculum in British Columbia: Intended, Implemented, and Realized; and (11) Report of a Meeting on the Goals of Mathematics Education | 677.169 | 1 |
The program allows you to solve algebraic equations in the automatic mode. You just enter an equation in any form without any preparatory operations. Step by step Equation Wizard reduces it to a canonical form performing all necessary operations. After that it determines the order of the equation, which can be any - linear, square, cubic or, for instance, of the 7-th power. The program finds the roots of the equation - both real and imaginary. You just enter an equation you see in your textbook or notebook and click one button! In an instant, you get the step-by-step solution of the equation with the found roots and the description of each step. The solution is completely automatic and does not require any math knowledge from you. Then just print the result or save it to a file. Besides, the program allows you to simplify math expressions with one variable. Use this feature to speed up your calculations. Equation Wizard is an indispensable assistant for students at university and at school allowing them to save their time and make their learning easier.
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Other Windows Software of Developer «DailySoft.ru»:
Tags4Files This software is intended to make finding a file on your computer much easier. It merges any file into a library and associates tags with it. Each tag is a simple text mark - "important", "personal", "children", "presentation", "our vacation", "2010" and sTags4Docs Tags4Docs introduces the handy Web 2.0 tag cloud functionality on your PC. You could see that later and though, why this isn't implemented for my local documents as well and here it is! With Tags4Docs you simply assign different tags to any document, which
The Geometer's Sketchpad The Geometer's Sketchpad is the world's leading software for teaching mathematics. Sketchpad gives students at all levels--from third grade through college--a tangible, visual way to learn mathematics that increases their engagement, understanding, and achHighRoad HighRoad is an interactive graphical application for the geometric design of roads and highways. Its intelligent object design and highly intuitive interface allows for fast and flexible design. Design roads as you always have - graphically. HighRoad uses bootCamp The early stages of learning math require the memorizing a lot of detailed information and this can be quite frustrating for some small children. Experts know that young children learn at a faster pace while they are having fun. That is especially true fo
GRE Calculator GRE Calculator is a simple application that will help you familiarize yourself with calculator which you'll be using on actual GRE revised General Test. Since the most important and crucial aspect on GRE is time management, there are also included some tip | 677.169 | 1 |
Geometric Etudes in Combinatorial Mathematics, Second edition
This second edition of Alexander Sofier's Geometric Etudes in Combinatorial Mathematics provides supplementary reading materials to students of all levels interested in pursuing mathematics, especially in algebra, geometry, and combinatorial geometry.
Within the text, the author outlines an introduction to graph theory and combinatorics while exploring topics such as the pigeonhole principle, Borsuk problem, and theorems of Helly and Szokefalvi—Nagy. The book introduces these ideas along with practical applications that will prepare young readers for the mathematical world.
Geometric Etudes in Combinatorial Mathematics is not only educational; it is inspirational. This distinguished mathematician captivates his readers, propelling them to search for solutions of life's problems- problems that previously seemed hopeless. | 677.169 | 1 |
MDSolids is educational software designed to assist engineering students in the introductory mechanics of materials course. MDSolids explains and solves a wide variety of engineering problems commonly found in the mechanics of materials course.
MDSolids is also very useful to the practicing engineer who is needs a quick, easy-to-use tool for basic engineering calculations.
MDSolids has routines pertaining to all of the topics taught in a typical mechanics of materials course. Twelve modules pertaining to a wide range of common textbook problems are presently available: basic stress and strain concepts, trusses, statically indeterminate axial structures, torsion, determinate beams, section properties, general analysis (of axial, torsion, and beam members), combined loadings, column buckling, pressure vessels, and Mohr's circle transformations. Within the modules, each routine solves types of problems typically found in all mechanics of materials textbooks. Each MDSolids routine features a picture, sketch, or plot that graphically depicts important aspects of the problem. Sketches are used to show the direction of internal stresses, applied loads, and reaction forces.
MDSolids is an 'electronic solutions manual,' giving not only the correct solution for a particular problem but also providing intermediate solutions that can be used to confirm the problem solving approach. Many of the MDSolids modules provide extra explanations to describe in words how the calculations are performed. These explanations can help students develop the thought processes used in solving mechanics of materials problems. The text explanations are dynamic and context-sensitive, tailored specifically to the particular problem in terms of the values and units entered for the problem.
MDSolids was a winner of the 1998 Premier Award for Excellence in Engineering Education Courseware.Knowledge NoteBook It ea
NoProb Planner NoProb Planner helps you manage all of your homework for each of your classes. With reminders, weekly and monthly views, and much more, staying on top of your homework has never been easier. Features: Reminders Weekly and monthly overviews of your homewo | 677.169 | 1 |
Product Information
Publisher's Description
Achieve the best possible standard with this bestselling book of practice and guidance - now in colour
First Aid in Mathematics is written in clear and concise language to provide all the help, support and practice needed to master the key topics in basic mathematics. The book contains over 1,000 carefully planned practice questions to help readers reinforce their knowledge. Revision tests are provided at the end of each chapter. First Aid in Mathematics is ideal for use in the classroom or as a reference and revision book for those wanting to improve their mathematical | 677.169 | 1 |
AO2 and AO3, Skills Book: Reason, interpret and communicate mathematically and solve problems is full of expertly written practice questions to help students succeed in mathematical reasoning and problem-solving.
• Structured by strand, it is ideal to be used alongside the Practice Book or Student Book • New, bespoke questions to focus on two of the main aims of the new curriculum (AO2, reason mathematically, and AO3, solve problems) • Longer questions can be tried in class to generate discussions • Tackle problems within and outside mathematics to improve literacy, thinking skills and investigative strategies • New questions requiring students to think and analyse their work encourages independence • Full mark schemes are provided online | 677.169 | 1 |
Investigating algorithmic methods based on machine learning in order to design sequential investment strategies for financial markets, this is a self-contained text suitable for researchers and graduate students in computer science, finance, statisti...
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For one-semester courses in Introductory/ Preparatory Chemistry. Written for students who have had little or no previous instruction in chemistry, or for those who need a thorough review, this interesting and informative text gives readers the backgr...
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How can a Christian brought up in the metropolis of Sao Paulo speak the gospel clearly to a Buddhist raised in the mountains of Tibet? Every missionary confronts the difficulty of cross-cultural communication. But missionaries from the Third World, B...
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The best way to practice for the SAT Subject Tests in Mathematics Every year nearly 300,000 high school students take an SAT Subject Test(TM) in Mathematics Level 1 or Mathematics Level 2. Many selective colleges recommend or require them for admissi...
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Covers the previously administered tests across all 20 SAT Subject Tests. This title offers enhanced information on tips and approaches from the test maker on selecting the right test to take, the best time to take the tests, and how best to be ready...
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"The Official SAT Study Guide with DVD "by The College Board is the only book that features official SAT(R) practice tests created by the test maker. With 1,000 pages and more than 20 chapters, it has everything you need to prepare for the SAT. The b...
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Developing logical thinking and fundamental mathematical ideas, and using problems that pique students' mathematical curiosity, this work aims to prepare readers for all upper-division mathematics courses and improve their skills in presenting cohere...
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"Understanding Psychology" presents introductory psychology as a science, providing engaging coverage of all traditional topics and giving readers a firm grasp of the scope, vocabulary, and concepts of the discipline. It integrates diversity and psyc...
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We are living in a defining moment, when the world in which teachers do their work is changing profoundly. In his latest book, Hargreaves proposes that we have a one-time chance to reshape the future of teaching and schooling and that we should seize...
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Data analysis and modeling are now integral components of high school courses and the Common Core State Standards. Fathom provides a dynamic, visually compelling environment for students to meet these standards as they explore, analyze, and model data.In addition to helping students understand algebra, precalculus, and statistics, Fathom's powerful data analysis capabilities make it an excellent tool for the physical and biological sciences, as well as for social science courses.
With Fathom's dynamic tools you can quickly represent data in a variety of graphs, including bar charts, scatter plots, function plots, and histograms, Plot values and functions on top of bivariate data and vary them dynamically with sliders to show the effects of variables,Build simulations that illuminate concepts of probability and statistics,Demonstrate standard statistical analyses, including multiple linear regression, and Set up populations and sample from them repeatedly to get a feel for the sampling process, and create sampling distributions of any statistic.Fathom's dynamic tools to:
Quickly represent data in a variety of graphs, including bar charts, scatter plots, function plots, and histograms
Plot values and functions on top of bivariate data and vary them dynamically with sliders to show the effects of variables
Build simulations that illuminate concepts of probability and statistics
Demonstrate standard statistical analyses, including multiple linear regression
Set up populations and sample from them repeatedly to get a feel for the sampling process, and create sampling distributions of any statistic
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Other Windows Software of Developer «Key Curriculum Press»:UniConv UniConv is a Windows-based program for converting dimensional e.g. litres to gallons, feet to cms, pounds to Kgs, square yards to square metres etc.It is completely user-customizable using simple text-based files, you don't need a new release to improve thYourConverter for Windows 8 With YourConverter, you can customize your favorite conversions for immediate access as soon as you open the app - no more messing around in order to convert those units you use on a regular basis. You can even hide those unit conversions that you'll never
3D Graph Draw 3D or 2D equation with this equation graphing free software. Visualizing equations is made easy with 3D Graph. To draw 3D or 2D graphs all required is filling in the equation of the form z=f(x, y).Requirements:Windows Me/2000/XP/2003 Server, .net fram
Insert Equation Editor Button This add-In inserts the Equation Editor Button on the Command bar of the Excel. The [Equation Editor] button does not exist in the Excel. When you want to insert the Equation object, you must select Insert-Object and select the Microsoft Equation 3.0 in th
MathsOwn Junior MathsOwn Junior is a free version of MathsOwn, designed as a child s first computer application, to assist teach the four maths operations of addition, subtraction, multiplication and division as well as basic computer and keyboard skills. MathsOwn is a so | 677.169 | 1 |
books.google.com - MATLAB is an interactive system for numerical computation that is widely used for teaching and research in industry and academia. It provides a modern programming language and problem solving environment, with powerful data structures, customizable graphics, and easy-to-use editing and debugging tools.... Guide | 677.169 | 1 |
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PRODUCT DESCRIPTION
Are you having trouble getting your students to take effective notes during math lessons? Are you looking for quick activities that can be completed in a math journal or at the end of a lesson as a check for understanding? If so, these activities are for you.
Teach your students how to solve systems of equations in two and three variables with this comprehensive note-taking pack. These activities for Algebra 1, Algebra 2, Pre-Calculus, or Integrated Math can be used as foldable interactive math journal activities, or as step-by-step note-taking aids. They can be used for introduction of topics or to reflect/review what you have covered in class discussion. Our students enjoy filling them out and organizing them in their math journals so that they can refer back to them throughout the year. These activities are a great way to promote writing in math. Keys are included for reference.
Scaffolded Note Sheets with Sample Teacher Keys are provided for the following:
Characteristics of Systems of Equations
What is a System of Equations?
How do you solve a system of equations by graphing?
Classifying Systems of Equations by Graphing
How do you solve a system of equations using substitution?
How do you solve a system of equations using elimination?
Multiply, Then Use Elimination
Graphing Inequalities
What is a system of inequalities?
Solving a System of Inequalities by Graphing
Solving a System of Equations (Graphing, Substitution,
and Elimination)
Systems of Equations in Three Variables
14 Steps for Solving a System of Equations in Three Variables with Elimination
9 Steps for Solving a System of Equations in Three Variables with Substitution
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How to earn TPT Credits:
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-NASA Collaborative 2007: Algebra-the basics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material CSUB-NASA Collaborative 2007: Algebra-the basics
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Discussion
Discussion for CSUB-NASA Collaborative 2007: Algebra-the basics
Dr. Troy Tenhet
(Teacher (K-12))
I'm amazed to see something as simple as this from our friends at Goddard. I am happy to see that you posted it...great work. I now challenge you to focus the learning object with an actual assignment (lesson plan). | 677.169 | 1 |
Textbooks Collection
Active Calculus, Matthew Boelkins, David Austin, Steven Schlicker
Open Textbooks
Active Calculus is different from most existing calculus texts in at least the following ways: the text is free for download by students and instructors in .pdf format; in the electronic format, graphics are in full color and there are live html links to java applets; the text is open source, and interested instructors can gain access to the original source files upon request; the style of the text requires students to be active learners — there are very few worked examples in the text, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problemsHonors Scholar Theses
Differential equations are equations that involve an unknown function and derivatives. Euler's method are efficient methods to yield fairly accurate approximations of the actual solutions. By manipulating such methods, one can find ways to provide good approximations compared to the exact solution of parabolic partial differential equations and nonlinear parabolic differential equations.Calculating The Time Constant Of An Rc Circuit, Sean Dunford
Undergraduate Journal of Mathematical Modeling: One + Two
In this experiment, a capacitor was charged to its full capacitance then discharged through a resistor. By timing how long it took the capacitor to fully discharge through the resistor, we can determine the RC time constant using calculus.
Faculty Work: Comprehensive List
Algebra deals with more than computations such as addition or exponentiation; it also studies relations. Calculus touches on this a bit with locating extreme values and determining where functions increase and decrease; and in elementary algebra you occasionally "solve" inequalities involving the order relations of < or ≤ , but this almost seems like an intrusion foreign to the main focus, which is making algebraic calculations.
Relational ideas have become more important with the advent of computer science and the rise of discrete mathematics, however. Many contemporary mathematical applications involve binary or n-ary relations in addition to computations. We began discussing this topic in the last chapter when we introduced equivalence relations. In this chapter we will explore other kinds of ...An Introduction To Boolean Algebras, Amy Schardijn
Electronic Theses, Projects, and Dissertations
This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular notation. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra. From this fundamental understanding, we were able to study atoms, Boolean algebra isomorphisms, and Stone's Representation Theorem for finite Boolean algebras. We also verified and proved many properties involving Boolean algebras and related structuresPlanar Graphs, Biplanar Graphs And Graph Thickness, Sean M. Hearon
Electronic Theses, Projects, and Dissertations
A graph is planar if it can be drawn on a piece of paper such that no two edges cross. The smallest complete and complete bipartite graphs that are not planar are K5 and K{3,3}. A biplanar graph is a graph whose edges can be colored using red and blue such that the red edges induce a planar subgraph and the blue edges induce a planar subgraph. In this thesis, we determine the smallest complete and complete bipartite graphs that are not biplanar.
Common Core In Tennessee: An Analysis Of Eighth Grade Mathematics Standards, Hayley Little
Honors Theses
Since their introduction in 2010, the Common Core State Standards (CCSS) have been a highly controversial topic in educational reform. Though the standards are not a product of the federal government and are not federally mandated, they do represent a push towards national academic standards in America. For states such as Tennessee, educational policies of the past pushed them to lower their academic standards in order to create the illusion of success. Those states are now some of the places that have seen the most change with the adoption of the CCSS. It still remains somewhat unclear, however, which changes ...
Newton'S Third Law In Karmic Warfare, Kazmier Maślanka
The STEAM Journal
A work entitled "Newton's Third Law in Karmic Warfare" is a mathematical visual poem which is a perfect example of a technique, that I call The Paradigm Poem. This piece makes a direct connection with the concept of karma and Newton's Third Law of motion. I will introduce the concept of "The Mathematical Paradigm Poem" to illuminate an example of how metaphor is used in mathematical visual poetry. I will also discuss much of the process in making this aesthetic expression.Electronic Journal of Linear Algebra
The singularity problem of the solutions of some particular Sylvester equations is studied. As a consequence of this study, a good choice of a Sylvester equation which is associated to a linear continuous time system can be made such that its solution is nonsingular. This solution is then used to solve an eigenstructure assignment problem for this system. From a practical point view, this study can also be applied to automatic control when the system is subject to input constraints.
Electronic Journal of Linear Algebra
An $n\times n$ sign pattern ${\cal A}$ is said to be potentially nilpotent (PN) if there exists some nilpotent real matrix $B$ with sign pattern ${\cal A}$. In [M.~Arav, F.~Hall, Z.~Li, K.~Kaphle, and N.~Manzagol.Spectrally arbitrary tree sign patterns of order $4$, {\em Electronic Journal of Linear Algebra}, 20:180--197, 2010.], the authors gave some open questions, and one of them is the following: {\em For the class of $4 \times 4$ tridiagonal sign patterns, is PN (together with positive and negative diagonal entries) equivalent to being SAP?}\ In this paper, a positive answer ...
Mathematical Optimization And Applications, Teodora Suciu
Mathematics Colloquium Series
This talk centers on mathematical optimization in the context of Calculus of Variations. Optimization involves choosing the best element from a set of choices, usually through mathematical approaches. Solving these kinds of problems is considered an essential tool in many areas of science and engineering. Additionally, various mathematics and business applications are discussed. Also explored is a real-life example with a detailed algorithm that is closely related to the Traveling Salesman problem.
Mathematical Relativity And The Nature Of The Universe, Priscila Reyes
Mathematics Colloquium Series
In this talk, I will be discussing certain space-times, which can be used to model celestial objects and events in the universe. These are solutions to Einstein's field equations, which roughly describe the relation between matter, energy and the geometry of the universe. The concept of time in relation to an observer will be demonstrated. I will also include some interesting phenomena that arise out of the unusual mathematical structure of space-times , such as Lorentz contraction, reverse Cauchy-Schwarz, and the twin paradox.
Doctor of Education in Teacher Leadership Dissertations
Abstract
The purpose of this mixed methods research study is to explore how teacher-made video tutorials in one middle school mathematics classroom are being used and what enables/impedes their effectiveness. A quantitative, quasi-experimental design investigates how video tutorial usage relates to students' self-efficacy and achievement in mathematics. The sample consisted of 55 students in the experimental group and 65 students in the control group. Results found parents wanted the tutorials even if they did not view them. Students did actually use the video tutorials for remediation and learning. Videos longer than ten minutes impeded the effectiveness of the videos ...
Leon Hall
"Functional analysis techniques are used to prove a theorem, analogous to the Harris-Sibuya-Weinberg theorem for ordinary differential equations, which yields as corollaries a number of existence theorems for holomorphic solutions of linear functional differential systems of the form zDy'(z) = A(z)y(z) + B(z)y(αz) + C(z)y'(αz) in the neighborhood of the singularity at z = 0"--Abstract, page 2.
Leon Hall
"In this thesis existence and uniqueness of solutions to certain second and third order boundary value problems for delay differential equations is established. Sequences of upper and lower solutions similar to those used by Kovač and Savčenko are defined by means of an integral operator, and conditions are given under which these sequences converge monotonically from above and below to the unique solution of the problem. Some numerical examples for the second order case are presented. Existence and uniqueness is also proved for the case where the delay is a function of the solution as well as the independent variable ...
Leon Hall
"Compiling and writing the history of the Missouri MAA Section has been time-consuming, but it has mainly been rewarding and a wonderful learning experience. Both the Monthly and the MAA began with strong Midwestern and Missouri influences, something which our section can look back on with well-deserved pride. Missouri MAA members have consistently advanced collegiate mathematics, mathematics education, mathematics research and scholarship, and public appreciation for and understanding of mathematics in both Missouri and the nation. Looking to the future, the MAA and the Missouri Section can continue to be a great source of opportunities for leadership and service for
Patrick R. Lowenthal
We describe the first phase of a study aimed at developing video-based instructional modules for secondary mathematics teachers. We began by consulting the literature on figural pattern tasks (c.f. Rivera, 2010) and teachers' ability to interpret student work (c.f. El Mouhayar & Jurdak, 2012). Interpreting student work on figural pattern tasks requires awareness of different problem solving strategies, such as recursive and constructive, and how students might use them with tasks that require different levels of generalization (El Mouhayar & Jurdak, 2012). | 677.169 | 1 |
About this product
Description
Description
McCaulay's Sample Math Questions for the ACT* Mathematics Test contains sample questions of the type that are most likely to appear in the mathematics section of the college entrance exam designed to measure the mathematical skills students have typically acquired in courses taken by the end of 12th grade. These practice questions will help improve scores on the ACT*. The categories of questions include Arithmetic; Algebra and Functions; Plane Euclidean Geometry and Measurement; Coordinate Geometry; Three-Dimensional Geometry; Trigometry; Data Analysis; Statistics; and Probability.
Author Biography
Philip Martin McCaulay is an actuary with a degree in mathematics from Indiana University. He has sold thousands of study guides and practice exam books in the fields of math, pensions, investments, finance, real estate, and massage therapy. He has also published books on card games, cooking, and military history. He volunteers to write, publish, and ship free copies of books to troops and military families through Operation Paperback. He is a Fellow of the Society of Actuaries, an Enrolled Actuary, a Member of the American Academy of Actuaries, and a Fellow of the Conference of Consulting Actuaries, with experience as a Vice Chair for the Society of Actuaries Education & Examination Committee. | 677.169 | 1 |
First Course in Algebra and Number Theory presents the basic concepts, tools, and techniques of modern algebra and number theory. It is designed for a full year course at the freshman or sophomore college level. The text is organized into four chapters. The first chapter is concerned with the set of all integers - positive, negative, and zero. It... more...
Algebra, Topology, and Category Theory: A Collection of Papers in Honor of Samuel Eilenberg is a collection of papers dealing with algebra, topology, and category theory in honor of Samuel Eilenberg. Topics covered range from large modules over artin algebras to two-dimensional Poincaré duality groups, along with the homology of certain H-spaces as... more...
Test Bank for Precalculus: Functions & Graphs is a supplementary material for the text, Precalculus: Functions & Graphs. The book is intended for use by mathematics teachers. The book contains standard tests for each chapter in the textbook. Each set of test focuses on gauging the level of knowledge the student has achieved during the course. The... more...
Intermediate Algebra focuses on the principles, operations, and approaches involved in intermediate algebra. The book first elaborates on basic properties and definitions, first-degree equations and inequalities, and exponents and polynomials. Discussions focus on the greatest common factor and factoring by grouping, factoring trinomials, special... more...
Elementary Algebra, Third Edition focuses on the basic principles, operations, and approaches involved in elementary algebra. The book first ponders on the basics, linear equations and inequalities, and graphing and linear systems. Discussions focus on the elimination method, solving linear systems by graphing, word problems, addition property of... more...
The History of Modern Mathematics, Volume I: Ideas and their Reception documents the proceedings of the Symposium on the History of Modern Mathematics held at Vassar College in Poughkeepsie, New York on June 20-24, 1989. This book is concerned with the emergence and reception of major ideas in fields that range from foundations and set theory, algebra... more...
Lectures in General Algebra is a translation from the Russian and is based on lectures on specialized courses in general algebra at Moscow University. The book starts with the basics of algebra. The text briefly describes the theory of sets, binary relations, equivalence relations, partial ordering, minimum condition, and theorems equivalent to the... more...
Modern Syllabus Algebra presents topics of traditional and modern algebra found in the Teachers Certificate and B.Ed, part I syllabuses of University Institutes of Education. It also contains additional exercises taken from examination papers of the University of London Institute of Education (the Teachers' Certificate). The book discusses several... more...
Some Modern Mathematics for Physicists and Other Outsiders: An Introduction to Algebra, Topology, and Functional Analysis, Volume 1 focuses on the operations, principles, methodologies, and approaches employed in algebra, topology, and functional analysis. The publication first offers information on sets, maps, and algebraic composition laws and... more...
Intermediate Algebra, Third Edition focuses on operations, principles, and approaches involved in intermediate algebra. The manuscript first ponders on basic properties and definitions, linear equations and inequalities in one variable, and exponents and polynomials. Discussions focus on factoring trinomials, special factoring, solving equations... more... | 677.169 | 1 |
Math XL was without a doubt one of the worst math online programs I ever had to work with in my life
this picture is a picture I took, and showed to my professor since this question broke my 100 on a text, making it a 96. I also emailed support for them, and they said that this was not an error on their side, and was most likely due to slow internet connection.
I mean, holy **** ******* tap dancing mother of all that is holy is this thing fiddly as ****.
I have to spend hours making sure my equation is damn near perfect just so the whole question doesn't change on me if I get it wrong the first time, and even if you get it right it still isn't a guarantee that you are going to be able to move on to the next question.
Mastering physics is also painful to use. I honestly wish my physics professor would stop ******* around and just use webassign like everyone else. (Web assign has the nice benefit of accepting whatever correct answer you give it).
"Ah, I see you brought your graphical calculators from last year, and learned how to use them and are eager to use them again? You have used computer programs as well? How quaint. There's the bin, you can throw that expensive **** in there, we're using pencil and paper in my classroom, **** your technological ******** and **** this so called smartboard that doesn't work half the time. I'm using chalk and a blackboard." Best math teacher I had in my life.
<<< This is how the ones who didn't specialise in math thought he looked like.
Funny. My Calculus and Discrete Math professors hate it to the core. My Pre-Cal professor allowed unlimited retries because she knew how bad it was. They praise it because it takes nearly all their time grading **** away.
the letter i is not algebra, it represents the square root of negative 1 - A NUMBER THAT DOESN'T EXIST. Everything past the 5 is called a complex number.
MFW i never have to do A-level further maths again.
this is actually early high school algebra... the letter i is supposed to represent the square root of negative 1 yes, but its also supposed to be used to add negatives to other square roots... i(squareroot)3 is just square root of 3...
also I dont get this. how can FJ not do maths but can exponentially be talented at politics and the like?
Complex numbers are used for extremely niche fields - like quantum physics or electrical engineering. While the numbers themselves don't exist in the real, physical world, they still can be used to solve complex equation. Like happyness isn't a physical THING, but it still has an effect.
>Be me doing math homework
>Systems of Linear Equations
>******* story problems
>"40 tickets are bought for a concert. The number of tickets purchased for seats in either Floor 2 or Floor 1 is the same as the number purchased for Floor 3. Floor 1 costs X, Floor 2 costs Y, Floor 3 costs Z. How many of each type of tickets were purchased?"
>Logically F1 = F3 and F2 = F3, therefore F1 = F2
>Book gives three different answers
>wut.png
>Read problem again
>Ask professor about it the next day, he explains that F1 + F2 = F3
>MFW mathematicians think "either X or Y" is the same as "X and Y."
Professor explained it wrong also. The key word was EITHER. So its F1 =F3 OR F2=F3. Doesnt matter which one you make the variable. SO lets give each one variables
F1=F3, F2=x, F3=Y
SO the equation becomes (40-x)/2=Y
F1=x, f2=Y, F3=F1
SO the equation becomes 40-2x= Y
And the third answer is prob one that has the variables switched around again.
Source: Its ******* Linear Equations( Thats straight lines dumbass) | 677.169 | 1 |
2014. GRADE 9. MATHEMATICS. EXEMPLAR QUESTIONS. This booklet ... GUIDELINES FOR THE USE OF ANNUAL NATIONAL ASSESSMENT (ANA) EXEMPLARS . 1.1 At the beginning of a lesson as a diagnostic test to identify learner | 677.169 | 1 |
Excerpted from A Mind for Numbers: How to Excel in Math and Science (Even if You Flunked Algebra), by Barbara Oakley, Penguin, July, 2014
1. Use recall.After you read a page, look away and recall the main ideas. Highlight very little, and never highlight anything you haven't put in your mind first by recalling. Try recalling main ideas when you are walking to class or in a different room from where you originally learned it. An ability to recall—to generate the ideas from inside yourself—is one of the key indicators of good learning.
3. Chunk your problems. Chunking is understanding and practicing with a problem solution so that it can all come to mind in a flash. After you solve a problem, rehearse it. Make sure you can solve it cold—every step. Pretend it's a song and learn to play it over and over again in your mind, so the information combines into one smooth chunk you can pull up whenever you want.
4. Space your repetition.Spread out your learning in any subject a little every day, just like an athlete. Your brain is like a muscle—it can handle only a limited amount of exercise on one subject at a time.
5. Alternate different problem-solving techniques during your practice.Never practice too long at any one session using only one problem-solving technique—after a while, you are just mimicking what you did on the previous problem. Mix it up and work on different types of problems. This teaches you both how and when to use a technique. (Books generally are not set up this way, so you'll need to do this on your own.) After every assignment and test, go over your errors, make sure you understand why you made them, and then rework your solutions. To study most effectively, handwrite (don't type) a problem on one side of a flash card and the solution on the other. (Handwriting builds stronger neural structures in memory than typing.) You might also photograph the card if you want to load it into a study app on your smartphone. Quiz yourself randomly on different types of problems. Another way to do this is to randomly flip through your book, pick out a problem, and see whether you can solve it cold.
6. Take breaks. It is common to be unable to solve problems or figure out concepts in math or science the first time you encounter them. This is why a little study every day is much better than a lot of studying all at once. When you get frustrated with a math or science problem, take a break so that another part of your mind can take over and work in the background.
7. Use explanatory questioning and simple analogies.Whenever you are struggling with a concept, think to yourself, How can I explain this so that a ten-year-old could understand it? Using an analogy really helps, like saying that the flow of electricity is like the flow of water. Don't just think your explanation—say it out loud or put it in writing. The additional effort of speaking and writing allows you to more deeply encode (that is, convert into neural memory structures) what you are learning.
8. Focus.Turn off all interrupting beeps and alarms on your phone and computer, and then turn on a timer for twenty-five minutes. Focus intently for those twenty-five minutes and try to work as diligently as you can. After the timer goes off, give yourself a small, fun reward. A few of these sessions in a day can really move your studies forward. Try to set up times and places where studying—not glancing at your computer or phone—is just something you naturally do.
9. Eat your frogs first. Do the hardest thing earliest in the day, when you are fresh.
10. Make a mental contrast. Imagine where you've come from and contrast that with the dream of where your studies will take you. Post a picture or words in your workspace to remind you of your dream. Look at that when you find your motivation lagging. This work will pay off both for you and those you love!
Ten Rules of Bad Studying
Excerpted from A Mind for Numbers: How to Excel in Math and Science (Even if You Flunked Algebra), by Barbara Oakley, Penguin, July, 2014
Avoid these techniques—they can waste your time even while they fool you into thinking you're learning!
1. Passive rereading—sitting passively and running your eyes back over a page.Unless you can prove that the material is moving into your brain by recalling the main ideas without looking at the page, rereading is a waste of time.
2. Letting highlights overwhelm you. Highlighting your text can fool your mind into thinking you are putting something in your brain, when all you're really doing is moving your hand. A little highlighting here and there is okay—sometimes it can be helpful in flagging important points. But if you are using highlighting as a memory tool, make sure that what you mark is also going into your brain.
3. Merely glancing at a problem's solution and thinking you know how to do it. This is one of the worst errors students make while studying. You need to be able to solve a problem step-by-step, without looking at the solution.
4. Waiting until the last minute to study. Would you cram at the last minute if you were practicing for a track meet? Your brain is like a muscle—it can handle only a limited amount of exercise on one subject at a time.
5. Repeatedly solving problems of the same type that you already know how to solve. If you just sit around solving similar problems during your practice, you're not actually preparing for a test—it's like preparing for a big basketball game by just practicing your dribbling.
6. Letting study sessions with friends turn into chat sessions. Checking your problem solving with friends, and quizzing one another on what you know, can make learning more enjoyable, expose flaws in your thinking, and deepen your learning. But if your joint study sessions turn to fun before the work is done, you're wasting your time and should find another study group.
7. Neglecting to read the textbook before you start working problems. Would you dive into a pool before you knew how to swim? The textbook is your swimming instructor—it guides you toward the answers. You will flounder and waste your time if you don't bother to read it. Before you begin to read, however, take a quick glance over the chapter or section to get a sense of what it's about.
8. Not checking with your instructors or classmates to clear up points of confusion. Professors are used to lost students coming in for guidance—it's our job to help you. The students we worry about are the ones who don't come in. Don't be one of those students.
9. Thinking you can learn deeply when you are being constantly distracted. Every tiny pull toward an instant message or conversation means you have less brain power to devote to learning. Every tug of interrupted attention pulls out tiny neural roots before they can grow.
10. Not getting enough sleep. Your brain pieces together problem-solving techniques when you sleep, and it also practices and repeats whatever you put in mind before you go to sleep. Prolonged fatigue allows toxins to build up in the brain that disrupt the neural connections you need to think quickly and well. If you don't get a good sleep before a test, NOTHING ELSE YOU HAVE DONE WILL MATTER.
okay so maybe these are mostly destiel but come on look how many times dean and cas are just staring into each others eyes - okay i'd like to point out to y'all that i'm happily taking requests to make sets of text posts based off a character or a ship or just the show in general (from more shows than just supernatural) so yeah - send away! :-)
Summary – Bucky's mind is finally
free from Hydra and he moves into the Avengers Compound.
Warnings – None
Word Count - 772
Notes – So, this
is my way of writing about the first long-haired brunette bad-boy that I fell
in love with from the MCU!! Obviously
this is strictly my theory on how I think the magic works. There was just something about that green
light when Loki's Magic dissipates that gave me the idea. Anyway, I hope you like this part! As always, your feedback is greatly
appreciated!!
Loki took a closer look at you, his head cocked to one side as his eyes
squinted in concentration. "Yes. I can see the energy you are emitting. I do believe I can help, but not from within
these confines."
"You will not be allowed out of the cell again, Loki," Thor
explained. "The three of us will join
you as you instruct Lady (Y/N)."
Loki took an exasperated breath as he resigned himself to the
arrangement. He backed away from the energy
field as Thor released the barrier. You
and Bucky began walking toward the opening, but Bucky stopped next to Thor.
"You might need this in there," Bucky told Thor as he reached down and
picked up the hammer, holding it out to the dumbfounded god.
Thor hesitantly took the hammer
from Bucky's hand while you looked on in shock.
Bucky turned toward you, a look of confusion coming over him as he saw
the expression on your face.
"Bucky," you whispered in
awe.
"What?" he asked, genuinely
confused.
"The hammer," you replied, still
too shocked to put a full sentence together.
"Mjolnir is not a common hammer,"
Thor explained as the shock finally wore off.
"Only those that are worthy to rule Asgard can wield it. You, my new friend, have been deemed worthy."
I was in a writing mood today, but am still burdened with writer's block. So, I made some drabble prompts. I wrote most of them, but I took a few from tv and movies as well. Please feel free to use them and go ahead and change any pronouns you want. *Edit: Please reblog this post. Do not take these prompts to compile your own list and then pass it off as your own.*
1 "Will you please just give me a hand?"
2 "Why is there a drunk man sleeping in the bathtub?"
3 "Where did that cat come from?"
4 "How could anyone be that cruel?
5 "Why choose me?"
6 Do you like me? Check yes or no.
7"I'm sorry, run that by me again."
8 "It's not nearly as bad as it looks, Darling."
9 "Frankly, I couldn't care less."
10 "How did you get that to stick to the ceiling?"
11 "I do the best I can."
12 "I don't know where she gets it from."
13 "Honestly, I'm just relieved."
14 "You broke what?!"
15 "Why don't you say that to my face?"
16 "Want to know a secret?"
17 "You've got to be kidding me!"
18 "Bring your pretty little butt over here."
19 "Is that what I think it is?"
20 "Is that real?"
21 "We have to pretend to be married."
22 "Excuse me if I don't believe your psychic friend."
23 "Why are you dressed like that?"
24 "I'll sleep under the sheets, you sleep on top of them."
25 "I thought we talked about this."
26 "Why are you lying?"
27 "That guy at the bar keeps staring at you."
28 "I need you to peel five pounds of potatoes!"
29 "It's been fun. We've had a good run, but you parked in my spot. I'm going to have to kill you now."
30 "It made a difference to me."
31 "I shouldn't have even been there!"
32 "I don't know why I'm crying."
33 "You were my new dream."
34 "I had a nightmare about you and just wanted to make sure you're okay."
35 "At what point did you think that was a good idea?"
36 "Look, I don't have much time, but I wanted to say I love you."
37 "You had me at 'free pizza!'"
38 "I am not losing to you again!"
39 "Why don't they just kiss already?"
40 "Are you okay?" "Why do you ask?" "You're wearing two different shoes."
41 "I think I picked up your coffee by mistake."
42 "Is that an apology?"
43 "Why do you always do that?"
44 "I can't get enough of the stuff!"
45 "Oh, why did I eat that? I think I'm dying!"
46 "All I wanted was your honesty."
47 "You have to make a choice."
48 "How did you get up there?"
49 "I found it!"
50 "Do you ever stop eating?"
51 "Why are you always pushing me away?"
52 "Just talk to me!"
53 "I can't get you out of my head."
54 "Not a day will go by that I won't think of you."
55 "I didn't ask for any of this!"
56 "The sign said not to push the button, so naturally I had to push it!"
57 "Will you just shut up for a moment so I can say something nice to you!"
58 "Where did you find this?"
59 "I'm so sorry! I will never doubt you again!"
60 "I can't explain right now, but I really need you to trust me."
61 "Well, this is awkward."
62 "That came out wrong."
63 "I never believed in soulmates until I met you."
64 "You deserve so much better."
65 "I think you're just afraid to be happy."
66 "I'm up to the challenge."
67 "I didn't know you were so competitive."
68 "I didn't know you could cook." "Oh, trust me I can't."
69 "You know my name?"
70 "I didn't realize I needed your permission."
71 "This is so going on Youtube!"
72 "That is not coming in this house!"
73 "Oh, you beautiful weirdo!"
74 He looked at her like she was the moon. She looked at him like he was the sun.
75 "Guess who's going to be a father?"
76 "The joke's on them."
77 "That is the most ridiculous thing I have ever heard!"
78 "I do not snore, do I?" "Like a chainsaw."
79 "I never meant for anyone to get hurt!"
80 "I've never felt this way about anyone before… and it scares the crap out of me."
81 "You weren't supposed to hear that."
82 "I love you more than anything in this world… which is why you have to stay here."
83 "I trusted you!"
84 "You have to remember!"
85 "Wake up! Please, please wake up."
86 "Please just leave."
87 "I don't want to hurt you."
88 "If I didn't know any better, I'd say you were trying to seduce me."
89 "Meet me on the roof in ten minutes."
90 "Meet me on the bridge in an hour."
91 "You are the best thing that's ever happened to me."
92 "You are nothing like them."
93 "You have the most amazing eyes."
94 "Are you hitting on her for me?"
95 "What other hidden talents do you have?"
96 "I can manage on my own."
97 "Since when do you drive a motorcycle?"
98 "You look good for your age."
99 "I can arrange that."
100 "When I come back, that better be exactly where you found it!"
101 "I love you. I've loved you since the moment I first laid eyes on you and – Oh, screw it!" ((This is preferably where they just go for it with a kiss.))
Ok I've been wanting to talk about this for a while and something's finally given me reason to so here we go, buckle up kids.
The accent market has gone to shit. There, I said it.
The "5-10 print customs" thing is crap. Limited run/retiring accents are one thing, but honestly with most of these it's just an expensive circlejerk status symbol. "Look how many customs from ____ I have! I will trade them but only for other 5-10 print customs so I feel exclusive!"
It leads to people taking advantage of artists, which is shitty. It's bad enough that a lot of the time accent artists don't see half the profit their works earn on the secondary market. It's stagnated the market; good luck trying to find stuff for less common breeds or that isn't the same boring #aesthetic. Personally, I feel like exclusive customs should stay as either 1-2 prints, or come with some sort of no-resale rider. If you change your mind and want to resell it one day, I mean, fine, but don't take that as a chance to make an extra 10kg because you've got another status symbol to pawn.
I have heard of accent artists having a commissioner auction a custom and then splitting the profits. And that's cool. I like that trend. I don't blame artists for wanting to make customs. They tend to be one of the few things on FR that command a fair price for the time investment. But when it gets to the point that people aren't buying regular-run accents, but will fall over themselves fighting for custom slots, there's a problem.
And don't get me started on the entitlement I see sometimes. Artists are not obligated to make customs for you, and they're certainly not obligated to put in a ton of work in for you to take your extra copies and make a ton of money off them.
Stop being shitty to artists. Stop trying to take advantage of custom makers. You know who you are, and I know who you are, the same little circle of users with too much real cash to spend on a dragon game, and yet still you feel the need to be greedy.
Yes, a lot of this is stemming from a series of incidents with a single person, but I'm vagueing about it because someone involved wants to stay out of drama. If I know you and you are that thirsty for details, I will discuss privately.
I went downstairs and decided to weigh myself just as a check point to make sure that I'm still headed in the direction I wanted to be headed in after I had a bad weekend and what not. OMG. OMG. LOOK.
1 FREAKING 94.4 AND I'M NOT EVEN JOKING
I am OFFICIALLY 10.6 pounds down from my starting weight of 205 which was on June 15th. I've been on my journey for 7 weeks now and I could NOT be more impressed with how much progress I've made, both mentally and physically.
Okay did everyone survive Day 1? It wasn't too bad right? Well I hope not because it's pretty much exactly the same with a slight increase in squats… but WE GOT THIS!!! I'll probably leave the pictures up for a while until we get familiar with the exercises. By request… I'm going to throw in some healthy eating tips from time to time so look out for that. Tag me or #JeanteFit to let me know how the challenge is going for you! Stay motivated and help motivate others!!
Repeat 1 more time for a total of 2 sets altogether excluding regular squats.
Squat Chop – Stand with feet shoulder width apart, arms overhead. You can use a 5 or 6 pound medicine ball or dumbbell. Squat down as you bring the ball down to the right side. Return to starting position and squat, bringing ball down to left side. Continue to alternate right and left.
External image
Squat Chop
Standing Oblique Crunch - Stand with feet shoulder width apart and left arm overhead; right hand on right hip. Pull left elbow down to side as you lift the left knee to the elbow. Repeat on the right side.
External image
Single Leg Lifts – Lie on your back with both arms overhead and legs straight out in front of you. As you sit up, alternate lifting your legs one at a time.
External image
Wide Leg Cross Sit Up - Lie on your back as shown in the picture below. As you sit up, reach the left hand to the right toe. Go back to starting position and sit up crossing left hand to right toe. Continue to alternate right and left.
Feels like just last week I made this list now it's done. Time to give the list one final assessment and cross stuff off to see how well I did. Or how well Arrow did. Hm. Depends on how you look at it. LOL.
My Top 10 Little Things for Oliver & Felicity (Season 2)
I want….
1. Oliver to confide something personal to Felicity
2. Felicity to visit Oliver at the Queen Mansion for some reason
3. A fancy party/gala type night with both Oliver and Felicity in attendance
4. Oliver to physically step in between Felicity and a threat/bad guy (purrrrrr!)
unless you're actually doing something new with the concept or have put thought into the satire, I'd say a solid 7.
Like, it's a bad idea not because you'll get backlash or anything, it's a bad idea if you don't actually put any thought into the darn thing. I mean, lookatallthisjunk. You can probably find even more just by googling "donald trump game" or searching "donald trump" on gamejolt, newgrounds, or any other free games site.
How are you going to parody him? Or are you making a satire? Are you parodying the way he acts, his wishy washy stance on ideas, his life choices - what part of him are you going to parody? Or are you just going to continuously insult him (which is not something I'm against ) and dress him up in a little clown costume and have a bunch of memes in the background? I highly discourage the latter.
How are you going to reflect it via the gameplay? If you're asking me, I assume you're making this on RPG Maker. Would you have the gameplay be an RPG where he runs around and fights his political opponents, with the game having a message on why he's wrong or it's the wrong thing to do? Are you going to have him just be this terrible overlord with a player character who disagrees with his ideals? You'll run into strawmanning if you do that, and might make your side look worse in the process (although if you want to make him be a strawman for comedy's sake, sure). Maybe you want to make an adventure game - you could simulate a day in the life of Trump, and show how you think his thought process works and parody/satirize his choices.
Basically, it's a bad idea if you're gonna be lazy and put memes everywhere, but not bad if you at least put some thought into it. And hey, if you're just making it to get some coding experience or for shits and giggles, well, there aren't any bad ideas for that, only practice. It's as bad of an idea as you make of it. ¯\_(ツ)_/¯
"Aster's a jackass. On a scale from 1-10? I hate him at a solid level of a six." He chews at the inside of his cheek, looking thoughtful.
"Felix likes him a lot, so he gets a bit of a pass. I can see when he's around people he, uh, he actually fucking likes, he's not really bad. He's petty as hell, and he treats me like garbage, but I understand why. It still gets under my skin and I still kinda hate him for it, though." | 677.169 | 1 |
(TXQ/MP)The shortest path between two truths in the real
domain passes through the complex domain.
Jacques HadamardSolving algebraic equations has been historically one of the favorite topics of mathematicians. While linear equations are always solvable in real numbers, not all quadratic
equations have this property. The simplest such equation is x 2 + 1 = 0. Until the 18th
century, mathematicians avoided quadratic equations that were not solvable over R.
√
Leonhard Euler broke the ice introducing the "number" −1 in his famous book Elements of Algebra as " . . . neither nothing, nor greater than nothing, nor less than nothing . . . " and observed " . . . notwithstanding this, these numbers present themselves to
the mind; they exist in our imagination and we still have a sufficient idea of them; . . .
nothing prevents us from making use of these imaginary numbers, and employing them
√
in calculation". Euler denoted the number −1 by i and called it the imaginary unit.
This became one of the most useful symbols in mathematics. Using this symbol one
defines complex numbers as z = a + bi, where a and b are real numbers. The study of
complex numbers continues and has been enhanced in the last two and a half centuries;
in fact, it is impossible to imagine modern mathematics without complex numbers. All
mathematical domains make use of them in some way. This is true of other disciplines
as well: for example, mechanics, theoretical physics, hydrodynamics, and chemistry.
Our main goal is to introduce the reader to this fascinating subject. The book runs
smoothly between key concepts and elementary results concerning complex numbers.
The reader has the opportunity to learn how complex numbers can be employed in
solving algebraic equations, and to understand the geometric interpretation of com-x
Preface
plex numbers and the operations involving them. The theoretical part of the book is
augmented by rich exercises and problems of various levels of difficulty. In Chapters 3 and 4 we cover important applications in Euclidean geometry. Many geometry
problems may be solved efficiently and elegantly using complex numbers. The wealth
of examples we provide, the presentation of many topics in a personal manner, the
presence of numerous original problems, and the attention to detail in the solutions to
selected exercises and problems are only some of the key features of this book.
Among the techniques presented, for example, are those for the real and the complex
product of complex numbers. In complex number language, these are the analogues of
the scalar and cross products, respectively. Employing these two products turns out to
be efficient in solving numerous problems involving complex numbers. After covering
this part, the reader will appreciate the use of these techniques.
A special feature of the book is Chapter 5, an outstanding selection of genuine
Olympiad and other important mathematical contest problems solved using the methods already presented.
This work does not cover all aspects pertaining to complex numbers. It is not a
complex analysis book, but rather a stepping stone in its study, which is why we have
not used the standard notation eit for z = cos t + i sin t, or the usual power series
expansions.
The book reflects the unique experience of the authors. It distills a vast mathematical
literature, most of which is unknown to the western public, capturing the essence of an
abundant problem-solving culture.
Our work is partly based on a Romanian version, Numere complexe de la A la . . . Z,
authored by D. Andrica and N. Bis¸boac˘a and published by Millennium in 2001 (see our
reference [10]). We are preserving the title of the Romanian edition and about 35% of
the text. Even this 35% has been significantly improved and enhanced with up-to-date
material.
The targeted audience includes high school students and their teachers, undergraduates, mathematics contestants such as those training for Olympiads or the W. L. Putnam Mathematical Competition, their coaches, and any person interested in essential
mathematics.
This book might spawn courses such as Complex Numbers and Euclidean Geometry for prospective high school teachers, giving future educators ideas about things
they could do with their brighter students or with a math club. This would be quite a
welcome development.
Special thanks are given to Daniel V˘ac˘aret¸u, Nicolae Bis¸boac˘a, Gabriel Dospinescu,
and Ioan S¸erdean for the careful proofreading of the final version of the manuscript. WePreface
xi
would also like to thank the referees who provided pertinent suggestions that directly
contributed to the improvement of the text.
Titu Andreescu
Dorin Andrica
October 2004
the set of integers
the set of positive integers
the set of rational numbers
the set of real numbers
the set of nonzero real numbers
the set of pairs of real numbers
the set of complex numbers
the set of nonzero complex numbers
the set of real numbers x such that a ≤ x ≤ b
the set of real numbers x such that a < x < b
the conjugate of the complex number z
the modulus or absolute value of complex number z
the vector AB
the open segment determined by A and B
the closed segment determined by A and B
the open ray of origin A that contains B
the area of figure F
the set of n th roots of unity
the circle centered at point P with radius n
Definition. The set R2 , together with the addition and multiplication operations, is
called the set of complex numbers, denoted by C. Any element z = (x, y) ∈ C is called
a complex number.
The notation C∗ is used to indicate the set C \ {(0, 0)}.
For algebraic manipulation it is not convenient to represent a complex number as an
ordered pair. For this reason another form of writing is preferred.
To introduce this new algebraic representation, consider the set R × {0}, together
with the addition and multiplication operations defined on R2 . The function
f : R → R × {0},
f (x) = (x, 0)
is bijective and moreover,
(x, 0) + (y, 0) = (x + y, 0) and (x, 0) · (y, 0) = (x y, 0).
The reader will not fail to notice that the algebraic operations on R × {0} are similar to the operations on R; therefore we can identify the ordered pair (x, 0) with the
number x for all x ∈ R. Hence we can use, by the above bijection f , the notation
(x, 0) = x.
Setting i = (0, 1) we obtain
z = (x, y) = (x, 0) + (0, y) = (x, 0) + (y, 0) · (0, 1)
= x + yi = (x, 0) + (0, 1) · (y, 0) = x + i y.
In this way we obtain
Proposition. Any complex number z = (x, y) can be uniquely represented in the
form
z = x + yi,
where x, y are real numbers. The relation i 2 = −1 holds.
The formula i 2 = −1 follows directly from the definition of multiplication: i 2 =
i · i = (0, 1) · (0, 1) = (−1, 0) = −1.
The expression x + yi is called the algebraic representation (form) of the complex
number z = (x, y), so we can write C = {x + yi| x ∈ R, y ∈ R, i 2 = −1}. From
now on we will denote the complex number z = (x, y) by x + i y. The real number
x = Re(z) is called the real part of the complex number z and similarly, y = Im(z)
is called the imaginary part of z. Complex numbers of the form i y, y ∈ R — in other
words, complex numbers whose real part is 0 — are called imaginary. On the other
hand, complex numbers of the form i y, y ∈ R∗ are called purely imaginary and the
complex number i is called the imaginary unit.
The following relations are easy to verify:
For a complex number z = x + yi the number z = x − yi is called the complex
conjugate or the conjugate complex of z.
Proposition. 1) The relation z = z holds if and only if z ∈ R.
2) For any complex number z the relation z = z holds.
3) For any complex number z the number z · z ∈ R is a nonnegative real number.
4) z 1 + z 2 = z 1 + z 2 (the conjugate of a sum is the sum of the conjugates).
5) z 1 · z 2 = z 1 · z 2 (the conjugate of a product is the product of the conjugates).
6) For any nonzero complex number z the relation z −1 = (z)−1 holds.
z1
z1
7)
=
, z 2 = 0 (the conjugate of a quotient is the quotient of the conjuz2
z2
gates).
8) The formulas
Re(z) =
Prove that there is a unique number z ∈ H such that |z| ≤ |w| for all w ∈ H14
1. Complex Numbers in Algebraic Form
Solution. Let ω = y − 1 + yi, with y ∈ R.
It suffices to prove that there is a unique number x ∈ R such that
(x − 1)2 + x 2 ≤ (y − 1)2 + y 2
for all y ∈ R.
In other words, x is the minimum point of the function
Using the same algebraic manipulation as in the case of real coefficients, we get
b 2
a
z+
− 2 = 0.
2a
4a
This is equivalent to
z+
b
2a
2
=
4a 2
or
(2az + b)2 = ,
where = b2 − 4ac is also called the discriminant of the quadratic equation. Setting
y = 2az + b, the equation is reduced to
y 2 = = u + vi,
where u and v are real numbers16
1. Complex Numbers in Algebraic Form
This equation has the solutions
y1,2 = ±
r +u
r −u
+ (sgn v)
i ,
2
2
where r = || and signv is the sign of the real number v.
The roots of the initial equation are
1
(−b + y1,2 ).
2a
Observe that the relations between roots and coefficients
b
c
z1 + z2 = − , z1 z2 = ,
a
a
as well as the factorization formula
z 1,2 =
Geometric Interpretation of the Algebraic
Operations
Geometric interpretation of a complex number
We have defined a complex number z = (x, y) = x + yi to be an ordered pair of
real numbers (x, y) ∈ R × R, so it is natural to let a complex number z = x + yi
correspond to a point M(x, y) in the plane R ×22
1. Complex Numbers in Algebraic Form
For a formal introduction, let us consider P to be the set of points of a given plane
equipped with a coordinate system x O y. Consider the bijective function ϕ : C → P,
ϕ(z) = M(x, y).
Definition. The point M(x, y) is called the geometric image of the complex number
z = x + yi.
The complex number z = x + yi is called the complex coordinate of the point
M(x, y). We will use the notation M(z) to indicate that the complex coordinate of M
is the complex number z.
Figure 1.2.
The geometric image of the complex conjugate z of a complex number z = x + yi
is the reflection point M (x, −y) across the x-axis of the point M(x, y) (see Fig. 1.2).
The geometric image of the additive inverse −z of a complex number z = x + yi is
the reflection M (−x, −y) across the origin of the point M(x, y) (see Fig. 1.2).
The bijective function ϕ maps the set R onto the x-axis, which is called the real axis.
On the other hand, the imaginary complex numbers correspond to the y-axis, which
is called the imaginary axis. The plane , whose points are identified with complex
numbers, is called the complex plane.
On the other hand, we can also identify a complex number z = x + yi with the
−−→
→
vector −
v = O M, where M(x, y) is the geometric image of the complex number z1.2. Geometric Interpretation of the Algebraic Operations
23
Figure 1.3.
Let V0 be the set of vectors whose initial points are the origin O. Then we can define
the bijective function
ϕ : C → V0 ,
−−→ →
−
→
−
→
ϕ (z) = O M = −
v =x i +y j ,
−
→ −
→
where i , j are the vectors of the x-axis and y-axis, respectively.
1.2.2
Geometric interpretation of the modulus
Let us consider a complex number z = x + yi and the geometric image M(x, y) in the
complex plane. The Euclidean distance O M is given by the formula
OM =
Polar Representation of Complex Numbers
Polar coordinates in the plane
Let us consider a coordinate plane and a point M(x, y) that is not the origin.
The real number r = x 2 + y 2 is called the polar radius of the point M. The direct
−−→
angle t ∗ ∈ [0, 2π) between the vector O M and the positive x-axis is called the polar
argument of the point M. The pair (r, t ∗ ) is called the polar coordinates of the point M.
We will write M(r, t ∗ ). Note that the function h : R×R\{(0, 0)} → (0, ∞)×[0, 2π ),
h((x, y)) = (r, t ∗ ) is bijective.
The origin O is the unique point such that r = 0; the argument t ∗ of the origin is
not defined.
For any point M in the plane there is a unique intersection point P of the ray (O M
with the unit circle centered at the origin. The point P has the same polar argument t ∗ .
Using the definition of the sine and cosine functions we find that
x = r cos t ∗ and y = r sin t ∗ .
Therefore, it is easy to obtain the cartesian coordinates of a point from its polar coordinates.
Conversely, let us consider a point M(x, y). The polar radius is r = x 2 + y 2 . To
determine the polar argument we study the following cases:
The roots of the equation Z n − 1 = 0 are called the n th roots of unity. Since 1 =
cos 0 + i sin 0, from the formulas for the n th roots of a complex number we derive that
the n th roots of unity are
εk = cos
2kπ
2kπ
+ i sin
,
n
n
k ∈ {0, 1, 2, . . . , n − 1}.
Explicitly, we have
ε0 = cos 0 + i sin 0 = 1;44
2. Complex Numbers in Trigonometric Form
ε1 = cos
2π
2π
+ i sin
= ε;
n
n
ε2 = cos
4π
4π
+ i sin
= ε2 ;
n
n
...
εn−1 = cos
2(n − 1)π
2(n − 1)π
+ i sin
= ε n−1 .
n
n
The set {1, ε, ε2 , . . . , εn−1 } is denoted by Un . Observe that the set Un is generated
by the element ε, i.e., the elements of Un are the powers of ε.
As stated before, the geometric images of the n th roots of unity are the vertices of a
regular polygon with n sides inscribed in the unit circle with one of the vertices at 1.
We take a brief look at some particular values of n.
i) For n = 2, the equation Z 2 − 1 = 0 has the roots −1 and 1, which are the square
roots of unity.
ii) For n = 3, the cube roots of unity, i.e., the roots of equation Z 3 − 1 = 0 are given
by
2kπ
2kπ
εk = cos
+ i sin
for k ∈ {0, 1, 2}.
3
3
Hence
√
3
2π
1
2π
ε0 = 1, ε1 = cos
+ i sin
=− +i
=ε
3
3
2
2
and
√
3
4π
4π
1
ε2 = cos
+ i sin
=− −i
= ε2 .
3
3
2
2
They form an equilateral triangle inscribed in the circle C(O; 1) as in the figure
below.
Figure 2.72.2. The n th Roots of Unity
45
iii) For n = 4, the fourth roots of unity are
εk = cos
2kπ
2kπ
+ i sin
for k = 0, 1, 2, 3.
4
4
In explicit form, we have
ε0 = cos 0 + i sin 0 = 1;
ε1 = cos
ε2 = cos π + i sin π = −1 and ε3 = cos
π
π
+ i sin = i;
2
2
3π
3π
+ i sin
= −i.
2
2
Observe that U4 = {1, i, i 2 , i 3 } = {1, i, −1, −i}. The geometric images of the
fourth roots of unity are the vertices of a square inscribed in the circle C(O; 1).
Figure 2.8.
The root εk ∈ Un is called primitive if for all positive integer m < n we have
= 1.
|z|2002 = |z 2002 | = |z| = |z|,
from which it follows that
|z|(|z|2001 − 1) = 0.
Hence |z| = 0, and (a, b) = (0, 0), or |z| = 1. In the case |z| = 1, we have
z 2002 = z, which is equivalent to z 2003 = z · z = |z|2 = 1. Since the equation
z 2003 = 1 has 2003 distinct solutions, there are altogether 1 + 2003 = 2004 ordered
pairs that meet the required conditions.
Problem 2. Two regular polygons are inscribed in the same circle. The first polygon
has 1982 sides and the second has 2973 sides. If the polygons have any common vertices, how many such vertices will there be?
Because n is odd, from the relation φ2n (x) = φn (−1) it follows that φn (−1) =
φ2n (1) = 1. Then
1 = φn (−1) =
(1 − εnk ) = (−1)ϕ(n)
(1 + εnk )
1≤k≤n−1
gcd(k,n)=1
= (−1)ϕ(n)
⎛
1≤k≤n−1
gcd(k,n)=1
2 cos
1≤k≤n−1
gcd(k,n)=1
⎜
= (−1)ϕ(n) 2ϕ(n) ⎝
cos
kπ
kπ
+ i sin
n
n
⎞
cos
1≤k≤n−1
gcd(k,n)=1
= (−1)
kπ
n
ϕ(n)
2
ϕ(n)
ϕ(n)
kπ ⎟
cos
π
+
i
sin
π
⎠
n
2
2
2ϕ(n)
cos
1≤k≤n−1
gcd(k,n)=1
kπ
,
n
yielding the desired identity.
2.2.3
Binomial equations
A binomial equation is an equation of the form Z n + a = 0, where a ∈ C∗ and n ≥ 2
is an integer.
Solving for Z means finding the n th roots of the complex number −a. This is in fact
a simple polynomial equation of degree n with complex coefficients. From the wellknown fundamental theorem of algebra it follows that it has exactly n complex roots,
and it is obvious that the roots are distinct.
Example. 1) Let us find the roots of Z 3 + 8 = 0.
We have −8 = 8(cos π + i sin π ), so the roots are
π + 2kπ
π + 2kπ
Z k = 2 cos
+ i sin
,
3
3
Let A and B be two distinct points with complex coordinates a and b. We say that the
point M with complex coordinate z is between the points A and B if z = a, z = b and
the following relation holds:
|a − z| + |z − b| = |a − b|.
We use the notation A − M − B.
The set (AB) = {M : A − M − B} is called the open segment determined by the
points A and B. The set [AB] = (AB) ∪ {A, B} represents the closed segment defined
by the points A and B.
Theorem 1. Suppose A(a) and B(b) are two distinct points. The following statements are equivalent:
1) M ∈ (AB);
2) there is a positive real number k such that z − a = k(b − z);
3) there is a real number t ∈ (0, 1) such that z = (1 − t)a + tb, where z is the
complex coordinate of M.
Proof. We first prove that 1) and 2) are equivalent. Indeed, we have M ∈ (AB) if and
only if |a − z| + |z − b| = |a − b|. That is, d(a, z) + d(z, b) = d(a, b), or equivalently
there is a real k > 0 such that z − a = k(b − z).
t
k
To prove that 2) ⇔ 3), set t =
∈ (0, 1) or k =
> 0. Then we have
k+1
1−t
1
k
z − a = k(b − z) if and only if z =
a+
b. That is, z = (1 − t)a + tb and
k+1
k+1
we are done.
The set (AB = {M| A − M − B or A − B − M} is called the open ray with endpoint
A that contains B.
Theorem 2. Suppose A(a) and B(b) are two distinct points. The following statements are equivalent:
1) M ∈ (AB;
2) there is a positive real number t such that z = (1 − t)a + tb, where z is the
complex coordinate of M;
Theorem 3. Suppose A(a) and B(b) are two distinct points. The following statements are equivalent:
1) M(z) lies on the line AB.
z−a
2)
∈ R.
b−a
3) There is a real number
t such that z = (1 − t)a + tb.
z−a z−a
4)
= 0;
b−a b−a
z z 1
5) a a 1 = 0.
b b 1
Proof. To obtain the equivalences 1) ⇔ 2) ⇔ 3) observe that for a point C such
that C − A − B the line AB is the union (AB âˆŞ {A} âˆŞ (AC. Then apply Theorem 2.
Next we prove the equivalences 2) ⇔ 4) ⇔ 5).
z−a
z−a
z−a
Indeed, we have
∈ R if and only if
=
.
b−a
b − a
b − a
z−a z−a
z−a
z−a
That is,
, or, equivalently,
=
= 0, so we obtain that
b−a
b−a b−a
b−a
2) is equivalent to 4).
Moreover, we have
z z 1
z−a z−a 0
a
1 =0
a a 1 = 0 if and only if a
b b 1
b−a b−a 0
The last relation is equivalent to
z−a
b−a
circumcenter of triangle A1 A2 A3 is the origin of the complex plane. Notice that point
A lies on the line A2 A3 , so A1 A = |z − z 1 | is greater than or equal to the altitude A1 B
of the triangle A1 A2 A3 .
Consider two distinct points A(a) and B(b). A point M(z) on the line AB divides the
segments AB into the ratio k ∈ R \ {1} if the following vectorial relation holds:
−→
−→
M A = k · M B.
In terms of complex numbers this relation can be written as
a − z = k(b − z) or (1 − k)z = a − kb.
Hence, we obtain
a − kb
.
1−k
Observe that for k < 0 the point M lies on the line segment joining the points A and
B. If k ∈ (0, 1), then M ∈ (AB \ [AB]. Finally, if k > 1, then M ∈ (B A \ [AB].
z=58
3. Complex Numbers and Geometry
As a consequence, note that for k = −1 we obtain that the coordinate of the mida+b
point of segment [AB] is given by z M =
.
2
Example. Let A(a), B(b), C(c) be noncollinear points in the complex plane. Then
a+b
the midpoint M of segment [AB] has the complex coordinate z M =
. The cen2
troid G of triangle ABC divides the median [C M] into 2 : 1 internally, hence its
complex coordinate is given by k = −2, i.e.,
zG =
3.1.4
c + 2z M
a+b+c
=
.
1+2
3
Measure of an angle
Recall that a triangle is oriented if an ordering of its vertices is specified. It is positively or directly oriented if the vertices are oriented counterclockwise. Otherwise, we
say that the triangle is negatively oriented. Consider two distinct points M1 (z 1 ) and
M2 (z 2 ), other than the origin of a complex plane. The angle M
1 O M2 is oriented if the
points M1 and M2 are ordered counterclockwise (Fig. 3.2 below).
Proposition. The measure of the directly oriented angle
z2
M
.
1 O M2 equals arg
z1
Proof. We consider the following two cases.
Consider four distinct points Mi (z i ), i ∈ {1, 2, 3, 4}. The measure of the angle deterz3 − z1
z4 − z2
or arg
. The proof is
mined by the lines M1 M3 and M2 M4 equals arg
z4 − z2
z3 − z1
obtained following the same ideas as in the previous subsection.
3.1.6
Rotation of a point
Consider an angle α and the complex number given by
ε = cos α + i sin α.
Let z = r (cos t + i sin t) be a complex number and M its geometric image.
Form the product zε = r (cos(t + α) + i sin(t + α)) and let us observe that |zε| = r
and
arg(zε) = arg z + α.
It follows that the geometric image M of zε is the rotation of M with respect to the
origin by the angle α.
Figure 3.6.
Now we have all the ingredients to establish the following result:
Proposition. Suppose that the point C is the rotation of B with respect to A by the
angle α.
If a, b, c are the coordinates of the points A, B, C, respectively, then
c = a + (b − a)ε, where ε = cos α + i sin α.
Proof. The translation with vector −a maps the points A, B, C into the points
O, B , C , with complex coordinates O, b − a, c − a, respectively (see Fig. 3.7). The
point C is the image of B under rotation about the origin through the angle α, so
c − a = (b − a)ε, or c = a + (b − a)ε, as desired62
3. Complex Numbers and Geometry
Figure 3.7.
We will call the formula in the above proposition the rotation formula.
Problem 1. Let ABC D and B N M K be two nonoverlapping squares and let E be the
midpoint of AN . If point F is the foot of the perpendicular from B to the line C K ,
prove that points E, F, B are collinear.
Solution. Consider the complex plane with origin at F and the axis C K and F B,
where F B is the imaginary axis.
Let c, k, bi be the complex coordinates of points C, K , B with c, k, b ∈ R. The
π
rotation with center B through the angle θ =
maps point C to A, so A has the
2
complex coordinate a = b(1 − i) + ci. Similarly, point N is obtained by rotating point
π
K around B through the angle θ = − and its complex coordinate is
2
n = b(1 + i) − ki.
The midpoint E of segment AN has the complex coordinate
e=
Denote by a lowercase letter the coordinate of each of the points denoted by an
uppercase letter, i.e., o1 is the coordinate of O1 , etc.
Ď€
Point M is obtained from point A by a rotation about B through the angle θ = ;
2
hence m = b + (a − b)i. Likewise,
n = c + (b − c)i,
Proof. Assume that the points are collinear. We can arrange four points on a circle in
(4 − 1)! = 3! = 6 different ways. Consider the case when M1 , M2 , M3 , M4 are given
in this order. Then M1 , M2 , M3 , M4 are concyclic if and only if
M
1 M2 M3 + M
1 M4 M3 ∈ {3π, π }.
That is,
arg
z3 − z2
z1 − z4
+ arg
∈ {3π, π }.
z1 − z2
z3 − z4
arg
z3 − z2
z3 − z4
− arg
∈ {3π, π },
z1 − z2
z1 − z4
We obtain
i.e., k < 0.
For any other arrangements of the four points the proof is similar. Note that k > 0
in three cases and k < 0 in the other three.
The number k is called the cross ratio of the four points M1 (z 1 ), M2 (z 2 ), M3 (z 3 )
and M4 (z 4 ).
Remarks. 1) The points M1 , M2 , M3 , M4 are collinear if and only if
z3 − z2
z3 − z4
∈ R∗ and
∈ R∗ .
z1 − z2
z1 − z4
2) The points M1 , M2 , M3 , M4 are concyclic if and only if
k=
d +c+ f
a+b+c
=
,
3
3
so triangles ABC and D E F have the same centroid.
Problem 2. Let M, N , P be the midpoints of sides AB, BC, C A of triangle ABC.
On the perpendicular bisectors of segments [AB], [BC], [C A] points C , A , B are
chosen inside the triangle such that
MC
N A
P B
=
=
.
AB
BC
CA
Prove that ABC and A B C have the same centroid.
Solution. Note that from
MC
N A
P B
=
=
AB
BC
CA
AB) = tan(
it follows that tan(C
A BC) = tan(
B C A). Hence triangles AC B, B A C,
C B A are similar and we can proceed as in the previous problem.
Problem 3. Let AB O be an equilateral triangle with center S and let A B O be another equilateral triangle with the same orientation and S = A , S = B . Consider M
and N the midpoints of the segments A B and AB .
Prove that triangles S B M and S A N are similar.
(30th IMO – Shortlist)
Solution. Let R be the circumradius of the triangle AB O and let
ε = cos
2π
2π
+ i sin
.
3
3
Consider the complex plane with origin at point S such that point O lies on the positive
real axis. Then the coordinates of points O, A, B are R, Rε, Rε 2 , respectively.
Let R + z be the coordinate of point B , so R − zε is the coordinate of point A . It
follows that the midpoints M, N have the coordinates
zM =
The concept of the scalar product of two vectors is well known. In what follows we
will introduce this concept for complex numbers. We will see that in many situations
use of this product simplifies the solution to the problem considerably.
Let a and b be two complex numbers.
Definition. We call the real product of complex numbers a and b the number given
by
a·b =
1
(ab + ab).
2
It is easy to see that
a·b =
1
(ab + ab) = a · b;
2
hence a · b is a real number, which justifies the name of this product.
The following properties are easy to verify.
Proposition 1. For all complex numbers a, b, c, z the following relations hold:
1) a · a = |a|2 .
2) a · b = b · a; (the real product is commutative).
3) a · (b + c) = a · b + a · c; (the real product is distributive with respect to addition).
4) (αa) · b = α(a · b) = a · (αb) for all α ∈90
4. More on Complex Numbers and Geometry
5) a ¡ b = 0 if and only if O A ⊼ O B, where A has coordinate a and B has
coordinate b.
6) (az) ¡ (bz) = |z|2 (a ¡ b).
Remark. Suppose that A and B are points with coordinates a and b. Then the real
product a ¡ b is equal to the power of the origin with respect to the circle of diameter
AB.
a+b
Indeed, let M
be the midpoint of [AB], hence the center of this circle, and
2
1
1
let r = AB = |a − b| be the radius of this circle. The power of the origin with
2
2
respect to the circle is
a + b 2 a − b 2
OM − r =
−
2
2
2
Remark. If the numbers a, b, c, o, h are the coordinates of the vertices of triangle
ABC, the circumcenter O and the orthocenter H of the triangle, then h = a+b+c−2o.
Indeed, taking A diametrically opposite A in the circumcircle of triangle ABC, the
quadrilateral H B A C is a parallelogram. If {M} = H A ∩ BC, then
zM =
Solution. Consider the complex plane with origin at point O and let Rεk be the
coordinate of vertex Ak , where εk are the n th -roots of unity, k = 1, . . . , n. Let m be
the coordinate of M.
Using the properties of the real product of the complex numbers, we have
n
M A2k =
k=1
=
n
k=1
n
(m − Rεk ) · (m − Rεk )
(m · m − 2Rεk · m + R 2 εk · εk )
k=1
= n|m|2 − 2R
n
k=1
2
n
εk · m + R 2
|εk |2
k=1
2
= n · O M + n R = n(O M + R ),
2
1. The Real Product of Two Complex Numbers
since
n
93
εk = 0.
k=1
Remark. If M lies on the circumcircle of the polygon, then
n
M A2k = 2n R 2 .
k=1
Problem 4. Let O be the circumcenter of the triangle ABC, let D be the midpoint of
the segment AB, and let E is the centroid of triangle AC D. Prove that lines C D and
O E are perpendicular if and only if AB = AC.
(Balkan Mathematical Olympiad, 1985)
Solution. Let O be the origin of the complex plane and let a, b, c, d, e be the coordinates of points A, B, C, D, E, respectively. Then
d=
Remark. The following generalization can be proved similarly.
Let A1 A2 · · · An be a polygon with the centroid G and let Ai j be the midpoint of the
segment [Ai A j ], i < j, i, j ∈ {1, 2, . . . , n}.
Then
n
(n − 2)
M A2k + n 2 M G 2 = 4
M Ai2j ,
i< j
k=1
for all points M in the plane. A nice generalization is given in Theorem 5, Section 4.11.
4.2
The Complex Product of Two Complex Numbers
The cross product of two vectors is a central concept in vector algebra, with numerous
applications in various branches of mathematics and science. In what follows we adapt
this product to complex numbers. The reader will see that this new interpretation has
multiple advantages in solving problems involving area or collinearity.
Let a and b be two complex numbers.
Definition. The complex number
1
(ab − ab)
2
is called the complex product of the numbers a and b.
Note that
1
1
a × b + a × b = (ab − ab) + (ab − ab) = 0,
2
2
so Re(a × b) = 0, which justifies the definition of this product.
The following properties are easy to verify:
a×b =
Proposition 1. Suppose that a, b, c are complex numbers. Then:
1) a × b = 0 if and only if a = 0 or b = 0 or a = λb, where λ is a real number.
2) a × b = −b × a; (the complex product is anticommutative).
3) a × (b + c) = a × b + a × c (the complex product is distributive with respect to
addition).
4) α(a × b) = (αa) × b = a × (αb), for all real numbers α.
5) If A(a) and B(b) are distinct points other than the origin, then a × b = 0 if and
only if O, A, B are collinear.
Remarks. a) Suppose A(a) and B(b) are distinct points in the complex plane, different from the origin.
The complex product of the numbers a and b has the following useful geometric
interpretation:
2i · area[AO B],
if triangle O AB is positively oriented;
a×b =
−2i · area[AO B], if triangle O AB is negatively oriented.
if triangle ABC is directly (positively) oriented.
To prove the above formula, translate points A, B, C with vector −c. The images
of A, B, C are points A , B , O with coordinates a − c, b − c, 0, respectively. Triangles ABC and A B O are congruent with the same orientation. If ABC is positively
It follows that m = tan A, n = tan B, p = tan C, and the coordinate of H is given
by
zH =
(tan A)a + (tan B)b + (tan C)c
.
tan A + tan B + tan C
Remark. The above formula can also be extended to the limiting case when the
π
triangle ABC is a right triangle. Indeed, assume that A → . Then tan A → ±∞
2
(tan B)b + (tan C)c
tan B + tan C
and
→ 0,
→ 0. In this case z H = a, i.e., the
tan A
tan A
orthocenter of triangle ABC is the vertex A.
4) The Gergonne1 point J is the intersection of the cevians A A , B B , CC , where
A , B , C are the points of tangency of the incircle to the sides BC, C A, AB, respectively. Then
1
s−γ
=
,
1
A C
s−β
B A
1
C B
s
−
α,
=
1
B A
s−γ
1
AC
s−β
=
,
1
C B
s−α
and the coordinate z J is obtained from the same proposition, where
zJ =
rα a + rβ b + rγ c
.
rα + rβ + rγ
Here rα , rβ , rγ denote the radii of the three excircles of triangle. It is not difficult to
show that the following formulas hold:
rα =
K
,
s−α
rβ =
K
,
s−β
rγ =
K
,
s−γ
1
(α + β + γ ).
2
5) The Lemoine2 point K is the intersection of the symmedians of the triangle (the
symmedian is the reflection of the bisector across the median). Using the notation from
where K = area[ABC] and s =
1 Joseph-Diaz Gergonne (1771–1859), French mathematician, founded the journal Annales de Math´ema-
Given a triangle ABC, choose its circumcenter O to be the origin of the complex plane
and let a, b, c be the coordinates of the vertices A, B, C. We have seen in Section 2.22,
Proposition 3, that the coordinate of the orthocenter H is z H = a + b + c.
Let us denote by A1 , B1 , C1 the midpoints of sides BC, C A, AB, by A , B , C
the feet of the altitudes and by A , B , C the midpoints of segments AH , B H , C H ,
respectively.
ysis, and revised almost all the branches of pure mathematics which were then known, adding proofs, and
arranging the whole in a consistent form. Euler wrote an immense number of memoirs on all kinds of mathematical subjects. We recommend William Dunham’s book Euler. The Master of Us All (The Mathematical
Association of America, 1999) for more details concerning Euler’s contributions to mathematics5 The Nine-Point Circle of Euler
109
Figure 4.6.
If G s is the midpoint of segment [I N ], then its coordinate is
zGs =
(γ + α)
(α + β)
1
(β + γ )
(z I + z N ) =
a+
b+
c.
2
4s
4s
4s
The point G s is called the Spiecker point of triangle ABC and it is easy to verify
that it is the incenter of the medial triangle A1 B1 C1 .
Problem 1. Consider a point M on the circumcircle of the triangle ABC. Prove that
the nine-point centers of the triangles M BC, MC A, M AB are the vertices of a triangle similar to triangle ABC.
Solution. Let A , B , C be the nine-point centers of the triangles M BC, MC D,
M AB, respectively. Take the origin of the complex plane to be at the circumcenter of
triangle ABC. Denote by a lowercase letter the coordinate of the point denoted by an
uppercase letter. Then
a =
m+b+c
,
2
b =
m+c+a
,
2
c =
m+a+b
,
2
since M lies on the circumcircle of triangle ABC. Then
b − a
a−b
b−a
=
=
,
c − a
a−c
c−a
and hence triangles A B C and ABC are similar.
Problem 2. Show that triangle ABC is a right triangle if and only if its circumcircle
and its nine-point circle are tangent.
Solution. Take the origin of the complex plane to be at circumcenter O of triangle
ABC and denote by a, b, c the coordinates of vertices A, B, C, respectively. Then the
Assume that the circumcenter O of the triangle ABC is the origin of the complex plane
and let a, b, c be the coordinates of the vertices A, B, C, respectively.
Lemma. The real products a · b, b · c, c · a are given by
a · b = R2 −
Corollary 5. (Euler's inequality.) In any triangle ABC the following inequality
holds:
R ≥ 2r.
We have equality if and only if the triangle ABC is equilateral.
Proof. From Theorem 4 we have O I 2 = R(R − 2r ) ≥ 0, hence R ≥ 2r . The
equality R − 2r = 0 holds if and only if O I 2 = 0, i.e., O = I . Therefore triangle
ABC is equilateral.
4.6.3
The distance ON
Theorem 6. If N is the Nagel point of triangle ABC, then
O N = R − 2r.
Proof. The coordinate of the Nagel point of the triangle is given by
β
γ
α
zN = 1 −
a+ 1−
b+ 1−
c.
s
s
s
Therefore
Therefore triangles G I O9 and G N O are similar. It follows that the lines I O9 and
1
1
O N are parallel and I O9 = O N . Applying Theorem 6 we get I O9 = (R − 2r ) =
2
2
R
− r = R9 − r , hence the incircle is tangent to the nine-point circle.
2
The point of tangency of these two circles is denoted by ϕ and is called the Feuerbach point of triangle.
4.6.4
The distance OH
Theorem 8. If H is the orthocenter of triangle ABC, then
O H 2 = 9R 2 + 2r 2 + 8Rr − 2s 2 .
Proof. Assuming that the circumcenter O is the origin of the complex plane, the
coordinate of H is
z H = a + b + c.
5 Karl Wilhelm Feuerbach (1800–1834), German geometer, published the result of Theorem 7 in 1822.
Distance between Two Points in the Plane of a
Triangle
Barycentric coordinates
Consider a triangle ABC and let α, β, γ be the lengths of sides BC, C A, AB, respectively.
Proposition 1. Let a, b, c be the coordinates of vertices A, B, C and let P be a
point in the plane of triangle. If z P is the coordinate of P, then there exist unique real
numbers μa , μb , μc such that
z P = μa a + μb b + μc c and μa + μb + μc = 1.
Proof. Assume that P is in the interior of triangle ABC and consider the point A
PA
A B
such that A P ∩ BC = {A }. Let k1 =
, k2 = and observe that
PA
AC
zP =
a + k 1 z A
,
1 + k1
z A =
b + k2 c
.
1 + k116
4. More on Complex Numbers and Geometry
Hence in this case we can write
zP =
k1 k2
1
k1
b+
c.
a+
1 + k1
(1 + k1 )(1 + k2 )
(1 + k1 )(1 + k2 )
Moreover, if we consider
μa =
1
,
1 + k1
μb =
k1
,
(1 + k1 )(1 + k2 )
μc =
k1 k2
(1 + k1 )(1 + k2 )
we have
μa + μb + μc =
1
k1
k1 k2
+
+
1 + k1
(1 + k1 )(1 + k2 ) (1 + k1 )(1 + k2 )
=
1 + k1 + k2 + k1 k2
= 1.
(1 + k1 )(1 + k2 )
We proceed in an analogous way in the case when the point P is situated in the
exterior of triangle ABC.
If the point P is situated on the support line of a side of triangle ABC (i.e., the line
determined by two vertices)
zP =
where k =
1
k
1
k
b+
c =0·a+
b+
c,
1+k
1+k
1+k
1+k
PB
.
PC
The real numbers μa , μb , μc are called the absolute barycentric coordinates of P
with respect to the triangle ABC.
The signs of numbers μa , μb , μc depend on the regions of the plane where the point
P is situated. Triangle ABC determines seven such regions.
Figure 4.8.
In the next table we give the signs of μa , μb , μc :7. Distance between Two Points in the Plane of a Triangle
I
−
+
+
μa
μb
μc
4.7.2
II
+
−
+
III
+
+
−
IV
+
−
−
V
−
+
−
VI
−
−
+
117
VII
+
+
+
Distance between two points in barycentric coordinates
In what follows, in order to simplify the formulas, we will use the symbol called "cyclic
/
f (x1 , x2 , . . . , xn ), the sum of terms considered in the cyclic order.
sum." That is,
cyc
Applications. 1) Let us use the formula in Theorem 3 to compute the distance G I ,
where G is the centroid and I is the incenter of the triangle.
We have m 1 = n 1 = p1 = 1 and m 2 = ι, n 2 = β, p2 = γ ; hence
If P j is the intersection point of lines A A j , B B j , CC j , j = 1, 2, 3, and triangles
ABC, P1 P2 P3 are of the same orientation, then from Corollary 3 we obtain
1 4 2
49
area[P1 P2 P3 ]
1
1
=
2 1 4 = 3 = .
7
area[ABC]
7¡7¡7
7
4 2 1
2) If the cevians A A , B B , CC are concurrent at point P, let us denote by K P the
area of triangle A B C . We can use the formula in Corollary 4 to compute the areas
of some triangles determined by the feet of the cevians of some remarkable points in a
triangle.
(i) If I is the incenter of triangle ABC we have
Îł
1+ ¡
β
KI =
Îł
1+
1+
β
=
If we proceed in the same way for the Gergonne point J we ďŹ nd the relation
KJ =
r
sr 2
area[ABC] =
.
2R
2R
Remark. Two cevians A A and A A are isotomic if the points A and A are symmetric with respect to the midpoint of the segment BC. Assuming that
A B
B C
C A
=
m,
=
n,
= p,
A C
B A
C B
then for the corresponding isotomic cevians we have
A B
1
= ,
A C
m
Orthopolar Triangles
The Simson–Wallance line and the pedal triangle
Consider the triangle ABC, and let M be a point situated in the triangle plane. Let
P, Q, R be the projections of M onto lines BC, C A, AB, respectively.
Theorem 1. (The Simson7 line8 ) The points P, Q, R are collinear if and only if M
is on the circumcircle of triangle ABC.
7 Robert Simson (1687–1768), Scottish mathematician.
8 This line was attributed to Simson by Poncelet, but is now frequently known as the Simson–Wallance
line since it does not actually appear in any work of Simson. William Wallance (1768–1843) was also a
Scottish mathematician, who possibly published the theorem above concerning the Simson line in 1799126
4. More on Complex Numbers and Geometry
Figure 4.10.
Proof. We will give a standard geometric argument.
Suppose that M lies on the circumcircle of triangle ABC. Without loss of generality
we may assume that M is on the arc BC. In order to prove the collinearity of R, P, Q,
it suffices to show that the angles B
P R and C
P Q are congruent. The quadrilaterals
P R B M and PC Q M are cyclic (since
BRM ≡ B
P M and M
PC + M
QC = 180◦ ),
hence we have B
PR ≡
B M R and C
PQ ≡ C
M Q. But
B M R = 90◦ −
AB M =
◦
90 − MC Q, since the quadrilateral AB MC is cyclic too. Finally, we obtain
BMR =
90◦ − MC
Q=C
M Q, so the angles B
P R and C
P Q are congruent.
To prove the converse, we note that if the points P, Q, R are collinear, then the
angles B
P R and C
P Q are congruent, hence
AB M +
AC M = 180◦ , i.e., the quadrilateral AB MC is cyclic. Therefore the point M is situated on the circumcircle of triangles ABC.
When M lies on the circumcircle of triangle ABC, the line in the above theorem is
called the Simson–Wallance line of M with respect to triangle ABC.
We continue with a nice generalization of the property contained in Theorem 1. For
an arbitrary point X in the plane of triangle ABC consider its projections P, Q and R
on the lines BC, C A and AB, respectively.
The triangle PQR is called the pedal triangle of point X with respect to the triangle
ABC. Let us choose the circumcenter O of triangle ABC as the origin of the complex
plane.
Theorem 2. The area of the pedal triangle of X with respect to the triangle ABC is
given by
area[ABC]
area[P Q R] =
|x x − R 2 |
(1)
4R 2
Remarks. 1) The formula in Theorem 2 contains the Simson–Wallance line property. Indeed, points P, Q, R are collinear if and only if area[P Q R] = 0. That is,
|x x − R 2 | = 0, i.e., x x = R 2 . It follows that |x| = R, so X lies on the circumcircle of
triangle ABC.
2) If X lies on a circle of radius R1 and center O (the circumcenter of triangle ABC),
then x x = R12 , and from Theorem 2 we obtain
area[ABC] 2
|R1 − R 2 |.
4R 2
It follows that the area of triangle P Q R does not depend on the point X .
The converse is also true. The locus of all points X in the plane of triangle ABC
such that area[P Q R] = k (constant) is deďŹ ned by
area[P Q R] =
|x x − R 2 | =
4R 2 k
.
area[ABC]129
This is equivalent to
|x|2 = R 2 Âą
4k
4R 2 k
= R2 1 Âą
.
area[ABC]
area[ABC]
1
If k > area[ABC], then the locus is a circle of center O and radius R1 =
4
4k
R 1+
.
area[ABC]
1
If k ≤ area[ABC], then the locus consists of two circles of center O and radii
4
1
4k
R 1Âą
, one of which degenerated to O when k = area[ABC].
area[ABC]
4
Theorem 3. For any point X in the plane of triangle ABC, we can construct a
triangle with sides AX ¡ BC, B X ¡ C A, C X ¡ AB. This triangle is then similar to the
pedal triangle of point X with respect to the triangle ABC.
Proof. Let P Q R be the pedal triangle of X with respect to triangle ABC. From
formula (2) we obtain
q−p=
Corollary 4. In the plane of triangle ABC consider the point X and denote by
the triangle with sides AX · BC, B X · C A, C X · AB. Then
A B C
area[A B C ] = area[ABC]|x x − R 2 |.
(7)
Proof. From formula (6) it follows that area[A B C ] = 4R 2 area[P Q R], where
P Q R is the pedal triangle of X with respect to triangle ABC. Replacing this result in
(1), we find the desired formula.
Corollary 5. (Ptolemy's inequality) For any quadrilateral ABC D the following
inequality holds:
AC · B D ≤ AB · C D + BC · AD.
(8)
Corollary 6. (Ptolemy's theorem) The convex quadrilateral ABC D is cyclic if and
only if
AC · B D = AB · C D + BC · AD.
(9)
Proof. If the relation (9) holds, then triangle A B C in Corollary 4 is degenerate;
i.e., area[A B C ] = 0. From formula (7) it follows that d · d = R 2 , where d is the
coordinate of D and R is the circumradius of triangle ABC. Hence the point D lies on
the circumcircle of triangle ABC.
If quadrilateral ABC D is cyclic, then the pedal triangle of point D with respect to
triangle ABC is degenerate. From (6) we obtain the relation (9).
Corollary 7. (Pompeiu's Theorem9 ) For any point X in the plane of the equilateral
triangle ABC, three segments with lengths X A, X B, XC can be taken as the sides of
a triangle.
Proof. In Theorem 3 we have BC = C A = AB and the desired conclusion follows.
The triangle in Corollary 7 is called the Pompeiu triangle of X with respect to the
equilateral triangle ABC. This triangle is degenerate if and only if X lies on the circumcircle of ABC. Using the second part of Theorem 3 we find that Pompeiu's triangle
of point X is similar to the pedal triangle of X with respect to triangle ABC and
√
CX
AX
BX
2R
2 3
=
=
=
=
.
(10)
PQ
QR
RP
α
3
Problem 1. Let A, B and C be equidistant points on the circumference of a circle of
unit radius centered at O, and let X be any point in the circle's interior. Let d A , d B , dC
be the distances from X to A, B, C, respectively. Show that there is a triangle with
9 Dimitrie Pompeiu (1873–1954), Romanian mathematician, made important contributions in the fields
of mathematical analysis, functions of a complex variable, and rational mechanics131
sides d A , d B , dC , and the area of this triangle depends only on the distance from X
to O.
(2003 Putnam Mathematical Competition)
Solution. The first assertion is just the property contained in Corollary 7. Taking into
account the relations (10), it follows that the area of Pompeiu's triangle of point X is
2
area[P Q R]. From Theorem 2 we get that area[P Q R] depends only on the distance
3
from P to O, as desired.
Problem 2. Let X be a point in the plane of the equilateral triangle ABC such that X
does not lie on the circumcircle of triangle ABC, and let X A = u, X B = v, XC = w.
Express the length side α of triangle ABC in terms of real numbers u, v, w.
(1978 GDR Mathematical Olympiad)
The last relation is equivalent to
(abc − x yz)(a − b)(c − x)(y − z) = 0
and finally we get abc = x yz, as desired.
Case 2. Point X is a vertex of triangle ABC. Without loss of generality, assume that
X = B.
Then the Simson–Wallance line of point X = B is the orthogonal line from B to
AC. It follows that B Q is orthogonal to Y Z if and only if lines AC and Y Z are parallel.
This is equivalent to ac = yz. Because b = x, we obtain abc = x yz, as desired.
Remark. Due to the symmetry of the relation abc = x yz, we observe that the
Simson–Wallance line of any vertex of triangle X Y Z with respect to ABC is orthogonal to the opposite side of the triangle X Y Z . Moreover, the same property holds for
the vertices of triangle ABC.
Hence ABC and X Y Z are orthopolar triangles if and only if X Y Z and ABC are
orthopolar triangles. Therefore the orthopolarity relation is symmetric.
Problem 1. The median and the orthic triangles of a triangle ABC are orthopolar in
the nine-point circle.
Solution. Consider the origin of the complex plane at the circumcenter O of triangle
ABC. Let M, N , P be the midpoints of AB, BC, C A and let A , B , C be the feet of
the altitudes of triangles ABC from A, B, C, respectively.
If m, n, p, a , b , c are coordinates of M, N , P, A , B , C then we have
m=
and the claim is proved.
Problem 2. The altitudes of triangle ABC meet its circumcircle at points A1 , B1 , C1 ,
respectively. If A1 , B1 , C1 are the antipodal points of A1 , B1 , C1 on the circumcircle
ABC, then ABC and A1 B1 C1 are orthopolar triangles.
bc
ca
ab
Solution. The coordinates of A1 , B1 , C1 are − , − , − , respectively. Indeed,
a
b
c
the equation of line AH in terms of the real product is AH : (z − a) ¡ (b − c) = 0.
bc
It sufďŹ ces to show that the point with coordinate −
lies both on AH and on the
a
bc
|b||c|
R¡R
circumcircle of triangle ABC. First, let us note that − =
=
= R,
a
|a|
R
hence this point is situated on the circumcircle of triangle ABC. Now, we show that
bc
the complex number − satisďŹ es the equation of the line AH . This is equivalent to
a
bc
+ a ¡ (b − c) = 0.
a
Using the deďŹ nition of the real product, this reduces to
bc
bc
+ a (b − c) +
+ a (b − c) = 0
a
a
or
abc
R2
a R2
(b − c)
+
a
−
−
a
bc
R2
a relation that is clearly true.
= 0,135
Figure 4.12.
It follows that A1 , B1 , C1 have coordinates
bc ca ab
, ,
, respectively. Because
a b c
bc ca ab
路
路
= abc,
a
c
c
we obtain that the triangles ABC and A1 B1 C1 are orthopolar.
Problem 3. Let P and P be distinct points on the circumcircle of triangle ABC such
that lines A P and A P are symmetric with respect to the bisector of angle B
AC. Then
triangles ABC and A P P are orthopolar.
Figure 4.13.
Solution. Let us consider p and p the coordinates of points P and P , respectively.
It is clear that the lines P P and BC are parallel. Using the complex product, it follows
Theorem. Consider M a point in the plane of triangle ABC, M the isogonal point
of M and A B C the antipedal triangle of M with respect to ABC. Then
area[ABC]
|ρ(M )|
|R 2 − O M 2 |
=
,
=
area[A B C ]
4R 2
4R 2
where ρ(M ) is the power of M with respect to the circumcircle of triangle ABC.
and consequently
ιa + βb + γ c = m,
since it is clear that = 0.
We note that ι, β, γ are real numbers and ι + β + γ = 1, so ι, β, γ are the
barycentric coordinates of point M.
Since
z O1 =
we have
where G is the centroid of triangle ABC. Assume that the circumcenter O of the
triangle ABC is the origin of complex plane.
12 Gottfried Wilhelm Leibniz (1646–1716) was a German philosopher, mathematician, and logician who
is probably most well known for having invented the differential and integral calculus independently of Sir
Isaac Newton. | 677.169 | 1 |
About this product
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This book is the first major study of advanced mathematical thinking as performed by mathematicians and taught to students in senior high school and university. Topics covered include the psychology of advanced mathematical thinking, the processes involved, mathematical creativity, proof, the role of definitions, symbols, and reflective abstraction. It is highly appropriate for the college professor in mathematics or the general mathematics educator. | 677.169 | 1 |
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Success in your calculus course starts here! James Stewarts His patient examples and built-in learning aids will help you build your mathematical confidence and achieve your goals in the course!
Table of Contents
Functions and Models
Four Ways to Represent a Function
Mathematical Models: A Catalog of Essential Functions
New Functions from Old Functions
Graphing Calculators and Computers
Review
Principles of Problem Solving
Limits
The Tangent and Velocity Problems
The Limit of a Function
Calculating Limits Using the Limit Laws
The Precise Definition of a Limit
Continuity
Review
Problems Plus
Derivatives
Derivatives and Rates of Change
Writing Project: Early Methods for Finding Tangents
The Derivative as a Function
Differentiation Formulas
Applied Project: Building a Better Roller Coaster
Derivatives of Trigonometric Functions
The Chain Rule
Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation | 677.169 | 1 |
Description of the book "Mathematical Physics: A Modern Introduction to Its Foundations":
The goal of this book is to expose the reader to the indispensable role that mathematics---often very abstract---plays in modern physics. Starting with the notion of vector spaces, the first half of the book develops topics as diverse as algebras, classical orthogonal polynomials, Fourier analysis, complex analysis, differential and integral equations, operator theory, and multi-dimensional Green's functions. The second half of the book introduces groups, manifolds, Lie groups and their representations, Clifford algebras and their representations, and fiber bundles and their applications to differential geometry and gauge theories. This second edition is a substantial revision PDF of the first one with a complete rewriting of many chapters and the addition of new ones, including chapters on algebras, representation of Clifford algebras and spinors, fiber bundles, and gauge theories. The spirit of the first edition, namely the balance between rigor and physical application, has been maintained, as is the abundance of historical notes and worked out examples that demonstrate the "unreasonable effectiveness of mathematics" in modern physics.
Einstein has famously said, "The most incomprehensible thing about nature is that it is comprehensible." What he had in mind was reiterated in another one of his famous quotes ePub concerning the question of how " ...mathematics, being after all a product of human thought, is so admirably appropriate to the objects of reality." It is a question that comes to everyone's mind when encountering the highly abstract mathematics required for a deep understanding of modern physics. It is the experience that Eugene Wigner so profoundly described as "the unreasonable effectiveness of mathematics in the natural sciences." Some praise for the previous edition: PAGEOPH [Pure and Applied Geophysics] Review by Daniel Wojcik, University of Maryland "This volume should be a welcome addition to any collection. The book is well written PDF and explanations are usually clear. Lives of famous mathematicians and physicists are scattered within the book. They are quite extended, often amusing, making nice interludes. Numerous exercises help the student practice the methods introduced...I have recently been using this book for an extended time and acquired a liking for it.
Among all the available books treating mathematical methods of physics this one certainly stands out and assuredly it would suit the needs of many physics readers." ZENTRALBLATT MATH Review by G.Roepstorff, University of Aachen, Germany "...Unlike most existing texts with the same emphasis and audience, which are ePub merely collections of facts and formulas, the present book is more systematic, self-contained, with a level of presentation that tends to be more formal and abstract. This entails proving a large number of theorems, lemmas, and corollaries, deferring most of the applications that physics students might be interested in to the example sections in small print. Indeed, there are 350 worked-out examples and about 850 problems...A very nice feature is the way the author intertwines the formalism with the life stories and anecdotes of some mathematicians and physicists, leading at their times. As is often the case, the PDF historical view point helps to understand and appreciate the ideas presented in the text...For the physics student in the middle of his training, it will certainly prove to be extremely useful."
THE PHYSICIST Review by Paul Davies, Orion Productions, Adelaide, Australia "I am pleased to have so many topics collected in a single volume. All the tricks are there of course, but supported by sufficient rigour and substantiation to make the dedicated mathematical physicist sigh with delight." EMS [EUROPEAN MATHEMATICAL SOCIETY] NEWSLETTER "This book is a condensed exposition of the mathematics that is met in most parts of physics. The ePub presentation attains a very good balance between the formal introduction of concepts, theorems and proofs on one hand, and the applied approach on the other, with many examples, fully or partially solved problems, and historical remarks. An impressive amount of mathematics is covered...This book can be warmly recommended as a basic source for the study of mathematics for advanced undergraduates or beginning graduate students in physics and applied mathematics, and also as a reference book for all working mathematicians and physicists ."
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MATH Documents
Showing 1 to 3 of 3
1
Mahabub Alam
AP Calculus BC
Mr. Husbands
06/04/16
Pythagorean Theorem Paper
Pythagoras is often referred as the first pure mathematician and is a master of
mathematics. Pythagoras was born on the island of Samos in 570 BC at Greece, and he
died in 495 B
Jaquasia Barrett Professor T. Thomas Math 136
ANNIE EASLEY
Jaquasia Barrett Professor T. Thomas Math 136
Annie Easley was known as a Computer Engineer and Mathematician. She has
accomplished many things during her lifetime. Although she was independently
Monique Marshall
meccuny mth202-001
WeBWorK assignment WWSet2 is due : 04/18/2010 at 11:59am EDT. The URL http:/bvans.net/mth202 takes you to the course website which contains a lot of important information. To find references for the related theoretical | 677.169 | 1 |
This teaching unit addresses four standards under the state of Florida's strand/Big Idea 3 in 6th Grade Mathematics: Write, interpret and use mathematical expressions and equations. In informal terms, the teaching unit will serve as the very first introduction for sixth grade math students to vocabulary, concepts and basic operations with "unknowns" in algebra.
Standards Covered/Assessed Within the Unit
This unit will specifically address four standards under the Big Idea 3. Key learning objectives within the four standards are outlined below:
* Standard 1 - M.A.6.A.3.1: Write and evaluate mathematical expressions that correspond to given situations. * Learning Objective 1 – Students can define and use correctly/conversationally the terms, "variable, constant, algebraic expression and evaluate". * Learning Objective 2 – Students can translate between words and math, creating mathematical expressions given situations expressed in language. * Learning Objective 3 – Students can evaluate single-variable expressions when given the value for the variable with 100% proficiency. * Standard 2 – MA.6.A.3.2: Write, solve and graph one and two-step linear equations and inequalities. * Learning Objective 1 – Students can determine if values for variables are or are not solutions to equations with 80% proficiency. * Learning Objective 2 – Students can solve one-step equations using inverse operations with 80% proficiency. * Standard 3 – MA.6.A.3.4: Solve problems given a formula. * Learning Objective 1 – Students can substitute values for multiple variables into a multi-variable equation, such as a commonly used formula, to derive the solution with 80% proficiency. * Learning Objective 2 – Students can find the area of a rectangle, given the formula and the measurements of the rectangle's sides. * Standard 4 – MA.6.A.3.5: Apply the commutative, associative and distributive properties to show that two expressions are equal. * Learning Objective 1 – Students can define the commutative, associative and distributive properties, and can verbalize which properties were used when viewing a set of equivalent expressions. * Learning Objective 2 – Students can use the distributive property to find the product of two numbers. * Learning Objective 3 – Students can use the commutative and associative properties to simplify numerical expressions using mental math.
Pretest
The Unit will begin with a pretest of students' background knowledge and familiarity with content in the standards being taught and assessed. The pretest and answer key follow.
Formative Assessments
Formative assessments are used so frequently in my classroom that they often "feel" like instruction. Following are just several of the "assessments in disguise" (formative assessments) that correspond to the learning objectives, standards and overall "Big Idea #3" of the Introduction to Algebra unit.
Formative Assessment 1
Standard #1 within Big Idea 3: MA.6.A.3.1: Write and evaluate mathematical expressions that correspond to given situations. Measureable objective being formatively assessed within standard #1: 1. Students can define and use correctly & conversationally the terms "variable, constant, algebraic expression and evaluate". a. Formative Assessment - Assessing this objective will include a "who am I?" exercise, where students will be given stickers on their backs containing the new vocabulary words (one per student). I'll give the kids about five minutes to walk around and talk to their classmates to try to find out "who they are" (which vocabulary word is on their backs). The students describing the term to the guessers are not allowed to use the term in their definition. Assessment occurs as I circulate & hear the conversations. Also – a...
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...Assessment in education can be carried out in numerous ways & also has many benefits for learners, teachers & educational establishments .According to Petty we assess "learner's needs for 2 main reasons to ensure they are on the right course & to support learners towards success on that programme" Petty.G. 2004.
There are 3 main types of assessment initial, summative & formative. Initial assessments aim is to create a profile of theIntroduction to Assessment Worksheet
Use the following website as well as the videos/PowerPoints in the assessment folder on Blackboard to assist you in completing the following assignment.
1. Match the term in the left column with the correct definition from the right column:
Enter correct LETTER here
Termteacher accept to measure the quality of schooling, then they must provide other, credible evidence that | 677.169 | 1 |
Descrizione:
Su questo libro:
Riassunto: All Math Words Dictionary is a dictionary designed for students of pre-algebra, algebra, geometry, and intermediate algebra in middle school and high school. It is written using the four 'C's of math writing: ? Concise: Definitions are compact, yet understandable. ? Complete: All words and phrases of interest to students of the target classes are included, plus a few just beyond the scope of the target classes. Tables of symbols and notation, formulas, and units of measurement, plus lists of properties of objects give the student all the information needed to understand the concepts and decipher many word problems. ? Correct: The definitions have been thoroughly reviewed for mathematical correctness. ? Comprehensible: The definitions are written to be understood by students in the target classes. Abundant illustrations aid in understanding. One of the difficulties many students experience in learning math skills has to do with the fact that an entire language has grown up around math. Students that acquire that language are successful in math studies. Students that do not acquire that language have serious problems with mathematics. This dictionary includes: ? over 2800 entries ? more than 190 mathematical notations defined ? in excess of 580 illustrations ? greater than 750 formulas, equations and identities ? conversion of measures ? pronunciation guide This dictionary is also available in a large print edition, a compact edition, and a dyslexia edition.
L'autore:
David McAdams made a successful career in software development for 35 years. Then, bored with computer programming, he turned his attention to how children learn mathematics. His books are the result of this effort.FD100009WM | 677.169 | 1 |
Math word problems are frequently used to gauge students' ability to decipher pertinent information and also to assess students' ability to use their analytical and mathematics skills to solve problems. Math word problems are often used to relate mathematics to real life situations. For instance, Physics applications, finance applications, ----------- Tanyaa Mod Edit - Please do not put advertising in your signature - Thanks
I am a firm believer in students learning to do their own research, so instead of a straightforward answer, I will give you a link for some resources. Make sure you go all the way to the bottom to view the additional links they provide!
I use to look outside my window and wonder why about a lot of things. Then I started studying math, particularly non-linear differential equations. I no longer wonder why about a lot of things. Not saying I know, just saying I no longer wonder why | 677.169 | 1 |
Fundamentals of Mathematics Test Bank Light shelf and edge wear Appearance of only slight previous use. Minor imperfections may exist. COAS Books, A Bookstore for Everyone. Buy with confidence-Satisfaction Guaranteed!
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About the Book
This text often a comprehensive review of all basic mathematical concepts and prepares students for further coursework. The arithmetic is presented with an emphasis on problem solving, skills, and concepts, with some introductory algebra integrated throughout the text. | 677.169 | 1 |
Common Core Math 1: Exponential Unit Test
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PRODUCT DESCRIPTION
This product contains an exponential unit test with an answer key. The test has a total of 37 problems that cover an entire exponential unit. The exponential unit covers basic exponent rules (multiplying, power to a power, dividing, zero, and negative), rational exponents, differences in linear and exponential functions, writing exponential function rules, graphing exponential functions, finding the rate of change given an exponential function, evaluating an exponential function, and solving exponential word problems with a calculator.
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Algebra and Hands On Equations
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Basic steps in performing Hands On Equations, algebra problems, along with important things to remember.
The product is on two full pages and should be copied as a two-sided document. It can then be cut, mounted on tag board or construction paper, and laminated.
Students can keep the card in their math folder, desk, or locker. Because I have several small groups throughout the day I keep the cards with our Hands On Equations materials and put two on each table | 677.169 | 1 |
This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity. Topics include sets, logic, counting, methods of conditional and non-conditional proof, disproof, induction, relations, functions and infinite cardinality | 677.169 | 1 |
Maths
Who is this course suitable for?
Apart from academic research, mathematics is vital for careers such as engineering, computer related industries, banking, financial trading, accountancy and insurance. Other possible careers include medicine, pharmacy, dentistry, meteorology, weather forecasting and teaching. The statistics module also provides good grounding for careers in physical or social sciences.
Entry Requirements
Course requirements are a minimum of GCSE Maths grade A*-C as well as being able to pass the entry level test at the beginning of the course. Students will be required to have a scientific calculator.
Course Content
AS – 3 modules – Core 1, Core 2, and Statistics 1
A2 – 3 further modules in addition to those studied for AS level – Core 3, Core 4, and Mechanics .
Assessed through examinations at the end of the academic year.
Opportunities and Enrichment
Weekly revision sessions after school on Thursdays. Drop in help sessions during lunchtime and staff non contact time. | 677.169 | 1 |
About this eBook
PREFACE TO THE REVISED EDITION The new model of teaching methodologies, content and Continuous and Comprehensive Evaluation CCE introduced by CBSE has been religiously followed and adhered to in the revised edition. Test Your Understanding exercises for immediate practice have been given almost after every topic. Exercises have been re-designed to include all types of questions, especially the Multiple Choice Questions MCQs and High Order Thinking Skills HOTS . Chapter Assessment based on CCE Pattern is provided at the end of each chapter to assess the child s understanding of the concepts given therein. Lab Activities as per CBSE guidelines have been incorporated at the end of the book. Formative Assessment is a part of the on-going scholastic evaluation of the child as per CCE . It is a kind of Unit Test A Self-assessment of the concepts taught in the class. Mental Maths exercises have been given to develop skills in rapid calculations. It also incorporates Maths Alert to warn against likely mistakes and misconceptions. Summative Assessment at the end of each term has been skillfully prepared incorporating all types of questions such as i Very Short Answer Type Questions Concept Review, True-False, Matching Questions . ii Short Answer Type Questions. iii Multiple Choice Questions MCQs . iv High Order Thinking Skills HOTS . AUTHORS A NOTE ON THE NEW REVISED EDITION The present edition is a thoroughly revised version of the last edition. The content has been thoroughly revised, reorganised and re-written at a number of places making the treatment simpler and easier to understand. In deference to feedback and suggestions received from learned faculty in schools, a sincere effort has been made to prune the exercises and reduce the number of problems without of course compromising the quality which remains the hallmark of this series. AUTHORS Disclaimer While the authors of this book have made every effort to avoid any mistakes or omissions and have used their skill, expertise and knowledge to the best of their capacity to provide accurate and updated information, the authors and S. Chand do not give any representation or warranty with respect to the accuracy or completeness of the contents of this publication and are selling this publication on the condition | 677.169 | 1 |
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Algebra One on One 4.0 Full and Algebra One on One 4.0 Full and free Crack a fiji.
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Algebra II
Module 1
Topic D
Lesson 40
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Algebra II Module 1, Topic D, Lesson 40
Student Outcomes
Students understand the Fundamental Theorem of Algebra; that all polynomial expressions factor into linear terms in the realm of complex numbers. Consequences, in particular, for quadratic and cubic equations are understood.
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Algebra II Module 1, Topic D, Lesson 40: Teacher Version (786.9 KB)
Algebra II Module 1, Topic D, Lesson 40: Student Version (577.93 KB)
Algebra II Module 1, Topic D, Lesson 40: Teacher Version (321.26 KB)
Algebra II Module 1, Topic D, Lesson 40: Student Version (225.93 KB)
Tags
Common Core Learning Standards
CCLS State Standard
N.CN.7
Solve quadratic equations with real coefficients that have complex solutions.
N.CN.9
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
A.REI.4
Solve quadratic equations in one variable.
Ratios and proportions and how to solve them (Algebra 1, How to) Complex or imaginary numbers - A complete course in algebra
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This worksheet reviews the major concepts from Algebra 1 (these are essential concepts for Algebra 2). These concepts are the first three sections in our Algebra 2 textbook.
Each student works with another student and they designate one person as partner A and one as partner B. They take turns: Partner A completes the problem as B watches and coaches if needed. Then, Partner B completes that next problem as Partner A watches and coaches.
When the students are done with the worksheet, I have them cut the paper vertically down the middle and they each get to keep their portion of the worksheet (or you can have them turn it in for a | 677.169 | 1 |
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ALGEBRA REPORT Algebra is a branch of mathematics in which letters are used to represent basic arithmetic relations As in arithmetic the basic operations of algebra are addition subtraction multiplication division and the extraction of roots Arithmetic cannot generalize mathematical relations such as the Pythagorean theorem which states that the sum of the squares of the sides of any right triangle is also a square Arithmetic can only produce specific instances of these relations for example 2 3 4 But algebra can make a purely general statement that fulfills the conditions of the theorem a b c Any number multiplied by itself is termed squared and is indicated by a superscript number 2 Classical algebra which is concerned with solving equations uses symbols instead of numbers and uses arithmetic operations to establish ways of handling symbols Modern algebra has evolved from classical algebra by increasing its attention to the structures within mathematics Mathematicians consider modern algebra to be a set of objects with rules for connecting them Algebra may be described as the language of mathematics HISTORY The history of algebra started in Egypt and Babylon where people learned how to solve linear and quadratic equations They also learned to solve indeterminate equations where several unknowns are involved Babylonians used basically the same procedures that are used today In Alexandrian times Diophantus book Arithmetica led the way in finding difficult solutions to indeterminate equations This early knowledge was embraced by the Islamic world where it was known as the science of restoration and balancing The Arabic word for restoration is al-jabru hence the word algebra In the 9th century the Arab mathematician Al-Khwarizmi wrote one of the first Arabic algebras a systematic expose of the basic theory of equations with both examples and proofs By the end of the 9th century the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities | 677.169 | 1 |
Math in Focus: Singapore Math: Solutions Key Course 1
This application also includes instructions for administering and scoring these early math assessments, as well as suggestions for using Early Math Fluency Probes in a school-wide RTI Universal Screening. Develop varied formative assessment practices and assess mastery of the same essential math concepts in different ways. The framework not only provides a way to introduce technology in mathematics education, but also provides a structure to discuss how students learn mathematics and whether some software is more meaningful and helpful for learning than others.
Pages: 168
Publisher: HOUGHTON MIFFLIN HARCOURT; 1 edition (June 27, 2011)
ISBN: 0547579101
Chain Rule in English: Table Method for differentiating nested functions (Math in English Book 1)
HSP Math: Assessment Guide Grade 4
Harcourt Math: Challenge Workbook, Grade 5, Teacher Edition
The Elementary Math Teacher's Book of Lists: With Ready-to-Use Patterns and Worksheets (J-B Ed: Book of Lists)
Several small globes were listed, and then #13. The cylindrical revolving Macerator's chart. [This seems to be a Mercator projection laid onto a cylinder for dispalay so that it could be rotated about the polar axis like a globe. In chapter three (pg 109) of The Autobiography and Other Memorials of Mrs Gilbert, Formerly Ann Taylor by Ann Taylor, edited by Josiah Gilbert Errors and Misconceptions in Maths at Key Stage 2: Working Towards Success in SATS. While there are presently excellent accounts of the history of algorithms such as [ Ch ], or the history of logic [ Da, Gr ], such texts do not focus on the needs of undergraduates and contain no curricular materials ready for classroom use Cambridge Primary Mathematics Stage 3 Teacher's Resource with CD-ROM (Cambridge International Examinations). You'll be introduced to typical scenarios in the classroom, such as teaching subtraction and regrouping, multi-digit multiplication, division by fractions, and the relationship between area and perimeter EZ Solutions - Test Prep Series - Math Practice - Advanced Workbook - PRAXIS (Edition: Updated. Version: Revised. 2015). Bring speakers into the classroom to explain how math has opened doors in their professions and careers. * Provide some activities that can be done independently or in groups based on student choice epub. Specific weaknesses are not defined within the context of math, therefore specific therapies are not prescribed to treat them Apex Maths 6 Pupil's Textbook: Extension for all through Problem Solving. And why a recent government study estimated that "the gap in reading deficiency [favoring girls] is ... equivalent to about 1 1/2 years of schooling" during a school career. By Laurie Willis and Laurie Willis,SUN STAFF January 14, 2002 Dontray Colbert figures he can be an engineer, if only he masters math. So the 15-year-old Edmondson/ Westside High School freshman has enrolled in an after-school program designed to help him calculate with comfort. "Right now I'm making C's, and my goal is B's and A's," said Dontray, who is taking algebra at Edmondson. "I want to try to bring my grades up."
The video game, that is built into the framework of the software, is the reward students earn by answering the mathematics exercises correctly. With well-designed review and practice software, the role of technology is to reinforce skills through an optimal sequence that assures certain predictable outcomes (Hooper & Hokanson, 2000). Technology used in this way is an extension of the behavioral approach How to Teach Math to Black Students. Read through some fun math books, such as Theoni Pappas books or puzzle books. Get to know some interesting math topics besides just schoolbook arithmetic. There are lots of story books (math readers) that teach math concepts - see a list here McDougal Littell Math Course 1 Georgia: Assessment Book Course 1. Make sure you keep the discovery topics independent from test based/performance-based topics. As much as you're tempted, don't connect the two. If the students connect them, acknowledge it briefly and keep moving. Keeping them separate keeps the "must get the right answer" attitude out of the discovery time Gcse Mathematics C for Ocr (Graduated Assessment) Stages 3 & 4. They later formulate a law or rule about that relationship based on their observations, experiences, inferences and conclusions read Math in Focus: Singapore Math: Solutions Key Course 1 online. See the Resources Pages if you need to order any of the items mentioned above. A VI teacher writes: I have a seventh grade braille student who will soon be studying a math chapter in a regular classroom Math in Focus: Singapore Math: Intermediate Virtual Manipulatives CD Grades 3-5. Elaborate: This phase of the 5 E's extends students' conceptual understanding and allows them to practice skills and behaviors. Through new experiences, the learners develop deeper and broader understanding of major concepts, obtain more information about areas of interest, and refine their skills. Evaluate: This phase of the 5 E's encourages learners to assess their understanding and abilities and lets teachers evaluate students' understanding of key concepts and skill development Windows on Literacy Fluent (Math: Math in Science): Rocks. The question was: "An army bus holds 36 soldiers. If 1,128 soldiers are being bused to their training site, how many buses are needed?" If you divide the first number into the second, you get 31 with a remainder of 12, meaning that 32 buses would be required to transport all the soldiers. Most students did the division correctly, but fewer than one out of four got the question right download Math in Focus: Singapore Math: Solutions Key Course 1 pdf.
Mastering Essential Math Skills Book 2 Middle Grades/High School New Redesigned Library Version with companion DVD
Math Warm Ups Ccss Series
Pre-Algebra (Mathskills)
Logic Posters, Problems & Puzzles (Grades 3-6)
More Minute Math Drills: Multiplication and Division, Grades 3-6
Harcourt School Publishers Math: Success For Ell Grade 6Ourt 2004
Early Childhood Math Centers
Houghton Mifflin Math Steps: Teachers Ed Level 3 2000
Nonetheless, questions should emanate from and be specific to worthwhile mathematical tasks. Many times these tasks can be embedded in contexts that encourage students to make important connections, e.g., connections to real life or to other ideas within mathematics, to broaden and deepen their understanding of the concept under study pdf. Cooperative learning is a specific type of group work that has the following defining elements: Positive interdependence. Learning activities are structured in such a way that students are required to depend on one another to successfully accomplish a task Calculus: Early Transcendentals, Books a la Carte Edition. Advanced speech recognition technology allows learners to engage in real-life dialogues and conversations. S. (Spoken Error Tracking System) technology which provides learners with immediate feedback and the opportunity to improve their pronunciation Harcourt Math: Harcourt Assessment System CD-ROM, 5-Computer Pack Grade 1. Teaching and learning of advanced mathematical ideas (algebra and calculus); improving teachers' ability to orchestrate and sustain inquiry-based and discussion-based instruction; technology in mathematics education The Handbook of Mathematics Teacher Education: Volume 3 (International Handbook of Mathematics Teachere Education, Volume 3). Tier 3 programs should include a data-based process for identifying specific causes of poor performance in mathematics, and individual interventions should be developed to target those specific deficits while monitoring both intervention-specific and generalized improvements in mathematics. Children who receive Tier 3 intervention may require specific training to learn how to apply learned skills under conditions that are required in the regular classroom Standards-Based Mathematics Assessment in Middle School: Rethinking Classroom Practice (Ways of Knowing in Science and Mathematics (Paper)). These students in Matamoros, Mexico, didn't have reliable Internet access, steady electricity, or much hope—until a radical new teaching method unlocked their potential. You can read a version of this story in Spanish here. Pueden leer una versión de esta historia en español aquí. José Urbina López Primary School sits next to a dump just across the US border in Mexico Experiencing School Mathematics: Teaching Styles, Sex, and Setting. Too much homework provides the students with unneeded stress, and can cause a lot of frustration and anxiety about math. The last point I would like to mention about teaching math and differentiating instruction is that it is important to make it fun. For some reason, students often have anxiety about math Getting Ready to Teach Math, Grade 3: For the New Teacher. In this way, all students are attentive and ready to work Teach Your Children Tables! The study of vector calculus and the calculus of space create an even greater challenge; however, I leave this to others Pre-Algebra: Grades 6-8: Squares, Square Roots, Pythagorean Theorem, and Irrational Numbers (Math Tutor Lesson Plan Series) (Volume 3). Antiderivatives, definite integrals, the Fundamental Theorem of Calculus, methods of integration, areas and volumes, separable differential equations. (No credit given if taken after or concurrent with Math 20B.) Prerequisites: AP Calculus AB score of 3, 4, or 5 (or equivalent AB subscore on BC exam), or Math 10A, or Math 20A. Introduction to functions of more than one variable Math Skills for the Redesigned SAT. In the second of this newsletter series, I write about Joy in Learning and Playing Games. My colleague Bob Albrecht has spent much of his professional career developing games for use in math education and a number of his games are available online (free). You may think that this short chapter is somewhat "off the wall" and outside the realm of preparing K-8 teachers of mathematics HSP Math: One-Stop Planner CD-ROM Grade 2. | 677.169 | 1 |
I would recommend this class to anyone who wants to learn Pre-Calculus. It is a great class that covers many topics. It may get hard at times but there is a great teacher and she will help you out.
Course highlights:
I learned how to graph the many basic functions and transformations, and how to graph sin, cos, and tan.
Hours per week:
3-5 hours
Advice for students:
My advice would be to do the weekly reviews no matter what. They are a great tool and if you do them correctly they will help your grade.
Course Term:Spring 2015
Professor:Kelley Felder
Course Tags:Great Intro to the Subject
Oct 10, 2016
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
It is a great class and the teacher is very helpful. You can ask her any question and she will answer it. She also has a lot of jokes and funny things to say to keep class interesting.
Course highlights:
In this class you will learn a variety of different topics. There's not one specific thing you will learn. Each chapter is something different. There is also a big project at the end of the year where you have to sum up all that you learned in the course in a video or other assignment.
Hours per week:
6-8 hours
Advice for students:
MAke sure you do your homework every night and also ask lots of questions if you are confused. | 677.169 | 1 |
Mathematics
1.
MATHEMATICS FUNDAE
2.
<ul><li>A game of numbers. </li></ul><ul><li>It is also a tool used in day to day life. </li></ul><ul><li>Needed in Physics and Chemistry also. </li></ul><ul><li>Helps in getting a good rank. </li></ul>What is Maths?
3.
HOW TO ENJOY MATHEMATICS <ul><li>Focus On Basics . </li></ul><ul><li>Try To Understand The Formulae. </li></ul><ul><li>Predict Its Application. </li></ul><ul><li>Match With The Applications As In Books. </li></ul><ul><li>Think WHY? </li></ul><ul><li>Write The Concepts Which You Are Going To Use On The Paper Before Starting The Solution Of Problem. </li></ul>
7.
Trigonometric Functions and Equations <ul><li>Widely used in other chapters of mathematics. </li></ul><ul><li>Requires lot of practice. </li></ul><ul><li>It has only formulae. </li></ul><ul><li>Try to see some pattern in the formulae and then learned. </li></ul>
17.
Radical Axis <ul><li>S – S' = 0 </li></ul><ul><li>Perpendicular to the line joining the centres. </li></ul><ul><li>Bisects the direct common tangents. </li></ul><ul><li>For 3 circles, taking 2 at a time, they are concurrent. </li></ul>
29.
Reflection Property : Ray passing through a focus, passes through the other focus after reflection.
30.
HYPERBOLA Asymptotes Tangent to the hyperbola at infinity <ul><li>Properties : </li></ul><ul><li>Difference between hyperbola and pair of asymptotes is constant. </li></ul><ul><li>Hyperbola and its conjugate hyperbola have the same asymptotes. </li></ul>x 2 - y 2 = 0 a 2 b 2
34.
FUNCTIONS <ul><li>Most Important Is Concept Of DOMAIN And RANGE . </li></ul><ul><li>Knowledge About Some Important Functions Like LOGARATHMIC, TRIGONOMETRIC, GREATEST INTEGER etc. </li></ul><ul><li>Focus On Domain And Range Of These Functions. </li></ul>
38.
LIMIT <ul><li>Condition For EXISTANCE OF LIMIT </li></ul><ul><li>Remember Some Important Expansions. </li></ul><ul><li>L'HOSPITAL RULE </li></ul><ul><li>Try To Simplify The Question If You Get A Hard And Tough Looking Problem. 95% Of Such Problems Becomes Easy After Simplification </li></ul>
39.
<ul><li>In Case When x Tends To Infinity In Algebraic /Algebraic Function Be Careful About Constants If Given In Question. </li></ul>
40.
CONTINUITY AND DIFFERENTIABILITY <ul><li>Concept Of LIMIT Should Be Clear Before Attempting The Questions Of Continuity. </li></ul><ul><li>Remember The Approach Of Continuity At End Points . </li></ul><ul><li>For Differentiability Of Function, Careful Where To Use Basic Funda And Where Direct Differntiation . </li></ul>
41.
<ul><li>Differentiability At End Points In A Closed Or Open Interval. </li></ul><ul><li>Differentiability Implies Contuinity . </li></ul>
42.
APPLICATIONS OF dy/dx <ul><li>ROLLE'S THEOREM . </li></ul><ul><li>LAGRANGE'S MEAN VALUE THEOREM. </li></ul><ul><li>Use Of dy/dx In Deciding The Nature Of Curve. </li></ul><ul><li>Try To Co-Relate Max. And Min. With The Help Of Graph. </li></ul>
45.
INDEFINITE INTEGRAL <ul><li>Get Some Important Results. </li></ul><ul><li>Use By Parts Method Whenever There Is Any Scope. </li></ul><ul><li>Rearrangement Is Your Motto . </li></ul><ul><li>Get The Approach Of Some Important Forms. </li></ul>
46.
DEFINITE INTEGRALS <ul><li>An Easy Version Of Indefinite Integrals. </li></ul><ul><li>Use Of Properties Are Very Useful. </li></ul><ul><li>No Need Of Gamma Function , Walli's Function etc. </li></ul><ul><li>Maximum And Minimum Value of Integral Are Very Helpful During Exams. </li></ul>
49.
DIFFERENTIAL EQUATION <ul><li>Order And Degree. </li></ul><ul><li>Degree Is Defined For Polynomial Equation Only. </li></ul><ul><li>First Step Is To Check If Equation Can Be Solved By Rearranging. </li></ul><ul><li>If Equation Is Of Form f(ax+by+c) , Then Solve It By Taking ax+by+c=t. </li></ul>
64.
PERMUTATION & COMBINATION 1. Permutation of n different things taking r at a time = n P r 2. Permutation of n things taken all at a time, p are alike of one kind, q are alike of 2 nd kind, r are alike of 3 rd kind, rest are different n! p! q! r! 3. Number of permutations of n different things taken r at a time, when each thing may be repeated any no. of times n r Circular Permutation <ul><li>When anticlockwise and clockwise are treated as different : </li></ul><ul><li>(n – 1)! </li></ul><ul><li>When anticlockwise and clockwise are treated as same : </li></ul><ul><li>(n – 1)! </li></ul><ul><li>2 </li></ul>
65.
COMBINATION 1. Combination of n different things taking r at a time : n C r 2 . Combination of n different things taking r at a time, when k particular objects occur is: n-k C r-k When k particular objects never occur : n-k C r 3. Combination of n different things selecting at least one of them : n C 1 + n C 2 + n C 3 + …………. + n C n = 2 n – 1 4. If out of (p+q+r+t) things, p are alike of one kind , q are alike of 2 nd kind, r are alike of 3 rd kind, and t are different, then the total number of combinations is : (p+1)(q+1)(r+1)*2 t – 1 5. Number of ways in which n different things can be arranged into r different groups is : n+r-1 P n
68.
VECTORS <ul><li>You Must Give More Concentration On Vectors As It Is Also Required In Physics. </li></ul><ul><li>You All Are Aware Of Simple Applications. </li></ul><ul><li>Linearly Dependent And Linearly Independent Vectors. </li></ul>
69.
<ul><li>Combination Of DOT And CROSS In Problems. </li></ul><ul><li>For Solving Vector Equation, Just Try To Simplify It And Use The Conditions Given. </li></ul><ul><li>If Sol. Requires r In Form Of Two Vectors a & b, Take </li></ul><ul><li>r = λ a + µ b + σ (a×b) </li></ul>
71.
3D-GEOMETRY <ul><li>Projection Of Segment Joining Two Points On Line. </li></ul><ul><li>Angle Between Two Lines. </li></ul><ul><li>Different Forms Of Straight Line Including Vector Form. </li></ul><ul><li>Perpendicular Distance Of A Point From Line. </li></ul><ul><li>Be Careful From Skew Lines . </li></ul>
72.
<ul><li>Shortest Distance Between Two Skew Lines(Its Better To Use Detailed Approach). </li></ul><ul><li>Equation Of Plane In Different Forms. </li></ul><ul><li>Angle Between Line And Plane.Angle Between Two Planes. </li></ul><ul><li>Just Try To Visualize Problem And Use Examination Hall To Put The Conditions In That Frame. </li></ul>
73.
<ul><li>Derive The Following Yourself: </li></ul><ul><li>Equation Of Plane Containing The Given Lines. </li></ul><ul><li>Shortest Distance Between Two Lines In All Three Forms. </li></ul><ul><li>Condition For Lines To Intersect. </li></ul>
75.
TIPS FOR EXAMINATION <ul><li>First Of All , Keep Faith In Yourself. </li></ul><ul><li>On Entering Examination Hall, Your Confidence Level Is Like That You Are JEE-2012 Topper But Don't Let This Confidence To Become Over-Confidence. </li></ul><ul><li>During Examination, First Make A Quick View On Q. Paper. </li></ul>
76.
<ul><li>Then Select The Question Which You Think ,You Will Solve Easily. </li></ul><ul><li>Don't Lose Your Confidence When You Are Not Able To Solve Any Problem. </li></ul><ul><li>Keep In Mind, Some Questions Of JEE Are Not Given For Solving But Are Given To Leave . You Must Develop A Sense About Selection Of Question. </li></ul> | 677.169 | 1 |
Algebra Word Problems
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In this packet, there are 8 word problems that can be used in a Pre-Algebra or Algebra classroom. The word problems translate into, one-step, two-step, and multi-step equations. This packet can be used to give notes or for extra practice. The packet is saved as a word document so you have the option to add or change the notes to fit your needs. It can be easily saved as a pdf then imported into ActivSoftware. ActivSoftware is the program that links to our Promethean Boards. I am a new shop | 677.169 | 1 |
Mathematics in the Real World is a self-contained, accessible introduction to the world of mathematics for non-technical majors. With a focus on everyday applications and context, the topics in this textbook build in difficulty and are presented sequentially, starting with a brief review of sets and numbers followed by an introduction to elementary statistics, models, and graph theory. | 677.169 | 1 |
Editors' Review: 04/14/2015 23:58:26 At Dummies.com, you can find the help you need to understand basic math, algebra, geometry, trigonometry, calculus, statistics, and probability.Looking for books about math? Browse our entire For Dummies online collection and find the perfect how-to book for you.Basic Math and Pre Algebra for Dummies was a great purchase. I was working through some online training at home and having a hard time with certain math problems in ...Basic Math and Pre-Algebra Workbook For Dummies, 2nd Edition helps take the guesswork out of solving math equations and will have you unraveling the mystery of …Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration ...10/6/2013 Download the ASVAB for Dummies free book here: Follow us on Google+: …6 Responses to How To: Knitting Pattern Tutorial for Hats with Math for Dummies (and everyone else too)Adding to the for dummies. The real spherical harmonics are orthonormal basis functions on the surface of a sphere. I'd like to fully understand that sentence and ...Dummies.com provides clear, focused how-to videos to make even the most complex topics easy to learn and understand. mathematics for dummies | 677.169 | 1 |
Factoring Review Day 3 for Geometry Students Spring 2014
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This worksheet is designed to replace a lecture on the topic of factoring trinomials. This is the third day of a three-day sequence. In my experience, students of very mixed and diverse ability levels all become successful with factoring trinomials with these three days of practice.
I start out class with a 15-minute "mini-lesson," giving my students some basic examples of what today's lesson will be about. Once the mini-lesson is over, I have them get to work within their groups on this worksheet | 677.169 | 1 |
The uses of programable calculators in the mathematics classroom are presented. A discussion of the "microelectronics revolution" that has brought programable calculators into our society is also included. Pointed out is that the logical or mental processes used to program the programable calculator are identical to those used to program any computer. A list and description of thirteen mathematical- and computer-related concepts that students can learn by working with programable calculators is presented. The report concludes with four additional uses of these electronic devices by teachers and pupils in the classroom. (MP) | 677.169 | 1 |
Hi, I may sound really dumb to all the math gurus here, but it's been 2 years since I am studying online monomial calculator , but I never found it appealing. In fact I always commit mistakes. I practise quite often, but still my grades do not seem to be improving.
Don't fret my friend. It's just a matter of time before you'll have no trouble in answering those problems in online monomial calculator . I have the exact solution for your math problems, it's called Algebrator . It's quite new but I assure you that it would be perfect in assisting you in your algebra problems. It's a piece of program where you can solve any kind of algebra problems easily. It's also user friendly and shows a lot of useful data that makes you learn the subject matter fully.
graphing, relations and conversion of units were a nightmare for me until I found Algebrator , which is really the best math program that I have ever come across. I have used it frequently through many math classes – Algebra 2, Algebra 1 and Remedial Algebra. Simply typing in the math problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my math homework would be ready. I highly recommend the program.
I have never been so confident with algebra before this. I will surely recommend Algebrator to all my friends.
P.W., Illinois
I use to scratch my head solving tricky arithmetic problems. I can recall the horrible time I had looking at the equations and feeling as if I will never be able to solve them but once I started with Algebrator things are totally different | 677.169 | 1 |
Technology Masters (Merrill Algebra 2 with Trigonometry)
It also features DoubleCheck, to ensure that all answers given are correct to 8 digits. Glencoe mathematics practice linear inequalities, mcdougal littell algebra 1 structure and method book answers, factoring on TI-83 Graphing Plus. Solutions in series of eigenfunctions, maximum principles, the method of characteristics, Green's functions, and discussion of well-posedness problems. When logarithms are used, the Law of Cosines is very inconvenient. Messerla is currently pursuing two different projects, one with Professor Gabriel Kerr and one with Professor David Auckly.
Graphing calculator lessons are scattered throughout the 7-program series and help students keep up with today's math education usage of graphing calculators The Civil Engineer's Pocket-Book. Tyler (1949) reasons that we must give students an opportunity to show they have acquired a behavior online. For each angle P or Q, there are six functions, each function is the ratio of two sides of the triangle. The only difference between the six functions is which pair of sides we use. a is the length of the side adjacent to the angle (x) in question. o is the length of the side opposite the angle. h is the length of the hypotenuse. "x" represents the measure of ther angle in either degrees or radians Student Solutions Manual for Dwyer/Gruenwald's College Algebra and Trigonometry. It's easy for me to keep track of what my son is studying at school. Thanks to the key on each odd number in the practice part and the illustration in the model part,I can step by step show my son how to study Trigonometry. This text was required for my university Trig class download. I guess 1984 didinfiltrateand immerseour society with thought control. The book was delivered as promised and on time. It was the book we desired, desperately needed, and it was the merchandise as advertised APPLIED ALGEBRA AND TRIGONOMETRY. I happened to debate with a friend during college days whether advanced mathematics is necessary for any veteran programmer. He said that programmers need only basic mathematical knowledge from high school or fresh year college math, no more no less, and that almost all of programming tasks can be achieved without even need for advanced math Algebra & Trigonometry Enhanced with Graphing Utilities, Math 148-150 Columbus State Community College. Note: Students must use a calculator capable of performing logarithms and exponents. A graphing calculator capable of plotting graphs is also desirable, but not necessary. To ease communication with a tutor, it is desirable that each student have an e-mail account Elements of Trigonometry, and Trigonometrical Analysis, Preliminary to the .... You can do this if you start with algebra in seventh grade. Your study sequence would then be: What Ought I to Be Learning Graphing Calculator Manual to Accompany Trigonometry, Graphs and Models, 1e? Some of the trigonometry questions are simply based on trigonometry formulae and are quite easy to crack while others may demand some trigonometry tricks. Thus IIT JEE trigonometry syllabus is a perfect blend of questions of all levels Given below is the trigonometry table that illustrates the values of the functions at different angles: It would be an added advantage if the aspirants could memorize all above trigonometry formulas but if not; they must at least grasp the major ones like: The graphs also constitute a vital component in Trigonometry download.
Students can even draw right on the flashcard! The lite version includes Flashcard Training for the iPhone, iPod Touch, and the iPad Algebra & Trigonometry with Applications. If the kid does not know the answer our intelligent tool will show the correct answer, it�s a learning curve! Downloadable Tutorials are easy to use, but you will need Acrobat Reader to use this format ( which can be freely downloadable at ) One can save the work sheet, even if required take print out or just simply e-mail it to teachers and friends Manual of Geometry and Conic Sections: With Applications to Trigonometry and Mensuration. Here is a short two and a half minute video which shows the Sine, Cosine, and Tangent Ratios. This next seventeen minute video goes through the Trig Ratios, and does working out several example triangles. In this first example, we are given a Right Triangle with the sides labelled, and some number values for these sides. We than use SOH-CAH-TOA to write the fraction and decimal values for Sin, Cos, and Tan for the 37 degree angle that is in the Right Triangle online.
An Introduction to the Theory and Practice of Plain and Spherical Trigonometry: And the Stereographic Projection of the Sphere : Including the Theory of Navigation ...
Using the formula for the sum of two sines ( above ): We get, by substituting in Equation 5.2: That is, Equation 5.1, which we wished to prove. ■ We can show these relationships are true by expanding the right-hand sides using the compound angle formulas, the result occurs immediately Trigonometry SparkCharts. Although its mathematics is simple and not calculus based this is the way theory was developed (and this is also touched upon in the book by Box, Hunter, and Hunter Modern elementary trigonometry,. With this information, we can use the tangent function to find the angle pdf. Just so I disappoint you now, no, there is no Tangent rule. There is a Cosine rule, however, which is a bit fiddly at times so I'll just go ahead and show you the two forms it's often seen in. As you can see, you need 3 sides or 2 sides and the angle between the sides to use the Cosine Rule. This works with all sides and angles: And that is that with finding sides and angles so there's just one more thing left Navigational applications of plane and spherical trigonometry. Trivedi (Author) Provides a comprehensive introduction to probability, stochastic processes and.. download Technology Masters (Merrill Algebra 2 with Trigonometry) pdf. We have the right to terminate your access to the Site if we determine that you have failed to comply with any of the provisions of these Terms and Conditions pdf. More in depth Trigonometry is sometimes taught in high download. However, we know that the trigonometric functions are analytic functions of a complex variable, so any proof in a restricted region can be extended to everywhere in the region of definition Spherical trigonometry,. A supplement to middle school geometry, Dr. Math Introduces Geometry has a light tone, clear layout, and humorous cartoons by Jessica Wolk-Stanley. The best part is that the explanations to math questions are accurate and clear. You'll find answers to dozens of real questions from students who needed help understanding the basic concepts of geometry College Algebra&Trigonometry. For example, if you have the square root of (1 - cosx)/2, then you can simply write this as sin(x/2). Applications of this and similar processes can be seen using all of the other trigonometric identities. Putting together the things that we now know about trigonometric functions and the Unit Circle, we can construct a very well-defined graphical image of the Unit Circle itself, shown below, which is considered by many to be the base from which all of trigonometry stems Algebra& Trigonometry: Graphs & Awl Tc Pkg. And, of course, no list of trigonometric relations could be complete unless the Laws of Cosines and Sines are mentioned. Trigonometry is a methodology for finding some unknown elements of a triangle (or other geometric shapes) provided the data includes a sufficient amount of linear and angular measurements to define a shape uniquely Judicial interpretation of political theory: a study in the relation of the courts to the American party system. | 677.169 | 1 |
Handbook on the History of Mathematics Education
4.11 - 1251 ratings - Source
This is the first comprehensive International Handbook on the History of Mathematics Education, covering a wide spectrum of epochs and civilizations, countries and cultures. Until now, much of the research into the rich and varied history of mathematics education has remained inaccessible to the vast majority of scholars, not least because it has been written in the language, and for readers, of an individual country. And yet a historical overview, however brief, has become an indispensable element of nearly every dissertation and scholarly article. This handbook provides, for the first time, a comprehensive and systematic aid for researchers around the world in finding the information they need about historical developments in mathematics education, not only in their own countries, but globally as well. Although written primarily for mathematics educators, this handbook will also be of interest to researchers of the history of education in general, as well as specialists in cultural and even social history.Each country could reform its mathematics teaching according to its own needs,
but it was recommended to establish as much cooperation as possible. ... Roman
numerals; and place value notation to the base five were all introduced before
the close of the third grade. ... Subtraction was presented as a search for a
missing addend and division as a search for a missing factor (Bundgaard 1969a€"
1972).
Title
:
Handbook on the History of Mathematics Education
Author
:
Alexander Karp, Gert Schubring
Publisher
:
Springer Science & Business Media - 2014-01-25
ISBN-13
:
Continue
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Study_Guide_for_MAS2103_MIDTERM_EXAM - existence of the...
STUDY GUIDE FOR MAS 2103 MID-TERM EXAM Be able to do each of the following: 1. Solve a system of equations by Gauss elimination and back-substitution. 2. Solve a system of equation by Gauss-Jordan elimination. 3. Use Gauss-Jordan elimination to find A-1 , starting with the augmented matrix [ A | I ]. 4. Identify the elementary row operations used in solving a system of equations. 5. Write down the elementary matrices E 1 , E 2 , …, E n such that E n , …, E 2 E 1 A = I . 6. Give A ! 1 = E n ... E 2 E 1 and A = E 1 ! 1 E 2 ! 1 ... E n ! 1 as products of elementary matrices. 7. Evaluate the determinant of a matrix A by a cofactor expansion along any row or column. Know the relationship between the determinant of a matrix and the
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Unformatted text preview: existence of the inverse of a matrix. 8. Find the norm of a vector. 9. Find the dot product and cross product between two vectors. 10. Find the cosine of the angle and sine of the angle between two vectors. 11. Show that two vectors are orthogonal. 12. Find the equation of the plane passing through a point where the plane has a normal vector. 13. Perform compositions of linear transformations, including reflection, projection, rotation, contraction, and dilation. 14. Justify whether a linear operator is one-to-one. 15. Find the standard matrix for a linear operator T from the images of the standard basis vectors....
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The use of mathematics is so woven into every area of everyday life
that seldom if ever does one fully realize how very helpless we would be
in the performance of most of our daily work without the knowledge of even
the simplest form of mathematics. Many persons have difficulty with relatively
simple computations involving only elementary mathematics. Performing mathematical
computations with success requires an understanding of the correct procedures
and continued practice in the use of mathematical manipulations.
A person entering the aviation field will be required to perform with
accuracy. The aviation mechanic is often involved in tasks that require
mathematical computations of some sort. Tolerances in aircraft and engine
components are often critical, making it necessary to measure within a
thousandth or ten-thousandth of an inch. Because of the close tolerances
to which he must adhere, it is important that the aviation mechanic be
able to make accurate measurements and mathematical calculations.
Mathematics may be thought of as a kit of tools, each mathematical operation
being compared to the use of one of the tools in the solving of a problem.
The basic operations of addition, subtraction, multiplication, and division
are the tools available to aid us in solving a particular problem. | 677.169 | 1 |
Math 311: Topics in Applied Math 1
5: Orthogonality
5.6: The Gram-Schmidt
Orthogonalization Process
Solution
2
5
and v2 =
.
1
10
They are linearly independent and hence form a
basis for the column space of A, although not an
orthonormal one.
The norm of
Math 311: Topics in Applied Math 1
5: Orthogonality
5.1: The Scalar Product in Rn
Summary
Let x, y
Solution
Since x and y be linearly independent, neither vector is a
multiple of the other. Hence the angle between them is
not a multiple of . Hence |cos |
Math 311: Topics in Applied Math 1
3: Vector Spaces
3.1: Denition and Examples
Notes
For us, the eld F is almost always R, the set of real
numbers. We speak of V as a real vector space.
(The eld could be C, the set of complex numbers.
Then V is a complex
Math 311: Topics in Applied Math 1
1: Matrices & Systems of Equations
1.5: Elementary Matrices
The system of n linear equations in n unknowns
Ax = b has a unique solution if and only if A is
nonsingular.
If A is nonsingular, a systematic method of
compu
Math 311: Topics in Applied Math 1
1: Matrices & Systems of Equations
1.4: Matrix Algebra
A square n n matrix A is invertible or
nonsingular if there is a matrix B such that
AB = BA = I. If such a matrix B exists, it is unique.
We denote this multiplicat | 677.169 | 1 |
Pearson Mathematics 7 Student Book
ISBN :
9781442522770
Publisher :
Pearson Education Australia
Author(s) :
Coffey, David
Publication Date :
7 Jun 2011
Edition :
1
Overview
Pearson Mathematics student book for Year 7 follows the Australian Curriculum for Mathematics. It has been strategically designed to attract maximum student engagement, develop a deep understanding of key concepts and skills, and to encourage inquiry and problem solving.
This student book provides you with extensive material, with a collection of maths games, investigations, problem solving tasks, revision activities, practice questions and technology explorations. Additionally, a mini, re-usable whiteboard has been provided, in the back of Pearson Mathematics student book for Year 7, to help encourage active participation from your students. All exercises within the student books are split into the Australian Curriculum proficiency strands: fluency, understanding and reasoning. You'll also find open-ended questions that encourage creative maths thinking. Accuracy has been observed by this series, with experienced teachers carefully checking every question within Pearson Mathematics - up to five times! | 677.169 | 1 |
Exemplar Problems Solutions for class 12
More about Exemplar Books
The main objective of the books on 'Exemplar Problems in Mathematics, Physics, Chemistry or in Biology' is to provide the teachers and students a large number of quality problems with varying cognitive levels to facilitate teaching learning of concepts in all subjects that are presented through the textbook for Classes 9th, 10th, 11th and 12th. It is envisaged that the problems included in these volumes would help the teachers to design tasks to assess effectiveness of their teaching and to know about the achievement of their students besides facilitating preparation of balanced question papers for unit and terminal tests.
The feedback based on the analysis of students' responses may help the teachers in further improving the quality of classroom instructions. In addition, the problems given in these books are also expected to help the teachers to perceive the basic characteristics of good quality questions and motivate them to frame similar questions on their own. Students can benefit themselves by attempting the exercises given in the book for self assessment and also in mastering the basic techniques of problem solving. Some of the questions given in the book are expected to challenge the understanding of the concepts of subject of the students and their ability to applying them in novel situations.
The NCERT Exemplar Problems Solutions included in these books were prepared through a series of workshops organised by the DESM for their development and refinement involving practicing teachers, subject experts from universities and institutes of higher learning and the members of the subject experts group of the DESM whose names appear separately. We gratefully acknowledge their efforts and thank them for their valuable contribution in our endeavour to provide good quality instructional material for the school system. | 677.169 | 1 |
Compressed Zip File
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0.12 MB | 2 pages
PRODUCT DESCRIPTION
This r squared creation document is a review of the topics that will be covered on the Chapter 10 Test. The materials covered in the review are standards based and are aligned to Chapter 10 of the Holt Algebra textbook. Concepts covered are how to simplify an already factored rational expression, how to simplify a rational expression that is not yet factored, how to multiply a rational expression, how to divide a rational expression, how to add and subtract rational expressions with like denominators, how to add and subtract rational expressions with unlike denominators, how to solve a rational equation by cross-multiplying, and how to solve a rational equation by multiplying by the Least Common Denominator.
The zip file contains the worksheet in both .doc format as well as in .pdf. If you would like to make any changes, you may need to download MathType | 677.169 | 1 |
Basic Mathematics Through Applications
9780321536310
032153660
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These lab- and classroom- friendly workbooks offer extra practice exercises for every section of the text, with ample space for students to show their work. The worksheets list the learning objectives and key vocabulary terms for every text section and provide extra vocabulary practice. | 677.169 | 1 |
set up and solve exponential growth and decay problems, given real world sets of data. | 677.169 | 1 |
Spectrum Grade 6 - 8 Algebra Paperback Book focuses on factors and fractions, equalities and inequalities, functions, graphing, proportion, interest while includes a complete answer key for better learning. It is designed with variety of activities to enhance skills on proficiency tests, problem solving and analytical ability. Math book has titles that offer grade-appropriate instruction and reinforcement in an effective sequence for learning success. Book features easy to understand directions while it is aligned to national and state standards. It is used at home, school, by parents, homeschoolers and teachers | 677.169 | 1 |
Polynomials I
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Practice with the basics of polynomials: listing terms, and identifying degree, name, and coefficient; identifying like terms; adding and subtracting terms, collecting like terms and distributing and simplifying. Also available as part of a combo | 677.169 | 1 |
Pre-Calc Exam Notes 138 - 138 Chapter 6 Additional Topics...
138 Chapter 6 • Additional Topics §6.2 programming language 3 (chosen here for simplicity) the decimal module can be used to set any level of precision. 4 Below we show how to get accuracy up to 50 decimal places using Heron's formula for the triangle in Example 2.16 from Section 2.4, by using the python interactive command shell: [1]: from decimal import * [2]: getcontext().prec = 50 [3]: a = Decimal("1000000") [4]: b = Decimal("999999.9999979") [5]: c = Decimal("0.0000029") [6]: s = (a+b+c)/2 [7]: K = s * (s-a) * (s-b) * (s-c) [8]: print Decimal(K).sqrt() 0.99999999999894999999999894874999999889618749999829 (Note: The bracketed line numbers are just command prompts, not part of the code.) Notice in this case that we do get the correct answer; the high level of precision eliminates the rounding errors shown by many calculators when using Heron's formula. Another software option is Sage 5 , a powerful and free open-source mathematics package based on Python. It can be run on your own computer, but it can also be run through a web
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MENTS OF MODERN ALGEBRA 7e, with its user-friendly format, provides you with the tools you need to get succeed in abstract algebra and develop mathematical maturity as a bridge to higher-level mathematics courses.. Strategy boxes give you guidance and explanations about techniques and enable you to become more proficient at constructing proofs. A summary of key words and phrases at the end of each chapter help you master the material. A reference section, symbolic marginal notes, an appendix, and numerous examples help you develop your problem solving skills. | 677.169 | 1 |
Effect of using mathematics teaching aids in teaching mathematics on the achievement of mathematics students
NAME: OGUNSHOLA STEPHEN ADESHINA
MATRIC NO : 081004105
SUPERVISOR: MR OPARA
CHAPTER ONE: INTRODUCTION
1.1 Background to the Study
Mathematics is the foundation of science and technology and the functional role of mathematics to science and technology is multifaceted and multifarious that no area of science, technology and business enterprise escapes its application (Okereke, 2006). Ukeje (1986) described mathematics as the mirror of civilization in all the centuries of painstaking calculation, and the most basic discipline for any person who would be truly educated in any science and in many other endeavours. Despite the importance placed on mathematics, researchers (Odili, 1986; Salau, 1995; Amazigo, 2000; Agwagah, 2001; Betiku, 2001; bioma, 2005; Maduabum and Odili, 2006; Okereke, 2006) had observed that students lack interest in the subject and perform poorly in it. Ukeje (1986) observed that mathematics is one of the most poorly taught, widely hated and abysmally under-stood subject in secondary school, students particularly girls run away from the subject. The West African Examination Council (WAEC) Chief Examiners [2003, 2004, 2005, and 2006] consistently reported candidates' lack of skill in answering almost all the questions asked in general mathematics. WAEC Chief Examiners [2003, 2005] further observed that candidates were weak in Geometry of circles and 3- dimensional problems. According to their reports, most candidates avoided questions on 3-dimensional problem, when they attempt geometry questions; only few of the candidates showed a clear understanding of the problem in their working. WAEC [2004] also reported candidates' weakness in Algebraic expression and word problems among others. Obioma (1985), Obodo (1993) and Okereke (2006) reported gender as a significant factor in mathematics...
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...Teaching is a challenging job, whether you are in charge of a traditional public school classroom or tutoring a single child in a language. Fortunately, there are many aids and accessories available to make the job easier. Though some cost money, others are available for free or easily manufactured from household items or on your computer.
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...The Impacts of TeachingAids usage on secondary school students' performance in biology: a study of some selected senior secondary school in Yewa South Local government area.
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Teaching is a demanding job that requires in-depth knowledge of subject content, age-specific pedagogy, and many varied skills such as patience, leadership, and creativity, just to name a few. If you are considering a career as a teacher, paraprofessional (teacher's aide), administrator, counselor, school nurse, or school Library Media Specialist, you may want to learn more about what the profession is all about. Click on any of the followingNigeria National Policy on education laid emphasis on the quality of education at the primary level and achieving the best educators and curriculum planner, searching for possible ways of making learning more internalizing to students. Teaching has passed from using traditional methods to scientific methods. (Avello 2005)
As policy makers and educational leaders today, we will discuss what we need to change in our education system, though some people see this | 677.169 | 1 |
Limits, Limits Everywhere: The Tools of Mathematical Analysis
Paperback | April 15, 2012, and series. Almost all textbooks on introductory analysis assume some background in calculus. This book doesn't and, instead, the emphasis is on the application of analysis to number theory. The book is split into two parts. Part 1 follows a standard university course on analysis and each chapter closes with aset of exercises. Here, numbers, inequalities, convergence of sequences, and infinite series are all covered. Part 2 contains a selection of more unusual topics that aren't usually found in books of this type. It includes proofs of the irrationality of e and p, continued fractions, an introductionto the Riemann zeta function, Cantor's theory of the infinite, and Dedekind cuts. There is also a survey of what analysis can do for the calculus and a brief history of the subject. A lot of material found in a standard university course on "real analysis" is covered and most of the mathematics is written in standard theorem-proof style. However, more details are given than is usually the case to help readers who find this style daunting. Both set theory and proof by inductionare avoided in the interests of making the book accessible to a wider readership, but both of these topics are the subjects of appendices for those who are interested in them. And unlike most university texts at this level, topics that have featured in popular science books, such as the Riemannhypothesis, are introduced here. As a result, this book occupies a unique position between a popular mathematics book and a first year college or university text, and offers a relaxed introduction to a fascinating and important branch of mathematics.
Pricing and Purchase Info,...
David Applebaum obtained his PhD at the University of Nottingham in 1984. After postdoctoral appointments in Rome and Nottingham, he became a lecturer in mathematics at Nottingham Trent University (then Trent Polytechnic) in 1987 and was promoted to reader in 1994 and to a chair in 1998. He
was Head of Department 1998-2001. He left NotI Approaching Limits1. A Whole Lot of Numbers2. Let's Get Real3. The Joy of Inequality4. Where Do You Go To, My Lovely5. Bounds for Glory6. You Cannot be SeriesII Exploring Limits7. Wonderful Numbers8. Infinite Products9. Continued Fractions10. How Infinite Can You Get?11. Constructing the Real Numbers12. Where to Next in Analysis? The Calculus13. Some Brief Remarks About the History of AnalysisFurther ReadingApendices1. The Binomial Theorem2. The Language of Set Theory3. Proof by Mathematical Induction4. The Algebra of NumbersHints and Selected Solutions | 677.169 | 1 |
By Theresa Gurl Implementing the Common Core State Standards through Mathematical Problem Solving: High School (First)
About the Book
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B00N4FJ7 Theresa Gurl Implementing the Common Core...
Publisher: National Council of Teachers of Mathemat | 677.169 | 1 |
Calculus I: Early Transcendental Functions / Edition 4
Overview define and demonstrate concepts and rules of calculus; and to design comprehensive teaching resources for instructors that employ proven pedagogical techniques and save time. The Larson/Hostetler/Edwards Calculus program offers a solution to address the needs of any calculus course and any level of calculus student. Calculus I: Early Transcendental Functions, 4/e, contains Chapters 1-6 of the full Calculus: Early Transcendental Functions, 4/e, text.Every edition from the first to the fourth of Calculus: Early Transcendental Functions, 4/e, has made the mastery of traditional calculus skills a priority, while embracing the best features of new technology and, when appropriate, calculus reform ideas. Now, the Fourth Edition is the first calculus program to offer algorithmic homework and testing created in Maple so that answers can be evaluated with complete mathematical accuracy. | 677.169 | 1 |
Extensive Mathematics For JEE 2014 Mains and Advanced
About the
Course
This course focuses on the key fundamental concepts in mathematics based on enhancing problem solving ability. The course focuses on stressing into new kinds of problem formation there by making you confident of solving any kind of problem thrown at you. The assignments and the lecture notes are provided at the end of every class.
Topics
Covered
Algebra: Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations.
Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots.
Arithmetic, geometric and harmonic progressions, arithmetic, geometric and harmonic means, sums of finite arithmetic and geometric progressions, infinite geometric series, sums of squares and cubes of the first n natural numbers.
Logarithms and their properties.
Permutations and combinations, Binomial theorem for a positive integral index, properties of binomial coefficients.
Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables.
Addition and multiplication rules of probability, conditional probability, Bayes Theorem, independence of events, computation of probability of events using permutations and combinations.
Trigonometry: Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles, general solution of trigonometric equations.
Relations between sides and angles of a triangle, sine rule, cosine rule, half-angle formula and the area of a triangle, inverse trigonometric functions (principal value only).
Analytical geometry:
Two dimensions: Cartesian coordinates, distance between two points, section formulae, shift of origin.
Equation of a straight line in various forms, angle between two lines, distance of a point from a line; Lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines, concurrency of lines; Centroid, orthocentre, incentre and circumcentre of a triangle.
Equation of a circle in various forms, equations of tangent, normal and chord. Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line.
Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal.
Locus Problems.
Three dimensions: Direction cosines and direction ratios, equation of a straight line in space, equation of a plane, distance of a point from a plane.
Differential calculus: Real valued functions of a real variable, into, onto and one-to-one functions, sum, difference, product and quotient of two functions, composite functions, absolute value, polynomial, rational, trigonometric, exponential and logarithmic functions.
Limit and continuity of a function, limit and continuity of the sum, difference, product and quotient of two functions, L'Hospital rule of evaluation of limits of functions.
Even and odd functions, inverse of a function, continuity of composite functions, intermediate value property of continuous functions.
Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions.
Derivatives of implicit functions, derivatives up to order two, geometrical interpretation of the derivative, tangents and normals, increasing and decreasing functions, maximum and minimum values of a function, Rolle's Theorem and Lagrange's Mean Value Theorem. | 677.169 | 1 |
ISBN 9789351410041
ISBN-10
9351410048
Binding
Paperback
Number of Pages
382 Pages
Language
(English)
Subject
Education
Prem Kumar's All in One Mathematics: A Complete Book in itself, published by Arihant Publications, is a comprehensive book for CBSE Class 10 students for Term II. It comprises of 9 chapters altogether, under titles like Quadratic Equations, Arithmetic Progressions, Circles, Constructions, Applications of Trigonometry, Probability, Coordinate Geometry, Areas Related to Circles and Surface Areas and Volumes. Every chapter contains complete explanation of all the concepts. This book has an exclusive segment of solutions for all the questions given in the NCERT textbook. Various concepts are explained with solved examples, figures, illustrations and useful notes. Every chapter has been divided into five parts focusing on learning & understanding the concepts, practicing different types of questions and summative and formative assessment of the concepts learned. A large number of MCQs, Short Answer Type, Long Answer Type and Value Based Questions are included. Each question is provided with step by step solution. Two solved and three unsolved sample question papers based on the syllabus of Term-II have been added at the end of the book, for a thorough practice for the examination. The book conforms to the latest syllabus and NCERT Textbook. | 677.169 | 1 |
ISBN: 9781486011193 Format: Paperback Number Of Pages: 0 Published: 11 June 2014 Country of Publication: AU Description: The Student Book provides an easy-to-use 'nuts and bolts' book at each year level. We've had the opportunity to build them from the ground up specifically for the Australian Curriculum, and we've developed it for the NSW syllabus. We found it important to create our resources from scratch for the Australian Curriculum to ensure we could include the latest research about how students learn Mathematics | 677.169 | 1 |
Browse related Subjects ...
Read More graphing calculators to solve problems more quickly. Perhaps most important-this book effectively prepares readers for further courses in mathematics.
Read Less
Very Good. 0534396437 Multiple copies available. No excessive markings and minimal highlighting. CD Roms, access cards/codes, and other supplemental materials may or may not be included based on availabilityPrecalculus: Functions and Graphs
by Earl William Swokowski
Not enough examples
This book moves too fast with poor examples. I also purchased the solutions guide which was also poor. The first example of a set of problems is shown with all stepps required to complete it but then the more difficult problems do not show the important steps so that I could see where I went wrong | 677.169 | 1 |
Elementary Algebra for College Students [With CD-158 013199457316.57New:
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$86 dynamic new edition of this proven series adds cutting edge print and media resources. An emphasis on the practical applications of algebra motivates learners and encourages them to see algebra as an important part of their daily lives. The reader-friendly writing style uses short, clear sentences and easy-to-understand language, and the outstanding pedagogical program makes the material easy to follow and comprehend. KEY TOPICS Chapter topics cover real numbers, solving linear equations and inequalities, formulas and applications of algebra, exponents and polynomials, factoring, rational expressions and equations, graphing linear equations, systems of linear equations, roots and radicals, and quadratic equations. For the study of Algebra. | 677.169 | 1 |
Description:
About this title:
Synopsis: "A rigorous and lively introduction . . . careful and lucid . . ."-- The American Mathematical Monthly. Excellent hardcover edition. This concise and idea-rich introduction to a topic of perennial interest in mathematics is written so clearly and lucidly, it is well within the reach of senior mathematics students. It covers mainly existence theorems for first-order scalar and vector equations, basic properties of linear vector equations, and two-dimensional nonlinear autonomous systems. Throughout, the emphasis is on geometric methods. Witold Hurewicz was a world-class mathematician whose untimely death in 1956 deprived the mathematics community of one of its leading lights. His contributions to dimension theory, homotopy and other topics are outlined by Professor Solomon Lefschetz in a prefatory article "Witold Hurewicz in Memoriam" included in this volume. Also included is a list of books on differential equations for those interested in further reading, and a bibliography of Hurewicz's published works. Unabridged Dover republication of the work originally published by MIT Press, 1958. Prefatory article "Witold Hurewicz in Memoriam" by Solomon Lefschetz. List of References. Index. 26 figures.
From the Back Cover:
A lively group of animal friends gather for hijinks in this compilation of three hard-to-find picture books: Johnny Crow's Garden, Johnny Crow's Party, and Johnny Crow's New Garden. Brimming with simple but charming rhymes ? "Even the Llamas Put on Pyjamas" ? pen-and-ink drawings, and luminous watercolor illustrations, the tales will captivate youngsters, especially 4- to 6-year-olds. Reprint of the Frederick Warne and Company editions.
Dover (2014) republication of the edition originally published by MIT Press, Cambridge, Massachusetts, 1958. See every Dover book in print at | 677.169 | 1 |
This book addresses the important role, now obligatory in HL, SL and Maths Studies, that calculators play in International Baccalaureate (IBDP) Maths courses and exams. It is based on the widely used TI-Series calculators. It uses a step-by-step approach and includes an excellent section on programming your calculator.
Author: Ian Lucas & Andy Kemp
Curriculum: SL/HL
Dimension: 257 x 297
Edition: 3rd Edition
Format: Print Book
Isbn 10: 190737468X
Language: English
Pages: 72 pages
Release date: March 31, 2013
Series: OSC IB Revision Guides for the International Baccalaureate Diploma | 677.169 | 1 |
homeschoolers, tutors, and students striving for classroom excellence. It features funky icons and lively cartoons by award-winning Santa Fe artist Sally Blakemore, a wraparound cover that closes the book with a string tie, and an "Algebra Wilderness" board game guaranteed to help students steer clear of "Negativeland" -- and have fun.This study aid meets the Curriculum and Evaluation Standards for School Mathematics developed by the National Council of Teachers of Mathematics....more
Paperback, 520 pages
Published
June 28th 1998
by Singing Turtle Press
(first published June 1998) | 677.169 | 1 |
Description
Make math homework a breeze! kmyAlgebra solves common formulas and functions encountered in Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus, and will now show step-by-step how it solved the quadratic formula using the values you entered.…..
Read more | 677.169 | 1 |
FORMATION EVALUATION
Lecture 7
Lithologies
1
Importance of Lithology
Without a good Lithology Interpretation we
cannot calculate
the correct porosity
OR the correct hydrocarbon
saturation
2
Importance of Lithology
We need to know lithology to determine ma
Lecture 6
Formation density and neutron
logs
Formation Density Log
Formation density logs
These are active gamma logs, i.e., there is a source of
gamma rays on the tool as well as 2 detectors
The density logs use a complex compensation to
remove mudcake e
0N1 Mathematics Lecture 12 Methods of Proof 64
Lecture 12
Methods of proof, continued
II Statements of the form (8x)(p(x) ! q(x)
An example is
For all x, if x > 2 then x2 > 4.
In practice such a sentence is often expressed as
If x > 2 then x2 > 4
where th
78
0N1 Mathematics Lecture 16 Polynomials
Lecture 16
Polynomials, continued
Division with remainder
If f (x) is a polynomial and ax + b is a linear polynomial we
can investigate whether ax + b is a factor of f (x) by long
division.
For example, if ax + b
0N1 Mathematics Lecture 1
7
Lecture Notes
Lecture 1
Sets
A set is any collection of objects, for example, set of numbers.
The objects of a set are called the elements of the set.
A set may be specied by listing its elements. For example, cfw_1, 3, 6 denot
0N1 Mathematics Arrangements for the Course
2
Arrangements for the Course
Aims of 0N1
A basic course in pure mathematical topics for members
of the foundation year.
Key ingredient: language of Mathematics, including specic use of English in Mathematics.
14
0N1 Mathematics Lecture 3
Lecture 3
Operations on Sets
A
A[B
A\B
B
Figure 2: Sets A and B and their intersection A \ B and union
A [ B.
Suppose A and B are sets. Then A \ B denotes the set of
all elements which belong to both A and B:
A \ B = cfw_ x :
23
0N1 Mathematics Lecture 4 Set Theory
Lecture 4
Set theory
The identities in (1)-(7) of the previous lecture are called the
laws of Boolean algebra. Several of them are obvious because * obvious = evident, self-evident
of the denitions of \, [ and 0 . T
0N1 Mathematics Exercises 7
1
1. Which of the following statements are true?
(i) 105 is divisible by 15.
(ii) 13 | 2971.
(iii) 99 is a divisor of 3960.
(iv) 75 and 192 have no common factors.
2. Find the greatest common divisors of the following pairs.
(i
0N1 Mathematics
Assignment 2
1.
(i) If A B is it necessarily true that B A?
(ii) If A B and A C is it necessarily true that B = C?
(iii) If A B and B C is it necessarily true that A C?
(iv) If A B is it necessarily true that A B?
(v) If A B is it necessar
0N1 Mathematics Assignment 4
1. Determine the truth value of each of the compound statements (a), (b)
and (c) given the following information:
(i)
(ii)
(iii)
(iv)
The statement Mr Black is taller than Mr Blue is true.
The statement Mr Green is shorter tha | 677.169 | 1 |
Find a Menlo Park AlgebraThe following topics from calculus 2 and calculus 3 can also be covered:
Partial differentiation, directional derivatives, total derivative, vector and scalar fields, tangent planes, matrix form of chain rule, line integrals, the gradient, multiple integrals, Green's theorem in the plane, surface... | 677.169 | 1 |
Math_105_Group_Project - Math 105 Group Project- I (50...
Math 105 Group Project- I (50 points) Brief Summary: You will work on this Group Project in groups of 3 or 4 (not more than 4 and not less than 2 students per group). On Tuesday October 20 you will turn in your group project for a grade. For this project, you will choose one of the WebQuests (see my website for these Webquests) and work on that. Specifics on what needs to be turned in are described in each of the WebQuests. For each of these WebQuests you will be using the concepts learned in this course and high school algebra. Purpose: The purpose of this assignment is to expose you to the mathematical concepts, and processes that are currently being used to answer some real world investigations, as well as exposing you to wide data and information available on the internet that can help you solve some interesting questions. Your own investigations, collection of information, decisions, and analysis will likely serve as much more powerful learning experiences than any mathematical lecture could. This assignment will also help you in becoming a self-sufficient learner by giving you independence in the choice of concepts and sources of information that you want to use in solving certain problems. It will also expose you one of the research-based, inquiry-oriented activity, WebQuests that you may plan to use in
This preview
has intentionally blurred sections.
Sign up to view the full version. | 677.169 | 1 |
2.1 Understand and use the concept of squares of numbers 2.2 Understand and use the concept of square roots of positive numbers. 2.3 Understand and use the concept of cube of numbers 2.4 Understand and use the concept of cube roots of numbers.
3 Understand and use the concept of ratio of three quantities to solve problems
14
15
.2011
9
3.2 Understand the concept of proportion to solve problems 5. RATIOS.2 Linear Equations in One Unknown 4. 3.2 Concept of Proportion to Solve Problems 5. 3. SMK Rusila
7 3.3 Concept of Algebraic Expressions 3.1 Understand the concept of ratio of two quantities 4.3 Concept of Ratio of Three Quantities to Solve Problem 6: PYTHAGORAS¶ THEOREM 6. LINEAR EQUATIONS 4.3 Solutions of Linear Equations One Unknown 5.3 Understand the concept of algebraic expressions. ALGEBRAIC EXPRESSIONS 3.03.02.1 Equality 4.2 Multiplication and Division of Two or More Algebraic Terms 8 3.Prepared by Noor Azam bin Nordin.2 Understand and use the concept of linear equations in one unknown.2011) 11 4.1 Understand the relationship betw een the sides of A right angled triangle 5.1 The Concept of Ratio of Two Quantities 13 5.1 Understand the concept of algebraic terms in two or More unknowns.3 Understand the concept of solutions of linear equations in one unknown
12
5.1 The Relationship Between the Sides Of a Right-Angled Triangle 6.4 Computations Involving Algebraic Expressions
3.2011 ± 23.02.03.
SCHOOL HOLIDAY (11.1 Understand and use the concept of equality.2 Perform computations involving multiplication and division of two or more terms
UJIAN SETARA 1 FORM 1 AND FORM 2 20.2011 ± 19. RATES AND PROPORTIONS 5.4 Perform computations involving algebraic expressions
10
4.1 Algebraic Terms in Two or More Unknowns 3. 4.
10.2.2 Understand and use the concept of scales for the coordinate axes 8.3 Understand and use the concept of arc of a circle to solve problems.2 Understand the concept of the intersection of two loci 7.1 Recognize and draw parts of a circle.05.2 Intersection of Two Loci 9.1 Understand and use the concept of coordinates 8.1 Perform constructions using straight edge (ruler and set square) and compass 6.
24
10.6.Prepared by Noor Azam bin Nordin.2 The Converse of Pythagoras¶ Theorem 16 7. OTI 1 (Mid Year Examination) FORM 1 AND FORM 2 (18.1 Parts of a Circle 10.2 Understand and use the concept of circumference to Solve problem 10.4 Understand and use the concept of midpoints.1 Constructions Using a Straight Edge and Pair of Compasses 17 8: COORDINATES 8.3 Distance Between Two Points in Cartesian Plane 8.1 Understand the concept of two dimensional loci SCHOOL HOLIDAY (3. SMK Rusila
6.4 Midpoint
19 20
23
10: CIRCLES I 10.3 Arc of a Circle
.1 Coordinates 8.6.1 Two Dimensional Loci 9.2 Scales for the Coordinates Axes 8.2011) 21 9: LOCI IN TWO DIMENSIONS 9.2 Circumference
25
10.2011 ± 26. GEOMETRICAL CONSTRUCTIONS 7.3 Understand and use the concept of distance between two points on a Cartesian Plane 8.2011) 22 9.05. Understand and use the converse of the Pythagoras¶ Theorem
18
8.2011 ± 18.
3 Understand the concept of surface area.2 Understand and use the concept of translations 11.5 Understand and use the concept of isometry.9.2011 ± 10. 12.4 Rotation
30
11.1 Concept of Data 13.1 Properties of Prisms.5 Understand and use the concept of area of sector of a circle to solve problems
27
10.4 Area of a Circle 10.3 Surface Area Chapter 13: STATISTICS I 13. SMK Rusila
26 10.Prepared by Noor Azam bin Nordin.2 Concept of Frequency 13.
.3 Reflection 11.7 Properties of Quadrilaterals 11. 12.5 Isometry 11. and Spheres 34 35 36 12.08.1 Understand the concept of data.4 Understand and use the concept of rotations 11. cylinders.1 Understand the concept of transformations 11.6 Understand and use the concept of congruence
29
11.4 Understand and use the concept of area of a circle to solve problems 10.2 Translation 11. 11.2011 ± 25. Pyramids. cones and spheres.1 Understand geometric properties of prisms.6 Congruence
31
UJIAN SETARA 2 FORM 1 AND FORM 2 21.7 Understand and use the properties of Quadrilaterals using concept of transformations
32
SCHOOL HOLIDAY (2.2 Understand the concept of nets.08. pyramids.2 Nets of Geometric Solids 12.2 Understand the concept of frequency 12. 13. Cones.2011 11.1 Transformation 11.9.2011) 33 12: SOLID GEOMETRY II 12.5 Area of a Sector of a Circle
28
11: TRANSFORMATIONS I 11.3 Understand and use the concept of reflections 11. Cylinders. | 677.169 | 1 |
This Volume II of Engineering Mathematics is a sequel to volume I of Engineering Mathematics written by the same authors. It is designed for the second-semester courses for undergraduate students of engineering. It is a lucid and effective presentation of linear algebra, its mathematical concepts, its tools and techniques that are essential to the solution of problems encountered most frequently in connection with engineering applications.
The book introduces students to the related concepts, rules and use of matrices and vector spaces in the solution of linear systems of equations which appear frequently as models of various engineering problems. Separate chapters are devoted to a thorough study of linear transformations, inner product spaces and eigenvalue problems in connection with matrices. The book makes liberal use of solved examples and provides plenty of exercises for homework in order that the students can apply these mathematical methods to the successful solution of real problems.
KEY FEATURES :
Presents a clear discussion of theory.
Incorporates more than 289 worked-out examples and 245 homework exercises | 677.169 | 1 |
Browse by
Algebra
Algebra is basically a fancy way to solve puzzles. Except you use mathematics, multiplication, square roots, addition, distributive property and lots of other confusing yet easy concepts. Algebra concerns the rules of operations and relations. Coming from this we get polynomials, equations, terms and stuff like that. In context, using variables, finding missing numbers, certain properties and confusing rules we get the helpful subject of algebra which is a branch of mathematics. Certain occupations use algebra daily. Architects make plans using algebra. The guys up in NASA planning expensive shuttle launches find algebra a necessity to accurate missions. Just by looking around you can see algebra all around you | 677.169 | 1 |
Product Media
Product Description
Beginning Algebra features a variety of exercises to help students become more comfortable using the distributive property, the order of operations, exponents, formulas, and other basic algebra skills. Aligned to Common Core. 60 pages, classroom-reproducible, softcover. Grades 6-8.
Product Information
Format: Paperback Vendor: Milliken Dimensions: 11.00 X 8.50 (inches)
ISBN: 0883359901 ISBN-13: 9780883359907 UPC: 0504870404 | 677.169 | 1 |
Course objectives:
A second course that is designed to give students a basic understanding
of Discrete Mathematics and its role in Computer Science. Computers handle
discrete data rather than continuous data. The course presents an
overview of some of the major theoretical concepts needed to analyze
this type of data.
Topics covered include:
Counting, sequences and sums, discrete probability, basic statistics,
recurrence relations, randomized algorithms. Material is illustrated
through examples from computing.
Textbook:
Additional material:
Discrete Mathematics and its Applications, by Rosen. This book
contains a large number of exercises as well as solutions.
Do not buy this book, because it is too expensive. We used
to use this book for COMP 1805, so you may still have an old
copy, or you may borrow a copy from a friend.
Important dates:
Assignment 1: posted September 21, due October 5
Assignment 2: posted October 5, due October 19
Assignment 3: posted November 9, due November 23
Assignment 4: posted November 23, due December 7
Midterm: Wednesday November 2 (in class)
Final exam: Wednesday December 14, 7-9pm
Grading scheme:
Assignments: 25%
Midterm: 25%
Final exam: 50%
Final exam:
Final exam: Wednesday December 14, 7-9pm.
Calculators are allowed.
The exam will be multiple-choice, 25 questions. It covers everything
we have done in class, assignments, and midterm. Below you find old
exams.
You are supposed to be familiar with the following topics
from COMP 1805: basic logical reasoning, sets and functions,
proof strategies (direct proof, proof by contradiction, proof
by induction), Sigma-notation for summations, basic graph
theory, Big-Oh, Big-Omega, Big-Theta.
You may take a look at Chapter 2 of the textbook and do some
of the exercises at the end of that chapter. | 677.169 | 1 |
irsch, Devaney, and Smale's classic Differential Equations, Dynamical Systems, and an Introduction to Chaos has been used by professors as the primary text for undergraduate and graduate level courses covering differential equations. It provides a theoretical approach to dynamical systems and chaos written for a diverse student population among the fields of mathematics, science, and engineering. Prominent experts provide everything students need to know about dynamical systems as students seek to develop sufficient mathematical skills to analyze the types of differential equations that arise in their area of study. The authors provide rigorous exercises and examples clearly and easily by slowly introducing linear systems of differential equations. Calculus is required as specialized advanced topics not usually found in elementary differential equations courses are included, such as exploring the world of discrete dynamical systems and describing chaotic systems. Classic text by three of the world's most prominent mathematicians Continues the tradition of expository excellence Contains updated material and expanded applications for use in applied studies | 677.169 | 1 |
Alfred S. Posamentier is Professor of Mathematics Education and Dean of the School of Education of The City College of the City University of New York. He is the author and co-author of many mathematics books for teachers and secondary school students. He believes that teachers should use methods and materials which build on their individual strengths, rather than on a prescribed "best way to teach" in general. This popular book is built on this philosophy, as is his book Tips for the Mathematics Teacher: Research-Based Strategies to Help Students Learn (Corwin Press, 1998), which complements this publication. Dr. Posamentier's recent book Problem Strategies for Efficient and Elegant Solutions: A Resource for the Mathematics Teacher (Corwin Press, 1998) elaborates on the chapter on problem solving, while his book 101 Great Ideas for Introducing Key Concepts in Mathematics (Corwin Press, 2001) provides a teacher with interesting alternatives to the traditional development of common topics and concepts in the curriculum.
After completing his A.B. degree in mathematics at Hunter College of the City University of New York, he took a position as a teacher of mathematics at Theodore Roosevelt High School in the Bronx (New York), where he focused his attention on the teaching process in general and the improvement of students' problem-solving skills in particular. He also developed the school's first mathematics teams (both at the junior and senior level) and established a special class whose primary focus was enrichment topics in mathematics and problem solving. He is currently involved in working with mathematics teachers both in the United States and internationally to help them better understand the ideas presented in this book, so that they can comfortably incorporate them into their regular instructional program.
After six years as a high school teacher, Dr. Posamentier joined the faculty of The City College (after having received his masters' degree there), and soon thereafter he began to develop inservice courses for secondary school mathematics teachers, focusing on practical classroom applications of educational research. In addition to the usual inservice offerings, these courses addressed such topics as the uses of new technology in mathematics instruction, efficient ways to teach weaker students, problem-solving strategies and the enrichment of mathematics through a variety of ways including, but not limited to, recreational mathematics.
Dr. Posamentier received his Ph.D. from Fordham University (New York) in mathematics education and since has extended his reputation to Europe. He is an Honorary Fellow at the South Bank University (London, England). He has been visiting professor at several European universities, including the Technical University of Vienna and the Humboldt University at Berlin, and a Fulbright Professor at the University of Vienna.
Dr. Posamentier is often cited for his outstanding teaching. He was named Educator of the Year ( I 993) by The City College Alumni Association and on May I , 1993 had a "Day" named in his honor by the City Council President of New York City. More recently, he was awarded the Grand Medal of Honor from the Federal Republic of Austria and the Medal of Distinction from the city of Vienna. In 1999 he was awarded the title of University Professor for Austrian Universities.
Now, after more than 32 years on the faculty of CCNY, he still exudes an ever-increasing energy and enthusiasm for mathematics and mathematics education. With his penchant for mathematics instruction, he has been especially concerned that during the recent years of mathematics teacher shortages, those who enter the classroom are as well prepared as possible. He enthusiastically believes that providing mathematics teachers with an appropriate knowledge base grid repertoire of teaching strategies enables them to fulfill an essential role in society: empowering our nation's youngsters to engage in the critical study of mathematics. This book is designed to provide the basis for this goal.
Jay Stepelman has had over twenty-five years of experience as a mathematics teacher and supervisor at George Washington High School and Yeshiva University High School in New York City. He is directly involved with on-the-job teacher training and curriculum development in a practical rather than a theoretical setting. His experience ranges from planning, supervising, and teaching regular, honors, and remedial mathematics programs to developing a unique, individualized algebra program that has found special success with inner-city students, pupils with emotional disabilities, youngsters with limited reading ability, and bilingual students.
He is currently teaching at Baruch College and New York City Technical College of the City University of New York.
Dr. Stepelman is the author of Milestones in Geometry (Macmillan, 1970) and the Algebra Tutor series for the Apple II family of computers (Learning Well, 1985). In addition, he is the author of Current Trends in Mathematics and State Mathematics Frameworks, published by Kraus International Publications' Curriculum Resource Handbook, 1993 edition. He has lectured at professional meetings of the National Council of Teachers of Mathematics, the Association of Teachers of Mathematics of New York State, and the New York City Mathematics Chairmen's Association. He has received numerous awards, including Exemplary Supervisory Service Award (New York City Division of High Schools), and has been named as the person most influential in the success of several Westinghouse Scholarship winners. He has also been honored by having many mathematics doctorate theses dedicated to him.
Dr. Stepelman has evaluated mathematics programs and teachers at various private schools. He has served with the Board of Examiners of the City of New York in the area of teacher certification. His academic training includes mathematics and education work at the City College of New York, Courant Institute of Mathematics, New York University, and Adelphi University.
Alfred S. Posamentier, The City College, The City University of New York
Beverly S. Smith, The City College, The City University of New York
Jay Stepelman
Essential for teachers in traditional or alternate route teaching programs, the revision of this highly successful book includes the latest changes in student assessment and how to use technology in the secondary mathematics classroom. With a focus on problem solving, the authors address how to craft rich and effective daily lesson plans, and how to use a variety of instructional tools and strategies to reach all students in a classroom. This must-have resource for your first mathematics classroom provides practical teaching methods for immediate use along with answers to typical questions readers have about teaching mathematics.
Practical Content for Today's Teachers
Chapter topics in section one include:
Challenges facing today's teachers
Importance of planning for instruction
Teaching more effective lessons
Role of problem solving in the mathematics classroom
Using technology to enhance mathematics instruction
Authentic assessment and grading strategies
Enriching mathematics instruction
Extracurricular activities in mathematics
Section two provides a collection of 125 enrichment units appropriate for the entire secondary school curriculum spectrum.
New to this Edition
Introduces the newest assessments for students based on the NCTM Standards.
Encourages the use of technology in the classroom and includes the latest developments in this rapidly changing field. | 677.169 | 1 |
An Investigation of Sequences: Linear vs Exponential
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
1.06 MB | 68 pages
PRODUCT DESCRIPTION
The following lessons were created for an 8th grade class. The lessons introduce students to arithmetic and geometric sequences. The first lesson introduces the students to the type of pattern and any new key vocabulary pertaining to the subject matter. Next, students get deeper into the sequences and derive explicit formulas. Students will not be given the abstract formulas at the beginning of these lessons. Instead they will work collaboratively to determine the patterns within the sequences to "create" the formula. This method gains the students understanding of the concept behind the formula and leads them to the abstract explicit formula for finding the nth term of a sequence.
Once students have completed their investigations they will then create x/y tables for their sequences, which will then result in ordered pairs for graphing! Students graph the ordered pairs for our sequences and will uncover in their investigations that arithmetic sequences result in linear graphs because they are linear functions. Also, that geometric sequences result in exponential changes, thus exponential functions. Students will understand the difference of exponential changes vs linear through a series of assignments and hands on activities.
This unit plan integrates literature and science and makes use of collaborative activities and oral presentations. This will assist students with further studies when calculating the partial sums and understanding the idea of limits when analyzing graphs | 677.169 | 1 |
McDougal Littell Geometry Ц Student Textbook (2007)
In Geometry, students will develop reasoning and problem solving skills as they study topics such as congruence and similarity, and apply properties of lines, triangles, quadrilaterals, and circles. They will also develop problem-solving skills by using length, perimeter, area, circumference, surface area, and volume to solve real-world problems. In addition to its geometry content, Geometry includes numerous examples and exercises involving algebra, data analysis, and probability. These math topics often appear on standardized tests, so maintaining familiarity with them is important. To help students prepare for standardized tests, Geometry provides instruction and practice on standardized test questions in a variety of formats – multiple choice, short response, extended response, and so on. Reading age for native speakers: High School students
NB: Neither Glossary nor Index nor Worked-Out Solutions are included in this pdf version of the textbook! | 677.169 | 1 |
Tree Diagram Generator is a mathematical tool, aimed at students, teachers, and examiners who wish to avoid the hassle of drawing complex probability tree diagrams by hand. With a real-time point-and-click interface, you can create any number of stems and
Calculate interest. This program is made for school, but you can use it for small office, too. You can calculate the time it takes to reach a value of money, the money you get after some years and so on. Furthermore you can save the calculated results in a
This is The Ultimate Calculator designed to suit all your needs. It has abilities to calculate the area, volume of all solids like cube, cuboid. Compute area and volume in the fill version.Requirements:Windows 95/98/2000/XPLimitations:See description
This software is still in early stages. This simple tool will allow you to convert numbers with many different built in formulas such as converting temperatures C to F , bytes to gigabytes, US to Metric and vice versa.Requirements:Windows 95/98/Me/NT/2000
This program quickly calculates the solutions to the quadratic equation. All you do is enter the values of A, B, and C. Hit calculate and then it gives you both answers. Requires Microsoft .Net Framework 1.1 to run.Requirements:Windows 2000/XP/2003 Server,
The4D Viewer displays 4 dimensional wireframes such as a hypercube and rotates them across any of the four axes. The program takes input from text files specifying 4-d points and lines and renders them into a stereographic (blue/red 3d glasses) perspective vi
This excellent program solves quadratic equations of the form ax^2 * bx * c = 0, with full customization and a section to take notes regarding your equations or variables. A must-have for any student or mathematician. New version fixes unneeded errors and
Celestial Navigation Program. Master's Mate handles most of the calculations involved in this traditional, but still widely used method of navigation at sea. The features are organised using a simple tab system and are integrated to allow continuous progre
From the developer: ""Kids effectively learn math due to quiz-oriented learning accompanied by friendly natural animals and/or worms. As a whole inclusive hardle, the comprehensive game that target to reach a holy cup deep in jungle is provided and as th | 677.169 | 1 |
Operads are mathematical devices which describe algebraic structures of many varieties and in various categories. Operads are particularly important in categories with a good notion of 'homotopy' where they play a key role in organizing hierarchies of higher homotopies.
The book presents the best thinking to date on teaching and learning undergraduate science and engineering. Focusing on the disciplines of astronomy, biology, chemistry, engineering, geosciences, and physics, this book is an introduction to strategies to try in your classroom or institution. The book is an essential resource for enriching instruction and better educating students.
Fork algebras are a formalism based on the relational calculus, with interesting algebraic and metalogical properties. Their representability is especially appealing in computer science, since it allows a closer relationship between their language and models. | 677.169 | 1 |
Showing 1 to 3 of 3
Chapter 2
2.1
Integration
STA2203
Indefinite Integration and Differential Equations
How can a known rate of inflation be used to determine future prices?
What is the velocity of an object moving along a straight line with known acceleration?
How can knowi
Chapter 1
1.1
Differentiation: Basic Concepts
STA2203
The Derivative
Calculus is the mathematics of change, and the primary tool for studying change is a procedure
called differentiation.
Slope and Rates of Change
Rate of change of a function
Rate of cha
Page 1 of 3
FHMM1314 Mathematics Business I
UNIVERSITI TUNKU ABDUL RAHMAN
ACADEMIC YEAR 2016/2017 (MAY 2016 TRIMESTER)
FHMM1314 MATHEMATICS FOR BUSINESS I
FOUNDATION IN ARTS
TUTORIAL 1A
Real Number
1.
Evaluate each of the following expressions.
(i)
(ii)
( | 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 |
SMP 11-16 Book Y1 (School Mathematics Project 11-16)
Retrieved from Mandel, S. (2006). The expectations that 21st century students face are completely transforming the approach that's being taken in classrooms across America. It's worth remembering: you might say to yourself, "Hey, I went to school, I remember what it was like, I know math, and I think that this new Common Core Math stuff is ridiculous." Reminder: you know absolutely nothing about teaching.
Even now, more than 35 years later, a great many preservice and inservice math teachers make and implement their own decisions about whether calculators are appropriate or not Designing Assessment for Mathematics. It is a teaching model that is based on seven components of teaching, seven components of instruction, and seven components of behavioral objectives SMP 11-16 Book Y1 (School Mathematics Project 11-16) online. Calling that constant D, find its exact value. Using your final solution to part c), make a reasonable argument that A(t) is never larger than 5. Notice that the above is still not an easy problem it wasn't supposed to be. However, the first part should be manageable for any student who understands what you have taught about exponential growth and decay Holt McDougal Larson Geometry: Teacher Edition Geometry 2008. Session 5: This session examines particular mathematical problems and shows how different representations of a situation can highlight different characteristics of a problem McDougal Littell Math Course 3: @Home Tutor CD-ROM. Occasionally, students will have personal problem. I discuss each such anomaly about fifteen- percent of the total -- with the individual instructor, and we come to some sort of consensus Bringing Science And Mathematics To Life For All Learners. Some textbooks have periodic mixed reviews, but that's an exception, not the rule. If textbooks interleaved problems systematically, students would always need to decide which formula or type of computation to use for the problem. One important consideration is that the standard format that most textbooks utilize might be so common because as teachers we like to have material organized into modular lesson plans Problem Solved: Bar Model Math Grade 4: Tackle Word Problems Using the Singapore Method. Teaching a 9 year old should not be rocket science. And you'd think that after 60 years, the educators would have figured it out by now. BTW, the people that put men on the moon were not exactly hamstrung by their elementary schooling in the 50's and 60's. But here's the thing, there's big bucks in modifying teaching materials year in and year out Math Expressions: Teacher Modeling Kit Grade 2.
At this site you'll find a database of materials designed to assist the instruction of a Chance course or a more standard introductory probability or statistics course. Math lesson plans in the following categories can be found on a New Brunsick Co-operative Learning Site online. This point was made by Brownell, 1928, pp. 199, 208-9; and also by Jean Lave, cited in Brown et al., 1989, p. 36. Unfortunately, students in this situation aren't being appropriately challenged (by the teacher or other students) to rethink and improve their initial ideas, so they probably won't learn as effectively as they would in a nontraditional classroom. For another teacher's description of how – and how well – this approach works, see Strachota, 1996, chap. 3 Harcourt Math Intervention Skills Teachers Guide with Copying Masters Grade 6. STT's have been replaced by cars and trucks and they can kill you just as quickly if not more so if you aren't paying attention Houghton Mifflin Harcourt Go Math New York: Student Reteach Workbook Grade 5.
This will be a random assortment of tips and tricks for a variety of programs and web applications. You can become a member of the "Secret Technology Club" by learning the secret technology handshakes Harcourt School Publishers Math West Virginia: Kindergarten Program Harcourt Grade K. After several problems, each student proposes a "rule" for how to add integers of various types. • Students to use spreadsheets or the data table on a graphing calculators to explore how a change in the function equation affects the data it produces Holt McDougal Larson Geometry: Teacher Edition Geometry 2008. They might even mathematically analyze their data. Find ways to personalize recognition, such as giving individualized awards for accomplishments download SMP 11-16 Book Y1 (School Mathematics Project 11-16) pdf. Following on Erik's point: teachers don't need authoritative declarations of The One True Method. What they get is training in the prevailing, for the time, "wisdom" of what works best most often. What they've not been getting for a very long time is the trust and simple courtesy to give the "most often" children the benefit of economy-of-scale — the aspect of the original public school model, called by some the "assembly line" model — and devoting the rest of the time to those students who are not served by it Castles: Fact and Fiction. Teachers must examine the area of mathematics in which the software will be used and develop lessons that promote the type of learning on which they will focus. General software often can be used for a wide range of grade levels and mathematical subjects pdf. S. curricula for students with learning disabilities, in Japan significant attention is given to important concepts such as place value. "Mathematics and Academic Diversity in Japan", Journal of Learning Disabilities, Volume 37, Number 1, February 2004, pp. 74-82 Word Problem Practice: Reproducible Grades 3-4. You have responsibilities to students: Don't discuss their individual grades in public, and don't compare the students to each other. It is one thing to say, "You're a very strong student"; quite another to comment, "I thought that Joe would be better than you [or vice versa], but..."
The ultimate goal of this approach is that learners learn how to learn rather than what to learn Eighth Grade Math Minutes. Everyone has their own unique way of gathering knowledge and storing it for later use Puzzles and Games for Math, Grade 1: Gold Star Edition (Home Workbooks). He was an academically proficient student back in the Ukraine; however, he is currently struggling in his English and social studies classes because his English language skills are limited. He dreams of becoming an orthopedic surgeon. Jasmine is a 16-year-old multiracial female student. Her father is a truck driver who travels away from home most of the time, and her mother works in the bakery at the local supermarket Modern Curriculum Press Mathematics, Level D (Grade 4). Ask kids to bring such interesting items for next day's lecture. Kids will love to do this and they will find math as the most interesting subject. Kids can even eat food items when lesson is over. Many kids find it difficult to learn math without seeing an example from real life. For example, to teach time to kids, you can use a small clock. Move its minute and hour hands to teach time to kids Harcourt School Publishers First Place Math: Program With Display Box Grade 6 (Without Manipulatives) (First Place Math 02). The book includes translations of Germain's letter and manuscripts, and ninety-four portraits, mosaics, artwork, facsimiles of handwritten manuscripts and letters, and figures Realizing Rigor in the Mathematics Classroom. Synthesis is involved when creating something new. Advanced students might be asked to create a new theory. Synthesis is tested via major projects, for example, which might be long term involving creativity and application of all that students have learned on a topic epub. Several useful card games, such as Close to 100 and Close to 0, build on the learner's sense and experience with numbers using 100 and 1000 as benchmarks (see box). Using only single-digit cards, deal players hands of six cards. numbers that add up to a number as close to 100 as possible Asymptotic Statistics (de Gruyter Textbook). The goal is to move them to be self-educating as soon as possible. Instructor's main job is to observe and mentor, no planned lessons or homework Focus on the worldview of America's founding fathers. Use the Notebook method (Research, Reason, Relate, Record) Goal is to Implant Christian character, virtuous leadership and a Biblical worldview As you continue homeschooling, you may move from one method to another until you find one you are comfortable with pdf. Members of the Division for Learning Disabilities receive printed color copies of the Alerts series as a benefit of membership Using the Standards - Number & Operations / Grade 6. Other examples include crafts and handwork that are integral to the curriculum from kindergarten through the grades. Children learn knitting and needlework and create many beautiful and useful objects. Coordination, fine motor skills, patience, perseverance, and imagination are schooled through this practical work. The Waldorf method of education, through the arts, awakens imaginative and creative powers, bringing vitality and wholeness to learning Teaching Math to First Graders. Mathematics teachers experience a schizophrenic situation when they attempt to utilize instructional resources that reflect the NCTM Standards and, at the same time, prepare students for the multiple-choice tests that make up many state assessment systems download. | 677.169 | 1 |
Calculator Limitations
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If you are obliged to follow a mandated math curriculum at your school, then I suggest you read no further because this site is probably not suitable for you. However, if you are looking for something that is different, yet might spark an interest in either you or your students, then it might be worthwhile for you to continue reading.
In 1992 I wrote an alternative curriculum for Pre Calculus students who enjoyed mathematics, but not the rigor of mathematics. I named the course Mathematical Investigations and it soon became recognized as the most enjoyable math course at our school for many of our students. One of topics that we delved into was titled Calculator Limitations and Extensions. The students' reactions were so positive that I made a similar presentation with the same title at the Wisconsin Mathematics Council Annual Meeting to 60 math educators. Of the five problems that were examined, none could be solved directly by use of a calculator without applying the appropriate mathematical concepts. Interestingly, many of the math teachers struggled with the problems as much as my students did.
Eventually, I wrote an enrichment booklet series called Ten Plus One for American Educational Products that contained similar type problems. Of the original six booklets, four remain in their catalog for sale, but the other two have been discontinued so I am free to sell them myself through TpT. This booklet is titled as Calculator Investigations | 677.169 | 1 |
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