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## Orientation

A “toolkit” is a resource; it has a lot of stuff in it. This toolkit will help you learn about math, which is different form learning math itself. We hope you will take a look at it now, but also come back to the toolkit when needed.  Our main hope is that you use it to learn, not more math, but more about math. Remember: you can use it to inspire students’ interest in math!

It has two compartments, following this brief orientation:

1. the development of mathematics,
2. different kinds of math people can learn.                                                                               (ratios and proportions, algebra, geometry, trigonometry, calculus).

Every subject in school—English, Spanish, Math, Biology—has content that students learn.

 English Math Parts of Speech Sentences Paragraphs Punctuation Spelling So on… Numbers Facts Math Ideas : equality, the idea of zero, variables Measurements Place Value So on…

## Section #1: The Development of Mathematics-Very Briefly

(numbers, 0, fractions, arithmetic [adding, subtracting, multiplying, dividing)

Math is born:  Humans thought about number and space long before they invented writing. It seems, though, that writing began its development about 6,000 years ago. And one of the first uses was to provide:

### A  Record of Things Counted.

This picture shows the total boulders and the five that the villagers took away from the total to make their fire pit.  It is a record of things counted.

In a sense, then, mathematics is a kind of writing. For thousands of years, the record of things counted was made in lots of different number systems. In some of these systems people also invented ways to add, subtract, multiply, and divide.

Now look at that depiction of the village boulders.

How could it be a subtraction equation?

The invention of zero (0), however, was a big improvement. It made adding, subtracting, multiplying, and dividing much easier and it was an important idea!

Another milestone in math is the work on geometry done about 2,400 years ago. Euclid (YOU-klid) wrote a textbook that remains a poster-child for what mathematical thinking looks like. Euclid’s approach is still an important part of most geometry classes in U.S. schools! Maybe you have heard the term Euclidean geometry. And there is much more. When you get a chance, explore the following resources online. There are also many other resources you can find online to help you learn about the development of math.

## Resource One

Two minute video without narration; so short it leaves out the really important idea of zero.

https://youtu.be/mBOShy-rxnw

## Resource Two

Seven-minute video with narration; it gives zero the attention it needs!

https://youtu.be/cy-8lPVKLIo

## Resource Three

To get more detail (in writing), explore: http://www.storyofmathematics.com/

and

Wikipedia’s entry http://en.wikipedia.org/wiki/History_of_mathematics

## Section#2: The Different Kinds of Mathematics Students Can Learn

This compartment includes an assortment of brief explanations about different kinds of math students can learn in school, plus a collection of brief videos and some short text resources to help you understand five different kinds of math:

#### Calculus

This list covers the kinds of math that many junior and high schools teach.

The following table shows the idea of 1 part in 8 parts as a fraction, decimal, percent, ratio, and in a proportion. They have a lot in common:

As you can see fractions, decimals, percents, ratios, and proportions are related ideas. Often these are referred to as rational numbers.

If a student is not comfortable enough to think with ratios and proportions,  they will find it difficult to understand algebra, geometry, or calculus.

QUESTION: Why can rations and proportions be so difficult?

ONE POSSIBLE ANSWER: Ratio and proportional thinking requires the mind to do something unusual which is identify the two following truths about 1/8 at one time:

1. 1/8 is a comparison of the part (1) and the whole (8ths)
2. It is also a number… 1/8
 Ordinary Life Task Connection to Ratios and Proportions Gardening Cooking Driving House-cleaning Construction Nursing

## Here are some online videos about ratios

These videos show how common and important ratios are in everyday life:

https://youtu.be/8SgyDCGRpRw

https://youtu.be/Pwv7oJ8jKig

And once again, we can turn to Wikipedia for more detail: http://en.wikipedia.org/wiki/Ratio

## Algebra

What is algebra?

You may have heard that “algebra is arithmetic with letters.” –What Algebra Looks like

“algebra is a web of statements that include unknowns and with algebra we use what is known to figure out what is unknown”- What Algebra Means

Algebra is difficult because it combines ratio thinking with webs of statements that include both known values and unknown values.

The unknowns are probably the most difficult part, as explained next.

##### What are “unknowns”?

The unknowns are all the x’s and y’s that appear in algebra.

Here are some equations that are statements with known and unknown components:

73 = 2x + 3
y = 5x – 2
y = x2 – 4x + 2

Algebra has developed methods for finding out what the unknowns are.

Here’s a simple statement with an unknown:

7x = 56

This means            7 times x= 56

Here are some other more challenging examples of algebra: x2 + y2 = 1; y = 3x2 + 2x + 6.

Algebra is the foundation for many, many fields but you rarely see it in its exact form in your day to day life.  One of the major benefits of algebra is it helps students become flexible thinkers. (from http://profkeithdevlin.org/2011/11/20/what-is-algebra/)

The difference between ALGEBRA and ARITHMETIC given by National Public Radio’s Keith Devlin, “the math guy:”

ARTITHMETIC———-REASON WITH NUMBERS (using calculations)

ARTiTHMETIC—–calculate a number—- work with the numbers you are given

ALGEBRA—-reason logically —————-find the value of an unknown number

By the way, the strange-sounding word algebra comes from an Arabic word, al-jabr, the name of the methods of dealing with unknowns described by the Muhammad al-Khwarizmi, who worked around the year 800 in one of the then-great centers of learning: Baghdad.

What is Algebra? (4 minutes, easy) https://youtu.be/QLFpK0ScmqA

Understanding the Vocabulary of Algebra (concentrated content, but it’s only 3 minutes long) https://youtu.be/BHtE3JyZ-UQ?si=h_ZtFy1ooHGakTUo

¿Qué es el Álgebra? (What is Algebra?) (in Spanish: could be useful if you work with a student whose first language is Spanish, or if you speak Spanish yourself! It’s 8 minutes long.) https://youtu.be/P8JArvOLMOA

Algebra (Wikipedia article)
http://en.wikipedia.org/wiki/Algebra

## Geometry

Do you remember that math is the science of number and space?

Geometry is the “space” part.

Humans (most of us) see and move in space, so we have a natural sense of what geometry makes up geometry.

Geometry is one of the oldest formal systems or models of mathematical thinking.

That famous and excellent old system was written down by Euclid 2,000 years ago.

Euclid’s system starts from a small number of things or axioms  that are accepted as true :

Axioms

1.A line can be drawn from a point to any other point.

1. A finite line can be extended indefinitely.
2. A circle can be drawn, given a center and a radius.
3. All right angles are ninety degrees.
4. If a line intersects two other lines such that the sum of the interior angles on one side of the intersecting line is less than the sum of two right angles, then the lines meet on that side and not on the other side. (also known as the Parallel Postulate)

The system works with these few axioms to build many, many other statements in a very logical, step-by-step process. The logic is so careful that these many, many other statements are considered proven: true for all time and space.

Actually, geometry developed further after the 1500s, but Euclid’s care for logic, for selecting starting points (axioms).

PROOF

The idea of “proof” is essential to mathematics. Mathematical proof is usually about deductions from a few basic ideas to a conclusion that must be true if the original ideas are true. And Euclid was the one who established this pattern firmly.

Geometry is traditionally taught in U.S. schools after algebra. Geometry is supposed to teach students one method of mathematical reasoning—reasoning about space. An appreciation of the sort of logic that geometry uses helps make students better thinkers in general.

The web has lots of resources on geometry:

What’s the point of geometry? (explains the spirit of “proof”) https://youtu.be/_KUGLOiZyK8

What is Euclidean Geometry? (12 minutes) https://youtu.be/H0pPq8M0KMw

Geometry (Wikipedia article; lots of information about geometry after Euclid)
http://en.wikipedia.org/wiki/Geometry

## Trigonometry

“Trigonometry” sounds a little like “geometry,” doesn’t it? (Another “-metry.”)

Trigon- Greek word for triangle                  Metron- Greek word for measure

#### Trigonometry has a single focus—the study of triangles.

-studies the relationships between the side lengths and angles in a triangle

So, what’s to study with triangles?  Triangles have been traditionally useful in a very practical field: navigation. That just sounds weird, right?

https://letstalkscience.ca/educational-resources/backgrounders/triangles-and-trigonometry

The relations of sides in right-angled triangles define a new class of mathematical relationships: trigonometric functions. These functions have names you might have heard before: sine, cosine, and tangent, for instance.

Are there everydayevery day uses and applications? The trigonometry functions are useful in just about everything that’s in your house: electric wiring and the power grid, televisions and computers, motors, the placement of satellite dishes, and so forth.

What is Trigonometry? (a little bit more technical than the text above)

https://youtu.be/T9lt6MZKLck

How to Learn Trigonometry (it’s math content, but gives an easy feel for how trig works)
http://betterexplained.com/articles/intuitive-trigonometry/

Trigonometry (Wikipedia article: it’s short and to the point, but more detailed than the others)
http://en.wikipedia.org/wiki/Trigonometry

These trigonometric functions have many applications in science, engineering, and, of course, navigation—

 Navigation use of Trigonometry Functions it estimates in what direction to place the compass to get a straight direction. With the help of a compass and trigonometric functions in navigation, it will be easy to pinpoint a location and also to find distance as well to see the horizon. Science Suppose you are considering having a tree removed for a better view out your window. You notice the contractor measuring the shadow of the tree. You ask her why she’s doing that, and she says that based on the time of day and the length of the shadow of the tree (14 ft), she can determine the height of the tree to be 14 / sin(16). https://study.com/academy/lesson/evaluating-trigonometric-functions-with-a-scientific-calculator.html Engineering Let’s supose we have a vehicle which drives on an incline road. https://x-engineer.org/trigonometry/

## Calculus

Calculus is usually the most “advanced” math taught in high schools. So, the very word scares lots of people. Like ratios and like algebra, and like geometric proof, calculus pushes students to think differently again. It has helped stretch the minds of generations of students. Like the rest of the math taught in high school, it was invented a long time ago (about the year 1700).

Calculus deals with change. One of its major contributions is the idea of the “limit.”

Web resources:

What is Calculus? (90 seconds) https://youtu.be/ismnD_QHKkQ

Calculus with Confidence (10 minutes) https://youtu.be/a9XNg-SOlUM

Calculus (Wikipedia article; mathy, but you will understand a lot anyhow)
http://en.wikipedia.org/wiki/Calculus

#### Module: Helping Students Do Math

• Unit 1: What is Math?
• Unit 2: Experiences with and Attitudes Toward Math
• Unit 3: Math Instruction: The Big Picture
• Unit 4: Learning from a Math Textbook
• Unit 5: Math and Technology
• Unit 6: Asking Questions That Help Students Think
• Unit 7: Problem Solving
• Unit 8: Math and the Common Core
• Unit 9: Number Sense (Math Facts, Memorization, and Practice)
• Unit 10: Visual Math
• Unit 11: Putting it All Together for Paraprofessionals
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