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Multiply by 1

A powerful mathematical tool

Wes Bruning

October, 2005

Contents

Introduction.......................................................................... i Make a 1...............................................................................1 Solve for a Variable .............................................................3 Divide by 1...........................................................................4 Introduction to Fractions......................................................5 Multiply Fractions................................................................7 Multiply by 1........................................................................8 Changing a Fraction's Denominator..................................10 Decimals to Fractions ........................................................11 Decimals to Percents..........................................................12 Multiply by 1 (Revisited)...................................................13 Comparing Fractions..........................................................15 Fractions to Percents ..........................................................17 Change Signs in a Fraction ................................................18 Simplify Fractions..............................................................19 Simplify Algebraic Expressions ........................................23 Simplify Algebraic Rational Expressions..........................24 Unit Conversions ...............................................................25 Unit Conversions (Revisited).............................................28 Long Division ....................................................................31 Negative exponents............................................................32 Negative exponents in the Denominator............................33 Roots in the Denominator ..................................................34 Complex Numbers .............................................................35 Imaginary Numbers in the Denominator ...........................36 Multiplication and Division Facts......................................37

Introduction

Wes Bruning Three Dog Night had a hit song in the sixties that stated that "One is the loneliest number that there ever was." I am not so sure I agree with that. I am more inclined to call One the Rodney Dangerfield of math: "it don't get no respect." The number one and, in particular, multiplying by 1 is one of the most useful techniques we have in math. As we shall see, much of what we do when solving a wide variety of problems involves multiplying by 1. As we begin our study of these topics, I would stress to the student that the key to mastering math and algebra is understanding basic principles. The principles we will use in our course of study are usually pretty simple. At first glance, the student's initial response often is Well, duh! Of course! The principle seems trivial. "How could something so easy be useful?" is often asked. But, think about E=mc2. This very simple formula (or equation) has very significant use: it was the basis for developing the atom bomb. Along the same lines, these simple appearing formulas are among the foundations of science and business: A=LW relates area to the length and width of a rectangle C=D relates a circle's circumference to its diameter through a constant value, D=RT relates distance traveled to rate of travel and time F=ma E=IR relates force, mass and acceleration in physics relates voltage, current and resistance in electricity

I=PRT relates the amount of interest to the amount of principle, the interest rate and time All of the examples above look pretty simple. But each one is a fundamental pillar of the science that uses it.

© Wes Bruning 2005, 2006. No portion of this document may be duplicated or otherwise distributed without the written consent of the author. All rights reserved.

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Multiplying by 1 has wide application to what we do and where we will go in our mathematics study. This book examines the technique as it is applied to different types of problems. By understanding the principle in depth then applying the principle we will be able to solve many types of problems. This allows us to not have to memorize the solutions of different types of problems. Actually, we do not even want to memorize the solutions of different types of problems. Memorizing all the solutions is impossible as there are an infinite number of different problems. Rather, we will understand a basic principles then learn how to apply it to a wide range of problem types. This, then, produces the solutions to the problems. About the methods The various topics presented herein may appear to be overly detailed. Each section examines a problemtype or procedure step by step in detail. No steps are skipped. Nor do they use shortcuts. I strongly advise the student to likewise learn the methods stepbystep and not use any shortcuts. Learn every method and procedure exactly like it is presented. Sometimes, the answer is obvious or can be readily guessed. However, bear in mind that we are NOT simply working to get right answers. We are learning methods that we can use when we cannot guess the answer. Therefore, learn every step of every method. No skipping steps. No shortcuts. A mathematical tool chest Multiplying by 1 is a step in building a mathematical tool chest stocked with mathematical problemsolving tools. Think terms of carpentry. There are quite a few different kinds of carpentry tools: hammers, nails, saws, chisels, squares, etc. If we have a hammer (a tool) and we understand the hammer's use (the principle) we can hammer any nail that comes along (the application of the principle). Of course it is not good enough to just know about the hammer. Using only a hammer we are limited in what we can build. Therefore we have a variety of tools about which we must learn. Then, through the appropriate use of each tool, we can build a wide variety of useful things: chairs, tables, houses, etc. We can think of our studies in math along the same lines. Just as we accumulated a carpentry tool chest, we can accumulate a mathematical tool chest. We add [mathematical] tools and we learn how to use them. Then, when we want to

© Wes Bruning 2005, 2006. No portion of this document may be duplicated or otherwise distributed without the written consent of the author. All rights reserved.

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solve a problem, we go to our tool chest and pull out the right tool and apply it appropriately. As you step through the various lessons, we will be identifying and examining a basic math principle (tools) then learning how to apply it. A few key points 1. Consider the carpentry example above. We are building a tool box of useful tools. However, if we owned the tools but always just told someone else how to use them, we would never really learn how to use the tools ourselves. We must use the tools to master them. Math is no different. We must use the math tools in order to really learn them. We must think through the problems and apply the tools we have in a logical sequence. This means thinking! Thinking is hard work. The main reason most people have problems with math is that they never really learned how to think through a problem. Students often try to memorize the process not understand the process. As stated above, there are an infinite number of problems. We cannot memorize them all. There is very little memorization in math (there is some, more on that below) but there is a huge amount of understanding required. Understanding is easier than memorization. And more useful, too! 2. There is some memorization and it is very important. The weakest math area most people have is the multiplication tables. If we do not know our multiplication tables we will always have trouble with math. Without knowledge of the multiplication tables we do not really understand how the numbers work. (If we do not understand something then it is essentially magic. We say a few incantations, wave our arms around, throw some eye of newt, bat wings and mouse tails into the cauldron and presto a magic formula appears!) If multiplication is magic then the rest of math will be magic also. So, the beginning math student must memorize the multiplication table from 1 x 1 (one times one) through 12 x 12 (twelve times 12). We should know these backwards and forwards. The multiplication tables are often discussed (but much less so now with the widespread use of calculators). The multiplication tables are 3x4=12, 5x8=40, etc. The student should be very familiar with the multiplication tables at the minimum through 12x12. However, in addition to the multiplication tables we must know the division table as well. What is the division table? No one ever talks about the division table but it is just as important as the multiplication table. The

© Wes Bruning 2005, 2006. No portion of this document may be duplicated or otherwise distributed without the written consent of the author. All rights reserved.

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division table is the reverse of the multiplication table: 4 divided by 2 is 2 (4 / 2 = 2), 45 divided by 9 is 5 (45 / 9 = 5), 72 divided by 8 is 9 (72 / 8 = 9), etc. We should know both division facts for every value in the multiplication table. (That is, for 3 x 9 = 27, we should know 27 / 9 = 3 and 27 / 3 = 9). Both multiplication facts and division facts can be found on page 37 of this document. Practice them until you know them instantaneously. An excellent way to learn something is to teach it to someone else. Corner your elementaryschool child, significant other (if he/she REALLY loves you they will help you here use the guilt option if necessary) or your dog, cat, gold fish, what ever, and drill them on the math facts. In the process of helping them learn you will also. 3. Once we have the multiplication table and the division table in our minds, we can significantly reduce our reliance on the calculator. For the beginning math student, being rid of the calculator is a necessity. Pushing buttons on a calculator is not learning math. It is a major waste of time that gives the illusion of learning math. The U.S. public school system has embraced the calculator with disastrous results. At this writing, the United States, the world power in economic and military strength is #26 in the world in math skills. Our nation's math literacy has declined significantly relative to the rest of the world over the last few decades. This author lays the blame in no small part to the widespread use of the calculator in basic arithmetic/math courses.1 While you are learning math, ditch the calculator and learn the math and division tables. There is a place for the calculator but learning math basics is not it. Now, let's get started.

I recognize that there are other contributors as well such as the breakdown of discipline in the public school system, inadequate teacher preparation, changing attitudes, the computer, etc. So please do not email me on this topic. The calculator destroys the understanding of underlying mathematics during calculations. As a result students do not develop a "feel" for how our number system works. This, then, creates barriers in our understanding of how math principles are applied and how those principles interact. Math becomes a magical process that results after a sequence of calculator buttons are pressed. In learning math, this is exactly the outcome one does not want.

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Make a 1

Wes Bruning Principle: "Make a 1" Any quantity divided by itself = 1

a = 1 a

( a cannot be equal to 0)

The principle stated above should be pretty obvious as

A fraction indicates a mathematical division operation.

4 =1 4

56 =1 56 857.3 = 1 857.3

3 7 is read "3 divided by 7"

We typically do not use the ÷ symbol. We would rather show it as a fraction or replace the division operation with a multiplication.

So, we can represent the number 1 in many, many different ways. Use the space below to write some of your own: Lets expand on this a bit. Notice that the top number (the numerator) and the bottom number (denominator) are the same. They do not have to be the same actual number! But they must be equivalent amounts. For instance, consider this: 7 days = 1 week This is true, is it not? Yes, it is.

Math Language In a fraction, the top number is the numerator and the bottom number is the denominator: Numerator Denominator

1

Seven days is the same as one week. (The numbers can be different as long as the units are correct. The units are the key to making the quantities equivalent.) These two amounts are the same. So, we could write:

7 days = 1 1 week

Notice here that the numerators and denominators are not the same number but that they are equivalent amounts or quantities. And that is really all that counts! (It's the units of measure! Pay attention to the units.) Write some more examples of 1 where the numbers are not the same but the numerators and denominators are equivalent. Considering the above example:

7 days = 1 1 week 1 week = 1 7 days

does it make any difference if we write:

No it does not! The quantity on the top (the numerator) is still equivalent to the quantity on the bottom (the denominator) and the fraction is still equal to 1. Similarly,

1 gallon = 4 quarts so

1 gallon = 1 4 quarts

and

4 quarts = 1 1 gallon 1 dollar = 1 4 quarters

4 quarters = 1 dollar

so

4 quarters = 1 1 dollar

and

When we are writing 1 we can put either value in the numerator (on top) and the other value in the denominator (on the bottom). We will use this idea later. Use the examples you made up, above, and rewrite them another way:

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Solve for a Variable

Wes Bruning We have seen how to make a 1: Divide a quantity by itself.

1 3 0.5 That is, = 1 , = 1 , and 2 =1 , etc. 1 3 0.5 2

We can use this idea to solve for an algebraic variable in an equation. Suppose we have 5 x = 15 . We are looking for some number (represented by x) that when multiplied by 5 is equal to 15. We should be able to rapidly come up with the solution that x must be equal to 3 because 5 i 3 = 15 . But we must have a better way of finding the unknown value because not all problems are this easy. So, let's make a 1 as the coefficient of x. We do this by dividing both sides of the equation by 5. So we would have

We must divide both sides of the equation by the same amount if the equation is to remain true. Consider: 8=8 If we divided only 1 side by 2 we would have

5 x 15 5 5 = or x = 3 and we recognize that = 1 so 5 5 5 5

x =3

4=8

This equation is not true. But dividing both sides of the equation by the same amount (2) keeps the equation true:

Dividing both sides of the equation by the coefficient of the unknown variable (in this case, x) will make the coefficient equal to 1. We have solved for the value of a variable by making a 1.

4=4

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Divide by 1

Wes Bruning Let's take a look at what happens if we divide by 1. We have a principle to consider:

Principle: "Divide by 1" Any quantity divided by 1 = itself

a = a 1

( a can be any number)

In this principle, the letter a represents any number. So you could substitute 4 or 279 or any other number you want for a. As examples:

4 1

=

4 279

Math language A quotient is the result of dividing two numbers. If we divide 6 by 3 the quotient is 2.

279 = 1

1 2 1

=

1 2

45.6

6 =2 3

45.6 = 1

Does the third example, above, seem a little strange to you? One half (½) divided by 1 is one half. As the principle states: Any quantity divided by 1 is equal to itself. That includes fractions and, as shown in the last example above, decimals also.

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Introduction to Fractions

Wes Bruning A fraction indicates a part of an entire amount or quantity.

1 For instance the fraction means 1 part of 4 parts. The number on the bottom, 4 the 4 in this case, is called the denominator and is the total number of parts the whole is divided into. The top number, 1 in this example, the numerator, is the number of parts you actually have.

Fraction = Numerator Denominator

Here, there are 4 parts that make the total or 1: The portion shaded below represents ¼ of the whole, or 1 part of the 4 parts: Write the fraction that represents 3 parts out of 4 parts?

3 (Answer: ) 4

Write the fraction that represents 4 parts out of 4 parts?

4 (Answer: ) 4

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So, if you had 4 parts out of 4 parts what portion of the whole would you have? (Answer: The whole thing or 1)

1 Consider: If you had of a gallon of water, what would you have relative to a 2 whole gallon?

Answer: The gallon of water is divided into 2 parts and you have an amount of water equivalent to 1 of the 2 parts.

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Multiply Fractions

Wes Bruning Multiplying Fractions: Multiply the numerators and multiply the denominators.

aic a c x = bid b d

(Neither b nor d can equal 0)

In the previous lesson we saw that a fraction represents a part of an entire amount or quantity. If we want to take a fraction of a fraction we multiply the two fractions together. Why, you ask, would we want to deal with a fraction of a fraction?

3 Consider the situation where you would want to divide of a gallon of gas 4 1 equally among four people. Each person would receive of the available 4 3 1 quantity of gas. We would do this mathematically by multiplying the by : 4 4 3 1 x 4 4

When we multiply fractions, we multiply straight across: numerator times numerator and denominator x denominator like this:

a c x b d

=

ac bd

Using numbers in our example above, this would look like this:

The "ac" and "bd" means that the value represented by "a" (some number) is multiplied by the value of "c" (some other number). Likewise for "b" and "d".

3 x1 3 1 3 x = 4 = 16 of a gallon of gas 4x4 4

What we have done is take ¼ of ¾ . We do this by multiplying the fractions.

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Multiply by 1

Wes Bruning In a previous lesson we learned how to make a 1. Now lets see how this might be useful. What can we do with the 1 we made? We will start by learning another principle: Multiply by 1 Principle: Any quantity multiplied by 1 is equivalent to the original quantity. This is stated mathematically as: a · 1 = a In this principle, a represents any quantity (we will stay with numbers for now). So you could substitute 4 or 279 or any other number you want for a. As examples: Let's put a couple of things together and see what happens. 4 279 ½ 45.6

The symbol "·" indicates multiplication or "times." The "·" is the same as the "x" many people use in multiplication. In algebra, "x" is commonly used for an unknown value. Using them both will be confusing. So we will use the "·". However, be careful, that the 4 · 1 does not morph into 4.1. Handwriting neatness counts!

·

· · ·

1 1 1 1

= = = =

4 279 ½ 45.6

3 5 5 Multiply times ( is equivalent to 1 so we 7 5 5 are multiplying by 1)

3 5 15 i = 7 5 35

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Our principle states: "Any number multiplied by 1 is equivalent to the original 5 number." Therefore, since all we did was multiply by 1 (i.e. ) we see that 5

3 15 = 7 35

3 What we actually accomplished here is to convert into an equivalent quantity 7 with a denominator of 35.

We have a method of changing a fraction's denominator by simply multiplying by 1!

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Changing a Fraction's Denominator

Wes Bruning We will use the "Multiply by 1" Principle to change the denominator of fractions.

5 1 Let's change the fraction into the equivalent number of ths. 8 16 5 So we want to change the denominator of the (that is, the 8) into a 8 denominator of 16.

We notice that if we multiplied the 8 by 2 the product is 16 ( 8 · 2 = 16 ).

5 2 So, we multiply like this: · 8 2 2 Notice that we are multiplying by . Why? (Answer: Because we can only 2 2 multiply by 1 and not change a quantity's value and = 1). 2

When we multiply fractions we multiply straight across, so:

5 2 10 i = 8 2 16 5 10 We have converted into the equivalent number of . These two fractions 8 16 represent the same value and are equal.

Let's try a few more. Convert each of the fractions in column (1) to an equivalent fraction with the denominator shown in column (2). Multiply by "1" to obtain the solution. Show your work in column (3) (1) (2) (3) Equivalent value

3 4 5 6

? 12 ? 24

7 8 4 11

? 64 ? 55

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Decimals to Fractions

Wes Bruning We multiply by 1 to convert decimals to fractions. Consider 0.125. Multiply 0.125 by

1000 1000

Note that the number of decimal places (in this case 3 decimal places for the 0.125) tells you the power of 10 to use when making your 1. For 3 decimal places we would use 103 or 1000 ( 10 i 10 i 10 ). Then we have:

0.125 1000 125 i = 1 1000 1000

From here we simplify:

125 1 i 125 1 = = 1000 8 i 125 8

In the above simplification, the student may not recognize that the greatest common factor between 125 and 1000 is 125. The simplification may involve a series of steps such as first factoring out a 25 then a 5. The good news is that no matter how many simplification steps you use, the final answer will be the same.

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Decimals to Percents

Wes Bruning We multiply by 1 to convert decimals to percents. "Per cent" means "per hundred." The word "cent" show up in the English language when 100 is meant: century = 100 years; 100 cents = 1 dollar; there are 100 centimeters in a meter, etc. The word "per" is mathematics code meaning "divide." So when we say "per hundred" we are really saying (after the decoding) divide by 100. Along the same lines, we define the percent symbol (%) as the fraction Thus, % =

1 . 100

1 100 100 100

0.523 can be changed to a percent by multiplying the decimal by 1 =

0.523 100 52.3 i = 1 100 100

We recognize that

52.3 1 = 53.2 i 100 100 1 then, 100

As stated above % =

53.2 i

1 = 53.2% 100

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Multiply by 1 (Revisited)

Wes Bruning Now we can convert a fraction to a different denominator. Lets see how to use this idea. Math Language Numerator Denominator

The Numerator is the number on top of the fraction; the Denominator is the number on the bottom of the fraction.

Is

4 16 ? = 9 36

If so, how can we demonstrate that it is?

We must represent both items 4 16 (the and the ) with a common 9 36 reference. To do this, both fractions must have the same denominator. That 1 would be a 36. We will determine how many ths each fraction represents. 36 Why did we pick 36?

4 Lets look at the . If we were to multiply the denominator by 4, we would then 9 have a fraction with a bottom number of 36. So, this fraction starts to look like the other fraction that also has a denominator of 36. Both fractions would be represented with a common reference (or common denominator). In amount, 1 both fractions would represent the number of sized pieces. 36

But, we cannot just arbitrarily multiply only one part of a fraction by some value without changing the value of the fraction. We have to end up with an equivalent amount. So, to not change the value of the number, we will multiply 4 by 1. (Why can we do this?) But we will pick a special 1: . (Principle: Any 4 number divided by itself is 1.) So if we multiply

4 4 4 by (where = 1) we get: 9 4 4

4 4 16 i = 9 4 36

When we multiply fractions, we multiply the numerators together and we multiply the denominators together

So then, the two numbers are equivalent even though they do not look the same.

4 How did we do this? We multiplied by 1 ( ). Observe that the "1" was 4 carefully selected. How did we select the one? We found a number we could use

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to multiply the smaller denominator (the bottom number of the fraction) to get the larger denominator. Then we used that number to multiply both the top and the bottom of the fraction [to make it equivalent to 1]. Lets look at another example:

2 13 Is = ? 3 18

Well we must represent both fractions with some common denominator. This 1 would be the number of ths that each fraction represents. The denominator 18 1 for both fractions must be 18 for us to compare them. Why ? 18 (Important point: the denominator for both fractions will always be equal to or larger than the larger of the two original denominators). So, how do we determine how many

1 2 ths are the equivalent of ? 18 3

2 We would multiply the by 1. What 1 would we pick? We would pick a 1 that 3 2 would change the denominator of the (the 3) to an 18. Of course, we would 3 multiply the 3 times a 6. But our rule states that we can only multiply by 1.

So we would have to multiply both the numerator and the denominator by 6. 6 Thus we would multiply by . 6

2 6 12 · = 3 6 18 Then looking at the original problem: Is The answer is "No". 2 12 13 not = 3 18 18 . 2 13 ? = 3 18

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Comparing Fractions

Wes Bruning We will use our fundamental principles to compare two (or more) fractions to determine if the fractions are equal or, if not equal, which fraction might be larger compared to the other. The fundamental principles are ...

"Making a 1"

Any quantity divided by itself = 1 a = 1 a ( a cannot be equal to 0)

and

"Multiply by 1" Any number multiplied by 1 is equivalent to the original number.

This is stated mathematically as: a · 1=a

Suppose we must determine if two fractions are equal, and, if not which of two 11 8 numbers is larger: or ? 12 9 We would go about this by establishing a common reference by which the numbers can be compared. What is meant by a "common reference"? Consider a situation where you have several types of currency British Pounds, French Francs and Dutch Marks and you must determine the relative value of each sum of money. How would you determine which currency represented more value?

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If you were used to U.S. Dollars, it would be reasonable to convert each of the currencies into Dollars then compare their values in this common reference. In a like manner, when we compare fractions we must transform them into a common reference. The common reference is that each fraction has the same denominator. In the example above, we would change both fractions to have the same denominator. Begin by picking a value for the denominator that both 12 and 9 will divide into evenly. Pick the value 36. (Both 12 and 9 will divide evenly into 36). So then, multiply both fractions by a specially chosen 1 to convert the denominators into 36.

11 3 11 i 3 33 i = = 12 3 12 i 3 36

and

8 4 8 i 4 32 i = = 9 4 9 i 4 36

Once we have both fractions with the same denominator, it is clear that the fractions are not equal to the same value and which fraction is larger.

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Fractions to Percents

Wes Bruning We multiply by 1 to convert fractions to percents. We will use the concept presented earlier of changing the value of a fraction's denominator. A fraction, as we have seen, is a portion of a whole: ¾ means three parts of a total of four. Likewise, a percentage is so many parts of 100. The basics of converting a fraction to a percentage is changing the fraction's denominator to 100. We accomplish this by multiplying by 1.

3 25 : To change to a percentage multiply by 1 where 1 = 4 25

3 25 3 i 25 75 i = = 4 25 4 i 25 100

Earlier we noted that % is the symbol representing Then

1 100

75 1 = 75 i = 75% 100 100

Other examples:

17 5 17 i 5 85 1 i = = = 85 i = 85% 20 5 20 i 5 100 100 7 12.5 7 i 12.5 87.5 12.5 i = = = 87.5% Notice that the 1 is . 8 12.5 8 i 12.5 100 12.5

When we convert a fraction to a percent we are really just converting the fraction's denominator to 100 by multiplying by 1.

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Change Signs in a Fraction

Wes Bruning a -a = b -b (b cannot be equal to 0) It is often useful to change the sign of a fraction's denominator or numerator. Using Multiply by 1, we can easily do this. Consider the fraction in the denominator. Using 1 = a (where b 0) and we desire to not have a negative sign -b

-1 we can multiply the fraction and change its signs: -1

Multiplying Signed Numbers

a -1 (-1) a -a i = = b -b -1 (-1)(-b)

Remember the simple rule: Multiplying like signs yield a + product Multiplying unlike signs yield a product. Or (+)(+) = + ()() = + (+)() = ()(+) = (+3)(+2) = +6 (3)( 2) = +6 (+3)( 2) = 6 (3)(+2) = 6

Using this in an example:

3 3 -1 -3 = i = -4 -4 -1 4

So we can change the signs in a fraction by multiplying by 1.

This applies to division in a like manner: Dividing like signs yield a + quotient Dividing unlike signs yield a quotient.

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Simplify Fractions

Wes Bruning At this time, we have learned two fundamental principles: Making a 1 Any quantity divided by itself = 1 a = 1 a ( a cannot be equal to 0) and Multiply by 1 Any number multiplied by 1 is equivalent to the original number. This is stated mathematically as:

a i1= a

We use these principles to simplify fractions. Suppose we have the number need to simplify it.. We start by factoring both the 21 and the 35. Factor 21 into 3 · 7 Factor 35 into 5 · 7

21 and 35

Math language: To factor means to find two or more numbers that when multiplied together produce the original number.

Example: we can factor 6 into 2 · 3, or 2 · 3 = 6.

We can factor 12 into 2 · 2 · 3 = 12 or 2 · 6 = 12 or 4 · 3 = 12

Using these numbers we can rewrite the fraction as

21 3 i 7 3 7 3 = = i = 35 5 i 7 5 7 5

Do you see that we are able to regroup the fraction multiplication?

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7 The Key: recognize that = 1 7

So then,

3 3 i 1 = (Any number times 1 = itself). 5 5 21 3 to 5 with the following process: 35

We simplified the fraction

1. Factor the numerator and denominator Identify the 1s (Any number divided by itself = 1) 2. Remove the 1s (Any number multiplied by 1 is equivalent to the original number.) _____________________ Lets try another one: simplify 1. Start by factoring 30 and 42. Factor 30 into 5 · 6 and Factor 42 into 7 · 6 Rewrite the fraction as

30 42 .

We can factor 30 a number of different ways: 2 · 15 or 3 · 10 or 5 · 6 Same thing with 42: 2 · 21 or 3 · 14 or 7 · 6 Notice that we picked factors that have a common number: 6. There is a reason for this. Why is this important?

30 5 i 6 = 42 7 i 6

2. Rewriting we get

5i6 5 6 = i 7i6 7 6

6 = 1 6

The Key: recognize that 3. So then,

5 5 i 1 = (Any number times itself is 1). 7 7 30 5 to 7 . 42 15 = 27

We have simplified the fraction Try simplifying these fractions: 28 = 32

Lets try looking at this type of problem in the general case. That is, what if we were to represent the numbers in the numerator and the denominator generically. Instead of writing a 3, for example, in the numerator,

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lets represent it with a. And instead of a 7, lets use a b and instead of a 5 lets use a c. The values of 3, 5, and 7 are just picked out of thin air. They can really be any numbers at all! So, using the above, lets consider the fraction ac bc a, b and c just represent numbers. So, the principles we have learned still apply. Lets rewrite the above as ac a i c a a = = i1= bc b i c b b Notice that in the above Try these: aic = aib c = 1 c

There are a couple of things to understand about the fraction as written here:

ac = a i c

That is, the number represented by "a" is multiplied by the number represented by "c". For example, if a = 3 and c = 5, then the value ac would = 15. But, in this generic case we do not know what the values of a and c are, so we cannot multiply them together to obtain a numerical product. Therefore, we leave the numerator as ac and, in the denominator, bc.

xi y = yiz

In the second example rewrite y · z to z · y then you can group the y's into

y . y

The examples above are referred to as algebraic fractions. But they follow the same rules as arithmetic. Important Note: Effectively learning to simplify fractions is extremely valuable. Generally, we can make our computations much easier and less error prone if we simplify first. For example. Consider this mathematical expression: 20 25 i 45 16 The way to go about this is NOT to start multiplying things together. (I.e. 20 x 25 and 45 x 16).

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Rather, we would simplify first. By combining our knowledge of Math Facts and factoring, we can simplify the fraction to: 20 = 4 · 5 45 = 9 · 5 25 = 5 · 5 16 = 4 · 4 So then we have

4i5i5i5 = 9i5i4i4 4i5i5i5 = 4i5i4i9 4 5 5i5 i i = 4 5 4i9

1i1i 5 5 25 i = 4 9 36

Did you understand how the process moved from one part to the next? We simply factor (using our Math Facts), then regroup, then find equivalents of 1: 4 5 and . When we finish, we know there are no more factors of 1 in the fraction 4 5 and we have simplified it as much as possible.

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Simplify Algebraic Expressions

Wes Bruning Simplifying algebraic expressions employ the same methods as those used to simplify arithmetic fractions. ab . Note that we have a common factor in both the ac numerator and the denominator. For example consider: So, we can consider this algebraic fraction as

a ib a b which can be rewritten as i aic a c

a Recognizing that =1 a Our original expression is simplified to 1 i b Then our final solution is c More complicated expressions are handled the same way. Starting with b c

ab + ac we use the Distributive Property to obtain ad + ae

a(b + c) a b+c b+c which can be rewritten as i or 1 i a ( d + e) a d +e d +e

Which then is simplified to our final simplified solution of

b+c d +e

Multiplying by 1 allows us to simplify algebraic expressions. The Distributive Property

ab + ac = a(b + c) As example: 2(3) + 2(5) = 2(3 + 5) 6 + 10 = 2(8) 16 =16

23

Simplify Algebraic Rational Expressions

Wes Bruning Expanding on the previous section, we use multiply by 1 to simplify more complicated algebraic expressions such as x2 + 6x + 8 . x 2 + 7 x + 10 As before, we begin by factoring the numerator a denominator: x2 + 6x + 8 ( x + 2)( x + 4) ( x + 2) ( x + 4) = = i 2 x + 7 x + 10 ( x + 2)( x + 5) ( x + 2) ( x + 5) We recognize that

( x + 2) = 1 ( x + 2)

So the expression simplifies to

( x + 2) ( x + 4) ( x + 4) ( x + 4) = 1i i or simply ( x + 2) ( x + 5) ( x + 5) ( x + 5)

Thus we have used multiply by 1 to simplify rational expressions.

24

Unit Conversions

Wes Bruning Lets see how to use the idea of multiplying by 1 with dimensions. Notice that Principle 2 says that the product is equivalent to the original number. This means that the products magnitude (a magnitude is an amount or quantity represented) is equal to the original number. But, equivalent amounts may not look the same. We use this sort of thing all the time. We know that 12 inches is equivalent (i.e. the same length as) 1 foot: 2 inches = 1 foot. We know that 3 feet is equivalent (i.e. the same length as) one yard: 3 feet = 1 yard. We know that 365 days is equivalent to one year: 365 days = 1 year. Notice that in the examples above, we are NOT saying that 12 = 1 or that 3 = 1 or 365 = 1. We must be very careful of our units here: inches, feet, yards, days, years. Part of learning math effectively is to realize that the units are part of the number. So we deal with this concept of two things having equivalent measures even though they do not look the same.2 Math is more than just numbers. The numbers actually represent something. So, just as we saw in the previous lesson, consider the three examples above, we can make a 1 in the following ways:

12 inches = 1 1 foot 3 feet = 1 1 yard 365 days = 1 1 year

The mathematical "sentences" to the left are equations. Notice that an equation has an = (equal sign) in it. If no equal sign is present, it is referred to as an expression.

Here is a fun thing you can pull on your friends: Bet them that you can prove that 1 = 7. Write it like this: 1 = 7 . Leave a space between the 1 and the =. Of course, they will dispute the fact that 1 = 7. After the appropriate amount of wrangling, in which they refuse to believe that 1 = 7, you simply write the words "week" and "days" in the appropriate spots: 1 week = 7 days. Therefore, with the appropriate units 1 does indeed equal 7. You can use this with days and years, gallons and quarts, dollars and dimes, etc.

2

25

How did we come up with these three equations? (Write your explanation here)

Lets take an example we all can understand. Consider US money. Four quarters are equivalent to one dollar. 4 quarters = 1 dollar In this case we have four coins and that are the same amount as one piece of paper (one dollar). Lets examine the case where 4 quarters is equal to 1 dollar to see how we really think about this. Clearly, 4 coins do not look like one paper dollar. They look different. But are they the same amount? How do we compare them to see if they are?

A U.S. cent is really /100 of a dollar. So we are putting both 1 values in terms of /100 dollars

1

100 cents =1 1 dollar and

We first think of them in a similar way: a quarter is 25 cents; a dollar is 100 cents. So we can describe them both with a common reference: cents.

1 dollar =1 100 cents

Now we think: if a quarter is equal to 25 cents ( 1 quarter = 25 cents ) then

25 cents = 1 1 quarter

and we have four quarters, so

4 quarters 25 cents 4 i 25 quarters cents = = i 1 1 quarter 1 quarter

100 quarter i cents 100 icents quarters = i 1 quarter 1 quarter

Notice what happened here. The multiplication was re-arranged and the units "quarters" were grouped.

Now we have an interesting thing. We have the units "quarters" in both the numerator and denominator. Units behave just like numbers. So,

quarters = 1 quarters

26

Therefore,

100 cents quarter 100 cents = i i 1 = 100 cents 1 quarter 1

So, using Principle 2, we did not change the value of the 4 quarters only the way it looked (the units)! How did Principle 2 come into play in this example?

27

Unit Conversions (Revisited)

Wes Bruning We have learned that: Any number divided by itself = 1. The product of any number and 1 is equivalent to the original number. We can use these ideas to convert from one set of units to another. As examples: How many quarts are in 3 gallons? How many Y (Japanese Yen) can we get for 4 USD (U.S. Dollars)? How many feet are in 29 yards? The three questions above are examples of unit conversions. There are many more daily examples. Lets look at how we can apply the idea of 1 to these types of problems. Consider: Any number divided by itself = 1. We saw earlier that if we have two quantities that equal the same amount they are equal even if they do not look the same. For instance 4 quarters = 10 dimes. Both 4 quarters and 10 dimes are each worth 100 cents. Both represent the same amount even though they do not look the same. So, we can say that because they represent the same amount

4 quarters = 1 10 dimes

To answer the question of how many quarters can we get for 30 dimes, we would start with the 30 dimes and multiply by our representation of 1:

30 dimes 4 quarters = i 1 10 dimes 30 dimes i 4 quarters 3 i 10 i 4 dimes quarters = 10 dimes 10 i dimes

Notice that

10 dimes = 1 and = 1 10 dimes

Dividing common factors and units we have:

28

3 i 4 quarters = 12 quarters

So, 30 dimes = 12 quarters Of course this makes sense because 30 dimes = $3.00 and 12 quarters = $3.00 In this process we have used unit conversions to convert from dimes to quarters.

______________________

Lets take a look at how many quarts are in three gallons: 4 quarts = 1 gallon, therefore these two quantities are equivalent and can be used to express 1:

4 quarts =1 1 gallon

Start with 3 gallons and multiply by our 1

3 gallons 4 quarts i 1 1 gallon

The gallons will divide just like the numbers do (

gallons = 1 ), so gallons

3 4 quarts gallons = 3 i 4 quarts = 12 quarts ( i i 1 1 gallons

We have used multiplying by 1 to convert 3 gallons into 12 quarts.

______________________

Lets determine how many Y (Japanese Yen) can we get for 4 USD (U.S. Dollars): As of this writing 1 USD = 106 Y (source: http://www.xe.com) These are equivalent amounts so So, multiplying by our special 1 4 USD · 1 =

106 ¥ = 1 1USD

4 USD 106 ¥ 4 i 106 ¥ USD = i i = 424 Y 1 1 1USD USD

So, 4 USD = 424 Y

USD =1 USD

______________________

29

Lets determine how many feet are in 29 yards: 1 yard = 3 feet So,

3 ft = 1 1 yd

Start with 29 yards and multiply by our 1

29 yds i 1 = 1 29 yds 3 ft 29 i 3 ft yd = = 87 ft i i 1 1 yd 1 yd

So 29 yards = 87 feet We have converted yards to feet using multiplying by 1.

yd =1 yd

30

Long Division

Wes Bruning Long division involves something like 23 1897 . We know how to do long division with whole numbers. However, consider if you are dividing by a nonwhole number such as

4.3 245

In order to divide, we must have a whole number for the divisor. The divisor is the number we are dividing by: the 4.3 . In this case the 4.3 has a decimal in it. We have to change it to be a whole number, i.e. 43. Let's consider this problem in light of what we know about fractions. We know that fractions are just division problems. So the division problem above can be written as

245 4.3

In order to have a whole number in the denominator, we will change the denominator by multiplying by 10 (in the case of long division such as this we will always multiply by a factor of 10: 10, 100, 1000, etc.). But, as before, we can only multiply by "1" (a · 1 = a) so we do not change the value of the number, only it's appearance. So when we multiply both numerator and denominator by 10 10 (that is, we are multiplying the original fraction by 1 written as ), we get 10

245 10 2450 = i 4.3 10 43

Now when we reform our long division problem we have

43 2450

Notice what happened. By multiplying both numbers involved in the division by 10 (effectively multiplying by 1) we move the decimal in the divisor to make the divisor a whole number. When we did that we also increased the number being divided into (the dividend) by a factor of 10. We generally take the short cut of simply moving the decimal in the divisor to the right to make a whole number then moving the decimal in the dividend the same number of places. But, as you see here, we are really just multiplying by 1 to achieve the results.

31

Negative exponents

Wes Bruning Let us first consider some exponent basics. Exponents are really shorthand notation for successive multiplying. An exponent tells us how many factors to use when a base is multiplied times itself. As examples: a2 = a · a a = a · a · a

3

Math Symbology

baseexponent

a4 = a · a · a · a To multiply exponential terms with like bases we simply add the exponents:

a m i a n = a m+n

To see how this works consider a 2 i a3 = a 2 + 3 = a5

a 2 = a i a and a3 = a i a i a a 2 i a 3 = (a i a ) i (a i a i a) = a i a i a i a i a = a 5

It is certainly possible to have a situation such as a -4 . So what does this mean? Using the multiplication rule for exponents indicated above we can change the 4 exponent into a positive value by multiplying by 1: a4 = 1 a4

a -4 = a -4 a -4 a 4 a -4 a 4 a -4 + 4 a0 1 = i 4 = = 4 = 4 = 4 4 1 1 a a a a a

(note that a0 = 1 )

So, a -4 =

1 1 . This can be generalized to a - m = m 4 a a

We converted a term with a negative exponent to a term with all positive exponents by multiplying by 1.

0 You might have noticed in the above example that a = 1 . Let's consider why. 1 Above, we showed that a - m = m a

a0 = a m+( - m) = a m i a - m = a m i

1 am = m = 1 am a

So then a 0 = 1

32

Negative exponents in the Denominator

Wes Bruning

1 . In this case we must deal with a -3 the negative exponent in the denominator. A negative exponent in the denominator is a mathematicallyundesirable thing. We do not allow negative exponents in the denominator or the numerator of a final answer.

Sometimes we run into a situation like this: We want only positive exponents in our terms. So we would multiply the fraction by a special "1": a4 = 1 a4 1 i 1 a -3 So 1 a4 1 i a4 i 4 = -4 a -4 a a i a4 Using our exponent rule for multiplication:

1 i a4 a4 a4 a4 = 4- 4 = 0 = = a 4 (remembering that a0 = 1 ). 4 -4 a ia a a 1

33

Roots in the Denominator

Wes Bruning Sometimes, we run across the situation where we have a root in the denominator 7 . of a fraction: 5 It is poor mathematical form to leave roots in the denominator. The thorough mathematician will "rationalize" the fraction to remove the offending root. We do this by multiplying by 1: So then,

5 5

7 5 7 5 7 5 = = i 5 5 5 5 5

We have eliminated the root in the denominator by multiplying by 1.

34

Complex Numbers

Wes Bruning Complex numbers are of the form a + bi where i is the imaginary number -1 . It is poor form to have a complex number in the denominator of a fraction such as 1 3 + 2i To eliminate the imaginary number from the denominator we would multiply by a specially selected 1 comprised of the complex conjugate of 3 + 2i which is 3 - 2i . The "1" would be

3 - 2i 3 - 2i

Complex Conjugate A complex conjugate is a complex number where the sign of the imaginary term is reversed. For example the complex conjugate of 6 - 5i is 6 + 5i. In general the complex conjugate of a + bi is a - bi .

So

1 3 - 2i 3 - 2i i = 3 + 2i 3 - 2i (3 + 2i )(3 - 2i ) 3 - 2i 3 - 2i = 2 9 + 6i - 6i - 4i 9 - 4(-1)

Evaluating the denominator we have:

Notice here that i 2 = -1 i -1 = -1

So

3 - 2i 3 - 2i = 9+4 13

From here, of course, we would simplify this fraction as much as possible. But this example requires no further simplification although we could write it as

3 2 - i to put it in the form a + bi 13 13

By multiplying by 1, we have eliminated the complex expression from the denominator.

35

Imaginary Numbers in the Denominator

Wes Bruning As in an earlier section, a problem also occurs when the root of the negative number is in the denominator of a fraction. Like before, we do not leave fractions with a root in the denominator. We must rationalize the fraction. We use Multiply by 1 to accomplish this. Consider the fraction We will multiply by 1:

3 . -5

-5 -5

3 i -5

Then

-5 3 -5 = -5 -5

3 -5 3 5 -1 3 5 i = = -5 -5 -5

We do not like to leave a negative sign in the denominator (5), so

-3 5 3 5 i -1 -3 5 i i = i or 5 5 -5 -1

36

Multiplication and Division Facts

Wes Bruning If I had eight hours to chop a cord of wood, Id spend six sharpening my axe.

Abraham Lincoln

One of the most beneficial things one can do to prepare for learning mathematics is to memorize the multiplication tables: from 1 x 1 through 12 x 12. The student must have instant recall of all combinations of the multiplication of numbers from 1 through 12. Notice that we only have to remember about one half of the table. This is because 3 x 8 = 8 x 3 = 24.3 This greatly reduces the number of multiplication facts we must memorize. In addition, we must know the division table as well. That is,

32 and 8

72 = 9 . One can think of it as the multiplication table in reverse ... sort of. 8

Consider 7 x 3. We know (or should know) 7 x 3 = 21. From this multiplication 21 21 fact, we can determine two division facts: = 3 and = 7 . 7 3 Write the two division facts for the following multiplication facts:

Multiplication fact

7 x 4 = 28 3 x 9 = 27

Division Facts

a) ___________________________________ b) ___________________________________ a) ___________________________________ b) ___________________________________

Above, we saw that multiplication is associative (The associative property of multiplication). Is there an associative property of division? Why or why not?4 Learn the tables completely. Before each mathematics lesson, spend time reviewing the math tables. In time, as the student reviews the tables and uses

This is the associative property of multiplication. It does not matter in what order we multiply numbers: 3 x 2 = 2 x 3. Similarly, addition is associative also: 3 + 2 = 2 + 3.

3 4

There is no associate property for division.

32 8 is not equal to . 8 32

Page 37

them, the student will learn them. Learning the tables cannot be over emphasized. They are one of the keys to success in math. Much of what we do in mathematics involves simplifying mathematical expressions. Being very familiar with the division processes as well as the multiplication processes will eliminate a lot of frustration. In the many years he as been teaching, Mr. Bruning has observed that not knowing the multiplication and division tables results in severe problems in a student's study of mathematics. Knowing the math tables inside and out is essential to a smooth learning process. Use Mr. Bruning' Multiplication Facts program to exercise and extend the student's knowledge of the multiplication and division tables.

Page 38

Multiplication Table

1 1 2 3 4 5 6 7 8 9 10 11 12 1 x 1 2 x 1 3 x 1 4 x 1 5 x 1 6 x 1 7 x 1 8 x 1 9 x 1 10 x 1 11 x 1 12 x 1 2 1 x 2 2 x 2 3 x 2 4 x 2 5 x 2 6 x 2 7 x 2 8 x 2 9 x 2 10 x 2 11 x 2 12 x 2 3 1 x 3 2 x 3 3 x 3 4 x 3 5 x 3 6 x 3 7 x 3 8 x 3 9 x 3 10 x 3 11 x 3 12 x 3 4 1 x 4 2 x 4 3 x 4 4 x 4 5 x 4 6 x 4 7 x 4 8 x 4 9 x 4 10 x 4 11 x 4 12 x 4 5 1 x 5 2 x 5 3 x 5 4 x 5 5 x 5 6 x 5 7 x 5 8 x 5 9 x 5 10 x 5 11 x 5 12 x 5 6 1 x 6 2 x 6 3 x 6 4 x 6 5 x 6 6 x 6 7 x 6 8 x 6 9 x 6 10 x 6 11 x 6 12 x 6 7 1 x 7 2 x 7 3 x 7 4 x 7 5 x 7 6 x 7 7 x 7 8 x 7 9 x 7 10 x 7 11 x 7 12 x 7 8 1 x 8 2 x 8 3 x 8 4 x 8 5 x 8 6 x 8 7 x 8 8 x 8 9 x 8 10 x 8 11 x 8 12 x 8 9 1 x 9 2 x 9 3 x 9 4 x 9 5 x 9 6 x 9 7 x 9 8 x 9 9 x 9 10 x 9 11 x 9 12 x 9 10 1 x 10 2 x 10 3 x 10 4 x 10 5 x 10 6 x 10 7 x 10 8 x 10 9 x 10 10 x 10 11 x 10 12 x 10 11 1 x 11 2 x 11 3 x 11 4 x 11 5 x 11 6 x 11 7 x 11 8 x 11 9 x 11 10 x 11 11 x 11 12 x 11 12 1 x 12 2 x 12 3 x 12 4 x 12 5 x 12 6 x 12 7 x 12 8 x 12 9 x 12 10 x 12 11 x 12 12 x 12

The student should be able to point to a random multiplication and instantly know the product. (A product is the result of a multiplication).

Page 39

Multiplication Facts

1 1 x 1 = 1 2 x 1 = 2 3 x 1 = 3 4 x 1 = 4 5 x 1 = 5 6 x 1 = 6 7 x 1 = 7 8 x 1 = 8 9 x 1 = 9 10 x 1 = 10 11 x 1 = 11 12 x 1 = 12 2 1 x 2 = 2 2 x 2 = 4 3 x 2 = 6 4 x 2 = 8 5 x 2 = 10 6 x 2 = 12 7 x 2 = 14 8 x 2 = 16 9 x 2 = 18 3 1 x 3 = 3 2 x 3 = 6 3 x 3 = 9 4 x 3 = 12 5 x 3 = 15 6 x 3 = 18 7 x 3 = 21 8 x 3 = 24 9 x 3 = 27 4 1 x 4 = 4 2 x 4 = 8 3 x 4 = 12 4 x 4 = 16 5 x 4 = 20 6 x 4 = 24 7 x 4 = 28 8 x 4 = 32 9 x 4 = 36 5 1 x 5 = 5 2 x 5 = 10 3 x 5 = 15 4 x 5 = 20 5 x 5 = 25 6 x 5 = 30 7 x 5 = 35 8 x 5 = 40 9 x 5 = 45 10 x 5 = 50 11 x 5 = 55 12 x 5 = 60 6 1 x 6 = 6 2 x 6 = 12 3 x 6 = 18 4 x 6 = 24 5 x 6 = 30 6 x 6 = 36 7 x 6 = 42 8 x 6 = 48 9 x 6 = 54 7 1 x 7 = 7 2 x 7 = 14 3 x 7 = 21 4 x 7 = 28 5 x 7 = 35 6 x 7 = 42 7 x 7 = 49 8 x 7 = 56 9 x 7 = 63 8 1 x 8 = 8 2 x 8 = 16 3 x 8 = 24 4 x 8 = 32 5 x 8 = 40 6 x 8 = 48 7 x 8 = 56 8 x 8 = 64 9 x 8 = 72 10 x 8 = 80 11 x 8 = 88 12 x 8 = 96 9 1 x 9 = 9 2 x 9 = 18 3 x 9 = 27 4 x 9 = 36 5 x 9 = 45 6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 9 x 9 = 81 10 x 9 = 90 11 x 9 = 99 10 1 x 10 = 10 2 x 10 = 20 3 x 10 = 30 4 x 10 = 40 5 x 10 = 50 6 x 10 = 60 7 x 10 = 70 8 x 10 = 80 9 x 10 = 90

10 x 10 = 100 11 x 10 = 110

1 2 3 4 5 6 7 8 9 10 11 12

11 1 x 11 = 11 2 x 11 = 22 3 x 11 = 33 4 x 11 = 44 5 x 11 = 55 6 x 11 = 66 7 x 11 = 77 8 x 11 = 88 9 x 11 = 99

10 x 11 = 110 11 x 11 = 121 12 x 11 = 132

12 1 x 12 = 12 2 x 12 = 24 3 x 12 = 36 4 x 12 = 48 5 x 12 = 60 6 x 12 = 72 7 x 12 = 84 8 x 12 = 96 9 x 12 = 108

10 x 12 = 120 11 x 12 = 132 12 x 12 = 144

10 x 2 = 20 10 x 3 = 30 10 x 4 = 40 11 x 2 = 22 11 x 3 = 33 11 x 4 = 44 12 x 2 = 24 12 x 3 = 36 12 x 4 = 48

10 x 6 = 60 10 x 7 = 70 11 x 6 = 66 11 x 7 = 77 12 x 6 = 72 12 x 7 = 84

12 x 9 = 108 12 x 10 = 120

Solutions to the Multiplication Table.

Page 40

Division Facts

Quotient 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 10 / 1 11 / 1 12 / 1 2/2 4/2 6/2 8/2 10 / 2 12 / 2 14 / 2 16 / 2 18 / 2 20 / 2 22 / 2 24 / 2 3/3 6/3 9/3 12 / 3 15 / 3 18 / 3 21 / 3 24 / 3 27 / 3 30 / 3 33 / 3 36 / 3 4/4 8/4 12 / 4 16 / 4 20 / 4 24 / 4 28 / 4 32 / 4 36 / 4 40 / 4 44 / 4 48 / 4 5/5 10 / 5 15 / 5 20 / 5 25 / 5 30 / 5 35 / 5 40 / 5 45 / 5 50 / 5 55 / 5 60 / 5 6/6 12 / 6 18 / 6 24 / 6 30 / 6 36 / 6 42 / 6 48 / 6 54 / 6 60 / 6 66 / 6 72 / 6 7/7 14 / 7 21 / 7 28 / 7 35 / 7 42 / 7 49 / 7 56 / 7 63 / 7 70 / 7 77 / 7 84 / 7 8/8 16 / 8 24 / 8 32 / 8 40 / 8 48 / 8 56 / 8 64 / 8 72 / 8 80 / 8 88 / 8 96 / 8 9/9 18 / 9 27 / 9 36 / 9 45 / 9 54 / 9 63 / 9 72 / 9 81 / 9 90 / 9 99 / 9 108 / 9 10 / 10 20 / 10 30 / 10 40 / 10 50 / 10 60 / 10 70 / 10 80 / 10 90 / 10 100 / 10 110 / 10 120 / 10 11 / 11 22 / 11 33 / 11 44 / 11 55 / 11 66 / 11 77 / 11 88 / 11 99 / 11 110 / 11 121 / 11 132 / 11 12 / 12 24 / 12 36 / 12 48 / 12 60 / 12 72 / 12 84 / 12 96 / 12 108 / 12 120 / 12 132 / 12 144 / 12 1 2 3 4 5 6 7 8 9 10 11 12

The student should be able to point to a random division and instantly know the quotient. (A quotient is the result of a division)

Page 41

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