multiply by 1 - clackamas community collegecs.clackamas.cc.or.us/bruning/math/fractions.pdf ·...

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

© 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.

‐ i ‐

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 ʺWell, duh! Of course!ʺ 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=πD relates a circle’s circumference to its diameter through a constant value, π

D=RT relates distance traveled to rate of travel and time

F=ma relates force, mass and acceleration in physics

E=IR 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.


‐ ii ‐

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 problem‐type 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 step‐by‐step 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 problem‐solving 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


‐ iii ‐

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 ʺmagicʺ 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


‐ iv ‐

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 elementary‐school 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.

1 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 e‐mail 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.


‐ 1 ‐

Make a 1

Wes Bruning

Principle: “Make a 1”

Any quantity divided by itself = 1

aa = 1

(ʺ ʺ cannot be equal to 0) a

A fraction indicates a mathematical division operation.

37

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.

The principle stated above should be pretty obvious as

44 = 1

5656 = 1

857.3857.3

= 1

So, we can represent the number 1 in many, many different ways.

Use the space below to write some of your own:

Letʹs expand on this a bit.

Math Language

In a fraction, the top number is the numerator and the bottom number is the denominator:

Numerator Denominator

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.


‐ 2 ‐

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:

71

daysweek

= 1

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: 71

daysweek

= 1

does it make any difference if we write: 17weekdays

= 1

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 14

gallonquarts

= 1 and 41

quartsgallon

= 1

4 quarters = 1 dollar

so 41quartersdollar

= 1 and 1

4dollar

quarters = 1

When we are writing ʺ1ʺ 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:


‐ 3 ‐

Solve for a Variable

Wes Bruning

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

That is, 3 13= , 0.5 1

0.5= , and

12 112= , etc.

We can use this idea to solve for an algebraic variable in an equation.

Suppose we have . 5 15x=

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 3 15=i

But we must have a better way of finding the unknown value because not all problems are this easy.

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

4 = 8 This equation is not true. But dividing both sides of the equation by the same amount (2) keeps the equation true:

4 = 4

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

5 15 5x=

5 or 5 35

x= and we recognize that 5 15= so

3x=

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 ‐

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

1a = a

(ʺ ʺ can be any number) a

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

As examples:

Math language

A quotient is the result of dividing two numbers.

If we divide 6 by 3 the quotient is 2. 6 23=

41 = 4

2791 = 279

1

21 = 1

2

45.61 = 45.6

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.


‐ 5 ‐

Introduction to Fractions

Wes Bruning

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

For instance the fraction 14 means 1 part of 4 parts. The number on the bottom,

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?

(Answer: 34)

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

(Answer: 44)


‐ 6 ‐

So, if you had 4 parts out of 4 parts what portion of the whole would you have?

(Answer: The whole thing or 1)

Consider: If you had 12 of a gallon of water, what would you have relative to a

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.


‐ 7 ‐

Multiply Fractions

Wes Bruning

Multiplying Fractions:

Multiply the numerators and multiply the denominators.

ab x c

d = a c

b dii

(Neither ʺbʺ nor ʺdʺ 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.

ʺWhy,ʺ you ask, ʺwould we want to deal with a fraction of a fraction?ʺ

Consider the situation where you would want to divide 34 of a gallon of gas

equally among four people. Each person would receive 14 of the available

quantity of gas. We would do this mathematically by multiplying the 34 by 1

4:

34 x 1

4

When we multiply fractions, we multiply straight across: numerator times numerator and denominator x denominator 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".

ab

x cd

= acbd

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

34 x 1

4 = 3 14 4

xx =

316 of a gallon of gas

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


‐ 8 ‐

Multiply by 1

Wes Bruning

In a previous lesson we learned how to make a 1.

Now letʹs see how this might be useful. What can we do with the 1 we made?

We will start by learning another principle:

ʺMultiply by 1ʺ Principle:

Any quantity multiplied by 1 is equivalent to the original quantity.

This is stated mathematically as:

a • 1 = a

In this principle, ʺaʺ represents any quantity (we will stay with numbers for now). So you could substitute 4 or 279 or any other number you want for ʺaʺ.

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!

As examples:

4 • 1 = 4

279 • 1 = 279

½ • 1 = ½

45.6 • 1 = 45.6

Let’s put a couple of things together and see what happens.

Multiply 37 times

55 ( 5

5 is equivalent to 1 so we

are multiplying by 1)

3 5 157 5 35

=i


‐ 9 ‐

Our principle states: “Any number multiplied by 1 is equivalent to the original

number.” Therefore, since all we did was multiply by 1 (i.e. 55) we see that

3 157 35=

What we actually accomplished here is to convert 37 into an equivalent quantity

with a denominator of 35.

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


‐ 10 ‐

Changing a Fraction’s Denominator

Wes Bruning

We will use the “Multiply by 1” Principle to change the denominator of fractions.

Let’s change the fraction 58 into the equivalent number of 1

16 ths.

So we want to change the denominator of the 58 (that is, the 8) into a

denominator of 16.

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

So, we multiply like this: 58 • 2

2

Notice that we are multiplying by 22. Why? (Answer: Because we can only

multiply by 1 and not change a quantity’s value and 22 = 1).

When we multiply fractions we multiply straight across, so:

5 2 10=8 2 16i

We have converted 58 into the equivalent number of 10

16. These two fractions

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

34

?12

56

?24

78

?64

411

?55


‐ 11 ‐

Decimals to Fractions

Wes Bruning

We multiply by 1 to convert decimals to fractions.

Consider 0.125.

Multiply 0.125 by 10001000

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 or 1000 (10 ). 310 10 10i i

Then we have:

0.125 1000 1251 1000 1000

=i

From here we simplify:

125 1 125 11000 8 125 8

= =ii

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.


‐ 12 ‐

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 1100

.

Thus, % = 1100

0.523 can be changed to a percent by multiplying the decimal by 1 = 100100

0.523 100 52.31 100 100

=i

We recognize that 52.3 153.2100 100

= i

As stated above % = 1100

then,

153.2 53.2%100

=i


‐ 13 ‐

Multiply by 1 (Revisited)

Wes Bruning

Now we can convert a fraction to a different denominator.

Letʹs see how to use this idea.

Is 4 169 36= ?

If so, how can we demonstrate that it is?

We must represent both items

(the 49 and the 16

36) with a common

reference. To do this, both fractions must have the same denominator. That

would be a 36. We will determine how many 136

ths each fraction represents.

Math Language

NumeratorDenominator

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

the bottom of the fraction.

Why did we pick 36?

Letʹs look at the 49. If we were to multiply the denominator by 4, we would then

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,

both fractions would represent the number of 136 sized pieces.

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

by 1. (Why can we do this?) But we will pick a special ʺ1ʺ: 44. (Principle: Any

number divided by itself is 1.) When we multiply fractions, we multiply the numerators together and we multiply the denominators together

So if we multiply 49 by 4

4 (where 4

4 = 1) we get:

4 4 169 4 36

=i

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

How did we do this? We multiplied by 1 ( 44 ). Observe that the “1” was

carefully selected. How did we select the one? We found a number we could use


‐ 14 ‐

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].

Letʹs look at another example:

Is 23 = 13

18 ?

Well we must represent both fractions with some common denominator. This

would be the number of 118

ths that each fraction represents. The denominator

for both fractions must be 18 for us to compare them. Why 118

?

(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 118

ths are the equivalent of 23 ?

We would multiply the 23 by 1. What ʺ1ʺ would we pick? We would pick a 1 that

would change the denominator of the 23 (the 3) to an 18. Of course, we would

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.

Thus we would multiply by 66 .

23 • 6

6 = 12

18

Then looking at the original problem: Is 2 133 18= ?

The answer is "No". 2 123 18= not 13

18 .


‐ 15 ‐

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"


aa = 1

(ʺa ʺ cannot be equal to 0)

and

"Multiply by 1"

Any number multiplied by 1 is equivalent

to the original number.


a • 1 = a

Suppose we must determine if two fractions are equal, and, if not which of two

numbers is larger: 11 or 12

89 ?

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?


‐ 16 ‐

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 3 3312 3 12 3 36

= =iii

and 8 4 8 4 329 4 9 4 36

= =iii

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.


‐ 17 ‐

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.

To change 34 to a percentage multiply by 1 where 1 = 25

25:

3 25 3 25 754 25 4 25 100

= =iii

Earlier we noted that % is the symbol representing 1100

Then 75 175 75%100 100

= =i

Other examples:

17 5 17 5 85 185 85%20 5 20 5 100 100

= = = =ii ii

7 12.5 7 12.5 87.5 87.5%8 12.5 8 12.5 100

= = =iii

Notice that the 1 is 12.512.5

.

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


‐ 18 ‐

Change Signs in a Fraction

Wes Bruning

a ab b

−=

−

(ʺbʺ 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 ab− (where b ≠ 0) and we desire to not have a negative sign

in the denominator.

Using 111

−=− we can multiply the fraction and change its signs:

1 ( 1)1 ( 1)( )

a ab b

− − −= =

− − − −i a

b

Multiplying Signed Numbers

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

This applies to division in a like manner:

Dividing like signs yield a + quotient

Dividing unlike signs yield a – quotient.

Using this in an example:

3 3 14 4 1 4

− −= =

− − −i 3

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


‐ 19 ‐

Simplify Fractions

Wes Bruning

At this time, we have learned two fundamental principles:

ʺMaking a 1ʺ


aa = 1

(ʺ ʺ cannot be equal to 0) a

and

ʺMultiply by 1ʺ

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


1a a=i

We use these principles to simplify fractions.

Suppose we have the number 2135 and

need to simplify it..

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

We start by factoring both the 21 and the 35.

Factor 21 into 3 • 7 Factor 35 into 5 • 7

Using these numbers we can rewrite the fraction as

21 3 7 3 7 335 5 7 5 7 5

= = =i ii

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


‐ 20 ‐

The Key: recognize that 77 = 1

So then, 3 15 5

=i 3 (Any number times 1 = itself).

We simplified the fraction 2135

to 35 with the following process:

1. Factor the numerator and denominator

Identify the ʺ1ʺs (Any number divided by itself = 1)

2. Remove the ʺ1ʺs (Any number multiplied by 1 is equivalent to the original number.)

_____________________

30◄ 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?

Letʹs try another one: simplify 42 .

1. Start by factoring 30 and 42.

Factor 30 into 5 • 6 and Factor 42 into 7 • 6

Rewrite the fraction as

30 5 642 7 6

=ii

5 6 5 67 6 7 6

=i ii

2. Re‐writing we get

66

The Key: recognize that = 1

5 517 7

=i3. So then, (Any number times itself is 1).

3042

57 to . We have simplified the fraction

Try simplifying these fractions: 2832

= 1527

=

Letʹs 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,


‐ 21 ‐

letʹs represent it with ʺaʺ. And instead of a 7, letʹs use a ʺbʺ and instead of a 5 letʹs use a ʺcʺ. 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, letʹs consider the fraction

acbc There are a couple of things to understand

about the fraction as written here:

ac a c= i

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.

ʺaʺ, ʺbʺ and ʺcʺ just represent numbers. So, the principles we have learned still apply.

Letʹs rewrite the above as

1ac a c a abc b c b b

= = =i ii

Notice that in the above 1cc=

Try these: a ca b

=ii

x yy z

=ii

In the second example re‐write y • z to z • y then you can group the y’s into yy

.

The examples above are referred to as ʺalgebraic fractionsʺ. 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 2545 16i

The way to go about this is NOT to start multiplying things together. (I.e. 20 x 25 and 45 x 16).


‐ 22 ‐

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

4 5 5 59 5 4 4

=i i ii i i

4 5 5 54 5 4 9

=i i ii i i

4 5 5 54 5 4 9

=ii ii

5 5 251 14 9 36

=i i i

Did you understand how the process moved from one part to the next? We simply factor (using our Math Facts), then re‐group, then find equivalents of 1: 44and 5

5. When we finish, we know there are no more factors of 1 in the fraction

and we have simplified it as much as possible.


‐ 23 ‐

Simplify Algebraic Expressions

Wes Bruning

Simplifying algebraic expressions employ the same methods as those used to simplify arithmetic fractions.

For example consider: abac . Note that we have a common factor in both the

numerator and the denominator.

So, we can consider this algebraic fraction as

a ba cii which can be rewritten as a b

a ci

Recognizing that 1aa=

Our original expression is simplified to 1 bci

Then our final solution is bc

More complicated expressions are handled the same way.

Starting with ab acad ae

++

we use the Distributive Property to obtain

(( )

a b ca d e

++

) which can be rewritten as a b ca d e

++

i or 1 b cd e++

i

Which then is simplified to our final simplified solution of b cd 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=


‐ 24 ‐

Simplify Algebraic Rational Expressions

Wes Bruning

Expanding on the previous section, we use multiply by 1 to simplify more complicated algebraic expressions such as

2

2

6 87 10

x xx x

+ ++ +

.

As before, we begin by factoring the numerator a denominator: 2

2

6 8 ( 2)( 4) ( 2) ( 47 10 ( 2)( 5) ( 2) ( 5)

x x x x x xx x x x x x

+ + + + + += =

+ + + + + +i )

We recognize that

( 2) 1( 2)xx+

=+

So the expression simplifies to

( 2) ( 4) ( 41( 2) ( 5) ( 5)x x xx x x+ + +

=+ + +

i i ) or simply ( 4)( 5)xx++

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


‐ 25 ‐

Unit Conversions

Wes Bruning

Letʹs see how to use the idea of multiplying by 1 with dimensions.

Notice that Principle 2 says that the product is ʺequivalentʺ to the original number. This means that the productʹs 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 ʺ1ʺ in the following ways:

12 11inchesfoot

= 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.

3 11

feetyard

=

365 11

daysyear

=

2 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.


‐ 26 ‐

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

Letʹs 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).

Letʹs 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?

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.

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

25 11

centsquarter

=

and we have four quarters, so

4 25 4 251 1 1

quarters cents quarters centsquarter quarter

= =ii

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

100 11

centsdollar

=

and 1 1

100 dollar

cents=

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

100 1001 1quarter cents cents quarters

quarter quarter=

i i i

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

So, 1quartersquarters

=


‐ 27 ‐

Therefore,

100 100 1 1001 1cents quarter cents cents

quarter= =i i

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?


‐ 28 ‐

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.

Letʹs 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 110quarters

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 41 10

dimes quartersdimes

=i

30 4 3 10 4dimes quarters dimes quarters10 10dimes dimes

=i i i

i

Notice that

10 110

= and 1dimesdimes

=

Dividing common factors and units we have:


‐ 29 ‐

3 4 12quarters quarters=i

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.

______________________

Letʹs 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 11

quartsgallon

=

Start with 3 gallons and multiply by our 1

3 41 1

gallons quartsgallon

i

The gallons will divide just like the numbers do ( 1gallonsgallons

= ), so

3 4 3 4 121 1

quarts gallons quarts quartsgallons

= =i i i (

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

______________________

Letʹs 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 106 11USD

=¥

So, multiplying by our special 1

4 USD • 1 = 1USD

USD=4 USD 106 4 106 USD

1 1USD US=

ii i¥ ¥1 D

= 424 Y

So, 4 USD = 424 Y

______________________


‐ 30 ‐

Letʹs determine how many feet are in 29 yards:

1 yard = 3 feet

So, 3 11

ftyd

=

Start with 29 yards and multiply by our 1

29 11yds

=i

1ydyd

=29 3 29 3 871 1 1yds ft ft yd ft

yd yd= =

ii i

So 29 yards = 87 feet

We have converted yards to feet using multiplying by 1.


‐ 31 ‐

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 non‐whole 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

2454.3

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 (that is, we are multiplying the original fraction by 1 written as 1010

), we get

245 10 24504.3 10 43

=i

Now when we re‐form 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.


‐ 32 ‐

5

Negative exponents

Wes Bruning

Let us first consider some exponent basics. Exponents are really short‐hand 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

a3 = a • a • a Math Symbology

exponentbase a4 = a • a • a • a

To multiply exponential terms with like bases we simply add the exponents: m n m na a a +=i

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

2a a= ia5=

and 3a a a a= i i2 3 ( ) ( )a a a a a a a a a a a a a= =i i i i i i i i i

It is certainly possible to have a situation such as 4a− . 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:

4

4 1aa

=

4 4 4 4 4 4 4 04

4 4 4 4

11 1

a a a a a a aaa a a a a

− − − − +− = = = = = =i 4 (note that 0 1a = )

So, 44

1aa

− = . This can be generalized to 1mma

a− =

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

You might have noticed in the above example that 0 1a = . Let’s consider why.

Above, we showed that 1mma

a− =

0 ( ) 1 1m

m m m m mm m

aa a a a aa a

+ − −= = = =i i =

So then 0 1a =


‐ 33 ‐

Negative exponents in the Denominator

Wes Bruning

Sometimes we run into a situation like this: 3

1a− . In this case we must deal with

the negative exponent in the denominator. A negative exponent in the denominator is a mathematically‐undesirable thing. We do not allow negative exponents in the denominator or the numerator of a final answer.

We want only positive exponents in our terms. So we would multiply the fraction by a special “1”:

4

4 1aa

=

3

1 1a− i

So 4 4

4 4 4 4

1 1a aa a a a− −=

iii

Using our exponent rule for multiplication: 4 4 4 4

44 4 4 4 0

11

a a a a aa a a a− −= = = =ii

(remembering that 0 1a = ).


‐ 34 ‐

Roots in the Denominator

Wes Bruning

Sometimes, we run across the situation where we have a root in the denominator

of a fraction: 75 .

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: 55

So then,

7 5 7 5 7 555 5 5 5

= =i

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


‐ 35 ‐

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

13 2i+

To eliminate the imaginary number from the denominator we would multiply by a specially selected 1 comprised of the complex conjugate of 3 which is

. 2i+

3 2i−

The “1” would be 3 23 2

ii

−−

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 2 3 23 2 3 2 (3 2 )(3 2 )

i ii i i

− −=

+ − + −i

i

Evaluating the denominator we have:

2

3 2 3 29 6 6 4 9 4( 1)

i ii i i− −

=+ − − − −

Notice here that 2 1 1i = − − = −i 1

So 3 2 3 29 4 13

i i− −=

+

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

3 213 13

i− to put it in the form a bi+

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


‐ 36 ‐

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 35−.

We will multiply by 1: 55−−

3 5 355 5

− −=

−− −i 5

Then

3 5 3 5 1 3 55 5 5

i− −= =

− − −

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

3 5 1 3 55 1 5

i i− −=

− −i or 3 5

5i−


Page 37

Multiplication and Division Facts

Wes Bruning

If I had eight hours to chop a cord of wood, Iʹd 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, 328 and

72 98= . One can think of it as the multiplication table in reverse … sort of.

Consider 7 x 3. We know (or should know) 7 x 3 = 21. From this multiplication

fact, we can determine two division facts: 21 37= and 21 7

3= .

Write the two division facts for the following multiplication facts:

Multiplication fact Division Facts

7 x 4 = 28 a) ___________________________________

b) ___________________________________

3 x 9 = 27 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

3 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.

4 There is no associate property for division. 328 is not equal to

832

.


Page 38

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.

Multiplication Table

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2 2 x 1 = 2 2 x 2 = 4 2 x 3 = 6 2 x 4 = 8 2 x 5 = 10 2 x 6 = 12 2 x 7 = 14 2 x 8 = 16 2 x 9 = 18 2 x 10 = 20 2 x 11 = 22 2 x 12 = 24

3 3 x 1 = 3 3 x 2 = 6 3 x 3 = 9 3 x 4 = 12 3 x 5 = 15 3 x 6 = 18 3 x 7 = 21 3 x 8 = 24 3 x 9 = 27 3 x 10 = 30 3 x 11 = 33 3 x 12 = 36

4 4 x 1 = 4 4 x 2 = 8 4 x 3 = 12 4 x 4 = 16 4 x 5 = 20 4 x 6 = 24 4 x 7 = 28 4 x 8 = 32 4 x 9 = 36 4 x 10 = 40 4 x 11 = 44 4 x 12 = 48

5 5 x 1 = 5 5 x 2 = 10 5 x 3 = 15 5 x 4 = 20 5 x 5 = 25 5 x 6 = 30 5 x 7 = 35 5 x 8 = 40 5 x 9 = 45 5 x 10 = 50 5 x 11 = 55 5 x 12 = 60

6 6 x 1 = 6 6 x 2 = 12 6 x 3 = 18 6 x 4 = 24 6 x 5 = 30 6 x 6 = 36 6 x 7 = 42 6 x 8 = 48 6 x 9 = 54 6 x 10 = 60 6 x 11 = 66 6 x 12 = 72

7 7 x 1 = 7 7 x 2 = 14 7 x 3 = 21 7 x 4 = 28 7 x 5 = 35 7 x 6 = 42 7 x 7 = 49 7 x 8 = 56 7 x 9 = 63 7 x 10 = 70 7 x 11 = 77 7 x 12 = 84

8 8 x 1 = 8 8 x 2 = 16 8 x 3 = 24 8 x 4 = 32 8 x 5 = 40 8 x 6 = 48 8 x 7 = 56 8 x 8 = 64 8 x 9 = 72 8 x 10 = 80 8 x 11 = 88 8 x 12 = 96

9 9 x 1 = 9 9 x 2 = 18 9 x 3 = 27 9 x 4 = 36 9 x 5 = 45 9 x 6 = 54 9 x 7 = 63 9 x 8 = 72 9 x 9 = 81 9 x 10 = 90 9 x 11 = 99 9 x 12 = 108

10 10 x 1 = 10 10 x 2 = 20 10 x 3 = 30 10 x 4 = 40 10 x 5 = 50 10 x 6 = 60 10 x 7 = 70 10 x 8 = 80 10 x 9 = 90 10 x 10 = 100 10 x 11 = 110 10 x 12 = 120

11 11 x 1 = 11 11 x 2 = 22 11 x 3 = 33 11 x 4 = 44 11 x 5 = 55 11 x 6 = 66 11 x 7 = 77 11 x 8 = 88 11 x 9 = 99 11 x 10 = 110 11 x 11 = 121 11 x 12 = 132

12 12 x 1 = 12 12 x 2 = 24 12 x 3 = 36 12 x 4 = 48 12 x 5 = 60 12 x 6 = 72 12 x 7 = 84 12 x 8 = 96 12 x 9 = 108 12 x 10 = 120 12 x 11 = 132 12 x 12 = 144

Solutions to the Multiplication Table.


Page 40

Division Facts Quotient

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

2 / 1 4 / 2 6 / 3 8 / 4 10 / 5 12 / 6 14 / 7 16 / 8 18 / 9 20 / 10 22 / 11 24 / 12 2

3 / 1 6 / 2 9 / 3 12 / 4 15 / 5 18 / 6 21 / 7 24 / 8 27 / 9 30 / 10 33 / 11 36 / 12 3

4 / 1 8 / 2 12 / 3 16 / 4 20 / 5 24 / 6 28 / 7 32 / 8 36 / 9 40 / 10 44 / 11 48 / 12 4

5 / 1 10 / 2 15 / 3 20 / 4 25 / 5 30 / 6 35 / 7 40 / 8 45 / 9 50 / 10 55 / 11 60 / 12 5

6 / 1 12 / 2 18 / 3 24 / 4 30 / 5 36 / 6 42 / 7 48 / 8 54 / 9 60 / 10 66 / 11 72 / 12 6

7 / 1 14 / 2 21 / 3 28 / 4 35 / 5 42 / 6 49 / 7 56 / 8 63 / 9 70 / 10 77 / 11 84 / 12 7

8 / 1 16 / 2 24 / 3 32 / 4 40 / 5 48 / 6 56 / 7 64 / 8 72 / 9 80 / 10 88 / 11 96 / 12 8

9 / 1 18 / 2 27 / 3 36 / 4 45 / 5 54 / 6 63 / 7 72 / 8 81 / 9 90 / 10 99 / 11 108 / 12 9

10 / 1 20 / 2 30 / 3 40 / 4 50 / 5 60 / 6 70 / 7 80 / 8 90 / 9 100 / 10 110 / 11 120 / 12 10

11 / 1 22 / 2 33 / 3 44 / 4 55 / 5 66 / 6 77 / 7 88 / 8 99 / 9 110 / 10 121 / 11 132 / 12 11

12 / 1 24 / 2 36 / 3 48 / 4 60 / 5 72 / 6 84 / 7 96 / 8 108 / 9 120 / 10 132 / 11 144 / 12 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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