MAKING CONNECTIONS: Course 1 Foundations for Algebra Toolkit Managing Editors / Authors Leslie Dietiker (Both Texts) Michigan State University East Lansing, MI

Evra Baldinger (Course 1)

Barbara Shreve (Course 2)

Phillip and Sala Burton Academic High School San Francisco, CA

San Lorenzo High School San Lorenzo, CA

Technical Assistants: Toolkit Sarah Maile

Aubrie Maize

Andrea Smith

Sacramento, CA

Sebastapool, CA

Placerville, CA

Cover Art Kevin Coffey San Francisco, CA

Program Directors Leslie Dietiker

Brian Hoey

Michigan State University East Lansing, MI

CPM Educational Program Sacramento, CA

Judy Kysh, Ph.D.

Tom Sallee, Ph.D.

Departments of Education and Mathematics San Francisco State University, CA

Department of Mathematics University of California, Davis

Notes:

MATH NOTES RATIOS A ratio is a comparison of two numbers, often written as a quotient; that is, the first number is divided by the second number (but not zero). A ratio can be written in words, in fraction form, or with colon notation. Most often in this class we will write ratios in the form of fractions or state them in words. For example, if there are 38 students in the school band and 16 of them are boys, we can write the ratio of the number of boys to the number of girls as: 16 boys to 22 girls

16 boys 22 girls

16 boys : 22 girls

MIXED NUMBERS AND FRACTIONS GREATER THAN ONE The number 3 41 is called a mixed number because it is composed of a whole number, 3, and a fraction, 41 . The number 13 is called a fraction greater than one because the numerator, 4 which represents the number of equal pieces, is larger than the denominator, which represents the number of pieces in one whole, so its value is greater than one. (Sometimes such fractions are called improper fractions, but this is just an historical term. There is nothing wrong with the fractions themselves.) As you can see in the diagram at right, the fraction 13 can be rewritten as 44 + 44 + 44 + 41 , which 4 shows that it is equal in value to 3 41 . Your choice: Depending on which arithmetic operations you need to perform, you will choose whether to write your number as a mixed number or as a fraction greater than one.

44

Making Connections: Course 1

Notes:

ADDING AND SUBTRACTING FRACTIONS To add or subtract two fractions that are written with the same denominator, simply add or subtract the numerators. For example, 15 + 25 = 53 . If the fractions have different denominators, rewrite them first as fractions with the same denominator (using the Giant One, for example). Below are examples of adding and subtracting two fractions with different denominators. Addition example:

1 5

+

2 3

3 + 10 = 13 ! 15 !!"!!! 33 !!!+ 23 !!"!!! 55 !!!! 15 15 15

Subtraction example:

5 6

!

1 4

7 " 56 !!#!!! 22 !!!! 14 !!#!!! 33 !!!" 10 ! 3 = 12 12 12

Using algebra to write the general method: a b

+

c d

! ba !!"!!! dd !!!+ dc !!"!!! bb !!!!

a"d b"d

b"c ! + b"d

a"d +b"c b"d

ADDING AND SUBTRACTING MIXED NUMBERS To add or subtract mixed numbers, you can either add or subtract their parts, or you can change the mixed numbers into fractions greater than one. To add or subtract mixed numbers by adding or subtracting their parts, add or subtract the whole-number parts and the fraction parts separately. Adjust if the fraction in the answer would be greater than one or less than zero. For example, the sum of 3 45 + 1 23 is calculated at right. It is also possible to add or subtract mixed numbers by changing them into fractions greater than one and then adding or subtracting as with fractions between zero and one. For example, the sum of 2 16 + 1 45 is calculated at right. 52

!!3 45 = 3 + 45 !!!!!! 33 !!!=!!3 12 15 +1 23 !=!1 + 23 !!!!!! 55 !!!= +1 10 !!!! 15 22 = 5 7 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!4 15 15

2 16 + 1 45 = 13 + 6

9 5

= 13 !!!!!! 55 !!!+ 95 !!!!!! 66 !!! 6 = = =

65 + 54 30 30 119 30 3 29 30

Making Connections: Course 1

Chapter 6: Similarity, Multiplying Fractions, and Equivalence

MULTIPLYING FRACTIONS

Notes:

You can find the product of two fractions, such as 23 and 43 , by multiplying the numerators (tops) of the fractions together and 6 dividing that by the product of the denominators (bottoms). So 23 ! 43 = 12 , which is equivalent to 12 . Similarly, 47 ! 53 = 12 . If we write this method in 35 a!c algebraic terms, we would say ba ! dc = b!d . The reason that this rule works can be seen using an area model of multiplication, as shown at right, which represents 23 ! 43 . The product of the denominators is the total number of smaller rectangles, while the product of the numerators is the number of the rectangles that are double shaded.

MULTIPLYING MIXED NUMBERS An efficient method for multiplying mixed numbers is to convert them to fractions greater than one, find the product as you would with fractions less than one, and then convert them back to a mixed number, if necessary. (Note that you may also use generic rectangles to find these products.) Here are three examples:

1 23 ! 2

3 4

= 53 ! 11 = 4

55 12

7 = 4 12

2 13 ! 4

Toolkit

1 2

1 53 ! 29 = 85 ! 29 = = 73 ! 92 =

63 6

16 45

= 10 63 = 10 12

45

Notes:

MULTIPLYING DECIMALS There are at least two ways to multiply decimals. One way is to convert the decimals to fractions and use your knowledge of fraction multiplication to compute the answer. The other way is to use the method that you have used to multiply integers; the only difference is that you need to keep track of where the decimal point is as you record each line. Here we show how to compute 1.4(2.35) both ways by using generic rectangles.

2

3 10

5 100

1

2

3 10

5 100

4 10

8 10

12 100

20 1000

2

0.3 0.05

1

2

0.3 0.05

0.4

0.8

0.12 0.020

If you carried out the computation as above, you can calculate the product in either of the two ways shown at right. In the first one, we write down all of the values in the smaller rectangles within the generic rectangle and add the six numbers. In the second example, we combine the values in each row and then add the two rows. We usually write the answer as 3.29 since there are zero thousandths in the product.

2.35 ! 1.4 0.020 0.12 0.8 0.05 0.3 2.0 3.290

2.35 ! 1.4 0.940 2.35 3.29

MULTIPLICATIVE IDENTITY If any number or expression is multiplied by the number one, the result is equal to the original number or expression. The number one is called the multiplicative identity. Formally, the identity is written:

1! x = x !1 = x for all values of x. One way the multiplicative identity is used is to create equivalent fractions using a Giant One. 2 !!!!!! 2 !!!=!! 4 3 2 6

Multiplying any fraction by a Giant One will create a new fraction equivalent to the original fraction.

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Making Connections: Course 1

Notes:

MULTIPLICATIVE INVERSES AND RECIPROCALS Two numbers with a product of 1 are called multiplicative inverses. 8 5 ! 5 8

=

40 40

=1

4 3 14 = 13 = 13 !4 = , so 3 14 ! 13 4 4 13

52 52

=1

1 7

!7 =1

In general a ! 1a = 1 and ba ! ba = 1 , where neither a nor b equals zero. We say that 1a is the reciprocal of a and ba is the reciprocal of ba . Note that 0 has no reciprocal.

DIVIDING BY FRACTIONS, PART 1 Method 1: Dividing Fractions Using Diagrams To divide any number by a fraction using a diagram, create a model of the first number using rectangles, a linear model, or some visual representation of it. Then break that model into the fractional parts named by the second fraction. For example, to divide ÷ , you can draw the diagram at right to visualize how many 12 -sized pieces fit into 87 . The diagram shows that one 12 fits, with 83 of a whole left. Since 83 is 43 of 12 , we can see that 1 43 12 -sized pieces fit into 87 , so 87 ÷ 12 = 1 43 . 7 8

For 45

÷

7 8

1 2

3 , you can draw a rectangle, 10

shown at right, divide it into five sections and cut each of them in half. The diagram 3 shows that there are two 10 ths in 45 with 2 2 2 3 ths left. 10 is 3 of 10 , so 45 ÷ 103 = 2 23 . 10

1 2 2 3

of

3 4

of

1 2

3 10

3 10

3 10

Method 2: Dividing Fractions Using Common Denominators To divide a number by a fraction using common denominators, express both numbers as fractions with the same denominator. Then divide the first numerator by the second. An example is at right. Method 3: Dividing Fractions Using a Super Giant One

2 5

4 ÷ 3 ÷ 103 = 10 10

=

4 3

= 1 13

To divide by a fraction using a Super Giant One, write the two numbers as a fraction, make the reciprocal of the super fraction’s denominator the fraction for the Super Giant One, then simplify as shown in the following examples.

6÷

4

3 4

4

3 4

60

÷

2 5

6! 4

= 63 !!!!!! 43 !!!= 13 = 6 ! 43 = 24 =8 3

=

3

3 5 3!5 4 !!!!!! 2 !!!= 4 2 2 5 1 5 2

=

3!5 4 2

= 15 = 1 87 8 Making Connections: Course 1

Notes:

MATH NOTES DIVIDING BY FRACTIONS, PART 2 Division of Fractions with the Invert and Multiply Method: In Chapter 8 you used a Super Giant One to divide fractions. Now that you have some experience with dividing fractions, you can generalize this process to simplify your work. Read the following example of dividing fractions using the Super Giant One method: 3 4

÷

2 5

=

3 5 3!5 4 !!!!!! 2 !!!= 4 2 2 5 1 5 2

=

3!5 4 2

= 15 8

Notice that the result of multiplying by the Super Giant One in this example, and all of the other examples in Chapter 8 that used a Super Giant One to divide, is that the denominator of the super fraction (also called a complex fraction) is always 1. In addition, the numerator is the product of the first fraction and the reciprocal of the second fraction (divisor). We can generalize division with fractions and name it the invert and multiply method. To use this method, take the first fraction and multiply it by the reciprocal of the second fraction. Some students prefer to say “flip” the second fraction and multiply it by the first fraction. If the first number is an integer, write it as a fraction over 1. Here is the same problem in the example above solved using this method: 3 4

÷

2 5

=

3!5 4 2

= 15 8

CALCULATING PERCENTS BY COMPOSITION Calculating 10% of a number and 1% of a number will help you to calculate other percents by composition.

1 10% = 10 1 1% = 100

To calculate 13% of 25, you can think of 10% of 25 + 3(1% of 25). 1 10% of 25 ! 10 of 25 = 2.5 and 1 1% of 25 ! 100 of 25 = 0.25 so

13% of 25 ! 2.5 + 3(0.25) ! 2.5 + 0.75 = 3.25 To calculate 19% of 4500, you can think of 2(10% of 4500) – 1% of 4500. 1 10% of 4500 ! 10 of 4500 = 450 and 1 1% of 4500 ! 100 of 4500 = 45 so

19% of 4500 ! 2(450) " 45 ! 900 " 45 = 855 66

Making Connections: Course 1