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.
46
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
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Making Connections: Course 1