Download Section 5-1 – The Set of Rational Numbers

Section 6-1 – The Set of Rational Numbers
The RATIONAL NUMBERS are numbers whose value can be expressed as the quotient or ratio of
a
two integers a and b, represented as
where b ≠ 0.
b
a is called the ___________________ and b is called the ______________________.
Proper Fractions vs. Improper Fractions
A fraction is called a proper fraction if the numerator is less than the denominator.
A fraction is called an improper fraction if the numerator is greater than or equal to the
denominator.
Examples: Proper or Improper?
a)
5
3
b)
To understand the meaning of a fraction
2
7
c)
5
5
a
, you must consider:
b
1. The whole
2. The number of b equal-sized parts into which the whole is divided
3. The number of a parts of the whole that we are selecting.
Understanding the fraction concept:
a) Divide each rectangle into 4 equal pieces
b) Divide each shape below into 3 equal pieces
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Examples:
represents 1, draw a rectangle that would represent 3/4.
1. If
2. If the following rectangle represents 2, shade the area equivalent to ½ .
represents 2/3, draw a rectangle that would represent 1.
3. If
4. If the following circle has a value of
3
, shade the area equivalent to 1.
2
5. Locate 5/6 on the number line.
-2
-1
0
1
2
2
6. Locate
-2
–4
/3 on the number line.
-1
0
1
2
7. In the following figure, represent each of the following as a fraction:
a) The dots outside the circle as a part of all the dots
b) The dots outside the rectangular region as a part of all the dots
c) The dots in the intersection of the circle and rectangle as a part of the dots in the union of the
circle and rectangle.
Two fractions are equivalent fractions if they have the same value.
“equi-valent = equal value”
Example: Show that 2/3 and 4/6 are equivalent using an area model:
3
Example: Show that 3/4 and 6/8 are equivalent using a number line:
0
1
Example: Show that 2/6 and 1/3 are equivalent using Pattern Blocks:
The Fundamental Law of Fractions:
a
a an

Let
be any fraction and let n be a non-zero whole number, then
.
b
b bn
Example: Find a value of x so that
9
x

42 126
Example: Find three fractions equivalent to
3
4
4
Simplifying Fractions:
A rational number
a
is in simplest form if a and b are relatively prime (ie, GCD(a, b) = 1).
b
Examples: Write each fraction in simplest form:
a)
64
300
b)
84
91
c)
36
144
d)
30 xy 2
16 x 2 y
e)
3a  3b
a 2  b2
f)
5a  10
10
Methods for determining if two fractions are EQUIVALENT:
1. Simplify both fractions to simplest form
Example:
60
/210
12
/42
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2. Rewrite both fractions to fractions with the same denominator. Compare numerators.
Example:
3. Cross multiply.
Example:
7
5
/10
a
c

b
d
3
/8
/8
ad  bc
iff
9
/24
Ordering Rational Numbers:
Theorem: If a, b, and c are integers and b > 0, then
a c

b b
iff
ac.
If the two fractions have different denominators, you can change them so that they have the
same denominator and then compare numerators.
Example:
Put in order from smallest to largest:
3
/4
2
/5
5
/6
7
/8
_________, _________, _________, _________
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Examples: Use pattern blocks and fraction bars to determine which fraction is larger.
Circle the fraction that is larger.
a)
3
2
or
6
3
b)
3
5
or
4
6
7
Rational numbers are called dense. This means that between any two rational numbers, there are an
infinite number of rational numbers.
How to find a fraction IN BETWEEN two other fractions:
1. Change both fractions so that they have the same denominator and choose a numerator in
between the other two.
2. Theorem: If
a
c
a
c
a
ac
c
and
are rational numbers where
< , then
<
<
b
b
b
bd
d
d
d
Example: Find two different fractions between 1/6 and 2/6
Example: Find two different fractions between 7/18 and 1/2
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Section 6-2 – Addition and Subtraction of Rational Numbers
ADDITION:
Example:
Add each of the following using Pattern Blocks and Fraction Bars:
a) 1/6 + 2/6 =
b)
1
/2 + 2/3 =
9
ADDITION:
To add two fractions that have different denominators, find the LCM of the denominators
(called the LCD), convert both fractions so they both have the LCD as their denominator, and
a c
ad  bc


then add the numerators.
b d
bd
Examples: Add the following:
a)
3
5

=
4
6
b)
2
1

=
3
25
c)
3
5
1


=
4
6
2
Mixed Numbers
2 1 is read “Two and one-third” and is the same as 2  13
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Examples: Convert the following mixed numbers to improper fractions
a)
21
b)
35
3
8
10
Examples: Convert the following improper fractions into a mixed numbers:
a)
31
4
b)
 29
25
Properties of Addition for Rational Numbers:
The Additive Inverse Property of Rational Numbers:
a
a
a
For any rational number , there exists a unique rational number,
, the additive inverse of ,
b
b
b
a a

0
such that
b
b
Examples: Find the additive inverse of the following:
a)
7
10
b)
2
5
c) 4 76
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SUBTRACTION:
To subtract two fractions that have different denominators, find the LCM of the
denominators (called the LCD), convert both fractions so they both have the LCD as their
a c
a c
ad  bc
  

denominator, and then subtract the numerators.
b d
b d
bd
Examples: Subtract the following:
a)
5
3

8
4
b)
25
2

=
4
5
=
3
2
c) 7 5  3 5 =
d) 12  4 78 =
12
Estimation with Rational Numbers:
Examples: Estimate the following quantities:
a)
b)
c)
3 15  6 87
5 2
1
 
6 5 11
4
 2 21  1 87
5
d) 2 21  1 41 
8
11
Example: A student suggests that to add 2 34 and 3 12 , you could first add the whole-number parts,
add the fraction parts, and add the results. Do you agree with the student? Justify your answer.
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Section 6-3 – Multiplication and Division of Rational Numbers
MULTIPLICATION:
Example: Multiply 4  32 using Pattern Blocks and repeated addition
1 1
Example: Multiply 3 2  3 using Pattern Blocks and repeated addition
Example: Multiply 5  3 using an area model.
2
Example:
2
1 3
Multiply 2  4 using an area model
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Definition of Multiplication of Rational Numbers:
a
c
a c
ac


If
and
are any rational numbers, then
.
b
b d
bd
d
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of the people in a certain city are college graduates and there are 6,000 people in
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the city, how many of the people in the city are college graduates?
Example: If
Example: Mary took
¾ of a pizza home in a box.
Her son ate
1
3
of what was in the box. How
much of a whole pizza did he eat?
1
because of an economic crisis
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in the state. American State College now has 350 faculty members. How many faculty members did
they have before the cut?
Example: American State College has had to reduce its faculty by
Example: A pair of shoes is on sale at ¾ of its original price. If the sale price is $66, what was the
original price?
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Properties of Multiplication for Rational Numbers:
Multiplicative Identity of Rational Numbers:
The number 1 is the unique number such that for every rational number
a
a
a
, 1 
b
b
b
Multiplicative Inverse of Rational Numbers
a
b
For any non-zero rational number , there exists a unique rational number, , the multiplicative
b
a
a
a b
inverse (or reciprocal) of , such that   1
b
b a
Examples: Find the multiplicative inverse of the following:
a)
7
10
c) 4 76
b)
2
5
d) 0
Multiplying Mixed Numbers:
Example: Multiply 3 14  4 23 using:
a) the distributive property
b) improper fractions
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Example: A student suggests that to multiply 2 12 and 3 14 , you can multiply and whole-number
parts, multiply the fraction parts, and add the results. Do you agree with the student? Justify your
answer.
DIVISION:
Recall the concept of division: 10  5 means “How many 5’s are in 10?”
Using this concept, what does 3  12 mean?
You can also think of division as repeated subtraction, ie, for 10  5 you would say, “How many
times can I subtract 5 from 10?”
So how would you think of 3  12 ?
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Example: Divide the following using the Pattern blocks:
a) 43  16 =
b)
1 32  31 =
1
c) 6
 13
=
Before, we did:
a – b = a + –b
Subtraction = Add the “additive inverse”
Now, we will do:
a ÷ b = a · b1
Division = Multiply by the “multiplicative inverse”
Note: The division problem a  b and the fraction
a
are equivalent.
b
Division Algorithm for Rational Numbers:
a
c
c
a c
a d

 
If
and
are any rational numbers and
is not zero, then
b
b d
b c
d
d
Examples:
1
a) 3
 34
Divide the following:
=
b)
3 14  2 34 =
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An Alternative Algorithm:
a c
ad bc



 ad  bc
b d
bd bd
Examples: Divide using the alternative algorithm:
1
 34
1
 83
2
 34
a) 3
b) 4
c) 5
Example: A small office purchases bottled drinking water. The water comes in containers that hold
5 gallons. Two-thirds of the office container was drained last week to put out a fire in the
wastebasket. How much water is left in the container?
19
Estimation and Mental Math with Rational Numbers:
Examples: Use mental math to find the exact answer:
a) 3 18  24
b)
2
 20
5
To estimate, use rounding or get upper and lower bounds for the answer.
Examples: Estimate the answer:
a) 3 13  2 34
b) 24 12  4 18
Examples: Suzy babysits on the weekends for extra money. She spends about 2/5 of her income
going to the movies with friends. If her monthly babysitting income is $88, how much money does
she spend on movies? (estimate the answer, then find the exact answer)
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Extending the Notion of Exponents
an = a · a ·… · a
Definition of an:
and a1 = a
Properties of Exponents:
am  an  amn
am  an  amn
a 
 amn
 ab 
 a m  bm
m n
a
 
b
m
m
b
 
a
m
a0  1
Examples: Write each of the following in simplest form using positive exponents in the final
answer.
a)
a5
=
a7
b) b5  b10 =
c) 2xy  =
3
d)
815
=
8 4
4
2
e)   =
7
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Section 6-4 – Ratios, Proportions, and Proportional Reasoning
Suppose that someone tells you that at a particular school, the ratio of students to teachers is 24:1.
What does that mean?
A ratio is a comparison of two quantities and is usually written
a
or a : b.
b
Ratios can represent part-to-part comparisons
ie, The ratio of the number of boys to girls is 3:2
Ratios can represent part-to-whole comparisons
ie. The ratio of the number of boys to the number of total students is 3:5
Notice that if the ratio of boys to girls is 3:2, that doesn’t tell you the total number of students in the
classroom.
Example: Find a ratio equivalent to 2:5.
Example: Iris has found some dinosaur bones and a fossil footprint. The length of the footprint
is 40 cm, the length of the thigh bone is 100 cm, and the length of the body is 700 cm.
a) What is the ratio of the footprint’s length to the thigh bone’s length?
b) What is the ratio of the footprint’s length to the dinosaur body’s length?
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Proportions
Two ratios are proportional if and only if the fractions representing them are equal.
Two equal ratios form a proportion.
Property of Proportions:
If a, b, c, and d are all real numbers and b ≠ 0 and d ≠ 0, then
a c
if and only if ad = bc.

b d
Example: Solve the following proportions:
a)
x 2

5 50
b)
7
4

15 3x
Examples: Set up and solve the following proportions
a) If there are 3 cars for every 8 students at a high school, how many cars are there for 1200
students?
b) If a recipe calls for 9 eggs to make 4 batches of cookies, how many eggs are needed to
make 30 batches of cookies?
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c) A Boeing-747 jet is approximately 230 feet long and has a wingspan of 195 feet.
If a scale model of the plane is about 40 cm long, what is the model’s wingspan?
d) Kirk, Valerie, and Mike worked mowing lawns. Mike worked 5 hours, Kirk worked 6
hours, and Valerie worked 3 hours. They were paid $252 and divided the money in
proportion to the number of hours worked. How much did each person earn?
e) A grasshopper can jump 20 times its length. If jumping ability in humans (height)
were proportional to a grasshopper’s (length), how far could a 6-foot-tall person jump?
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