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CHAPTER 3
Rational
Numbers
3.1 – WHAT IS A
RATIONAL NUMBER?
Chapter 3
WHAT ARE SOME NUMBERS BETWEEN 11 AND -12?
-13
-12
-11
-10
EVALUATE
18
6
-18
6
18
-6
It doesn’t matter if the negative sign is on the numerator, or on the
denominator—the fraction is still negative. They are equivalent.
-18 -1 18
´
=
6
-1 -6
The same way that positive integers all have negative counterparts (or opposites),
each fraction has a negative opposite as well.
RATIONAL NUMBERS
A rational number is any number that can be written in the
form of a/b where a and b are integers, and b ≠ 0.
In other words, the set of rational numbers includes all integers, fractions
and terminating or repeating decimals.
Rational Numbers
Non-Rational Numbers
EXAMPLE
Find three rational numbers between each pair of numbers.
a) –0.25 and –0.26
b) –1/2 and –1/4
a) Remember, we can always add a zero to the end of a decimal, without changing
the value. So, –0.25 and –0.26 can also be written as –0.250 and –0.260.
What numbers are between 250 and 260?
 251, 252, 253, etc…
 So possible answers are –0.251, –0.252, –0.253.
b) Try it!
What are some different ways that we could solve this?
EXAMPLE
Order these rational numbers from least to greatest:
1.13, –10/3, –3.4, 2.777… , 3/7, –2 2/5
Putting all of the numbers into decimal form may be the easiest way to do these
types of questions.
–10/3 = –3.333…
3/7 = 0.429
–2 2/5 = –2.4
Remember, negatives are smaller than positives:
 –3.4, –3.333…, –2.4, 0.429, 2.777…
Now, put the original numbers back in:
 –3.4, –10/3, –2 2/5, 3/7, 2.777…
BINGO
Number Line Handout
PG. 101-103, # 8, 9, 10,
16, 21, 22, 24
Independent
practice
3.2 – ADDING RATIONAL
NUMBERS
Chapter 3
3.3 – SUBTRACTING
RATIONAL NUMBERS
EXAMPLE
Evaluate:
Method 1:
1
5
-3 + 2
3
6
+1
+1
-3
-2
 So, the answer is –3/6 = –1/2.
Method 2:
1
5
10 17
20 17
3
1
-3 + 2 = - +
=- + =- =3
6
3 6
6
6
6
2
+5/6
-1
0
EXAMPLE
Evaluate:
3.1 + (–1.2)
Or:
3.1 + (–1.2) = 3.1 – 1.2 = 1.9
EXAMPLE
A diver jumps off a cliff that is 14.7 metres above sea level. After hitting the water, he
plunges 3.8 metres below the surface of the water before returning to the surface.
a) Use rational numbers to represent the difference in heights from the top of the cliff
to the bottom of his dive. Sketch a number line.
b) The water is 5.6 metres deep. What is the distance from the ocean floor to the
bottom of the dive?
a)
b)
14.7
 –3.8 – (–5.6) = –3.8 + 5.6
= 1.8 m
 14.7 – (–3.8) = 14.7 + 3.8
= 18.5 m
What would have happened if
we had subtracted –3.8 from
–5.6 instead? What would our
answer be?
0
–3.8
–5.6
PG. 111, # 3AC, 4AC, 9,
11, 12, 15, 20.
PG. 119, # 11, 12, 14,
15, 16
Chapter 3
3.4 – MULTIPLYING
RATIONAL NUMBERS
Chapter 3
MULTIPLICATION
6 ´ 8 = 48
(-7) ´ 9 = -63
6 8 48
16
´ =
=
-7 9 -63 -21
MULTIPLYING RATIONAL NUMBERS
When multiplying rational numbers:
• use the procedures for determining the sign of the
product of 2 integers
• for fractions, use the procedures you already know
about multiplying fractions
• for decimals, use the procedures you already know
about multiplying decimals
What happens when we multiply two negative numbers?
How about a positive and a negative?
EXAMPLE
Evaluate:
a)
-11 -21
´
7
44
a) Look for common factors to
cancel.
-11 -21 -11 ´ -(3 ´ 7)
=
´
7 ´ (4 ´ 11)
7
44
-1 ´ -(3)
=
4
3
=
4
b)
2
3
(2 ) ´ (-1 )
3
4
b) Turn them into improper fractions.
2
3
8 -7
(2 ) ´ (-1 ) = ´
3
4
3 4
(2 ´ 4) -7
=
´
3
4
2 -7
= ´
3 1
-14
=
3
DIVIDING RATIONAL NUMBERS
When dividing rational numbers:
• use the procedures for determining the sign of the
quotient of 2 integers
• for fractions, use the procedures you already know about
dividing fractions
• for decimals, use the procedures you already know about
dividing decimals
HANDOUT
Answer the questions on the handout to
the fullest of your ability, because this is a
summative assessment.
EXAMPLE
Simplify, and represent as a mixed fraction.
1
1
(-4 ) ¸ (-3 )
5
3
1
1
21
10
(-4 ) ¸ (-3 ) = (- ) ¸ (- )
5
3
5
3
21
3
= (- ) ´ (- )
5
10
63
=
50
13
=1
50
EXAMPLE
Solve for x:
5
8
b) (- ) ¸ x = -
a) x ÷ (–2.6) = 9.62
15
56
a) x ÷ (–2.6) = 9.62
 x ÷ (–2.6) × (–2.6) = 9.62 × (–2.6)
 x = –25.012
b)
5
15
(- ) ¸ x = 8
56
5
15
Þ (- ) ¸ x ´ x = - ´ x
8
56
5
15
Þ (- ) = - x
8
56
5
Þ (-56)( ) = -15x
8
Þ (-7)(5) = -15x
Þ
-35
=x
-15
7
Þx=
3
PG. 127, # 6, 10, 11,
12, 14, 15, 18
PG. 134, # 6, 9, 11, 12,
17, 19, 21
Independent
Practice
3.6 – ORDER OF
OPERATIONS WITH
RATIONAL NUMBERS
ORDER OF OPERATIONS
What’s the acronym used to remember the order of operations?
BEDMAS
Brackets
Exponents
Division
Multiplication
Addition
Subtraction
EXAMPLE
Evaluate:
(-3.2) - 0.9 ¸ [0.7 - (-1.2)]2
(-3.2) - 0.9 ¸ [0.7 - (-1.2)]2 = (-3.2) - 0.9 ¸ 1.9 2
Brackets
= (-3.2) - 0.9 ¸ 3.61
Exponents
= -3.2 - 0.249
Division
= -3.4
Subtraction
EXAMPLE
To convert a temperature in degrees Fahrenheit to degrees Celsius, we use the
formula:
F - 32
C=
1.8
In Fort Simpson, the mean temperature in December is –9.4°F. What is this
temperature in degrees Celsius?
C=
F - 32
1.8
ÞC =
-9.4 - 32
1.8
ÞC =
-41.4
1.8
In cases like this, it’s as if
there are invisible brackets
around the numerator and
denominator.
Þ C = -23
The mean temperature in Fort Simpson in December
is –23° Celsius.
PG. 140, # 4, 6, 7, 9, 11,
12, 16, 18, 19, 21
Independent
Practice