Table Of Content32014_01_ch1_p002-069 7/23/07 3:22 PM Page 3
Chapter
1
Rational
Numbers
GOALS
You will be able to
• Relate rational numbers to decimals,
fractions, and integers
• Evaluate expressions involving rational
numbers
? What might the negative values
in the photo represent?
3
NEL
32014_01_ch1_p002-069 7/23/07 3:22 PM Page 3
Chapter
1
Rational
Numbers
GOALS
You will be able to
• Relate rational numbers to decimals,
fractions, and integers
• Evaluate expressions involving rational
numbers
? What might the negative values
in the photo represent?
3
NEL
32014_01_ch1_p002-069 7/20/07 2:26 PM Page 4
1
Getting Started
WORDS YOU NEED to Know
1. Match each term with the example that most closely represents it.
a) opposite integers d) lowest common denominator g) power
b) numerator e) mixed number h) base of a power
c) denominator f) improper fraction i) exponent
34 2 10
(cid:3) (cid:2) (cid:1)
i) iv) 5 and 5 vii)
9 3 15
1 3
(cid:1)
5 15
2
ii) 1 v) 73 viii)73
3
5 5
iii) 73 vi) ix)
7 7
SKILLS AND CONCEPTS You Need
Study Aid
• For more help and practice, Addition of Fractions
see Appendix A-6 and A-7.
You can add two fractions using fraction strips, number lines, or by using a
common denominator.
EXAMPLE
1 2 Solution: fraction strips
(cid:3)
3 5
Solution: common denominator
(cid:4) (cid:4)
1 5 2 3
(cid:1) (cid:3)
(cid:4) (cid:4)
3 5 5 3
5 6 Solution: number line
(cid:1) (cid:3)
1 (cid:1) 5 2 (cid:1) 6
15 15 3 15 5 15
11
(cid:1)
15 11
0 15 1
2. Determine each sum.
1 1 3 3
(cid:3) (cid:3)
a) c)
2 3 10 5
3 1 2 2
(cid:3) (cid:3)
b) d)
4 8 5 3
4
Chapter 1 NEL
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Getting Started
Subtraction of Fractions
You can subtract two fractions using fraction strips, number lines, or by
using a common denominator.
EXAMPLE
2 1 Solution: fraction strips
(cid:2)
3 5 2 (cid:1)10
3 15
Solution: common denominator 1 (cid:1) 3
5 15
(cid:4) (cid:4)
(cid:1) 2 5 (cid:2) 1 3 10 3 (cid:1) 7
15 15 15
(cid:4) (cid:4)
3 5 5 3
10 3 Solution: number line
(cid:1) (cid:2)
3 10
15 15
15 15
7
(cid:1)
15
0 7 1
15
3. Determine each difference.
1 1 3 1 3 3 6 1
(cid:2) (cid:2) (cid:2) (cid:2)
a) b) c) d)
2 3 4 8 5 10 7 2
Multiplication of Fractions
You can use an area model to help you visualize the product of two fractions.
EXAMPLE
5 2 Solution: area model
(cid:4)
8 3 5
8
Solution: multiplying 2
3
(cid:4)
5 2
(cid:1)
(cid:4)
8 3
10
(cid:1)
24
5
(cid:1)
12
4. Determine each product.
1 1 2 3
(cid:4) (cid:4)
a) c)
2 4 5 10
2 3 1 4
(cid:4) (cid:4)
b) d)
3 4 6 5
5
NEL Rational Numbers
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PRACTICE
5. Complete the equation. Use a diagram to show how you got the answer.
Study Aid
(cid:1)
7
• For help, see the Review (cid:1)
5 6
of Essential Skills and 4 ˛4
Knowledge Appendix.
6. The number line represents the sum of two numbers. Write an
Question Appendix
equation to show the
6 and 7 A-6 numbers being added
and their sum. 0 1 2
8 A-7
1 2
9 and 10 A-3
7. a) Use a number line to determine how much greater 4 is than 1 .
3 3
11 A-4
b) Write the subtraction equation that gives the same result as in part a).
12 A-9
8. Evaluate.
13 A-1
2 9 3 9
(cid:4) (cid:5)
a) b)
14 A-2 5 8 4 10
9. Evaluate.
1(cid:2) 2 (cid:3) (cid:2) 1(cid:2) 2
a) 3 7 c) 11 4
1(cid:2) 2 (cid:3) 1(cid:2) 2 (cid:2) (cid:2) 1(cid:2) 2
b) 5 2 d) 3 8
10. Tiger Woods finished a golf tournament with a score of five under par.
Another player finished with a score of two over par. How many shots
behind Tiger Woods was the other player?
11. Evaluate.
(cid:2) (cid:4) (cid:5) 1(cid:2) 2
a) 4 6 c) 20 1
1(cid:2) 2 (cid:4) 1(cid:2) 2 1(cid:2) 2 (cid:5) 1(cid:2) 2
b) 5 3 d) 12 6
12. Evaluate.
(cid:3) (cid:4)
a) 0.32 3.9 c) 0.6 1.1
(cid:2) (cid:5)
b) 15.4 3.91 d) 24 1.2
13. Express each power as repeated multiplication, and then, evaluate.
a) 82 b) 5.23
14. Follow the order of operations to determine the value of each expression.
(cid:2) (cid:3)
12 3
a) b) 72 (cid:2) 1(cid:2)6 (cid:3) 22 (cid:4) 4
(cid:3)
4 5
15. Copy and complete the Definition What do you know
chart to show what you about them?
know about mixed Mixed
numbers. Numbers
Examples Non-examples
6
Chapter 1 NEL
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Getting Started
APPLYING What You Know
Fraction Patterns
Andrew and Kim are exploring a sequence of fractions.
1 1 1 1 1 1
— — — — — —
1 2 3 4 5 6 ...
+ + + + +
3 5
— —
2 6
? How can you use patterns to predict the 20th sum,
difference,product, and quotient using these fractions?
A. Describe the pattern in Andrew and Kim’s original sequence of
fractions.
B. Add consecutive terms (pairs of neighbouring fractions) until five sums
1 1 3 1 1 5
(cid:3) (cid:1) (cid:3) (cid:1)
have been calculated. For example: , , etc.
1 2 2 2 3 6
C. Describe the pattern and use it to predict the 20th sum.
D. Copy Andrew and Kim’s original fraction sequence.
Subtract consecutive terms instead of adding them.
E. Describe the pattern and use it to predict the 20th difference.
F. Repeat parts D and E using multiplication.
G. Repeat parts D and E using division.
H. Explain why the values in the pattern decrease for each operation.
I. Which patterns do you think were the easiest to predict? Why?
7
NEL Rational Numbers
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1.1
Addition and Subtraction
of Mixed Numbers
YOU WILL NEED
GOAL
• fraction strips
Add and subtract mixed numbers using a variety of methods.
LEARN ABOUT the Math
London,
England
5—3 h 2—1 h
4 3
Toronto,
Ontario
Rome,
Italy
Alisa and Greg plan to travel from Canada to Italy for March break.
Dawn is driving to her aunt’s in Ottawa for March break. According to a
travel website, Dawn’s trip should take 4 h 36 min.
? How many hours longer is Alisa and Greg’s trip than Dawn’s trip?
EXAMPLE 1 Selecting a strategy to add mixed numbers
Calculate the time to travel from Toronto to Rome by plane.
Alisa’s Solution: Using a strategy involving equivalent
fractions
3 1
5 (cid:3) 2 I estimated the answer to be
4 3
between 8 h and 9 h because
3 1
5(cid:3)2(cid:1)7and (cid:3) is greater
4 3
than 1 but less than 2.
8
Chapter 1 NEL
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1.1
I knew that I needed a common
9 4
(cid:1) (cid:3)
5 2 denominator when adding
12 12
fractions. I chose 12 since it is
1 the least common multiple (LCM) least common multiple (LCM)
1 of 4 and 3. the least whole numberthat
1 has all given numbers as factors
(e.g., 12 is the least common
1
multiple of 4 and 6)
1
1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 4 I used fraction strips to determine
3 9 1 4
that (cid:1) and (cid:1) .
1 4 12 3 12
1
1 1 1 1 1 1
12 12 12 12 3 3
13 I added the whole parts, and
(cid:1)
7
12 then, I added the twelfths.
1
1
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 12 12 12 13 twelfths was the same as
1
12 1 whole and 1 twelfth, so I had
8 wholes and 1 twelfth.
1
(cid:1) (cid:3)
7 1
12
(cid:1) 1 1
8 8 was between my estimate
12 12
of 8 h and 9 h, so I thought that
1
So, our trip will take 8 h. my answer was reasonable.
12
9
NEL Rational Numbers
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Greg used a different strategy from Alisa. He changed the mixed numbers
into improper fractions in order to add them.
Greg’s Solution: Using a strategy involving improper fractions
3 1
(cid:3) 1 1
5 2 is a bit more than .
4 3 3 4
3 1
That meant that 5 (cid:3)2 had to
4 3
be a bit more than
3 1
5(cid:3)2(cid:3) (cid:3) (cid:1)8.
4 4
I estimated our trip would take a
bit longer than 8 h.
23 7
(cid:1) (cid:3) I renamed the mixed numbers as
4 3 improper fractions.
(cid:4) (cid:4)
23 3 7 4
(cid:1) (cid:3) I used a common denominator of
(cid:4) (cid:4)
4 3 3 4 12 to add the fractions. I changed
69 28 the numerators so that the
(cid:1) (cid:3)
12 12 fractions would be equivalent,
(cid:3) and then, I added them.
69 28
(cid:1)
12
97
(cid:1)
I renamed the improper fraction
12
as a mixed number so that I
1
(cid:1)
8 could see how many hours the
12
trip would take.
This was close to my estimate,
1 so I thought that it was a
So, our trip will take 8 h.
12 reasonable answer.
Both Greg and Alisa have determined the time it will take to fly from
Toronto to Rome. Now, they must calculate how much longer this is than
Dawn’s trip to Ottawa.
10
Chapter 1 NEL
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1.1
EXAMPLE 2 Selecting a strategy to subtract
mixed numbers
Calculate the difference in time between the trip to Rome and the trip
to Ottawa.
Alisa’s Solution: Using a strategy involving equivalent
fractions
Dawn’s travel time:
Since there are 60 min in an
4 h 36 min hour,
36 36
(cid:1) 4 h 36 min(cid:1) h.
60
60
(cid:1) 3 I subtracted to determine the
4 h
5 difference in travel times.
3
8(cid:2)4(cid:1)4, and is more than
1 3
(cid:2) 5
8 4
12 5 1
, so I estimated the difference
2
to be between 3 h and 4 h.
1 (cid:4) 5 3 (cid:4) 12 I chose 60 as the common
(cid:1) 8 (cid:2) 4 Communication Tip
(cid:4) (cid:4) denominator since it is the
12 5 5 12
LCM is an abbreviation for
LCMof 12 and 5.
5 36 Least Common Multiple.
(cid:1) (cid:2)
8 4
60 60
60 (cid:3) 5 36 I noticed that if I subtracted
(cid:1) (cid:2)
7 4 5 36
60 60 (cid:2) , the numerator would
60 60
65 36 5
(cid:1) 7 (cid:2) 4 be negative. I renamed 8 to
60
60 60
make the first numerator greater
65 36
(cid:1) 1 (cid:2) 2 (cid:3) a (cid:2) b than the second.
7 4
60 60
1 (cid:2) 2
65 36 I subtracted the whole numbers,
(cid:1)
3
60 and then, the fractions.
29
(cid:1) 3 29
60 3 h was within my estimate of
60
between 3 h and 4 h, so I
29
So, we will take 3 h longer than thought that my answer was
60
reasonable.
Dawn will to arrive at our destination.
11
NEL Rational Numbers
