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A chapter from a mathematics textbook focusing on rational numbers. It covers topics such as comparing and ordering rational numbers, performing operations on rational numbers in decimal and fraction form, determining square roots of perfect square rational numbers, and finding approximate square roots of non-perfect square rational numbers. The document also includes examples, exercises, and problem-solving strategies.
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When you think of your favourite game, what comes to mind? It may be a computer game or video game. You may also enjoy playing games that have been around a lot longer. These may include the use of a game board and may involve cards, dice, or specially designed playing pieces. Examples of these games include chess, checkers, dominoes, euchre, bridge, Monopoly™, and Scrabble®.
In this chapter, you will learn more about games and about how you can use rational numbers to describe or play them. You will also design your own game.
Web Link For more information about board games invented by Canadians, go to www.mathlinks9.ca and follow the links.
Did You Know? Canadians have invented many popular board games, such as crokinole, Yahtzee®, Trivial Pursuit®, Balderdash™, and Scruples™.
42 Chapter 2 NEL
NEL
A Frayer model is a tool that can help you understand new terms. Create a Frayer model into your math journal or notebook.
1. Write a term in the middle. 2. Define the term in the first box. The glossary on pages 494 to 500 may help you. 3. Write some facts you know about the term in the second box. 4. Give some examples in the third box. 5. Give some non-examples in the fourth box.
Definition Facts
Examples Non-examples
Rational Number
rational number perfect square
non-perfect square
NEL Chapter 2^^43
Math Link
Problem Solving With Games
Millions of Canadians enjoy the challenge and fun of playing chess. Early versions of this game existed in India over 1400 years ago. The modern version of chess emerged from southern Europe over 500 years ago.
In this chapter, you will describe or play other games by solving problems that involve decimals, fractions, squares, and square roots. You will then use your skills to design a game of your own.
NEL Math Link^^45
2.
Focus on…
After this lesson, you will be able to…
Comparing and Ordering
Rational Numbers
The percent of Canadians who live in rural areas has been decreasing since 1867. At that time, about 80% of Canadians lived in rural areas. Today, about 80% of Canadians live in urban areas, mostly in cities. The table shows changes in the percent of Canadians living in urban and rural areas over four decades.
Decade
Change in the Percent of Canadians in Urban Areas (%)
Change in the Percent of Canadians in Rural Areas (%) 1966 - 1976 +1.9 - 1. 1976 - 1986 +1.0 - 1. 1986 - 1996 +1.4 - 1. 1996 - 2006 +2.3 - 2.
How can you tell that some changes in the table are increases and others are decreases?
Did You Know? An urban area has a population of 1000 or more. In urban areas, 400 or more people live in each square kilometre. Areas that are not urban are called rural. What type of area do you live in?
Cape Dorset, Nunavut
46 Chapter 2
Link the Ideas
Example 1: Compare and Order Rational Numbers
Compare and order the following rational numbers.
__ 5 - __^7 8
You can estimate the order.
__ 5 is a little less than - 0.5.
__ 5 , __^45 ,^7 __ 8.
Express all the numbers in the same form. You can write the numbers in decimal form.
__ 5 = - 0.555… -^7 __ 8
Place the numbers on a number line.
7 8
_ 4 5
_ 7 8
The numbers in ascending order are - 1.2, -^7 __ 8
__ 5 ,^4 __ 5
, and^7 __ 8
The numbers in descending order are __^7 8
__ 5, - __^7 8
, and - 1.2.
Compare the following rational numbers. Write them in ascending order and descending order.
__ 3 - 0.6 - __^3 4
Show You Know
Draw a Diagram
Strategies
What number is the opposite of - __^78? How does the position of that number on the number line compare with the position of -^7 __ 8?
48 Chapter 2
Example 2: Compare Rational Numbers
Which fraction is greater, -^3 __ 4
or - __^2 3
Method 1: Use Equivalent Fractions You can express the fractions as equivalent fractions with a common denominator.
A common denominator of the two fractions is 12.
× 3 × 4
× 3 × 4
When the denominators are the same, compare the numerators.
, because - 8 > - 9.
-^2 __ 3
is the greater fraction.
Method 2: Use Decimals You can also compare by writing the fractions as decimal numbers.
-^3 __ 4
__ 6
__ 6 > - 0.
-^2 __ 3
is the greater fraction.
Literacy Link The quotient of two integers with unlike signs is negative. This means that
and
Web Link For practice comparing and ordering rational numbers, go to www. mathlinks9.ca and follow the links.
Which fraction is smaller, - ___^7 10
or - __^3 5
Show You Know
How do you know 12 is a common denominator?
How does the number line show the comparison? − 12 __^9 = − 34 _ − 12 __^8 = − _^23
− 1 0
2.1 Comparing and Ordering Rational Numbers 49
Check Your Understanding
incorrectly on a number line, as shown.
− 3 − 2 − 1 0 How could you use the idea of opposites to show Laura how to plot - 21 __ 2
correctly?
as a decimal, and then she compared the two decimals.
a) Which method do you prefer? Explain. b) Which is greater, - 0.9 or -^7 __ 8
? Explain how you know.
For help with #4 to #9, refer to Example 1 on page 48.
− 3 − 2 − 1 0 + 1 + 2 + 3
A B C D E
a) 3 __ 2
b) - 0.7 c) - 21 __ 5 d) 14 ___ 5
e) - 11 __ 3
− 2 − 1 0 + 1 + 2
V W X Y Z
a) - 12 __ 5
b) 3 __ 4
c) 1 ___^1 20 d) - 13 __ 5
e) - 0.
__ 4
a) 8 __ 9
b) - 1.2 c) 2 ___^1 10
d) -^11 ___ 3
a) - 4.
__ 1 b) 4 __ 5
c) - 53 __ 4
d) 9 __ 8
, - 0.1, 1.9, and -^1 __ 5
Write the numbers in ascending order.
__ 8 , __^9 5
, and - 1. Write the numbers in descending order.
2.1 Comparing and Ordering Rational Numbers 51
For help with #10 to #13, refer to Example 2 on page 49.
b) 10 ___ 6 c) - ___^9 12
d) - ___^4 3
b) - ___^4
d) ___^7
a) 1 __ 3
b) - ___^9 10
c) -^1 __ 2
d) - 21 __ 8
a) 4 __ 7
b) -^4 __ 3
c) - ___^7 10
d) - 13 __ 4
For help with #14 to #17, refer to Example 3 on page 50.
b) -^1 __ 2
c) -^5 __ 6
, 1 d) -^17 ___ 20
b) - 22 __ 3
c) 1 __^3 5
d) - 3 ____^1 100
b) growth of 2.9 cm
c) 3.5 m below sea level
d) earnings of $32.
e) 14.2 °C below freezing
52 Chapter 2
Play the following game with a partner or in a small group. You will need one deck of playing cards.
5
5 represents - 5
4
4
represents 4
With a five of clubs and a four of hearts, you can make ___^4
___^4
and ___^15
__ 3 and 0. c) - 0.7 and - 0. d) - 0.66 and - 0.
__ 6
and 0.
__ 6? Explain.
c) - ___^3 ■
d) - 1.5■ 2 > - 1.
e) -^3 __ 4
< - 0.7■ f) - 51 __ 2
g) - 23 __ 5
h) __^8 ■
= ____ x
b) x __ 3
c) 5 __ x = -^20 ___ 12
d) - ___^6
= 30 ___ x
54 Chapter 2
2.
Focus on…
After this lesson, you will be able to…
Problem Solving With Rational
Numbers in Decimal Form
In Regina, Saskatchewan, the average mid-afternoon temperature in January is - 12.6 °C. The average mid-afternoon temperature in July is 26.1 °C. Estimate how much colder Regina is in January than in July.
Explore Multiplying and Dividing Rational Numbers in Decimal Form
Did You Know? As Canada’s sunniest provincial capital, Regina averages almost 6.5 h of sunshine per day. That is over 2 h per day more sunshine than St. John’s, Newfoundland and Labrador. St. John’s is the least sunny provincial capital.
2.2 Problem Solving With Rational Numbers in Decimal Form 55
Estimate and calculate. a) - 4.38 + 1. b) - 1.25 - 3.
Show You Know
Example 2: Multiply and Divide Rational Numbers in Decimal Form
Estimate and calculate.
a) 0.45 × (-1.2) b) - 2.3 ÷ (-0.25)
a) Estimate. 0.5 × (-1) = - 0.
Calculate. Method 1: Use Paper and Pencil You can calculate by multiplying the decimal numbers. 0.45 × 1.2 = 0. Determine the sign of the product. 0.45 × (-1.2) = - 0.
Method 2: Use a Calculator C 0.45 × 1.2 +^ -^ =^ –0.
b) Estimate.
Calculate. C 2.3 +^ -^ ÷^ 0.25 +^ -^ =^ 9.
Estimate and calculate. a) - 1.4(-2.6) b) - 2.76 ÷ 4.
Show You Know
Literacy Link Parentheses is another name for brackets. They can be used in place of a multiplication sign. For example,
How do you know what the sign of the product is?
2.2 Problem Solving With Rational Numbers in Decimal Form 57
Example 3: Apply Operations With Rational Numbers in Decimal Form On Saturday, the temperature at the Blood Reserve near Stand Off, Alberta decreased by 1.2 °C/h for 3.5 h. It then decreased by 0.9 °C/h for 1.5 h. a) What was the total decrease in temperature? b) What was the average rate of decrease in temperature?
a) The time periods can be represented by 3.5 and 1.5. The rates of temperature decrease can be represented by - 1.2 and - 0.9.
Method 1: Calculate in Stages You can represent the temperature decrease in the first 3.5 h by 3.5 × (-1.2) = - 4.2.
You can represent the temperature decrease in the last 1.5 h by 1.5 × (-0.9) = - 1.35.
Add to determine the total temperature decrease.
The total decrease in temperature was 5.55 °C.
Method 2: Evaluate One Expression The total temperature decrease can be represented by 3.5 × (-1.2) + 1.5 × (-0.9).
Evaluate this expression, using the order of operations. 3.5 × (-1.2) + 1.5 × (-0.9) = - 4.2 + (-1.35) = - 5.
You can also use a calculator. C (^) 3.5 × (^) 1.2 + - + (^) 1.5 × (^) 0.9 + - = (^) -5. The total decrease in temperature was 5.55 °C.
b) The average rate of decrease in temperature is the total decrease divided by the total number of hours. The total number of hours is 3.5 + 1.5 = 5.
The average rate of decrease in temperature was 1.11 °C/h.
4 × (-1) = - 4 1.5 × (-1) = - 1.
Literacy Link Order of Operations
4 × (-1) + 1.5 × (-1) = - 5.
Why are the time periods represented by positive rational numbers? Why are the rates of temperature decrease represented by negative rational numbers?
58 Chapter 2
For help with #4 and #5, refer to Example 1 on page 56.
For help with #6 and #7, refer to Example 2 on page 57.
For help with #8 to #11, refer to Example 3 on page 58.
Literacy Link In - 1.1[2.3 - (-0.5)], square brackets are used for grouping because - 0.5 is already in parentheses.
Literacy Link A share is one unit of ownership in a corporation.
60 Chapter 2
2.5 is 25 ___ 10 or 25 tenths. -43 tenths + 25 tenths is -18 tenths. -18 tenths is -18___ 10 or -1.8. So, -4.3 + 2.5 = -1.8. a) Use Bella’s method to determine 6.1 + (-3.9). b) How could you modify Bella’s method to determine 1.25 - 3.46?
0.83 m
12 cm
Web Link To find out prices of gas in Calgary, go to www.mathlinks9.ca and follow the links.
2.2 Problem Solving With Rational Numbers in Decimal Form 61