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GOALS

You will be able to

• interpret, compare, and order rationalnumbers

• add, subtract, multiply, and dividerational numbers using the correct orderof operations

• solve problems using a guess-and-teststrategy

1Chapter

RationalNumbers

NEL 1

What might the negative valuesin the photograph represent?

MF9SB_CH01_p1-30 pp6.qxd 11/17/08 11:48 AM Page 1

CHAPTER 1 Getting Started

Fraction Patterns

NEL2 Chapter 1 Rational Numbers

David and Jia-Wen are exploring a sequence of fractions.

How can you use patterns to predict the 20th sum, difference,product, and quotient using this sequence of fractions?

A. Describe the pattern in David and Jia-Wen’s original sequence offractions.

B. Add consecutive terms (pairs of neighbouring fractions) until five sums have been calculated. For example, and so on.

C. Describe the pattern and use it to predict the 20th sum.

D. Subtract consecutive terms of David and Jia Wen’s original sequenceinstead of adding them, until five differences have been calculated.

12 1

13 5

56,1

1 112 5

32,

?

11 �

�

12

13

14

15

16 ...�

��

��

��

32

56

� � � � ��

11

12

13

14

15

16 ...


E. Describe the subtraction pattern and use it to predict the 20thdifference.

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 addition,subtraction, and multiplication.

I. Why might it have been harder to predict what the division patternwould be?

J. Which patterns do you think were the easiest to predict? Why?

WHAT DO You Think?Decide whether you agree or disagree with each statement. Be ready toexplain your decision.

1. You can add and only by using a common denominator.45

12

NEL 3Getting Started

�

2. When you divide two negative numbers, the quotient must begreater than

3. To subtract one negative number from another, just subtract theiropposites and then take the opposite of the difference.

4. The only number between and is 13.14

12

28 4 (24) 5 ??

11.

10


Interpreting Rational Numbers1.1

Relate rational numbers to fractions and integers.

GOAL


rational number

a number that can beexpressed as the quotient oftwo integers, where the divisoris not 0; it can be written infraction, mixed-number, ordecimal form or as an integer,

e.g., 3.25, 211422,2

3,25.8,

opposites

two numbers with oppositesigns that are the samedistance from 0, e.g., and

or and 20.510.52212

LEARN ABOUT the MathRachel looked at the thermometer outside. The temperature was between and

What might the temperature be?

A. What is the value of the rational number located at P ? Explain.

B. What other form of that number could you use to describe P ?

C. What is the value of the rational number located at R ? Explain.

D. Why are the values at P and R both possible solutions to Rachel’sproblem?

E. List three other possible solutions to the problem. Make sure one iswritten in decimal form, one as the opposite of an improperfraction, and one as the opposite of a mixed number.

Reflecting

F. How could thinking about the opposites of and help yousolve Rachel’s problem?

G. How do you know that between any two rationals there are alwaysmany other rationals?

H. How do you know that any integer or fraction is also a rationalnumber?

219218

R�19 �18

?

219 °C.218 °C

Sometimes rational numbersare called rationals. Rational

numbers like and are

read as “negative six tenths”and “positive five eighths“ orjust “five eighths.”

5820.6

Communication Tip

–30

–40

–20

–10

0

P


NEL 51.1 Interpreting Rational Numbers

WORK WITH the Math

EXAMPLE 1 Recognizing equivalent rational numbers

Which of the following represent the same rational number?

Thomas’s Solution

21.52112,2

64,26

24,322,23

2 ,

I will label the numbers to make sure that I use them all.

Since then

Since then 232 5 211

2.32 5 11

2,

264 5 2

32.6

4 532,

The number halfway between and should be

but it’s also since 12 5 0.5.21.5

2112,2221

A B C D E F

�2 –1

�1.5

�112

21.52112,2

64,26

24,322,23

2 ,

so E 5 F2112 5 21.5

so E 5 D21

12 5 2

64

232 5 21

12

264 5 2

32

C is different from D, E, and F.I know that When you divide two negatives,

the result is positive, so this cannot be the same as 21.5.

2624 5 (26) 4 (24).

The only ones left to think about are and

If there is a loss of $3 that 2 people share, each loses $1.50, so it

makes sense that 232 5 21.5.

322.23

2

A 5 F 5 21.5

232 5 (23) 4 2

Since theequivalent rational numbers are

and 21.5.2112,2

64,3

22,232 ,

A 5 B 5 D 5 E 5 F, All five of these numbers are located at the same place on anumber line.

To figure out the quotient, I think about what number tomultiply by to get It has to be negative to get apositive product. It has to be since 1.5 3 2 5 3.21.5

13.(22)

B 5 F 5 21.5

322 5 3 4 (22)



List five different rational numbers between and

Larissa’s Solution

213.2

15

EXAMPLE 2 Locating a rational number between two other rational numbers

–1 10– 1

3– 1

515

13

– 14

14

It helps to place and on a number line first. I did this by

placing their opposites on the number line, and then reflectingacross zero.Opposites must be the same distance from 0. Since is between

and I know that must be between and 213.2

152

14

15,1

3

14

2132

15

Since 0.3 and 0.28 are between 0.2and 0.333..., then and must be between and 21

3.215

20.2820.3

215 5 20.2

213 5 20.3333… I changed the fractions to decimals to get other in-between

rational numbers.

and

so and are between them.28302

730

213 5 2

1030,2

15 5 2

630

I used equivalent fractions.

Five rational numbers between and are

and 2 830.2

730,20.28,20.3,2

14,

2132

15

could be a temperature betweenand

m could be the number of metres belowsea level that a scuba diver is.

could be the number of seconds beforethe launch of a spacecraft, if you call the launchof the spacecraft time 0.

23.4

23.4

24 °C.23 °C23.4 I know that you can

have negativetemperatures, negativedistances, and negativetimes (times before anevent), so I’ll use thoseideas.

Name three situations that you might describe using the rationalnumber

Sam’s Solution

23.4.

EXAMPLE 3 Using a rational number to describe a situation


In Summary

Key Idea

• Rational numbers include integers, fractions, their decimal equivalents,and their opposites.

Need to Know

• Rational numbers can be positive, negative, or zero.• Every integer is a rational number because it can be written as the quotient

of two integers.

For example, some ways can be written are , and

• Just as with fractions, there are many ways to write the same rationalnumber.

For example,

• Every rational number (except 0) has an opposite. For example,

and are opposites, since they are both the same distance from 0 on a

number line.

2

34

22

34

22.5 5252 5 2

52 5

522 5 22.50 5 2

104 .

2102 .5

21,25125

Checking

1. Write the following rational numbers as quotients of two integers.

a) b) c) d) 7.2

2. Write the following rational numbers in decimal form.

a) b) c) d)

3. List three rational numbers between each pair.

a) and b) and c) 0.6 and

Practising

4. Multiple choice. Which values describe the positions of X and Y ?

A. and C. and

B. and D. and

5. Multiple choice. Which of these rational numbers are equivalent?

A. X, Z, and W B. X and Z C. X and W D. X and Y

6. Identify the values represented by B, D, K, M, and T as quotientsof two integers.

W: 2410Z: 242

10Y: 22.4X: 24.2

322

29421.222.1

21

1222

3421.522.2

1

18

142

122

452

23

4

31027

5622

382

14

25

6722

1420.5

�2 �1 0

X Y

�3

T M K D B

�4 �2 0 2

NEL 71.1 Interpreting Rational Numbers

Reading Strategy

Visualizing

Use number lines tovisualize your answersto question 3.


7. Write each as the quotient of two rational numbers.a) b) c) d)

8. Write each in decimal form.a) b) c) d)

9. Draw a number line. Mark integers from to 2 on it. Estimateand mark the location of each of these on your number line.a) c) e) g)

b) d) f ) h)

10. a) Describe a real-life situation that involves the number b) Describe a real-life situation that involves the number

11. Explain why

12. a) Name three fractions between and b) How would your answers in part a) help you name three

rational numbers between and c) Are your answers in part a) rational numbers? Explain.

13. Agree or disagree with each. Explain why.a) The opposite of every mixed number can be written as a

rational number in decimal form.b) If one number is greater than another, so is its opposite.

14. The natural numbers are the numbers 1, 2, 3, 4,...The whole numbers are the numbers 0, 1, 2, 3,...Complete this Venn diagram to show the relationship betweenthese sets of numbers: integers, whole numbers, natural numbers,and rational numbers.

Closing

15. If you were describing rational numbers to someone without justrepeating the definition, what is the most important thing you couldsay to help them quickly understand what rational numbers are?

Extending

16. a) Explain why can be written as the repeating decimal (or ).

b) How would you write as a decimal?

c) How would you write as a repeating decimal?d) How are your answers to parts b) and c) related?

17. Design an appropriate symbol for the term rational number.Explain why your symbol is appropriate.

2590

259

0.50.555....5

9

238?2

28

38.2

8

234 5

234 5

324.

223.

21.2.

28.42326.912

3

20.7823

2327.22

52

210

213252

19327

245

25

9

2323.0224

155.1


Naturalnumbers

Whole numbers


How Many Rational Numbers Are There?You might think there are twice as many rational numbers between 0 and asbetween 0 and There are actually the same number. To show that this is true, youneed to show that you can match each rational between 0 and with exactly onerational number between 0 and and vice versa.

Write each rational number between 0 and on the bottom number line as

Match it with on the top number line.

1. Choose other rational numbers between 0 and Use the matching rule matches to show that each number you chose matches one number

between 0 and

2. List different rational numbers between 0 and Show that each matchesone number between 0 and

3. How do you know that if is between 0 and then is between 0 and

4. Why does it make sense that if there is a one-to-one match between two setsof numbers, then the sets are the same size?

5. How many rational numbers are there between 0 and or between 0 and 22?

21

21?2

a2b22,2

ab

22.21.

21.2

ab2

a2b

22.

–2 0–1– 8

5– 4

3– 7

6– 2

3

– 810

– 712

– 46

– 26

0–1

2a

2b

2ab.22

2122

21.22

Curious MATH

NEL 9Curious Math



Comparing and OrderingRational Numbers

1.2

Order a set of rational numbers.

GOAL

LEARN ABOUT the MathLarissa described going east from her house as moving in a positivedirection and going west as moving in a negative direction.

On five different days, she described how far she was from home:

km km km 5.1 km km

On which day was Larissa farthest from home?

A. Draw a number line from to 10. Estimate and mark the positionof each of Larissa’s locations on the number line if her home is at 0.

B. List the location numbers from least to greatest.

C. Which position was farthest from home?

Reflecting

D. How does your answer to part C show that just because a number isfarthest from 0 in a list of numbers, it might not be the greatest?

E. How could you have looked at the location numbers and orderedthem without drawing them on the number line?

WORK WITH the Math

210

?

21.425.325.223.7

First I wrote as a mixed number. It’s between 2 and 3, so

it’s less than 12.1 and

Since these are the only positives, I know they are the greatest.

15

23.

83

Order these numbers from least to greatest.

12.1

David’s Solution

8325

3415

2325

1223.4

EXAMPLE 1 Ordering rationals in fraction and decimal form

The three greatest numbers are the positives.83 5 2

23 , 12.1 , 15

23


NEL 111.2 Comparing and Ordering Rational Numbers

I know that is the greatest. I decided to write theother numbers in fractional form as decimals, since Ifind decimals easier to work with.

15

23

I know that and that the other two negativesare less than so is the greatest negative.23.424,

23.4 . 24and 25

3423.4 . 25

12

I divided the part of the number line between andinto 4 equal sections to place the last two numbers.

is farther to the left, so it’s least.25

34

2625

�6 �5

�5 34

�5 12

The order of the numbers from least togreatest is

12.1 15

23

8323.425

1225

34

The five numbers that are less than are

, 12.1, and 1.67.25.75,25.5,23.4

15

23

83 5 1

23 8 1.67

25

34 5 25.75

25

12 5 25.5

I know that is greater than the other negativevalues and that 1.67 and 12.1 are greater than all theother decimals.

I know that since the opposite of 5.5 iscloser to 0 than the opposite of 5.75.

I thought of the numbers on a number line.

25.75 , 25.5

23.4

The order of all the numbers except as

decimals, is 1.67, 12.1.23.4,25.5,25.75,

15

23,

The order of the numbers from least togreatest is

12.1 15

23

8323.425

1225

34


�6�5.5

�5.75

645.5

5.75

20�4 �2



I know that but To be sure which wasleast, I compared both numbers to a benchmark of 6.

34 .

14.5 , 8,

Rachel invented an “adventurousness” scale. She and her friends arescoring themselves on how adventurous they are. Zero represents aboutaverage.

The seven friends’ scores are and Order the friends’ scores from least to greatest.

5

34.2

510,22

34,2

58,22

14,8

14,24

12,

0more unadventurous more adventurous

EXAMPLE 2 Comparing rationals that describe situations

and so is greater than 5

34.8

148

14 . 6,5

34 , 6

I know that negative values are less than positive ones,

so the five negative values are less than and 8

14.53

4

These are the two greatest values.

I compared three of the negatives to a benchmark ofone was to the left and two were to the right. 24;

which is greater than

which is greater than

So, and 24

12 , 22

34.24

12 , 22

14

24.22

34 . 23,

24.22

14 . 23,

24

12 , 24

When the denominators of two fractions are the same,you can compare numerators.

since 1 , 3.14 ,

34,

If a positive fraction is closer to 0 than another, itsopposite is also closer to 0, and so it is greater.

So, 234 , 2

14

Rachel’s Solution

and are each exactly 2 to the left of and

so is farther back from 0 than 22

14.22

342

14,

23422

1422

3422

34 , 22

14

and are both between 0 and so they are

greater than 21.

21,25

10258and are both greater than

and 24

12.22

14,

22

34,2

5102

58


Checking

1. List three rationals between and

2. Explain why

3. You and your friends are standing at a point called 0. You usepositive numbers to describe going east and negative numbers todescribe going west. One friend moves to Another moves to

A third moves to Which of your friends moved theleast distance? In what direction?

20.9.22.3.13.2.

23

34 , 23

18.

24

23.25

23

I realized I could have just compared the integer partsfor the rational numbers with different integer parts:24 , 22 , 0 , 5 , 8.

From least to greatest, the scores are

and 8

14.5

34,2

510,2

58,22

14,22

34,24

12,

and have the same numerator, but 10 is a greater

denominator. is farther to the right of zero than so

is farther to the left of zero than 2 510.2

58

510,5

8

58

510so 25

8 , 25

10.58 .

510,

In Summary

Key Ideas

• Rationals can be compared in the same ways as integers and fractions.• Negatives are always less than positives. A negative farther from 0 is

always less than a negative closer to 0.

Need to Know

• To compare rationals in fraction form, it helps to use mixed numberrepresentations, equivalent fractions with common denominators orcommon numerators, or benchmarks. For example,

since

since or since

since and

• You can also express all the rational numbers as decimals and comparethem in decimal form.

215 . 2

122

23 , 2

122

23 , 2

15

22135 . 2

21242

2440 . 2

35402

35 . 2

78

23 . 2423

12 . 24

13




Practising

4. Use , , or to make true statements. Explain how you knowparts b) and d) are true.

a) c) e)

b) d) f )

5. Replace the with a value to make each true.a) c) e)b) d) f )

6. Order each from least to greatest.

a) c)

b) 8.8, d) 14.2, 10.1

7. Draw X and Y on a number line to make each true.a) X is between and but greater than b) Y is greater than X but less than

8. Multiple choice. Which number sentence describes how Pcompares to Q ?

A. C.B. D.

9. Multiple choice. Which correctly orders

and from least to greatest?

A. 0.9, 8.1 C. 0.9, 8.1

B. 0.9, 8.1 D. 8.1, 0.9

10. Scientists sometimes use negative values to describe the time beforea major event. Suppose a scientist observed one event at sand another event at s. Which event occurred first? How doyou know?

11. Why would you probably use a different strategy to compare

and than to compare and 24.3?18.124

13

24.3

210.4212.2

24.3,24

13,2

245 ,2

245 ,24.3,24

13,

24.3,24

13,2

245 ,2

245 ,24

13,24.3,

2245

10.9,24

13,24.3,18.1,

25.3 . 24.425.3 , 24.425.6 . 24.825.6 , 24.8

�6 �4�5

P Q

22.2.22.8.2322

22

34,22.9,8

23210.3,210.2,28.4,

2315

315,2

38,3

8,24

233

14,214

12,212.2,

■.8 . 4.■1.9 , ■.421.■ , 2■.821.■ . 21.■20.■ , 0.■22.3 , 2■.4

■

22

310 ■ 2.324

12 ■ 2

9224.3 ■ 23.4

5.6 ■ 5

3521

25 ■ 1

250 ■ 20.5

5,.


12. a) Describe three different strategies you might use to order theserational numbers.

b) Use your strategies to order the numbers from least to greatest.

13. Point Q, is between point P at and point R at a) Copy the number line. Draw a point S between P and Q. Label

it with a value.b) Draw a point T between P and S. Label it with a value.c) Repeat two more times, each time drawing a new point between

P and the last point and labelling it with a value.d) How can you be sure there is always another rational number

between two given ones? Use an example to explain.e) What does this tell you about how many rational numbers there

must be?

14. If a and b are positive numbers and how do and compare? Explain why.

15. a) Write down six rational numbers, three positive and threenegative, in a mixed-up order.

b) Exchange lists with a partner and arrange the numbers fromleast to greatest.

c) Check each other’s work and discuss any corrections that mightbe necessary.

Closing

16. Agree or disagree with the following statement and explain yourreasons. “If you can order integers and you can order fractions, youhave all the skills needed to order rational numbers.”

Extending

17. Order these from least to greatest:

18. The same digit is substituted into each blank. The order from leastto greatest is

What could the digit be? Explain.

19. Let x represent a value that could be anywhere between andLet y represent a value that could be anywhere between

and About what fraction of the time will it be true that x , y ?23.5.2423.8.

22.4

24

2■

.24

■11,27

1■

,2■

35,

224982

24100,2

14,20.242424...,

2b2aa , b,

21.3.21.221.25,

1100

58,22

12,21

12,2

75,27.5,2

112 ,

�1.5 �1

R Q P

�1.3�1.25

�1.2



Adding and SubtractingRational Numbers

1.3

Solve problems that involve adding and subtracting rational numbers.

GOAL

LEARN ABOUT the MathJia-Wen’s father started working for a new company and was givenshares in the company. Jia-Wen paid attention to how those share priceswere changing.

Changes in share prices are sometimes reported using rational numbers.A positive number describes an increase in price and a negative numbera decrease.

The dollar value of a single share in Jia-Wen’s father’s company changedas follows:

Week 1: Week 2: Week 3:

By how much did the increase in week 3 make up for the lossesin weeks 1 and 2?

?

11.3420.2520.68

YOU WILL NEED

• a calculator with fraction capability


I used a number line to estimate the total change.I modelled each loss by going to the left. Then, Iadded the second loss to the first one.

I drew an arrow 1.3 units long and moved it tobegin at the end of the second loss.

EXAMPLE 1 Adding and subtracting rationals

Jia-Wen’s Solution: Representing sums and differences with a number line

I can tell the gain was greater than the combined loss since thepositive arrow is longer than the two negative ones together.

–1 10

–1 10

–1 10

–1 10


NEL 171.3 Adding and Subtracting Rational Numbers

To see exactly how much more, I compared thetotal loss to the gain.

I figured that if I am adding two negativenumbers, the sum would be negative, just as with integers.

I compared the length of the gain arrow to thecombined lengths of the loss arrows.

So, the combined loss is 20.68 1 (20.25) 5 20.93

0.68 1 0.25 5 0.93

The positive gain is 1.34. The difference is

The gain is 0.41 more than the loss. 1.34 2 0.93 5 0.41

Rachel’s Solution: Using reasoning to add and subtract

5 20.93 5 2(0.93)

(20.68) 1 (20.25) 5 2(0.68 1 0.25)

To add the gain of 1.34, I could subtract 0.93from 1.34.

The final increase overcame the total lossby 0.41.

5 0.41 20.93 1 1.34 5 1.34 2 0.93

I decided I was adding the two negative numbersand one positive number. I decided to use thezero principle.

(20.68) 1 (20.25) 1 1.34 5 x

I added the same amounts to both sides of theequation so that I could use the zero principle.I added 0.68 1 0.25.

I know that a number plus its opposite is 0.

5 x 1 0.68 1 0.25320.68 1 0.684 1 320.25 1 0.254 1 1.34

Sam’s Solution: Using the zero principle

0.41 5 x 1.34 2 0.93 5 x

1.34 5 x 1 0.93 0 1 0 1 1.34 5 x 1 0.93

Reflecting

A. How were the three solutions alike?

B. Why might one person think of the problem as an addition and another person think of it as a subtraction?

C. How is adding rationals like adding integers?



I realized I just needed to use thedecrease in hour 2 to estimatethe total decrease.

The final temperature was about 4 °C or 5 °C below the original temperature.

WORK WITH the Math

I estimated first. is a little more

than 3 and is about

I used to estimate.3 1 (22)22.

273 5 22

13

3

14is about 3 1 (22) 5 1.3

14 1

273

I wrote as Then, I wroteeach rational using equivalentfractions with the samedenominator, 12.

134 .3

14

273 5 2

73 5 2

2812

3

14 5

134 5

3912

I added the rationals by addingthe numerators.

5 1112

3

14 1

273 5

3912 1 Q228

12R

I compared my result to my estimate to check.

is close to 1, so my answer is reasonable.

1112

EXAMPLE 3 Adding rationals in fraction form

Calculate


3

14 1

273 .

I noticed that the temperaturesin hour 1 and hour 3 were closeto opposites.

is almost the opposite of 2.3, so those two temperatures add to avalue close to 0 °C.

22.7

EXAMPLE 2 Estimating the sum and difference of rational numbers

A temperature changed over the course of 3 h as follows:

Hour 1: Hour 2: Hour 3:

About how much higher or lower than the original temperature was the final temperature?

Thomas’s Solution

22.7 °C24.5 °C12.3 °C


–2 1 20–1

– 25

43

EXAMPLE 4 Subtracting rationals in fraction form

I know that to calculate a difference, I can thinkabout the distance from the second value to thefirst one.

Calculate

David’s Solution: Using a number line to visualize

1

13 2

225.

I added to get from to 0 and to get from

0 to 43.

432

25

25

is the distance from to on a number line.

1

13 5

43

225 5 2

251

13 2

225

and

or 1

1115 615 1

2015 5

2615

43 5

2015 25 5

615

To subtract a negative, I used the – key for thesubtraction and the ( ) key to change the valueto a negative.

2

Thomas’s Solution: Using a calculator

5 1

1115

2615

In Summary

Key Ideas

• Adding and subtracting rational numbers in the form of decimalscombines the rules for adding and subtracting positive decimals withthe rules for adding and subtracting integers.

For example,

• Adding and subtracting rational numbers in the form of fractionscombines the rules for adding and subtracting positive fractions withthe rules for adding and subtracting integers.

For example,

Need to Know

• It is useful to estimate sums and differences to verify calculations ofsums and differences.

• You can visualize a number line and use a combination of locations anddistances to estimate and calculate sums and differences of rationals.

5

34 2 Q22

13R 5 5

34 1 2

13.

5.25 2 4.3 5 0.95.24.3 1 5.25 5




Checking

1. Evaluate each.a) c) e)

b) d) f )

2. You lose $1.20 on each share you own and then gain back $0.65.Write the total loss on each share as a rational number.

Practising

3. Estimate the sums or differences. Explain your thinking.a) d)b) e)c) f )

4. Calculate exact answers for question 3.

5. Multiple choice. Which sum or difference is about A. C.B. D.

6. Multiple choice. Yaroslav takes h to cut his family’s front lawn

and h to cut the back lawn. How much longer does it take

Yaroslav to cut the back lawn than the front?

A. h B. h C. 35 min D. 45 min

7. Consider these numbers: Which numbers havea) a sum of 5.2? c) a sum of 4.038?b) a difference of 6.567? d) a difference of

8. Determine the missing digits for each. Use a calculator to help you.a)b)c)d)

9. a) How could using the zero principle help you add b) Why would the zero principle not help you add

What other strategy could you use instead?

10. Calculate. Show your work.

a) c) e)

b) d) f ) 528 2 Q211

3 R1

34 1 Q23

25R25

12 1 2

23

23

23 2 4

15

65 2

322

38 1 1

34

23.4 1 (28.9)?3.4 1 (28.9)?

25.1■ 2(2■.8) 2 7.■ 5 29.2122.45■ 2(25.■63) 5 ■.70521■.382 2(4.17■) 1 8.■3 5 27.■2723.5■2 1 ■.42■ 5 1.846

23.578?

11.20519.4,25.362,28.94,24.2,

1

131

12

1

13

34

23.98 1 (28.9)14.1 1 (22.1)

24.1 2 (219.8)22.3 2 18.4116?

25.1 1 (25.82) 1 5.0129.37 2 5.933.42 2 (25.6) 1 11.3212.2 2 (218.9)

0.47 2 (221.6)3.64 2 72.9

23

45 2 1

232.5 2 5.67

8 1 Q223R

243 2 Q23

4R2

15 1 Q21

2R24.2 1 (23.8)


11. The daily changes in price for a share during a week were $2.78,$ $0.38, $1.38, and $2.12. The price of the share was $58.22

at the start of the week. What was the price at the end of the week?

12. How do you know that is

a) greater than b) about 1?

13. Determine the value that makes each equation true.

a) b)

14. James finished the Manitoba marathon in a time of 3:57:53.3 (hours: minutes: seconds). The winner of the marathon finished in a time of 2:25:55.6. Determine how muchlonger James took to complete the marathon than the winner did.

15. Evaluate each expression for the given values.a) when and b) when and

c) when and

d) when and

16. To recreate the work of the voyageurs during the fur trade, a relayon the Red River near St-Boniface, MB, saw participants canoe tospecific points to find a message like those at right, which led themto a fur cache. What rational number operations would you use todetermine each of the following?a) the distance of the last leg b) the total distance paddled

17. List two rational numbers a and b that are not integers and thatwould make each statement true.a) is negative, but is positive.b) is positive, but is negative.c) and are both negative.

18. Describe a real-world problem where you might calculateSolve your problem.

Closing

19. Describe a strategy for calculating the sum and a strategy for calculating the difference of and

Extending

20. The sum of two rational numbers is Their difference is What are the numbers?

21. The sum of two rationals is 17.4 less than the difference. Whatcould the rationals be?

21

340.23

40.

15.005.23.4

23.2 2 (24.5).

a 2 ba 1 ba 2 ba 1 ba 2 ba 1 b

y 5 2

14x 5 21

12x 1 y

y 5 23

34x 5 22

12x 2 y

z 5 24.1y 5 27.8,x 5 2.5,x 1 y 1 zy 5 23.2x 5 24.1x 2 y

21

34 2 ■ 5 121

34 1 ■ 5 1

22.3 1 Q23

14R ?

22.3 2 Q23

14R

5.45,22

Message 1: 1.5 km southMessage 2: 0.68 km northMessage 3: 2.3 km southMessage 4: north to thestarting point




Multiplying and DividingRational Numbers

1.4

Solve problems that involve multiplying and dividing rationalnumbers.

GOAL

LEARN ABOUT the MathThomas was talking to his cousin in the United States. Thomas said thetemperature was His cousin said that the temperature where helived was However, Thomas was using the Celsius scale and hiscousin was using the Fahrenheit scale.

Thomas found this conversion formula:

Which temperature was lower?

A. What would Thomas’s temperature be in Fahrenheit, if it hadactually been instead of

B. Why does it make sense that is the opposite of

C. Thomas’s actual temperature was What was his temperature in degrees Fahrenheit?

D. What was his cousin’s temperature in degrees Celsius?

E. Which temperature was lower?

Reflecting

F. Why were your answers to parts A and C not opposites?

G. How does knowing how to multiply and divide integers help youcalculate the values in parts C and D?

H. How does knowing how to multiply and divide fractions help youcalculate the values in parts C and D?

25.5 °C.

95 3 5.5?9

5 3 (25.5)

25.5 °C ?15.5 °C

?

F 595C 1 32

21.5°.25.5°.

YOU WILL NEED

• a calculator


NEL 231.4 Multiplying and Dividing Rational Numbers

WORK WITH the Math

These temperatures were recorded at noon on January 1 from 2000 to2008 at Edmonton International Airport. Determine the mean noontemperature on January 1 for the years given.

I used my calculator and added thenegative temperatures together andthen added the one positive temperature. I divided the sum by 9, the number of temperaturereadings, to determine the meannoon temperature.

This answer seems reasonable sincethere were four temperatures abovethis amount and five temperaturesbelow it.

The mean noon temperatureat Edmonton InternationalAirport on January 1 is about210.6 °C.

EXAMPLE 1 Reasoning about the mean of a set of rational numbers

Year Temperature ( C)

2000

2001

2002

2003

2004

2005

2006

2007

2008 212.5

23.5

22.6

219.5

216.2

25.3

221.2

1.2

215.7

8

Rachel’s Solution

Source: Environment Canada



a) Use the numbers 1 and in the blanks so that hasthe least possible value.

b) Use the numbers 1 and in the blanks so that hasthe greatest possible value.

Jia-Wen’s Solution

2●5 4 ■

342

●◆

3 2

3425

The product had to be negativesince the fraction involved apositive and negative value.

a) The missing number is either

or 251 5 2

51

125 5 2

15

EXAMPLE 2 Solving a problem involving rational numbers

If you multiply a negative by apositive, the answer is less if thenegative is less.

so

251 3 2

34 , 2

15 3 2

34

215 . 2

51

15 ,

51,

I checked by multiplying.

I was right, since is almost

and is not even 21.21120214

2554

The least product is .251 3 2

34

5 21120

125 3 2

34 5 2

15 3

114

5 2554

251 3 2

34 5 25 3

114

The only possibilities are

To divide,

I converted the mixed numberand multiplied by the reciprocal.

is about .

is about which is less.214,2

835

21

122455

215 4 2

34 or 22

5 4 1

34.

b)

The greatest quotient is 225 4 1

34.

5 2835

5 225 3

47

225 4 13

4 5 225 4

74

5 2455

5 215 3

411

215 4 2

34 5 2

15 4

114


In Summary

Key Ideas

• Multiplying and dividing rational numbers in decimal form combines therules for multiplying and dividing positive decimals with the rules formultiplying and dividing integers. For example,

• Multiplying and dividing rational numbers in the form of fractionscombines the rules for multiplying and dividing positive fractions withthe rules for multiplying and dividing integers. For example,

Need to Know

• You can divide rational numbers in the form of fractions by using acommon denominator and dividing the numerators. For example,

• You can also divide by multiplying by the reciprocal. For example,

21225 4

35 5 2

1225 3

53

21225 4

35 5 2

1225 4

1525

5

34 3 Q22

13R 5 2Q23

4 373R

(23.2) 4 1.2 5 2(3.2 4 1.2)

Checking

1. Evaluate.a) c)

b) d)

2. How much less is than

3. A water tank lost of its volume of water one day and then of what was left the next day. What rational number describes thevolume of water after the second day as compared to the originalvolume?

Practising

4. Calculate.

a) c) e)

b) d) f ) 23 4

582

58 4

232

23 3

258

223 4 Q25

8R23 3

2852

23 3

58

12

13

234 3

56?23

4 456

25 4 Q25

8R247 3

625

(28) 4 (0.5)(22) (9.5)




5. Multiple choice. Which expression is about

A. B. C. D.

6. Multiple choice. Without evaluating, determine which expressions have the same product as

W: X: Y: Z:

A. X and Y C. X, Y, and ZB. X and Z D. all of these expressions

7. Use the numbers and 8 in the blanks so that hasa) the least possible valueb) the greatest possible value

8. Consider the numbers and 7.3.a) Which two have a product of 35.28?b) Which two have a quotient of about

9. The temperatures at Fort Nelson, BC, at 5:00 a.m. on December 25from 2002 to 2007 are shown in the table. Determine the meantemperature at 5:00 a.m. on December 25 for these years.

21.75?

28.4,21.3,24.2,

■■

3 ■

2323,21,

Q234 RQ

528RQ23

8RQ254R2Q34RQ2

58RQ23

24RQ58R

Q34RQ58R.

223 4

54

45 4 Q1

12R

89 4 Q21

2R223 3

18

212?

Source: Environment Canada

Year Temperature ( C)

2002

2003

2004

2005

2006

2007 216.3

210.5

29.3

215.8

27.6

220.4

8

10. Calculate. Show your work or use a calculator.

a) c) e)

b) d) f ) 7.2 4 (20.6)24

23 4

712Q3

67RQ28

13R

(23.2) 4 (28.4)1516 4 Q21

124RQ 5

212RQ2815 R

11. The formula to convert temperatures between degrees Fahrenheit anddegrees Celsius is Use this formula to convert thefollowing.a) Miami, Florida’s record high of to degrees Celsiusb) Anchorage, Alaska’s record low of to degrees Celsiusc) to degrees Fahrenheit

12. Two tanks hold the same amount of water. Tank 1 loses of its volume. Tank 2 gains of its volume. What is the final ratio of water volume, comparing tank 1 to tank 2?

13. An investment loses of its value and then loses another of thenew value.

a) What fraction of its original value is the final value?

b) Can you multiply to calculate the answer to

part a)? Explain.

212 3 Q22

3R

23

12

14

23

0 °C237 °F

98 °F

C 559(F232).


14. A pail of water has been sitting for a while, and of the water hasevaporated.

a) What could describe about this situation?

b) What could describe about this situation?

15. The product of two rationals is What might their quotient be?

16. The product of and two other rationals is The quotient of the

two other rationals is

a) How do you know that one unknown rational is positive andone is negative?

b) What could the unknown rationals be?

17. Evaluate each expression for the given values. Use a calculator.a) when and

b) when and c) when and

d) when and

Closing

18. Create a brief “instruction manual” to help someone with the rulesfor multiplying and dividing rational numbers.

Extending

19. Calculate each product.a) b)

20. The product of a positive and a negative rational number is 2greater than their sum. What could they be?

21. The width of a rectangle is of the length. If you increase the widthby 12 m and double the length, you obtain a perimeter of 60 m.Determine the dimensions of the original rectangle.

14

299 3 0.232323c29 3 0.2222c

y 5 2

14x 5 21

12

xy 1

yx

y 5 3

34x 5 22

12x(x 1 y)

y 5 27.8x 5 2.5(x 1 y) (x 2 y)

y 5 3.2x 5 29.78x 2 2y

●◆

4▲■

5 234

234 3

●◆

3▲■

514

234.

14.2

34

21225.

218 4 Q21

4R

34 3 Q21

8R

18




FREQUENTLY ASKED QuestionsQ: How do you place a negative rational on a number line?

A: You place the negative rational as far to the left of 0 as its opposite is to the right. For example, place to the left of 0, the same

distance that is to the right of 0. Notice that is the same as

which is not equal to

Q: How can you determine whether one rational numberis greater than the other?

A: Numbers can be compared on a number line. If a number isfarther to the right, it is greater. Therefore, any negative rational isless than any positive one.

Initially it helps to look at the integer part of a rational to decidewhere to place it. Sometimes writing all the numbers in the sameform, either as decimals or as fractions with a common positivenumerator or denominator, makes comparisons easier.

For example, since

since or since

Q: How can you add and subtract rational numbersto solve problems?

You add in situations where you combine. For example, suppose

the level of water in a full pail goes down to the mark and then

of another identical pail is added back in. The new level, compared

to the original level, is

This is lower than the original amount.

You subtract in situations where you have a loss or reduction. Forexample, suppose the level of water in a pail goes down by of thepail’s capacity, and then the remaining water is poured into a new

identical pail to the mark. The new level in the first pail, compared to its original level, is 25

8 :

38

14

18

5 218

234 1

58 5 2

68 1

58

218:

58

14

22

510 , 22

410.22.5 , 22.422

12 , 22

25

23 , 21.23

12 , 21

23

22 158.22 1 Q25

8R,22

582

58

22

58

CHAPTER 1 Mid-Chapter Review

• See Lesson 1.1, Example 2.• Try Mid-Chapter Review

questions 1 and 2.

Study Aid


questions 4, 5, and 7.

Study Aid

• See Lesson 1.3,Examples 1, 3, and 4.

• Try Mid-Chapter Reviewquestions 9 and 10.

Study Aid

–3 30

–2 58

2 58


NEL 29Mid-Chapter Review

The new level is of a pail lower than the original level.

Q: How can you multiply and divide rational numbers to solveproblems?

A: You multiply and divide rationals by combining the rules youknow for multiplying and dividing integers with those formultiplying and dividing positive fractions or decimals.

For example, suppose one investment loses of its value. Then, it

loses of that new value. What fraction of the value of the originalinvestment is the final value?

is the final value. That’s because you end up with of

the value after the first loss and then of that after the second loss.

For example, suppose you have two investments of equal value. The first investment loses of its value; the second loses of its value. How much more is the loss per dollar on the first investmentthan on the second one?

For each dollar lost on the second investment, you lose of adollar, or $1.20, on the first investment.

PracticeLesson 1.1

1. What rational numbers describe the points A, B, and C ?

2. Locate each rational on a number line from .

a) b) c) d)

3. On a February day, the daytime high temperature in Saddle LakeFirst Nations Reserve, AB, was The temperature inPortage la Prairie, MB, on the same day was Whichplace was colder? Explain.

212.8 °C.24.5 °C.

28

13

2323

218422.6

210 to 110

65

5 65

234 4 Q25

8R 568 4

58

58

34

38

14

14 3

38 5

332

58

34

58

5 258

214 2

38 5 2

28 2

38


questions 12, 13, and 14.

Study Aid

A B C

–5 20


Lesson 1.2

4. Use , , or to make true statements. Explain how you knoweach statement is true.

a) b) c) d)

5. Write these rational numbers in order from least to greatest.

a) c) 0.7,

b) d) 0,

6. List four rational numbers between and

7. A rational number of the form is between and What isthe numerator of the rational number?

Lesson 1.3

8. Calculate.

a) d)

b) e)

c) f )

9. Kristen walked 5.7 km east and then 9.1 km west. How far east orwest was Kristen from her original position?

10. The difference of two rational numbers is The sum is What arethe rational numbers?

Lesson 1.4

11. The daily changes in selling price for shares in Robots Inc. duringa week were $0.25, and Whatwas the mean daily change in selling price for the share duringthis week?

12. a) Create two other expressions that give the same answer as

b) Describe a situation that might represent.

13. Noah had 1.5 times as much savings as Kelly had debt. Then, Kellydoubled her debt. What rational number division describes Noah’smoney as compared to Kelly’s?

14. Create and solve your own problem involving multiplying ordividing rational numbers.

Q21

34RQ5

13R

Q21

34RQ5

13R.

2$3.72.2$2.36,2$0.95,2$4.50,

13.3

5.

27

114 1 Q21

4R27.2 2 (24.8)

23.5 1 (27.7)35 1 Q28

9R2

58 1 Q21

3R214 2 51

3

212.2

34

■6

212.2

38

2221.5,4522

15,22

5 ,

20.320.3,21

13

123,2

35,

225 ■ 2

31022

14 ■ 2

94

23 ■ 582

23 ■ 2

56

5,.



NEL 311.5 Order of Operations with Rational Numbers

Order of Operations withRational Numbers

1.5

Extend the order of operations rules to rational numbers.

GOAL

LEARN ABOUT the MathSam is solving a puzzle where he has to put in the operations signs

to make this statement true.

Where should Sam put each sign?

A. How do you know that the sign before the 5 cannot be a sign?

B. Why is it more likely that you multiply by the 5 than divide by it?

C. How does knowing that the value is help you know that the sign between and cannot be a or sign?

D. Where does each sign go?

Reflecting

E. Why would you not need to know the order of operations rulesif the sign were last?

F. Why do you need to know the order of operations rules if the sign is not last?

WORK WITH the Math

1

1

4323222

216

1

?

12 ■ (22) ■ Q23

2R ■ 12 ■ 5 5 216

3, 2, 4, and 1

YOU WILL NEED

• a calculator with fractioncapability

The convention for order ofoperations is BDMAS:B: First calculate anythinginside brackets.DM: Divide and multiply fromleft to right.AS: Add and subtract fromleft to right.

Communication Tip

EXAMPLE 1 Using the order of operations to evaluate a rational-number expression

Evaluate on a calculator when and

Does your calculator follow the rules for order of operations?


y 5 21

79.x 5 5

1322

12 1 x 4 y

I substituted the given valuesfor the variables.

22

12 1 x 4 y S 25

2 1163 4 Q216

9 R

MF9SB_CH01_p31-40 pp6.qxd 11/16/08 6:02 PM Page 31


I checked using a calculator and also got 2.4.

The actual average is 2.4.

I calculated by hand to see if thecalculator used the order ofoperations. I followed the order ofoperations the same way I would ifthe numbers were all integers orall fractions.

The answer on my calculator waswhich is 25

12.25.5,

Thomas had been texting daily temperatures to Sam.

Thomas then told Sam how to calculate the mean. He texted, “Add the numbers and divide by 4.” Sam wrote

2.775 is the average temperature.What is wrong with Sam’s calculation? What is the actual average?

Thomas’s Solution

5 2.77520.2 1 5.8 1 (25.1) 1 9.1 4 4 5 20.2 1 5.8 1 (25.1) 1 2.275

Sam should have used brackets aroundbefore dividing by 4.20.2 1 5.8 2 5.1 1 9.1

Sam’s error was that he divided only the 9.1 by 4.

First I did the addition since the values are in brackets.

Correction:320.2 1 5.8 1 (25.1) 1 9.14 4 4

I added the positives and the negatives separately.

5 2.45 9.6 4 45 325.3 1 14.94 4 4

EXAMPLE 2 Solving a problem using the order of operations

5 25

12

5 22

12 1 (23)

5252 1 Q29

3 R5

252 1

163 4 Q216

9 R22

12 1 5

13 4 Q21

79R

The calculator did follow the rules for order of operations.

Temperatures 19.125.115.820.2


NEL 331.5 Order of Operations with Rational Numbers

Checking

1. Which operation would you perform first in each situation?a)b)

2. Share values went up and down over the course of a week as shownat right. What was the mean daily change?

Practising

3. Which operation would you perform last in each situation?a)b)

c)

4. Temperatures went up and down over the course of five days as follows:, , , ,

What was the mean daily change?

5. Multiple choice. What is the value of

A. 11.09 B. 17.23 C. 17.83 D. 16.23

6. Multiple choice. What is the value of

A. 0.4 B. 1.7 C. D.

7. Mika calculated as Is Mika correct? Explain.

8. Evaluate.

a) when and

b) when and y 5 27

10x 5 21

342x 2

53 y

y 5 23

34x 5 2

1223

23x 4 y

24

14.Q23

4R1Q223R4

13

21.6090909...20.46(20.15) 3 0.2 1 2?20.7 2 0.3 4

325.1 2 (21.8) 4 1 0.2?32.4 2 5.74 3

21.8 °C13.7 °C11.9 °C21.4 °C24.2 °C

23 3

54 2

34 3

95

8 4 (21.3) 1 (26.8 2 3.4)

3.733.5 1 (22.9) 4 4 32.5 1 (21.8) 4

(25.2 2 1.4) 4 (23) 3 (2.1)

2.4 2 33.5 1 (22.9) 4 3 (25)

In Summary

Key Idea

• The rules for order of operations with rationals are the same as withintegers:• Do what is in brackets first.• Multiply and divide from left to right.• Add and subtract from left to right.

Day Share Value

Day 1 0.005

Day 2

Day 3

Day 4

Day 5 0.051

20.12

20.115

20.135

1



9. The temperature on Friday went up from Thursday. After aloss of on Saturday, the temperature doubled on Sunday.How did the temperature on Sunday compare to the temperatureon Thursday?

10. For which calculations would you not need to know the order ofoperations rules?

a) c)

b) d)

11. a) Create an expression, involving both positive and negativerational numbers that are not integers, for which you wouldneed to know the order of operations to calculate it correctly.Explain why you would need to know the order of operationrules for your expression.

b) Check to see if your calculator follows those rules for order ofoperations when you enter rational numbers. How do you know?

12. During the last heat for the 800 m run at the Jeux de la Francophoniein Edmonton, Xavier gained s over his nearest opponent during the

first 100 m. He then lost s during the second 100 m, lost a further

s during the third 100 m, but gained s during the fourth 100 m.

How much time will Xavier have to gain on his opponent during thelast 400 m to win the heat?

Closing

13. Why is it essential that the rules for order of operations for rationalnumbers be the same as the order of operations for integers?

Extending

14. Create an expression involving rational numbers where you performan addition before you perform any multiplications or divisions.Calculate the value of that expression.

15. Where would you add brackets in each expression to make it true?

a)

b)

16. You subtract a rational number from double the answer, and

then divide by The result is 8. What was the rational number?214.

223,

21.2 1 23 4 1.5 1 1.5 215 3 13 5 24

3

12 1 4 4 0.75 1 (28.1) 5 1.9

12

18

23

14

Q223R4

58 3 Q24

5RQ223R 3

58 2 Q24

5RQ22

3R 258 2 Q24

5R 3 2Q223R 2

58 2 Q24

5R

21.2 °C3.1 °C


NEL 351.6 Calculating with Rational Numbers

1.6 Calculating with RationalNumbers

Use number sense to compare the results of adding, subtracting,multiplying, and dividing rationals.

GOAL

EXPLORE the MathJia-Wen noticed that, for and

The order from least to greatest is

But for and

The order from least to greatest is

How can you choose pairs of rational numbers to get fourdifferent possible orderings as the result of adding,subtracting, multiplying, and dividing them?

?

3124

Q223R4

13 , Q22

3R 213 , Q22

3R 113 , Q22

3R 313

13,2

23

1342

Q223R 2

45 , Q22

3R445 , Q22

3R 345 , Q22

3R 145

45,2

23



1.7 Solve Problems UsingGuess and Test

Solve problems involving rational numbers using aguess-and-test strategy.

GOAL

LEARN ABOUT the MathEight students rated their video game enjoyment on a scale from to 10. The mean score was 1.625.

Three scores were negative and added to

Two positive scores were double two other ones.

There were no negative doubles.

What were the eight scores??

27.

210

Thomas’s Solution

EXAMPLE 1 Using guess and test to determine data values

1. Understand the ProblemI know the numbers all have to be between and 10.I know there have to be eight numbers.I know the mean of the eight numbers is 1.625.I know that three numbers are negative and that they add to I know that exactly two numbers are the double of two other onesand they are positive.

2. Make a PlanI will figure out the total of all the numbers, and then the total ofthe positive numbers.I will figure out what the five positive numbers add to. Then, I willmake sure that they add to the correct total but also make sure thattwo numbers are double the other two.

27.

210


NEL 371.7 Solve Problems Using Guess and Test

3. Carry Out the PlanSince the mean is based on 8 numbers,

First, I tried 8 and 4 and 6 and 3 as the doubles:

This is not possible since is already 21.

Then, I tried 6 and 3 and 2 and 1 as the doubles:

These values worked because 8 is not the double of any otherpositive. The positives are 6, 3, 2, 1, and 8.The negatives had to add to I tried and but is double and there can be no negative doubles.Next, I tried and

My solution is that the scores are 3, 6, 1, 2, 8, and

4. Look BackI checked by adding and dividing by 8 to make sure that the meanwas correct. 33 1 6 1 1 1 2 1 8 1 (25) 1 (21) 1 (21) 4 4 8 5 1.625

21.21,25,

25.2121,22

2424,22,21,27.

20 2 (6 1 3 1 2 1 1) 5 20 2 12 5 8

8 1 4 1 6 1 38 1 4 1 6 1 3 1 another positive 5 20

Total of positives 5 20Total of positives 1 (27) 5 13Total of negatives 5 27

Total of all 8 numbers 5 8 3 1.625 5 13

Reflecting

A. How did Thomas use logical reasoning in solving the problem?

B. How did using guess and test help Thomas to solve the problem?

C. Could Thomas have found a different solution? Explain.

WORK WITH the Math

The number of kilometres in the perimeter of this park is about 6 lessthan the number of square kilometres in its area. What is the width?

Sam’s Solution

1. Understand the ProblemI have to determine the width so that the number of units in theperimeter is about 6 less than the number of units in the area.

EXAMPLE 2 Using guess and test to solve a measurement problem

10.875 km

.125 km



In Summary

Key Idea

• You can use guess and test in combination with other strategies to helpyou solve problems. Once you guess, you can use the results of yourguess to improve your next guess.

2. Make a PlanI will use estimates for the width and then determine the perimeterand the area.

3. Carry Out the PlanFirst, I estimated the length as 11 km.I tried a width of 5 km.

The perimeter is about The area is about

So, the difference is too much.

I tried a width of 3 km.The perimeter is about The area is about

The difference is not quite enough, but it is close.

The diagram said the width was actually ■.125 km and the lengthwas actually 10.875 km. Since I was so close, I used 3.125 km.

(rounded tothe nearest thousandth)

The difference is 5.984, rounded to the nearest thousandth. This isvery close to 6. Therefore, the width is 3.125 km.

4. Look BackI checked the perimeter and area for widths of 2.125 km and4.125 km. For 2.125 km, and For 4.125 km, and So, themissing digit could only be 3.

area 8 45 km2.perimeter 5 32 kmarea 8 23 km2.perimeter 5 24 km

Area 5 (3.125 3 10.875) km2 5 33.984 km2Perimeter 5 2 3 14.000 km 5 28.000 km

(3 3 11) km2 5 33 km2.2 3 14 km 5 28 km.

55 2 32 5 23(5 3 11) km2 5 55 km2.

2 3 16 km 5 32 km.


33 2 28 5 5

NEL 391.7 Solve Problems Using Guess and Test

Checking

1. You subtract a rational number from its triple. The result is What is the rational number?

Practising

2. The mean of five numbers is The sum of the positive numbersin the set is 37 greater than the sum of the negative numbers in theset. What could the numbers be?

3. The number of kilometres in the perimeter of this park is 8 greaterthan the number of square kilometres in its area. What is the width?Use a calculator to help.

4. The sum of three rationals is 0. Two have denominators of 8. The product of two of them is The quotient of two of them is What are the rationals?

5. The product of two opposites is What are the numbers?Use a calculator to help you guess and test.

6. The sum of five numbers is The numbers include two pairs ofopposites. The quotient of two values is 2. The quotient of two

different values is What are the values?

7. A share price increased in value one day, doubled that increase the next day, and then decreased $0.12 in value the following day. If the total change was $0.33, by how much did the share go up thefirst day?

8. A rational number with a denominator of 9 is divided by The

result is multiplied by and then is added. The final value is

What was the original rational?

9. Two rational numbers are added. The sum is more than theproduct. What are the rational numbers?

Closing

10. Why is guess and test a good strategy to use for questions likequestion 3? Why might you also work backward?

Extending

11. Create a problem involving rational numbers that would bereasonable to solve with a guess-and-test strategy. Then, solve itusing that strategy.

14

110.2

56

45

Q223R.

234.

214.

21.8225.

25.116.

25.

258.

6.0 km

? km



Math GAME

Compare SixNumber of players: 2 to 4

How to play

1. Choose two rational number cards.

2. Decide whether and in what order to add, subtract, multiply, or divide thenumbers to get the greatest value possible.

3. The player with the greatest value wins a point. In the event of a tie, no pointsare scored.

4. The first player to reach 6 points wins.

Sample Turn

Thomas’s Draw

I’ve drawn and .

If I add, multiply, or divide, the answer is

negative. I’ll subtract from

5 1112

14 2 Q223R 5

312 1

812

14.2

23

223

14

Larissa’s Draw

I’ve drawn and If I multiply or divide, the answer is negative. If I add,

it’s only but if I subtract, I get 1; I win the point!

15,

225.3

5

YOU WILL NEED• Rational Number

Cards


NEL 41Chapter Self-Test

CHAPTER 1 Chapter Self-Test

1. Multiple choice. Which value is equivalent to

A. B. C. D.

2. Multiple choice. Which set of numbers is arranged in order fromleast to greatest? Explain.

A. C.

B. D.

3. Which value is farther from zero: or 4.3? Explain.

4. List three rationals between and


a) d)

b) e)

c) f )

6. The high temperatures, in degrees Celsius, for a city during a five-day period increased or decreased as follows:

, , a) What was the total increase or decrease from the start of the

period until the end?b) What was the average daily change in temperature?

7. The height of the water level in a tank dropped by Katia used

of what was left. What rational number describes the final height as a fraction of the original height?

8. The following are true about two rational numbers.• The product is greater than the quotient.• The sum is less than the difference.

List three things that you know about the numbers and explainhow you know those things.

WHAT DO You Think Now?Revisit What Do You Think? on page 3. How have your answers and explanations changed?

35

14.

15.2°22.3°,211.7°,11.7°14.2°

5

13 2 Q2 3 4

12R 1

83 4 223

15 1 Q24

3 R 1 4.5

3.2(24.1) 4 (2 1 3.4) 3 322

23 4

34

Q245RQ

83RQ2

56R

247 2 Q22

12R

2278 .23

14

24

13

2115

21125 ,22

25,22

252

115 ,211

25 ,

211252

115 ,22

25,22

25

21125 ,2

115 ,

45

4252

45

245

2425?



CHAPTER 1 Chapter Review

FREQUENTLY ASKED QuestionsQ: How do you apply the order of operations when working

with rational numbers?

A: You use the same rules as you would use if working with integers orpositive fractions or decimals. In order,• computations in brackets• multiplication and division from left to right• addition and subtraction from left to right

For example,

Q: How can you predict whether the sum, difference, product,or quotient of two rational numbers will be greatest?

A: You have to consider whether the numbers are positive or negative.You also have to consider whether they are between and 1 or not.

For example,

• If you add or subtract a negative, the difference will be greaterthan the sum.

since

• If you multiply or divide a negative by a negative fractionbetween 0 and the product will be less than the quotient.

since

• If you multiply or divide a negative by a negative less than the product will be greater than the quotient.

since 6 .32.(23) (22) . (23) 4 (22)

21,

32 , 6.23Q21

2R , (23) 4 Q212R

21,

5

12 . 4

12.5 2 Q21

2R . 5 1 Q212R

21

5 20.55 23 2 (22.5)

5 23(1) 2 5 4 (22)

23S13 2 Q22

3RT25 4 (22)

• See Lesson 1.5, Examples 1and 2.

• Try Chapter Review questions 17, 18, and 19.

Study Aid

• Try Chapter Review question 20.

Study Aid


NEL 43Chapter Review

PracticeLesson 1.1

1. Where is each value located on a number line?

a) b)

2. Which rational is between and or How do youknow?

3. Write each as a quotient of integers.

a) b) c)

4. Describe a situation that the number might represent.

5. What is wrong with this Venn diagram?

Lesson 1.2

6. Order from least to greatest.

a) 0.3, 1.2, b)

7. List three rational numbers between each pair of rationals.

a) and b) and c) and

8. Explain why even though

Lesson 1.3

9. Ann started gym h before lunch. Her art class began an hour and

a half after lunch. Lunch lasted h. Use a rational expression to tell how many hours before art class her gym class started.

34

23

1 , 8.212 . 2

82

2232

4525.00625.0124

3424

13

214

47,28

9 ,223,3

5,2151

283 ,125.1,

2123

23814

524.2

2313 ?229

329:210

224522.6

Integers

Rationalnumbers



10. Estimate. Show your reasoning.

a) c)

b) d)

11. Calculate the sums and differences in question 10. Show your work.

12. A share price increased by $0.05 one day, decreased by $0.02 thenext day, and decreased again by $0.01 the following day. What wasthe total change?

13. The sum of two rational numbers is The difference is What are the two rational numbers?

Lesson 1.4


a) c) e)

b) d) f )

15. One share lost $0.25. Another share lost $0.03. What is the ratio ofthe losses? Write the ratio as a rational number.

16. The quotient of two rationals is The product is

a) What are the rationals?b) How do you know there has to be another possible answer?

Lesson 1.5


a) c)

b) d)

18. Aaron calculated as Is thiscorrect? Explain.

19. Use a calculator to determine how much less is than

Lesson 1.6

20. Determine two rational numbers a and b so that

Lesson 1.7

21. The sum of three numbers is 1. One number is times another.The quotient of another pair of the numbers is 4. What are thenumbers? Explain.

(22)

a 3 b . a 4 b . a 2 b . a 1 b.

Q24 135R 3

23.

Q24 135R4

23

29.8.26.2 4 (3.1 1 1.9) 3 (22)

2112 1

2122 2 Q23

5R256 1 Q22

3R434

Q18 1223 R 3

1213

25 4 Q23

5 1110R

23

32.21.5.

(23) 4 Q21

23R

27 4 Q2 9

14R23Q2

65RQ2

53R

Q265R4 Q22

3Ra123b a2

49b2

52Q2

45R

21110.2

12.

23 1 Q216

5 R83 1 Q217

5 R23.7 1 (217.1)23

12 2 Q28

34R


NEL 45Chapter Task

CHAPTER 1 Chapter Task

Creating a GameTo review for a test, Jia-Wen and her friends are creating a game for playerswho know rational numbers. They decide to use a card game where youkeep cards that match. They want a game with at least 24 cards.

What cards can you create for a game like Jia-Wen’s?

A. Why might these cards match?

B. What card might match this card? Explain why it matches.

C. What pair of cards might you make to review equivalence ofrational numbers? Explain why your pair works.

D. What pair of cards might you make to review creating situations involving rational numbers?

E. What pairs of cards might you use to compare the results of addingand subtracting two rationals?

F. What pairs of cards might you use to compare the results of multiplying and dividing two rationals?

G. What pairs of cards might you use to review order of operationsinvolving calculations with rationals?

H. Create the cards and rules for a game like Jia-Wen’s.

Thesenumbers are

in order:

�2 , , 13

�53

, �1.2

–0.78

A numberbetween

and– 45

– 34

?

✔ Did you check to makesure that you considereddifferent ideas aboutrational numbers?

✔ Did you use some“creative” matches?

✔ Did you explain your rulesclearly?

✔ Did you explain why yourmatches make sense?

Task Checklist


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