NSCC SUMMER LEARNING SESSIONS NUMERACY SESSION Module 2 – Integers

Acknowledgement

Large portions (pages 9-71) of these modules were created using Level III and Level IV ALP locally developed math resources. These ALP resources are the intellectual property of the NS Department of Labour and Advanced Education (LAE). David Pilmer, the author and LAE Curriculum Consultant, has given permission to NSCC for the use of his materials in the creation of these learning modules.

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 2

Welcome! The Numeracy session has six modules. This is module number 2 - Integers. In this package you will find everything you need to complete this module.

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 3

Contents Acknowledgement ............................................................................................................................................. 2 Welcome! ............................................................................................................................................................ 3 LEARNING OUTCOMES – What will I learn? .............................................................................................. 6 Introduction to Signed Numbers ..................................................................................................................... 9 Questions...................................................................................................................................................... 11 Answers ........................................................................................................................................................ 15 Adding Signed Numbers - Part 1 .................................................................................................................. 17 Questions...................................................................................................................................................... 20 Answers ........................................................................................................................................................ 29 Adding Signed Numbers - Part 2 .................................................................................................................. 31 Questions...................................................................................................................................................... 32 Subtracting Signed Numbers - Part 1 .......................................................................................................... 34 Questions...................................................................................................................................................... 35 Answers ........................................................................................................................................................ 38 Questions...................................................................................................................................................... 41 Answers ........................................................................................................................................................ 43 Multiplying Signed Numbers - Part 1............................................................................................................ 44 Questions...................................................................................................................................................... 46 Answers ........................................................................................................................................................ 50 Multiplying Signed Numbers - Part 2............................................................................................................ 51 Questions...................................................................................................................................................... 58 Dividing Signed Numbers - Part 1 ................................................................................................................ 64 Questions...................................................................................................................................................... 65 SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 4

Answers ........................................................................................................................................................ 67 Dividing Signed Numbers - Part 2 ................................................................................................................ 68 Questions...................................................................................................................................................... 69 FINAL STEPS: Finishing up the module ..................................................................................................... 72 Coming up next… Module 3 – Fractions ................................................................................................. 72

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 5

LEARNING OUTCOMES – What will I learn? In this module you will learn and practice… • • • •

Reading and writing integers Adding, solving, multiplying and dividing integers Solving problems with integers Completing a study sheet about integers

This is an important part of working towards the session learning objectives:  

Students will be expected to demonstrate an understanding of number meanings with respect to integers and use integers as a foundation for more advanced math concepts Complete a math study sheet to summarize the learning for integers

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 6

EXTRA RESOURCES In case you’d like to explore more resources, here about some to check out: Level III Whole Number Word Problem Bridging Unit by David Palmer (School of Access).

http://www.gonssal.ca/documents/MathInTheRealWorldUnit.pdf http://www.youtube.com/user/MathMeeting?feature=chclk http://www.shodor.org/interactivate/activities/

http://www.khanacademy.org http://www.kutasoftware.com http://www.bbc.co.uk/skillswise/maths

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 7

INTEGERS + Adding +

X Multiplying X

Real Life examples

-Subtracting –

÷ Division ÷

Key Words

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 8

Introduction to Signed Numbers Up to this point in your math classes, you have only worked with positive numbers (i.e. numbers greater than zero). These positive numbers are all the numbers to the right of the zero on a number line.

Positive

0

1

2

3

4

5

6

There are also negative numbers. These are numbers that are less than zero, and found to the left of zero on a number line.

Negative

-6 -5 -4

-3

-2

-1

0

1

2

3

4

5

6

You may not have realized that you have been exposed to negative numbers for years. Consider the following examples. •

"The temperature outside today is -5oC." This means that the temperature is 5 degrees below zero.

•

"The elevation of New Orleans, Louisiana is -2 m." This means that this location is 2 metres below sea level.

•

"The balance on my bank account is -37 dollars." When your balance is 0 dollars, you have no money in the account. When the balance is a negative number, then you have overdrawn the account and you now owe the bank money. In this case you owe the bank 37 dollars.

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 9

•

"T minus 30 seconds." Launch time for a rocket is 0 seconds. In this case, we are 30 seconds before the launch or when the time equals -30 seconds.

•

"The company's profit for 2012 was -1.2 million dollars." If a company breaks even then their profit is reported as zero dollars. When a company loses money, their profits are recorded using negative numbers. If this case, the company lost 1.2 million dollars in 2012.

•

"Mike Weir's score on the hole is -1." When a golfer is below the recommended number of strokes for a hole (i.e. par), the score is reported as a negative number. In this case, Mike's score is one under par.

All signed numbers have magnitude (i.e. how far you move to the right or left) and direction (i.e. sign: positive - move to right, negative - move to left). • For example, the number -4 has a negative direction (to the left of 0 on the number line) and a magnitude of 4. • For example, the number -3.2 has a negative direction (to the left of 0 on the number line) and a magnitude of 3.2. • For example, the number +5 or 5 has a positive direction (to the right of 0 on the number line) and a magnitude of 5. • For example, the number +7.8 or 7.8 has a positive direction (to the right of 0 on the number line) and a magnitude of 7.8. Understanding the terms, magnitude and direction, is important when trying to explain the rules for addition, subtraction, multiplication and division of signed numbers.

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Questions 1. (a) Represent each situation with a signed number. Some answers will be positive numbers, while others are negative numbers. (i) a loss of $7 Number: _____ (ii) 8 metres above sea level Number: _____ (iii) a golf score of par Number: _____ (iv) 2oC below freezing Number: _____ (v) a golf score of 5 under par Number: _____ (vi) a gain of $3 Number: _____ (vii) 9 metres below sea level Number: _____ (viii) a golf score of 1 over par Number: _____ (ix) T minus 4 minutes Number: _____ (b) You generated nine signed numbers in part (a) of this question. Using an arrow for each number, identify each numbers position on the number line below. (Do not worry about working with different units of measure.)

-5

0

5

(c) In parts (a) and (b) of this question, you encountered nine signed numbers. In the blanks below, order those eight numbers from smallest to largest. ______, ______, ______, ______, ______, ______, ______, ______, ______

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2. With each question, we are provided with two numbers. Circle the larger number. (Hint: Consider the position of the numbers on the number line. The larger number is always the number to the right of the other number on the number line.) (a) -6

7

(b) 8

(d) -5

-4

(e) -9

(g) -3 -7

-5 -1

(h) 8 10

(c) 0 (f) -4

-2 -6

(i) -9 0

3. In the blanks below, order the numbers 3, -4, -7, 0 from smallest to largest. ______, ______, ______, ______ 4. In the blanks below, order the numbers 8, -2, 0, -6, -9, 4 from smallest to largest. ______, ______, ______, ______, ______, ______ 5. In the blanks below, order the numbers 6, -12, -1, 3, -4, -15, 10 from smallest to largest. ______, ______, ______, ______, ______, ______, ______ 6. In the blanks below, order the numbers -45, 6, -201, 517, -9, 28 from smallest to largest. _________, _________, _________, _________, _________, _________ 7. Would you rather invest in a company whose monthly profits are approximate 1000 dollars or one whose profits are -2000 dollars? Why?

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8. (a) What is the magnitude of the number -9?

_______

(b) What is the direction of the number 8?

_______

(c) What is the magnitude of the number 6?

_______

(d) What is the direction of the number -10?

_______

9. To answer the following question, you may want to look at the number line on the first page of this section, or create a mental image of that number line. e.g. What number is 3 more than -1? Answer: Start at -1 on the number line and move 3 units to the right. The answer is 2.

-6 -5 -4

-3

-2

-1

0

1

2

3

4

5

6

e.g. What number is 2 less than -3? Answer: Start at -3 on the number line and move 2 units to the left. The answer is -5.

(a)

-6 -5 -4

-3

-2

-1

0

1

2

3

4

than 6? SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 13

5

6

What number is 2 more _____

(b) What number is 3 less than 0?

_____

(c) What number is 4 more than -6?

_____

(d) What number is 2 less than -4?

_____

(e) What number is 5 more than -7?

_____

(f) What number is 4 less than -1?

_____

(g) What number is 3 more than -2?

_____

(h) What number is 5 less than 3?

_____

(i) What number is 6 more than -1?

_____

(j) What number is 4 less than 1?

_____

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 14

Answers Introduction to Signed Numbers 1. (a) (i) -7 (iv) -2 (vii) -9

(ii) 8 or +8 (v) -5 (viii) 1 or +1

(iii) (vi) (ix)

0 3 or +3 -4

(b)

-9

-7

-5 -4

-2

-5

0

1

3

0

8

5

(c) -9, -7, -5, -4, -2, 0, 1, 3, 8 2. (a) 7 (d) -4 (g) -3

(b) 8 (e) -1 (h) 10

(c) 0 (f) -4 (i) 0

3. -7, -4, 0, 3 4. -9, -6, -2, 0, 4, 8 5. -15, -12, -4, -1, 3, 6, 10 6. -201, -45, -9, 6, 28, 517 7. Invest in the company whose monthly profits are approximate $1000. The other company is actually losing money. 8. (a) 9 (b) positive SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 15

(c) 6 (d) negative 9. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

8 -3 -2 -6 -2 -5 1 -2 5 -3

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 16

Adding Signed Numbers - Part 1 In this section we are going to use the number line to add signed numbers. You will probably find this section fairly easy and may be wondering why we are bothering with it. We could just provide the rules for adding signed numbers but you would not understand where those rules come from and likely forget them in a short period of time. Math is only interesting when it makes sense. That is what we are trying to do; make sense of the rules for adding signed numbers. Example 1 Add the following signed numbers using a number line. (a) (c)

( +2 ) + ( +3) ( −5) + ( +4 )

(b) ( +6 ) + ( −4 )

(d) ( −4 ) + ( −2 )

Answers: (a) We obviously know the answer to this question; 2 plus 3 is equal to 5. The only reason this question has been included is so that we can see how the number line can be used to solve it. This will help us when we encounter harder questions. • We are going to start at 2 (or +2) and then move +3 units (meaning we move 3 units to the right). That means that we end up at 5 (or +5). • Therefore ( +2 ) + ( +3) =+5 (No big surprise!)

+

-7 -6 -5

-4

-3

-2

-1

0

1

2

3

4

(b) ( +6 ) + ( −4 ) SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 17

5

6

7

• We are going to start at 6 (or +6) and then move -4 units (meaning we move 4 units to the left). That means that we end up at 2 (or +2). • Therefore ( +6 ) + ( −4 ) =+2

-4

-7 -6 -5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

(c) ( −5 ) + ( +4 ) • We are going to start at -5 and then move +4 units (meaning we move 4 units to the right). That means that we end up at -1. • Therefore ( −5 ) + ( +4 ) =−1

+

-7 -6 -5

-4

-3

-2

-1

0

1

2

3

4

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 18

5

6

7

(d) ( −4 ) + ( −2 ) • We are going to start at -4 and then move -2 units (meaning we move 2 units to the left). That means that we end up at -6. • Therefore ( −4 ) + ( −2 ) =−6

-4

-7 -6 -5

-4

-3

-2

-1

0

1

2

3

4

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 19

5

6

7

Questions 1. Complete the following questions using the number line technique by drawing the appropriate arrow on the number line and filling the blanks in the explanation. (a)

( +1) + ( +5) • We are going to start at ______ and then move _____ units (meaning we move _____ units to the ______________). That means that we end up at _____. • Therefore ( +1) + ( +5 ) = ______

-7 -6 -5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

(b) ( −4 ) + ( +2 ) • We are going to start at ______ and then move _____ units (meaning we move _____ units to the ______________). That means that we end up at _____. • Therefore ( −4 ) + ( +2 ) = ______

-7 -6 -5

-4

-3

-2

-1

0

1

2

3

4

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 20

5

6

7

(c)

( −1) + ( −4 ) • We are going to start at ______ and then move _____ units (meaning we move _____ units to the ______________). That means that we end up at _____. • Therefore ( −1) + ( −4 ) = ______

-7 -6 -5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

2. Complete the following questions using the number line technique by drawing the appropriate arrow on the number line. A written explanation is not required. (a) ( +7 ) + ( −3) = ______

-7 -6 -5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-3

-2

-1

0

1

2

3

4

5

6

7

(b) ( +4 ) + ( −7 ) = ______

-7 -6 -5

-4

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 21

(c) ( −2 ) + ( −5 ) = ______

-7 -6 -5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-3

-2

-1

0

1

2

3

4

5

6

7

(d) ( −2 ) + ( +6 ) = ______

-7 -6 -5

-4

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 22

3. (a) Using the number line technique twice, show that the answers to ( −5 ) + ( +7 ) and ( +7 ) + ( −5 ) are the same.

( −5) + ( +7 ) = ______

-7 -6 -5 (b)

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-2

-1

0

1

2

3

4

5

6

7

( +7 ) + ( −5) = ______

-7 -6 -5

-4

-3

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 23

3. (a) Using the number line technique twice, show that the answers to ( −1) + ( −6 ) and ( −6 ) + ( −1) are the same.

( −1) + ( −6 ) = ______

-7 -6 -5

(b)

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-3

-2

-1

0

1

2

3

4

5

6

7

( −6 ) + ( −1) = ______

-7 -6 -5

-4

(c) When working with whole number, you learned that 3 + 5 = 5 + 3 (or a + b = b + a ). This is referred to as the commutative property. Do you think the commutative property applies to the addition of signed numbers? Yes

No

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 24

4. In this question, we are going to organize our results from the four sample questions, and the eleven additional problems you encountered in questions 1, 2, and 3. Each problem is going to be organized into one of the two categories in the chart below; addition problems where the numbers have the same sign, and addition problems where the numbers have different signs. We have already included the four sample questions in their appropriate categories. You must fill in the remaining eleven problems and their solutions. Addition Problems where the Numbers have the Same Sign

Addition Problems where the Numbers have the Different Signs

( +2 ) + ( +3) =+5 ( −4 ) + ( −2 ) =−6

( +6 ) + ( −4 ) =+2 ( −5) + ( +4 ) =−1

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 25

5. Four learners looked at their results in the first category of the chart above. From this, each tried to come up with the rule for adding numbers with the same sign. Their rules are listed below; only one of these rules is correct. Identify the correct rule using a check mark (). First Learner's Rule: To add two numbers with the same sign, add the magnitudes, and the resulting sign will be positive. Second Learner's Rule: To add two numbers with the same sign, add the magnitudes, and the resulting sign will be negative. Third Learner's Rule: To add two numbers with the same sign, add their magnitudes, and keep the common sign. Fourth Learner's Rule To add two numbers with the same sign, subtract the smaller magnitude from the larger magnitude, and use the sign of the number with the smaller magnitude.

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 26

6. Four learners looked at their results in the second category of the chart in question 4. From this, each tried to come up with the rule for adding numbers with different signs. Their rules are listed below; only one of these rules is correct. Identify the correct rule using a check mark (). First Learner's Rule: To add two numbers with different signs, add the magnitudes, and the resulting sign will be positive. Second Learner's Rule: To add two numbers with different signs, add the magnitudes, and the resulting sign will be negative. Third Learner's Rule: To add two numbers with different signs, subtract the smaller magnitude from the larger magnitude, and use the sign of the number with the smaller magnitude. Fourth Learner's Rule To add two numbers with different signs, subtract the smaller magnitude from the larger magnitude, and use the sign of the number with the larger magnitude.

7. Based on the rule you identified in question 5, answer each of the following questions without using the number line technique. (a) ( +4 ) + ( +6 ) =______

(b) ( −5 ) + ( −4 ) =______

(c) ( −5 ) + ( −7 ) =______

(d) ( +9 ) + ( +5 ) =______

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 27

8. Based on the rule you identified in question 6, answer each of the following questions without using the number line technique. (a) ( −5 ) + ( +8 ) =______

(b) ( +4 ) + ( −9 ) =______

(c) ( −10 ) + ( +3) =______

(d)

( +11) + ( −6 ) =______

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 28

Answers Adding Signed Numbers - Part 1 1. (a) We are going to start at +1 and then move +5 units (meaning we move 5 units to the right). That means that we end up at +6. Therefore ( +1) + ( +5 ) =+6 (b) We are going to start at -4 and then move +2 units (meaning we move 2 units to the right). That means that we end up at -2. Therefore ( −4 ) + ( +2 ) =−2 (c) We are going to start at -1 and then move -4 units (meaning we move 4 units to the left). That means that we end up at -5. Therefore ( −1) + ( −4 ) =−5 2. (a) ( +7 ) + ( −3) =+4 (b) ( +4 ) + ( −7 ) =−3 (c) ( −2 ) + ( −5 ) =−7 (d) ( −2 ) + ( +6 ) =+4 3. (a) ( −5 ) + ( +7 ) =+2 (b) ( −1) + ( −6 ) =−7 (c) Yes

( +7 ) + ( −5) =+2 ( −6 ) + ( −1) =−7

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 29

4.

Addition Problems where the Numbers have the Same Sign

Addition Problems where the Numbers have the Different Signs

( +2 ) + ( +3) =+5 ( −4 ) + ( −2 ) =−6 ( +1) + ( +5) =+6 ( −1) + ( −4 ) =−5 ( −2 ) + ( −5) =−7 ( −1) + ( −6 ) =−7 ( −6 ) + ( −1) =−7

( +6 ) + ( −4 ) =+2 ( −5) + ( +4 ) =−1 ( −4 ) + ( +2 ) =−2 ( +7 ) + ( −3) =+4 ( +4 ) + ( −7 ) =−3 ( −2 ) + ( +6 ) =+4 ( −5) + ( +7 ) =+2 ( +7 ) + ( −5) =+2

5. Third Learner's Rule: To add two numbers with the same sign, add their magnitudes, and keep the common sign. 6. Fourth Learner's Rule To add two numbers with different signs, subtract the smaller magnitude from the larger magnitude, and use the sign of the number with the larger magnitude. 7. (a) 10 (or +10) (c) -12

(b) -9 (d) 14 (or +14)

8. (a) 3 (or +3) (c) -7

(b) -5 (d) 5 (or +5)

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 30

Adding Signed Numbers - Part 2 In the previous section you discovered the two rules for adding signed numbers. These two rules and have provided below with worked examples. We have included questions with decimals and fractions. Remember that with fractions, whether they are positive or negative numbers, we need a common denominator to add them. To keep things simple, we are only going to deal with fractions that have a common denominator. • To add two numbers with the same sign, add their magnitudes, and keep the common sign.

e.g.

(+ 4) + (+ 6) = +10

e.g.

(− 5) + (− 3) = −8

e.g.

(− 2) + (− 7 ) = −9

e.g.

( −10 ) + ( −80 ) =−90

e.g.

( −400 ) + ( −200 ) =−600

e.g.

(− 7.2) + (− 1.3) = −8.5

• To add two numbers with different signs, subtract the smaller magnitude from the larger magnitude, and use the sign of the number with the larger magnitude.

e.g.

(− 4) + (+ 7 ) = +3

e.g.

(+ 5) + (− 3) = +2

e.g.

(+ 3) + (− 9) = −6

e.g.

(− 8) + (+ 1) = −7

e.g.

( −70 ) + ( +50 ) =−20

e.g.

( +600 ) + ( −100 ) =+500

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 31

Questions Do not use a calculator for any of these questions. 1. (a) ( −6 ) + ( −9 ) = ________

(b) ( −8 ) + ( +2 ) = ________

(c) ( +7 ) + ( −10 ) = ________

(d) ( −12 ) + ( −2 ) = ________

(e) ( +9 ) + ( −2 ) = ________

(f)

( −7 ) + ( +8) = ________

(g)

( −11) + ( −4 ) = ________

(h)

( −13) + ( +4 ) = ________

(i)

( +5) + ( +6 ) = ________

(j)

( +6 ) + ( −18) = ________

(k)

( −8) + ( −8) = ________

(l)

( +16 ) + ( −5) = ________

(m) ( −10 ) + ( +8 ) = ________

(n)

( −10 ) + ( +11) = ________

(o) ( −20 ) + ( −60 ) = ________

(p) ( −50 ) + ( +30 ) = ________

(q) ( +70 ) + ( −20 ) = ________

(r)

( +70 ) + ( +40 ) = ________

(s) ( −90 ) + ( −60 ) = ________

(t)

( +40 ) + ( −50 ) = ________

(u) ( +600 ) + ( −100 ) = ________

(v)

( −400 ) + ( −300 ) = ________

(w) ( −800 ) + ( +200 ) = ________

(x)

( −100 ) + ( +900 ) = ________

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 32

Answers Adding Signed Numbers - Part 2 1. (a) -15 (c) -3 (e) +7 (g) -15 (i) +11 (k) -16 (m) -2 (o) -80 (q) +50 (s) -150 (u) +500 (w) -600

(b) (d) (f) (h) (j) (l) (n) (p) (r) (t) (v) (x)

-6 -14 +1 -9 -12 +11 +1 -20 +110 -10 -700 +800

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 33

Subtracting Signed Numbers - Part 1 If you are given the numbers 2, 7 and 9, and asked to generate the corresponding fact family, you would come up with the following.

2+7 = 9

7+2= 9

9−2 = 7

9−7 = 2

This particular fact family involves the operations of addition and subtraction. There are other fact families that involve the operations of multiplication and division, but we will not be examining these types of families in this section. We can apply this fact family knowledge to explore the subtraction of signed numbers. •

If we know that ( −5 ) + ( −3) =−8 , which is one member of a fact family, then we can determine the other three members of that family. Those members would be the following.

( −3) + ( −5) =−8

•

( −8) − ( −3) =−5

If we know that ( −6 ) + ( +2 ) =−4 , which is one member of a fact family, then we can determine the other three members of that family. Those members would be the following.

( +2 ) + ( −6 ) =−4

•

( −8) − ( −5) =−3

( −4 ) − ( +2 ) =−6

( −4 ) − ( −6 ) =+2

If we know that ( −5 ) + ( +11) =+6 , which is one member of a fact family, then we can determine the other three members of that family. Those members would be the following.

( +11) + ( −5) =+6

( +6 ) − ( +11) =−5

( +6 ) − ( −5) =+11

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 34

Questions 1. In each row of the chart below you are going to create a fact family. The first two columns will be for the addition members of the fact family, and the last two columns will be for the subtraction members of the family. The three examples from the previous page have already been placed in the chart to give you some guidance. Complete the chart. Addition Members of the Fact Family

e.g . e.g . e.g . (a) (b) (c) (d) (e) (f)

Subtraction Members of the Fact Family

( −8) − ( −5) =−3

( −8) − ( −3) =−5

( +2 ) + ( −6 ) =−4

( −4 ) − ( +2 ) =−6

( −4 ) − ( −6 ) =+2

( +11) + ( −5) =+6

( +6 ) − ( +11) =−5

( +6 ) − ( −5) =+11

( −5) + ( −3) =−8

( −3) + ( −5) =−8

( −6 ) + ( +2 ) =−4 ( −5) + ( +11) =+6 ( −6 ) + ( −1) =−7 ( −9 ) + ( +7 ) =−2 ( −5) + ( +8) =+3 ( −2 ) + ( −3) =____ ( +9 ) + ( −1) =____ ( +3) + ( −7 ) =____

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 35

2. Four learners looked at their results in the last two columns (subtraction members) of the chart above. From this, each tried to come up with the rule for subtracting signed numbers. Their rules are listed below; only one of these rules is correct. Identify the correct rule using a check mark (). First Learner's Rule: To subtract one signed number from another, change the question from a subtraction question to an addition question, and change the sign of the number that was originally being subtracted (i.e. the second number). Once these changes have been made, follow the rules for adding signed numbers. Second Learner's Rule: To subtract one signed number from another, change the question from a subtraction question to an addition question, and change the sign of the first number. Once these changes have been made, follow the rules for adding signed numbers. Third Learner's Rule: To subtract one signed number from another, subtract the smaller magnitude from the larger magnitude, and keep the sign of the number with the larger magnitude. Fourth Learner's Rule: To subtract one signed number from another, subtract the smaller magnitude from the larger magnitude, and keep the sign of the number with the smaller magnitude.

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 36

3. Using the rule you discovered in question 2, solve each of the following. Space has been provided so that you can show your work. (b) ( −7 ) − ( −6 )

(a) ( −6 ) − ( +9 )

4.

(c)

( +2 ) − ( −1)

(d)

( −5) − ( +4 )

(e)

( +8) − ( −3)

(f)

( −5) − ( −8)

No works needs to be shown with these questions. (a) Geoff owes his sister $30. What is his balance with his sister? (b) If his sister generously takes away $20 worth of that debt, what is Geoff's new balance with his sister? (c) Which one of these math sentences best describes this situation?

−30 + ( −20 ) =−50

+30 + ( −20 ) =+10

−30 − ( −20 ) =−10

(d) Is taking away debt effectively the same as giving someone money? Yes

No

(e) Is subtracting -20 effectively the same as adding +20? Yes

No

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 37

Answers Subtracting Signed Numbers - Part 1 1.

Addition Members of the Fact Family

(a) (b) (c) (d) (e) (f)

( −6 ) + ( −1) =−7 ( −9 ) + ( +7 ) =−2 ( −5) + ( +8) =+3 ( −2 ) + ( −3) =−5 ( +9 ) + ( −1) =+8 ( +3) + ( −7 ) =−4

( −1) + ( −6 ) =−7 ( +7 ) + ( −9 ) =−2 ( +8) + ( −5) =+3 ( −2 ) + ( −3) =−5 ( +9 ) + ( −1) =+8 ( +3) + ( −7 ) =−4

Subtraction Members of the Fact Family

( −7 ) − ( −1) =−6 ( −2 ) − ( +7 ) =−9 ( +3) − ( +8) =−5 ( −5) − ( −2 ) =−3 ( +8) − ( +9 ) =−1 ( −4 ) − ( +3) =−7

( −7 ) − ( −6 ) =−1 ( −2 ) − ( −9 ) =+7 ( +3) − ( −5) =+8 ( −5) − ( −3) =−2 ( +8) − ( −1) =+9 ( −4 ) − ( −7 ) =+3

2. First Learner's Rule: To subtract one signed number from another, change the question from a subtraction question to an addition question, and change the sign of the number that was originally being subtracted (i.e. the second number). Once these changes have been made, follow the rules for adding signed numbers. 3. The middle step has been shown to assist learners. (a) ( −6 ) − ( +9 ) (b) ( −7 ) − ( −6 )

(c)

= ( −6 ) + ( −9 )

= ( −7 ) + ( +6 )

= −15

= −1

( +2 ) − ( −1) = ( +2 ) + ( +1) = +3

(d)

( −5) − ( +4 ) = ( −5 ) + ( −4 ) = −9

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 38

(e)

( +8) − ( −3) = ( +8 ) + ( +3) = +11

4. (a) (b) (c) (d) (e)

(f)

( −5) − ( −8) = ( −5 ) + ( +8 ) = +3

-30 dollars -10 dollars

−30 − ( −20 ) =−10

Yes Yes

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 39

Subtracting Signed Numbers - Part 2 In the previous section, you discovered the rule for subtracting signed numbers. This rule has been provided below with worked examples. We have included questions with decimals and fractions. Remember that with fractions, whether they are positive or negative numbers, we need a common denominator to subtract them. • To subtract one signed number from another, change the question from a subtraction question to an addition question, and change the sign of the number that was originally being subtracted (i.e. the second number). Once these changes have been made, follow the rules for adding signed numbers. e.g.

( +4 ) − ( −6 ) = ( +4 ) + ( +6 )

e.g.

= −8

= +10 e.g.

( −8) − ( −5) = ( −8 ) + ( +5 )

e.g.

( +500 ) − ( −200 ) = ( +500 ) + ( +200 ) = +700

( +10 ) − ( +60 ) = ( +10 ) + ( −60 ) = −50

= −3 e.g.

( −5) − ( +3) = ( −5 ) + ( −3)

e.g.

( −4.5) − ( −1.3) = ( −4.5 ) + ( +1.3) = −3.2

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 40

Questions Solve each of the following. Show your work. Do not use a calculator. 1. (a) ( +7 ) − ( −2 )

(c) ( −8 ) − ( −7 )

(b) ( −2 ) − ( +9 )

(d) ( +6 ) − ( +10 )

(e)

( −11) − ( +7 )

(f)

(g)

( +3) − ( −8)

(h) ( +7 ) − ( −6 )

(i)

( −5) − ( +4 )

(j)

( −6 ) − ( −6 )

(k)

( +50 ) − ( −40 )

(l)

( +10 ) − ( +70 )

(m) ( −90 ) − ( −20 )

( −5) − ( −2 )

(n) ( −60 ) − ( +80 )

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 41

(o) ( +20 ) − ( −80 )

(p) ( −400 ) − ( −600 )

(q) ( +500 ) − ( +900 )

(r)

( −100 ) − ( −900 )

(s) ( −500 ) − ( +700 )

(t)

( +300 ) − ( +800 )

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 42

Answers Subtracting Signed Numbers - Part 2 1. (a) +9 (c) -1 (e) -18 (g) +11 (i) -9 (k) +90 (m) -70 (o) +100 (q) -400 (s) -1200

(b) (d) (f) (h) (j) (l) (n) (p) (r) (t)

-11 -4 -3 +13 0 -60 -140 +200 +800 -500

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 43

Multiplying Signed Numbers - Part 1 When you first started working with multiplication questions like 3 × 2 , you learned that this question could be translated into "3 sets of 2." When this was represented on a number line, you started at 0 and then jumped 3 times along the line by increments of 2 each time. You ended up at 6, so you concluded that 3 × 2 = 6. This is shown on the number line below.

What about the question 2 × 3 ? It would be translated into "2 sets of 3." When this was represented on a number line, you started at 0 and then jumped 2 times along the line by increments of 3 each time. You ended up at 6, so you concluded that 2 × 3 = 6 . This is shown on the number line below. -12 -11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-12 -11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

So 3 × 2 and 2 × 3 are both equal to 6. You learned that the order when multiplying two numbers doesn't make any difference to the final answer (i.e. a × b = b × a ). This is an example of the commutative property at work. So how can we apply this knowledge to the multiplication of signed numbers? Let's consider the question ( +2 ) × ( −4 ) or 2 × ( −4 ) . This question could be translated into "2 sets of -4." When this was represented on a number line, you started at 0 and then jumped 2 times along the line by increments of -4 each time (i.e. jumping 4 units to the left each time). You ended up at -8, so you can conclude that ( +2 ) × ( −4 ) =−8 . This is shown on the number line below.

-12 -11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 44

6

7

8

9

10

11

12

What about the question ( −4 ) × ( +2 ) ? It would be translated into "-4 sets of 2." This doesn't make any sense. How can we have -4 sets? We can't. Our way around this issue is to use the commutative property. Rather than expressing the question as ( −4 ) × ( +2 ) , we express it as ( +2 ) × ( −4 ) . This form makes far more sense, especially when we are trying to find the answer using a number line. If ( −4 ) × ( +2 ) = ( +2 ) × ( −4 ) and ( +2 ) × ( −4 ) =−8 , then ( −4 ) × ( +2 ) =−8 .

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 45

Questions 1. Use the numbers lines provided to illustrate how the answer to each of these multiplication problems can be found. In a few cases, you may have to use the commutative property (i.e. a × b = b × a ) to express the question in a form that will make sense when using the number line technique. (a)

( +5) × ( +2 )

-12 -11

(b)

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

( +4 ) × ( −3)

-12 -11

(d)

-9

( +3) × ( +4 )

-12 -11

(c)

-10

( +3) × ( −2 )

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 46

-12 -11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 47

5

6

7

8

9

10

11

12

(e)

( −2 ) × ( +6 )

(f) -12 -11

-10

( −4 ) × ( +3)

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

2. Look at the answers you obtained in question 1, and the four worked examples below. e.g. ( +8 ) × ( −3) =−24 e.g. ( +5 ) × ( +3) =+15 e.g. ( +9 ) × ( +2 ) =+18 e.g. ( −6 ) × ( +7 ) =−42 Use this information to complete the following statements. Fill in the blanks with either the words "positive" or "negative." (a) When a positive number is multiplied by another positive number (e.g. ( +5 ) × ( +3) ), the resulting product is always a ___________________ number. (b) When a positive number is multiplied by negative number, or vice versa (e.g. ( +8 ) × ( −3) or ( −3) × ( +8 ) ), the resulting product is always a ___________________ number.

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 48

3. Based on the rules you completed in question 2, answer each of the following. No work needs to be shown. (a)

( +4 ) × ( +7 ) = _______

(b)

( +4 ) × ( −5) = _______

(c)

( −6 ) × ( +3) = _______

(d)

( +5) × ( +8) = _______

(e)

( +9 ) × ( −3) = _______

(f)

( −10 ) × ( +2 ) = _______

(g)

( +40 ) × ( +2 ) = _______

(h)

( +7 ) × ( −20 ) = _______

(i)

( −500 ) × ( +5) = _______

(j)

( −2 ) × ( +300 ) = _______

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 49

Answers Multiplying Signed Numbers - Part 1 Answer: ( +5 ) × ( +2 ) =+10

1. (a) Hint: 5 sets of +2 (b) Hint: 3 sets of +4

Answer: ( +3) × ( +4 ) =+12

(c) Hint: 4 sets of -3

Answer: ( +4 ) × ( −3) =−12

(d) Hint: 3 sets of -2

Answer: ( +3) × ( −2 ) =−6

(e) Hints: ( −2 ) × ( +6 ) = ( +6 ) × ( −2 ) , 6 sets of -2

Answer: ( −2 ) × ( +6 ) =−12

2. (a) positive (b) negative 3. (a) (c) (e) (g) (i)

( +4 ) × ( +7 ) =+28 ( −6 ) × ( +3) =−18 ( +9 ) × ( −3) =−27 ( +40 ) × ( +2 ) =+80 ( −500 ) × ( +5) =−2500

(b) (d) (f) (h) (j)

( +4 ) × ( −5) =−20 ( +5) × ( +8) =+40 ( −10 ) × ( +2 ) =−20 ( +7 ) × ( −20 ) =−140 ( −2 ) × ( +300 ) =−600

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 50

Multiplying Signed Numbers - Part 2 In the last section we used the number line technique to discover that: • A positive number multiplied by another positive number, results in a positive product. • A positive number multiplied by a negative number (or vice versa), results in a negative product. But what do we get when we multiply a negative number by another negative number? That is what we will learn about in this section. Investigation, Part 1 Consider the following. We have arranged a series of multiplication questions, where in all cases we are multiplying -2 by some number. Notice that we start by multiplying -2 by +4, and then go down by increments of 1 (i.e. +4, +3, +2,…). Look at the resulting products (-8, -6, -4, -2, …). What is happening to those resulting products? There is a pattern here that we can use to fill in the blanks. Fill in the blanks.

We started by multiplying -2 by +4. Then we multiplied 2 by +3. Each time we multiply -2 by a number that is one less than in the previous question (i.e. going down by increments of 1).

( +4 ) × ( −2 ) =−8 ( +3) × ( −2 ) =−6 ( +2 ) × ( −2 ) =−4 ( +1) × ( −2 ) =−2 0 × ( −2 ) =0 ( −1) × ( −2 ) = ( −2 ) × ( −2 ) = ( −3) × ( −2 ) = ( −4 ) × ( −2 ) =

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 51

There is a pattern between these products that we can use to predict the missing d t

Check your answers to the questions above before attempting the questions below. If you do not understand how those answers were obtained, see your instructor before proceeding. Again, fill in the blanks using the product pattern.

( +3) × ( −3) =−9 ( +2 ) × ( −3) =−6 ( +1) × ( −3) =−3 0 × ( −3) =0 ( −1) × ( −3) = ( −2 ) × ( −3) = ( −3) × ( −3) =

( +3) × ( −4 ) =−12 ( +2 ) × ( −4 ) =−8 ( +1) × ( −4 ) =−4 0 × ( −4 ) =0 ( −1) × ( −4 ) = ( −2 ) × ( −4 ) = ( −3) × ( −4 ) =

( +3) × ( −7 ) =−21 ( +2 ) × ( −7 ) =−14 ( +1) × ( −7 ) =−7 0 × ( −7 ) =0 ( −1) × ( −7 ) = ( −2 ) × ( −7 ) = ( −3) × ( −7 ) =

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 52

Complete this statement by filling in the blank with either the word "positive" or "negative." By completing the part 1 of the investigation on the previous page, you should have discovered that a negative number multiplied by another negative number results in a _______________ product. We learned this using a pattern that we saw when given a specific series of products. But why is this so? Below we have supplied an explanation. You do not have to know this, but it would be improper for us not to supply this explanation. Consider ( −2 ) × ( +4 ) + ( −4 )  According to order of operations (BEDMAS), we would start by completing the operations in the brackets. In this case, that means adding the +4 and -4; this gives 0. After that we multiply the -2 by the 0. This gives us a product of 0.

( −2 ) × ( +4 ) + ( −4 ) =( −2 ) × 0 =0

Now there is another way to evaluate this expression; it involves using the distributive property (i.e.

a × ( b + c ) = a × b + a × c ). With this property, we multiply the number outside of the brackets (i.e. -2)

by both of the numbers inside the brackets before doing the addition. That means the -2 must first be multiplied by the +4, then by the -4.

( −2 ) × ( +4 ) + ( −4 ) = ( −2 ) × ( +4 )  + ( −2 ) × ( −4 ) 

From our previous work (when we followed the order of operations), we know that this should equal 0.

( −2 ) × ( +4 )  + ( −2 ) × ( −4 )  =0

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 53

We know the rules for multiplying a positive number and a negative number (or vice versa); the product is a positive number. Therefore -2 multiplied by +4 is -8. However, we are pretending that we don't know what ( −2 ) × ( −4 ) is equal to. We placed a question mark in the expression to indicate that we do not know this product.

( −2 ) × ( +4 )  + ( −2 ) × ( −4 )  =0 ( −8) + ( −2 ) × ( −4 ) =0

( −8) + ? =0

So what is the value of the question mark (?). If that value is being added to -8, and the resulting sum is 0, the only possible value is +8. That means that ( −2 ) × ( −4 ) must be equal to +8. In this one case, we have shown that a negative number multiplied by another negative number results in a positive product. The formal proof of this involves algebra, which might be too confusing at this time. Your instructor may wish to supply it to you when you are taking Level IV Academic Math. From the work we have done in this section of the resource and the previous section, we now have all the rules for multiplying signed numbers. • To multiply two numbers with the same sign, multiply the magnitudes, and the resulting sign will be positive (i.e. positive × positive = positive, negative × negative = positive). e.g. (+ 6 ) × (+ 3) = +18 e.g. (− 5) × (− 4 ) = +20 e.g. (− 2 ) × (− 12 ) = +24

e.g. ( −8 ) × ( −60 ) =+480

e.g. ( −200 ) × ( −8 ) =+1600

e.g. ( +6 ) × ( +0.7 ) =+4.2

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 54

• To multiply two numbers with the different signs, multiply the magnitudes, and the resulting sign will be negative (i.e. negative × positive = negative). e.g. (− 6 ) × (+ 2 ) = −12

e.g. (+ 8) × (− 4 ) = −32

e.g. ( +30 ) × ( −7 ) =−210

e.g. ( −50 ) × ( +90 ) =−4500

Example 1 Find the two integers that have: (a) a sum of +7, and a product of +12. (b) a sum of -2, and a product of -15. (c) a sum of -7, and a product of +6. Answers: The word "sum" indicates that we are dealing with the operation of addition. The word "product" indicates that we are dealing with the operation of multiplication. Remember that integers (i.e. …, -3, -2, -1, 0, 1, 2, 3,…) are a specific type of signed number; they do not include decimals or fractions. (a) With this question we are trying to find the two integers that add to +7, and when multiplied, produce +12. ___ + ___ = +7 and ___ × ___ = +12 Most people usually start this type of question by thinking of all the possible pairs of integers that produce a product of +12. We have listed these below.

( +2 ) × ( +6 ) =+12

( +3) × ( +4 ) =+12

( +1) × ( +12 ) =+12

( −2 ) × ( −6 ) =+12

( −3) × ( −4 ) =+12

( −1) × ( −12 ) =+12

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 55

So we have six possible choices; only one of these satisfies the other condition that the sum must be +7. • Let's consider the first pair: ( +2 ) × ( +6 ) =+12 . When we add +2 and +6, we obtain +8, not 7. These are not the integers we are looking for. • Let's consider the second pair: ( +3) × ( +4 ) =+12 . When we add +3 and +4, we obtain +7, the desired sum. We have found the two integers. The two integers are +3 and +4. (b) 15.

With this question we are trying to find the two integers that add to -2, and when multiplied, produce -

___ + ___ = -2 and ___ × ___ = -15 List all the possible pairs of integers that produce a product of -15.

( −3) × ( +5) =−15 ( +3) × ( −5) =−15 ( −1) × ( +15) =−15 ( +1) × ( −15) =−15 We have four possible choices; only one of these satisfies the other condition that the sum must be 2. • Let's consider the first pair: ( −3) × ( +5 ) =−15 . When we add -3 and +5, we obtain +2, not -2. These are not the integers we are looking for. • Let's consider the second pair: ( +3) × ( −5 ) =−15 . When we add +3 and -5, we obtain -2, the desired sum. We have found the two integers. The two integers are +3 and -5.

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 56

(c) With this question we are trying to find the two integers that add to -7, and when multiplied, produce +6. ___ + ___ = -7 and ___ × ___ = +6 List all the possible pairs of integers that produce a product of +6.

( +2 ) × ( +3) =+6

( −2 ) × ( −3) =+6

( +1) × ( +6 ) =+6

( −1) × ( −6 ) =+6

We have four possible choices; only one of these satisfies the other condition that the sum must be 7. • Let's consider the first pair: ( +2 ) × ( +3) =+6 . When we add +2 and +3, we obtain +5, not -7. These are not the integers we are looking for. • Let's consider the second pair: ( −2 ) × ( −3) =+6 . When we add -2 and -3, we obtain -5, not -7. These are not the integers we are looking for. • Let's consider the third pair: ( +1) × ( +6 ) =+6 . When we add +1 and +6, we obtain +7, not -7. These are not the integers we are looking for. • Let's consider the fourth pair: ( −1) × ( −6 ) =+6 . When we add -1 and -6, we obtain -7, the desired sum. We have found the two integers. The two integers are -1 and -6. With the last few examples, we have shown the thought process one would follow to find the unknown integers that produce a specific product and sum. You do not have to show all this work; you can just figure the answer out in your head or on scrap paper.

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 57

Questions 1. Complete the following. No work needs to be shown. (a)

( +4 ) × ( +7 ) = _______

(b)

( −9 ) × ( +4 ) = _______

(c)

( −8) × ( −5) = _______

(d)

( +3) × ( −9 ) = _______

(e)

( −2 ) × ( +8) = _______

(f)

( −6 ) × ( −6 ) = _______

(g)

( +3) × ( −5) = _______

(h)

( +6 ) × ( +8) = _______

(i)

( −80 ) × ( −4 ) = _______

(j)

( −5) × ( +70 ) = _______

(k)

( +20 ) × ( +40 ) = _______

(l)

( +6 ) × ( −700 ) = _______

(m) ( −30 ) × ( −40 ) = _______

(n) ( −100 ) × ( −9 ) = _______

(o) ( −90 ) × ( +60 ) = _______

(p) ( +80 ) × ( −30 ) = _______

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 58

4. Complete the following. Notice that you are not limited to multiplication questions; there are also questions concerned with addition and subtraction. (a)

( −4 ) × ( +9 )

(b) ( −7 ) + ( +2 )

(c)

( +3) − ( +8)

(d)

( −8) × ( −5)

(e) ( +6 ) − ( −7 )

(f)

( +9 ) × ( −7 )

(g) ( −8 ) + ( −4 )

(h)

( −5) − ( +8)

(i)

( −30 ) × ( +5)

(j)

( −6 ) × ( −200 )

(k)

( −50 ) − ( −30 )

(l)

( −600 ) + ( +800 )

(m) ( +80 ) × ( −70 )

(n) ( −70 ) + ( −60 )

(o) ( +700 ) − ( −400 )

(p) ( −50 ) × ( −50 )

(q) ( +300 ) + ( −900 )

(r)

( +40 ) − ( +90 )

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 59

5. Find the two integers that have: (a) a sum of +6, and a product of +8.

_______ and _______

(b) a sum of -7, and a product of +10.

_______ and _______

(c) a sum of -1, and a product of -12.

_______ and _______

(d) a sum of -4, and a product of -12.

_______ and _______

(e) a sum of -8, and a product of +15. (f) a sum of -3, and a product of -4.

_______ and _______ _______ and _______

(g) a sum of +1, and a product of -20.

_______ and _______

(h) a sum of 0, and a product of -25.

_______ and _______

(i) a sum of -9, and a product of +8.

_______ and _______

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 60

Answers Multiplying Signed Numbers - Part 2 Investigation

( +4 ) × ( −2 ) =−8 ( +3) × ( −2 ) =−6 ( +2 ) × ( −2 ) =−4 ( +1) × ( −2 ) =−2 0 × ( −2 ) =0 ( −1) × ( −2 ) =+2 ( −2 ) × ( −2 ) =+4 ( −3) × ( −2 ) =+6 ( −4 ) × ( −2 ) =+8

It is hoped that learners recognize that the products are increasing by increments of 2 each time. Therefore if we extended the pattern, the missing products should be +2, +4, +6, and +8. (i.e. -8, -6, -4, -2, 0, +2, +4, +6, +8)

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 61

The products below are increasing by increments of 3.

The products below are increasing by increments of 4.

The products below are increasing by increments of 7.

( +3) × ( −3) =−9 ( +2 ) × ( −3) =−6 ( +1) × ( −3) =−3 0 × ( −3) =0 ( −1) × ( −3) =+3 ( −2 ) × ( −3) =+6 ( −3) × ( −3) =+9

( +3) × ( −4 ) =−12 ( +2 ) × ( −4 ) =−8 ( +1) × ( −4 ) =−4 0 × ( −4 ) =0 ( −1) × ( −4 ) =+4 ( −2 ) × ( −4 ) =+8 ( −3) × ( −4 ) =+12

( +3) × ( −7 ) =−21 ⊕ ( +2 ) × ( −7 ) =−14 ( +1) × ( −7 ) =−7 0 × ( −7 ) =0 ( −1) × ( −7 ) =+7 ( −2 ) × ( −7 ) =+14 ( −3) × ( −7 ) =+21

Fill in the Blank: positive

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 62

Questions 1. (a) +28 (c) +40 (e) -16 (g) -15 (i) +320 (k) +800 (m) +1200 (o) -5400

(b) (d) (f) (h) (j) (l) (n) (p)

-36 -27 +36 +48 -350 -4200 +900 -2400

4. (a) -36 (c) -5 (e) +13 (g) -12 (i) -150 (k) -20 (m) -5600 (o) +1100 (q) -600

(b) (d) (f) (h) (j) (l) (n) (p) (r)

-5 +40 -63 -13 +1200 +200 -130 +2500 -50

5. (a) (b) (c) (d) (e) (f) (g) (h) (i)

+2 and +4 -2 and -5 +3 and -4 -6 and +2 -3 and -5 +1 and -4 -4 and +5 -5 and +5 -8 and -1

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 63

Dividing Signed Numbers - Part 1 If you are given the numbers 2, 5 and 10, and asked to generate the corresponding fact family, you would come up with the following.

2×5 = 10

5× 2 = 10

10 ÷ 2 = 5

10 ÷ 5 = 2

This particular fact family involves the operations of multiplication and division. There are other fact families that involve the operations of addition and subtraction, but we will not be examining these types of families in this section. We can apply this fact family knowledge to explore the division of signed numbers. •

If we know that ( −5 ) × ( −3) =+15 , which is one member of a fact family, then we can determine the other three members of that family. Those members would be the following.

( −3) × ( −5) =+15

•

( +15) ÷ ( −3) =−5

If we know that ( −4 ) × ( +7 ) =−28 , which is one member of a fact family, then we can determine the other three members of that family. Those members would be the following.

( +7 ) × ( −4 ) =−28

•

( +15) ÷ ( −5) =−3

( −28) ÷ ( +7 ) =−4

( −28) ÷ ( −4 ) =+7

If we know that ( +5 ) × ( −9 ) =−45 , which is one member of a fact family, then we can determine the other three members of that family. Those members would be the following.

( −9 ) × ( +5) =−45

( −45) ÷ ( +5) =−9

( −45) ÷ ( −9 ) =+5

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 64

Questions 1. In each row of the chart below you are going to create a fact family. The first two columns will be for the multiplication members of the fact family, and the last two columns will be for the division members of the family. The three examples from the previous page have already been placed in the chart to give you some guidance. Complete the chart. Multiplication Members of the Fact Family

e.g . e.g . e.g . (a) (b) (c) (d) (e) (f)

Division Members of the Fact Family

( −5) × ( −3) =+15

( −3) × ( −5) =+15

( +15) ÷ ( −5) =−3

( +15) ÷ ( −3) =−5

( −4 ) × ( +7 ) =−28

( +7 ) × ( −4 ) =−28

( −28) ÷ ( +7 ) =−4

( −28) ÷ ( −4 ) =+7

( +5) × ( −9 ) =−45

( −9 ) × ( +5) =−45

( −45) ÷ ( +5) =−9

( −45) ÷ ( −9 ) =+5

( +6 ) × ( +5) =+30 ( −4 ) × ( +3) =−12 ( −8) × ( −7 ) =+56 ( +2 ) × ( −3) =____ ( −5) × ( −6 ) =____ ( −1) × ( +9 ) =____

Once you have completed the chart use a highlighter or colored pencil to identify division questions where you divide a number by another number of the same sign (e.g. ( −28 ) ÷ ( −4 ) and ( +30 ) ÷ ( +5 ) ). Doing so will help you on question 2.

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 65

2. Four learners looked at their results in the last two columns (division members) of the chart above. From this, each tried to come up with the rule for dividing signed numbers. Their rules are listed below; two of these rules are correct. Identify the correct rules using a check mark (). First Learner's Rule: To divide two numbers with the different signs, divide the magnitude of the first by the magnitude of the second, and the resulting sign will be positive. Second Learner's Rule: To divide two numbers with the different signs, divide the magnitude of the first by the magnitude of the second, and the resulting sign will be negative. Third Learner's Rule: To divide two numbers with the same sign, divide the magnitude of the first by the magnitude of the second, and the resulting sign will be negative. Fourth Learner's Rule: To divide two numbers with the same sign, divide the magnitude of the first by the magnitude of the second, and the resulting sign will be positive.

3. Using the two rules you identified in question 2, complete the following. No work needs to be shown. (a)

( +40 ) ÷ ( +5) = ______

(b)

( −20 ) ÷ ( +4 ) = ______

(c)

( +16 ) ÷ ( −2 ) = ______

(d)

( −18) ÷ ( −3) = ______

(e)

( −28) ÷ ( +4 ) = ______

(f)

( −36 ) ÷ ( −4 ) = ______

(g) ( −70 ) ÷ ( −7 ) = ______

(h)

( +32 ) ÷ ( −8) = ______

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 66

Answers Dividing Signed Number - Part 1 1.

Multiplication Members of the Fact Family (a) (b) (c) (d) (e) (f)

( +6 ) × ( +5) =+30 ( −4 ) × ( +3) =−12 ( −8) × ( −7 ) =+56 ( +2 ) × ( −3) =−6 ( −5) × ( −6 ) =+30 ( −1) × ( +9 ) =−9

Division Members of the Fact Family

( +5) × ( +6 ) =+30 ( +3) × ( −4 ) =−12 ( −7 ) × ( −8) =+56 ( −3) × ( +2 ) =−6 ( −6 ) × ( −5) =+30 ( +9 ) × ( −1) =−9

( +30 ) ÷ ( +5) =+6 * ( −12 ) ÷ ( +3) =−4 ( +56 ) ÷ ( −7 ) =−8 ( −6 ) ÷ ( −3) =+2 * ( +30 ) ÷ ( −5) =−6 ( −9 ) ÷ ( +9 ) =−1

( +30 ) ÷ ( +6 ) =+5 * ( −12 ) ÷ ( −4 ) =+3 * ( +56 ) ÷ ( −8) =−7 ( −6 ) ÷ ( +2 ) =−3 ( +30 ) ÷ ( −6 ) =−5 ( −9 ) ÷ ( −1) =+9 *

* - identifies division questions where a number is being divided by another number of the same sign.

2. Second Learner's Rule: To divide two numbers with the different signs, divide the magnitude of the first by the magnitude of the second, and the resulting sign will be negative. Fourth Learner's Rule: To divide two numbers with the same sign, divide the magnitude of the first by the magnitude of the second, and the resulting sign will be positive. 3. (a) (c) (e) (g)

+8 -8 -7 +10

(b) (d) (f) (h)

-5 +6 +9 -4

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 67

Dividing Signed Numbers - Part 2 In the previous section you discovered the following two rules for dividing signed numbers. • To divide two numbers with the same sign, divide the magnitude of the first by the magnitude of the second, and the resulting sign will be positive. e.g. (+ 6 ) ÷ (+ 2 ) = +3 e.g. (− 8) ÷ (− 4 ) = +2 e.g. ( −350 ) ÷ ( −7 ) =+50

e.g. ( −540 ) ÷ ( −90 ) =+6

• To divide two numbers with the different signs, divide the magnitude of the first by the magnitude of the second, and the resulting sign will be negative. e.g. (+ 16 ) ÷ (− 4 ) = −4 e.g. (− 45) ÷ (+ 9 ) = −5 e.g. ( +6300 ) ÷ ( −7 ) =−900

e.g. ( −5600 ) ÷ ( +80 ) = − − 70

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 68

Questions 1. Complete the following. No work needs to be shown. (a)

( +15) ÷ ( +5) = ______

(b)

( −12 ) ÷ ( +4 ) = ______

(c)

( −18) ÷ ( −3) = ______

(d)

( +28) ÷ ( −7 ) = ______

(e)

( −42 ) ÷ ( +6 ) = ______

(f)

( −36 ) ÷ ( −9 ) = ______

(g)

( +40 ) ÷ ( −8) = ______

(h)

( −16 ) ÷ ( +4 ) = ______

(i)

( −35) ÷ ( −5) = ______

(j)

( +63) ÷ ( +7 ) = ______

(k)

( −9 ) ÷ ( +9 ) = ______

(l)

( +800 ) ÷ ( −4 ) = ______

(n)

( −120 ) ÷ ( −4 ) = ______

(m) ( −60 ) ÷ ( +2 ) = ______ (o)

( +3600 ) ÷ ( −6 ) = ______

(p)

( +1200 ) ÷ ( +30 ) = ______

(q)

( −720 ) ÷ ( +80 ) = ______

(r)

( −500 ) ÷ ( −50 ) = ______

(s)

( −4500 ) ÷ ( −900 ) = ______

(t)

( +900 ) ÷ ( −300 ) = ______

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 69

2. Complete the following. Notice that you are not limited to division questions; there are also questions concerned with addition, subtraction, and multiplication. (a) ( −9 ) + ( +5 )

(b)

( −36 ) ÷ ( −4 )

(c)

( −3) × ( +7 )

(d)

( +8) − ( −6 )

(e)

( −22 ) ÷ ( +2 )

(f)

( −7 ) − ( +3)

(h)

( −4 ) × ( −5)

(g) ( −2 ) + ( +8 )

(i)

( −10 ) − ( −5)

(j)

( +7 ) × ( −7 )

(k)

( +27 ) ÷ ( −3)

(l)

( −6 ) + ( −14 )

(m) ( −30 ) × ( −5 )

(n) ( −70 ) + ( +30 )

(o) ( −140 ) ÷ ( +7 )

(p) ( +200 ) − ( −400 )

(q) ( −300 ) + ( −500 )

(r)

( +500 ) − ( +700 )

(s) ( +600 ) × ( −9 )

(t)

( −180 ) ÷ ( −60 )

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 70

Answers Dividing Signed Numbers - Part 2 1. (a) +3 (c) +6 (e) -7 (g) -5 (i) +7 (k) -1 (m) -30 (o) -600 (q) -9 (s) +5

(b) (d) (f) (h) (j) (l) (n) (p) (r) (t)

-3 -4 +4 -4 +9 -200 +30 +40 +10 -3

2. (a) -4 (c) -21 (e) -11 (g) +6 (i) -5 (k) -9 (m) +150 (o) -20 (q) -800 (s) -5400

(b) (d) (f) (h) (j) (l) (n) (p) (r) (t)

+9 +14 -10 +20 -49 -20 -40 +600 -200 +3

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 71

FINAL STEPS: Finishing up the module Well done! You’ve made it to the end of this module. In this module you’ve: • Reading and writing integers • Adding, solving, multiplying and dividing integers • Solving problems with integers • Completing a study sheet about integers This is an important part of your work towards these session learning objectives:  

Students will be expected to demonstrate an understanding of number meanings with respect to integers and use integers as a foundation for more advanced math concepts Complete a math study sheet to summarize the learning for integers

Coming up next… Module 3 – Fractions

SUMMER LEARNING SESSIONS | NUMERACY – Module 2 - Integers | Page 72