COSER @MUNI 2024 - 2. WORKING WITH INTEGERS

COSER @MUNI 2024

Topic 2:

WORKING WITH INTEGERS

Key Words

positive, negative, BODMAS, LCM, HCF

By the end of this topic, you should be able to:

i) identify, read and write natural numbers as numerals and

words in million, billion and trillion.

ii) differentiate between natural numbers and whole

numbers/integers.

iii) identify directed numbers.

iv) use directed numbers (limited to integers) in real life situations.

v) use the hierarchy of operations to carry out the four

mathematical operations on integers.

vi) identify even, odd, prime and composite numbers.

vii) find the prime factorisation of any number.

viii) relate common factors with HCF and multiples with LCM.

ix) work out and use divisibility tests of some numbers.

Introduction

Sarah was sent to a shop up the hill to buy 1kg of sugar, a packet of salt and a packet of tealeaves. She was given UGX. 5,000 note but all items cost her UGX. 6,500. How much money did Sarah owe the shopkeeper?

In your day-to-day life, you use numbers to count items, to keep information, to transact business and many others. Since you use numbers in your day-to-today situations, knowledge of integers will be helpful to you.

Subtopic 2.1: Natural Numbers

In lower primary, you learnt counting items using numbers one, two, three ---. In mathematics these numbers are called counting or natural numbers.

When zero is included in the set of natural numbers, they become whole numbers.

For example: N = {1,2,3,4,5,- - - - - -}This is a set of natural numbers.

W ={0,1,2,3,4,5. - - - - - -}This is a set of whole numbers.

Activity 2.1: Natural numbers

There is a box and a board. In the box, there are number cards: some have numbers in figures and others in words. While the board has two sections: one section for natural and the other for non-natural

numbers.

In groups, pick a card and place it in the appropriate section of the board.

Is it possible for a number to belong to two sections?

What can you say about the two categories of the numbers picked?

Where in real-life situations do we find such numbers?

Activity: 2.2: Writing and reading numbers

There are two boxes. In one box, number cards are written in figures and the others in words.

In groups, a member picks one card from one of the boxes. After all the cards have been picked, one member displays his/her card; then the others check their cards, and the matching card is displayed.

Exercise

Write the following in words:

1. 3,800

2. 8,008,008

3. 606,520,060

4. 9,000,909,800

5. 4,629,842,003

6. 1,629,284,729,000

Write the following in figures:

7. Six hundred two million eight thousand and eight

8. Two billion eighty-nine million four thousand seven

9. One trillion two hundred fifty billion eight hundred seventy-five

million three hundred sixty thousand

10. State the value of digit four in the following numbers.

i) 7,462,300,800

ii) 24,629,293,005

Sub topic 2.2: Differentiating between natural numbers and whole numbers/integers

Activity 2.3: Relating natural numbers and integers

In groups, read the text below and answer the questions that follow:

Two learners—Mary and Joy—went to the school canteen to buy some snacks for their breakfast. Joy bought 3 pancakes at UGX.200 each and 1 ban at UGX. 300.

Mary checked her bag and found out that her money was stolen. She borrowed some money from Joy. She bought four 4 pancakes and 2 bans.

Questions

i) Which of the two learners had more money?

ii) How much money did Mary borrow from Joy?

iii) Peter said that Mary had negative UGX. 1400. Was he correct?

Give reasons for your answer.

Sub-topic 2.3: Use Directed Numbers (Limited to Integers) in Real-life Situations

Activity 2.4: Integers in real-life situations

In groups, read the story below and answer the questions.

Once upon a time, there lived an old woman. She had hot and cold stones and a big pot of water. If she put one hot stone in the water, the temperature of the water would rise by 1 degree. If she took the hot stone out of the water again, the temperature would go down by 1 degree.

Question 1

If the temperature of the water is 24 degrees and the old woman adds 5 hot stones, what is the new temperature of the water?

Now imagine that the temperature of the water is at 29 degrees. The old woman takes a spoon and takes out 3 of the hot stones from the pot.

Question 2

What is the temperature of the water when the old woman removes 3 hot stones? Explain your answer.

The old woman also had cold stones. If she adds 1 cold stone to the water, the temperature goes down by 1 degree. The temperature of the water was 26 degrees. Then the old woman added 4 cold stones.

Question 3

What is the temperature of the water after the old woman added 4 cold stones? Give a reason for your answer.

Just like the old woman could remove the hot stones and the temperature would decrease she could also remove the cold stones.

Question 4

Imagine that the temperature of the water was 22 degrees and the old woman removes 3 cold stones. What happens to the temperature of the water?

What is the new temperature of the water? Explain your answer.

Activity 2.5: Real-life situations

In groups, get a cup of hot water and a thermometer. Identify a timekeeper in your group. One of you reads the temperature on the thermometer and the other members record in their notebooks. Put

the thermometer back into the hot water and after 5 minutes take the reading on the thermometer. Repeat this at same interval of 5 minutes for duration of 25 minutes.

Question 1

What is the change in temperature between the first reading and the second reading?

Question 2

What is the change in temperature between the 2nd and 3rd reading?

Question 3

What is the change in temperature between the 3rd and the 4th reading?

Question 4

What is the difference in temperature between the 4th and the 5th reading?

Sub-topic 2.4: Use the Hierarchy of Operations to Carry out the Four Mathematical Operations on Integers

Activity 2.6: Operations on integers

In groups, read the text below and answer the questions after.

Sarah moved 5 steps to the right from a fixed point. Then she moved 9 steps to the left.

Question 1

How far is Sarah from the fixed point?

Question 2

Peter gave his answer as 4 steps to the left of the fixed point and John as -4 (negative 4). Who is correct? Give reasons for your answer.

Example 1

Example 2: A group of learners of Geography went for a tour to Kabale. They found out that the temperature at one time was 130C. At around mid-night the temperature was 100C. By how many degrees had the temperature dropped?

Solution: 100C - 130C = - 30C

Example 2: Using a number line work out:

a) – 4 + +6

b) +5 + - 9

c) -6 - 4 = -6 + - 4

-6 - 4 = -6 + -4 = -10

Exercise

1. Work out the following in degrees:

2. Work out the following:

a) 8 + -6

b) 61 + + 7

c) 49 - - 30

d) 77 - + 50

e) -15 + + 20

f) -3 - - 13

3. 2.Using a number line work out:

a) -2 ++ 3

b) +5 + - 6

c) – 8 - -5

4. A national park guide on one of the mountains in East Africa recorded the temperature as 150C one day. At midnight the

temperature was -70C. By how many degrees had the temperature fallen?

5. Write down the next 3 terms in the sequence - 9, -7, - 5, -3, - , - ; -

6. Look at the sequence of the numbers:

-1, 3,-9, +27, -, - , -

Alex said the next three terms are +9, -36 and -729. Is Alex correct? Give

reasons for your answer.

Multiplication and Division

Multiplication such as +4 × + 3 or -4 × + 3 are interpreted as repeated

addition of positive or negative numbers.

+4 × + 3 = + 4 + +4 + +4 = +12

-4 × +3 = -4 + -4 + -4 = -12

Negative

3 × 3 = 9

3 ×- 3 = -9

-3 × -3 =9

Justification of the above is as follows:

3 × 3 = 9

3 × 2 = 6

3 × 1 = 3

3 ×0 = 0

3 × -1 = -3

3 × -2 = -6

3 × -3 = -9

Now reduce the first multiplier

3 × -3 =-9

2 × -3 = -6

1 × -3 =-3

0 × -3 =0

-1 × -3 = 3

-2 × -3 = 6

-3 × -3 = 9

The justification shows that any number multiplied by zero is zero; that a positive number multiplied by a positive number is a positive; a negative number multiplied by a positive number is a negative, and a negative number multiplied by a negative is a positive.

Multiplication and division have the same rules:

A negative number divided by a positive and a positive number divided by a negative number is a negative, Also a negative number divided by a negative is a positive.

Example

+ 4 × -3 = -12

-12 ÷ - 3 = +4

- 12 ÷ +4 = - 3

Note: Rules of integers

a) Positive number multiplied by a positive number is a positive.

b) Negative number multiplied by a positive number is a negative.

c) Negative number multiplied by a negative is a positive.

d) Negative number divided by a positive is a negative

e) Positive number divided by a negative is a negative.

f) Negative number divided by a negative is a positive.

Exercise

Work out

1. - 2 ×+ 4 × -3

2. -4 ×+2 × - 3

3. -3× -5 × +2

4. -12 ×-5 ÷ +6

5 -15 ÷ 5 × -4

6. -24 × + 4 ÷+2

7. In a certain test a correct answer scores 3marks and an incorrect answer, the child gets a penalty of two marks deducted. Joy guessed all the answers. She got 6 correct and 4 wrong. Work out her total marks.

8. Simplify +6 - +7 ÷ +4 + + 6 × +7

9. Work out 7 of 13 – (18 ÷ 6 +3) ÷ (9 × 3 -25)

10. 56 - (38 - 35 ÷5 +2)

11. 69 ÷ (6 + (3 × 8 -7))

12. 4 of (5 + 2) - 2 (3 + 7) ÷ 5

Sub-topic 2.5: Identify Even, Odd, Prime and Composite Numbers

Natural numbers can be classified into various groups of numbers. In your primary education, you learnt numbers such as even, odd, prime and composite.

Activity 2.6: Identifying even, odd, prime and composite numbers

Each group is given a box containing number cards. In your groups pick the card and read the number. Identify which group of numbers it belongs to by filling the table below.

Question 1

Are there numbers that belong to more than one group?

Question 2

How do you identify that a number is:

a) odd

b) even

c) prime

d) composite

Sub-topic 2.6: Find the Prime Factors of any

Number

In your primary education you studied multiples and factors of

numbers. When two numbers are multiplied together, the product is

called multiple. The two numbers multiplied together are called

factors of the multiple.

Note: A multiple has two or more factors.

For example: The factors of 12 are (1 × 12), (2 × 6 ) , and ( 3 × 4 );

hence, the factors of 12 are {1,2,3,4,6,12}= F12 = {1,2,3,4,6,12}

The multiples of 3 are {3,6,9,12,15,18,21- - -} = M3 =

{3,6,9,12,15,18,21- -}

Exercise

Find the factors of the following:

1. 42

2. 56

3. 36

4. 108

Find the multiples of the following:

5. 7

6. 12

7. 9

8. 5

Note: A factor of a number which is a prime number is called its prime factor. For example the factors of 36 are {1,2,3,4,6,9,12,36}

Qn. What are the prime factors of 36?

Qn. Write 36 as a product of its prime factors.

Answer:

This approach of determining prime factors is called prime factorization.

This can be written in power notation.

Exercise

Find the prime factors of the following numbers. Give your answer in power form.

1. 108

2. 288

3. 180

4. 1232

5. 993

6. 2145

Sub-topic 2.7: Relate Common Factors with HCF and Multiples with LCM

A number can have one or more common factors; for example, 2 and 4 are common factors of 8 and12. However, the highest common factor

is 4. Therefore, the highest common factor (HCF) of 8 and 12 is 4.

Activity 2.7: Highest common factor

In groups, find the HCF of the following:

i) 54, 48

ii) 42 ,63 ,105

iii) 132, 156,204,228

Sub-topic 2.8: Work Out and Use Divisibility Tests of Some Numbers

Activity 2.8: Identifying divisibility tests for some numbers

1. In your groups, pick a number card and determine which numbers on the chart divides it. Write a number under its divisor.

2. What can you say about the numbers under each divisor? Give reasons for your answers.

3. The relationship between the dividend and the divisor leads to divisibility tests.

Exercise

Given the following numbers:

12, 132, 1212, 3243, 1112, 81, 18, 27, 279, 2580, 5750

Find out which of them are divisible by:

a) 3 b) 4 c) 6 d) 9 e) 10

Exercise

Find the HCF the following:

1. 2 × 2× 3 × 3× 3 ×3 × 5 × 5 ×5 × 11

2. 2 × 2 ×2 ×2 ×2 ×3 ×3 ×5 ×7 ×13

3. 2. 23 ×32×5 2, 25 × 35 ×52

4. 36, 60, 84

4. A rectangular field measures 616m by 456m. Fencing posts are placed along its sides at equal distances. What will be the distance between the posts if they are placed as far apart as possible? How many posts are required?

Sub-topic 2.9: Least Common Multiple (LCM)

In the previous section of multiples and factors you learnt about multiples of numbers. For example, the multiples of 5 are 5, 10, 15,20,25,30, 35, 40, 45, 50, 55, 60, 65, 70, 75 ------. The multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77 ------. From the above example, 35 and 70 are common multiples of 5 and at the same time of 7. However, 35 is smaller than 70, therefore, 35 is the least common multiple of 5 and 7.

There is another approach of getting LCM of numbers without listing the multiples of the numbers.

Example

Find the LCM of 8 and 12

The LCM of 8 and 12 is 24.

Activity 2.9: In your groups, find the LCM of the following:

a) 28, 42 ,98

b) 35,48 ,56, 70

Exercise

Find the LCM of the following numbers:

1. 14, 21

2. 18, 24, 96

3. 49, 84, 63

4. 60, 72, 84, 112

5. Determine the smallest sum of money out of which a number of men, women and children may receive UGX. 75, Ush.90 and Ush.120 each.

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