Topic 2: Working with Integers

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

• identify, read and write natural numbers as numerals and words in millions, billions, and trillions
• differentiate between natural numbers and whole numbers/integers
• identify directed numbers and use them in real-life situations
• use the hierarchy of operations to carry out the four mathematical operations on integers
• identify even, odd, prime, and composite numbers
• find out the prime factors of any number
• relate common factors with HCF and multiples with LCM
• work out and use the divisibility test of some numbers

Keywords

• composite number
• directed number
• divisibility test
• even number
• factor
• highest common factor
• integers
• multiple
• natural number
• numeral
• odd number
• operator
• prime number
• whole number

Introduction

Imagine you are planning to go for camping on a mountain, say Mt. Rwenzori. The tour guide may tell you that:

1. Temperature during day ranges between ‘1°C to ’12°C.

2. Temperature during the night ranges between -5°C to ‘3°C.

3. Altitude is 5,109 m or 16,762 ft above sea level.

When you climb up a mountain, temperatures keep reducing (source: New Vision)

In real life, we use negative and positive numbers in instances such as: Sea level: Wherever you go, you are either above or below sea level. Floors in a building: You can either be on the 3rd floor (positive) or in an underground floor (negative).

AD & BC Time: This shows the timeline being part of integers. Other areas with integers are money, temperature, and music. Can you identify any other area where integers are used? In this topic therefore, you will learn about positive and negative integers and how they help you solve everyday life problems.

Identifying, Reading and Writing Natural Numbers as Numerals and Words

In this section, you will learn to write natural numbers as numerals and words in million, billion and trillion.

Activity 2.1 Reading and writing natural numbers as numerals and words (work in pairs)

What you need: ruler, pen, paper and a pencil.

What to do:

1. Draw a place value chart of 5 periods like the one below.

2. Insert a digit of your choice in each of the small boxes. 3. State the figures in the ones, tens and hundreds place value position.

4. From the figure formed, how many trillions, billions, millions, thousands and units have you formed? Fill them in the table below:

5. Exchange your work and mark yourselves.

Example 1

Write UGX 21,479,215,631,800 in words.

21 trillions UGX 21,479,215,631,800 is Twenty

479 billions one trillion, four hundred seventy

215 millions nine billion, two hundred fifteen

631 thousands million, six hundred thirty one

800 ones thousand, eight hundred shillings.

Example 2

Write in numerals; One hundred forty trillion, six hundred eighty eight billion, two hundred fifty million, nine hundred thousand, one hundred twelve.

140 trillions = 140,000,000,000,000

688 billions = 688,000,000,000

250 millions = 250,000,000

963 thousands = 963,000250 millions

One hundred twelve = 112

Total = 140,688,250,963,112

Exercise 2.1

1. Write the following numerals in words.

a) 260,117,888

b) 36,677,709,285

c) 9,238,457,002

d) UGX 341,583,440,000

2. Write the following in numerals.

a) Fifty six trillion, one hundred forty eight billion, two hundred twenty two million, seven hundred ninety nine thousand one hundred thirty one.

b) Seven trillion, two hundred seven billion, one hundred fifty six million, one hundred fifty six thousand, one hundred sixty six.

c) Nine hundred eight billion, three hundred thirty eight thousand, one hundred shillings.

Differentiating Between Natural and Whole Numbers

Activity 2.2 Identifying counting and whole numbers (work in groups)

What you need: pens, notebook What to do:

1. Copy and complete the table below.

2. Find the product of the first 4 consecutive counting numbers.

3. List all the first 20 natural numbers.

Identifying directed numbers

Directed numbers have size and direction

Activity 2.3 Ex Identifying directed numbers (work in groups) .

What you need: rope, ash. What to do:

1. Get outside the classroom and draw a number line on the ground using a rope and ash.

2. Identify the origin (point) then the negative and positive side;

3. Give direction to a friend to face to the positive, move some steps, then some steps to the negative.

4. Record the number of steps as direct numbers.

5. Present your work to the whole class. Numbers with direction and size are directed numbers (or integers).

Exercise 2.3

Study the number line below and identify the directed numbers marked a, b, c and d.

Applying Integers in Everyday Life

Cheptegei runs uphill to 200 m above sea level. If he slopes down to 50m below sea level, how can you write such figures using directed numbers? How else do we use directed numbers?

Example (Discuss in groups)

Mugisha had UGX 200,000. He bought a radio at UGX 280,000 from Ouma. If Mugisha already had a debt of UGX 30,000 against Ouma, how much money does Ouma owe Mugisha altogether?

Tip

Money is not always written in negatives. Just put the figure in brackets to mean it is negative.

Exercise 2.4

1. A frog jumps 5 steps forward from point K, 6 steps backward and then 8 steps forward. At what step will it be forward?

2. Robert had a debt of UGX 16,000 at the school canteen. On visitation day, his parents gave him UGX 9,500. If he used all the money to pay all the debt, how much did he remain with?

3. In an examination, 3 marks are awarded for a correct answer, 1 mark is deducted for a wrong answer, for a question attempted. If a student gets 6 correct responses from 10 questions attempted,

a) how many marks will the student get?

b) find the percentage mark the student will get.

4. Okware was born in 15 BC and died 68 AD. How old was he by the time he died?

Using the Hierarchy of Operations to Workout the Four Mathematical Operations on Integers

Activity 2.4 Using the hierarchy of operations to workout the four mathematical operations on integers (work in groups)

What you need: pen, notebook, calculator.

What to do:

1. Use the hierarchy of operations to complete the table below.

2. Exchange your work and mark your selves. Do not write anything like V. Good or Otherwise on your friends book.

Exercise 2.5

Evaluate the following

1. 18 +-2(7–3)/ 10 x -1

2. 78 of (8x-3) ÷ 5-6 2

3. -8-3(-8-16) x-3 ÷ 7

4. 16/ 4 of (3 x 2 ÷ 1)-3+5

Identifying Even, Odd, Prime and Composite Numbers

Identifying Odd and Even Numbers

Activity 2.5 Identifying odd and even numbers (work in groups) What you need: counters

1. Get counters from 1 to 20 and divide them by 2 as you note down the answer and the remainder, for example: 1+2=0r1 3+2=1r1 2÷2=1r0 4÷2=2r0

2. Record your observation?

3. Present your work to the class.

Exercise 2.6

1. List the first 6 odd numbers.

2. Find the sum of all even numbers between 20 and 30.

Identifying Prime and Composite Numbers

Activity 2.6 Identifying prime and composite numbers (work in groups)

What you need: papers, pens, ruler What to do:

1. Draw a 10 x 10 grid and write in numbers from 1 to 100.

2. Cross out one and circle all numbers which have two factors (1 and its self).

3. List all the numbers with two factors and those with more than two factors.

4. Present your work to the whole class.

Example 1

What is the sum of all the prime numbers below 10?

The numbers are: 2, 3, 5, 7

Their sum = 2+3+5+7

= 5 + 12 = 17

Exercise 2.7

1. List all the first 8 odd numbers.

2. Given that set Q = (all even numbers below 10). Find n(Q).

3. Find the sum of all prime numbers between 20 and 30.

4. What is the quotient of the 6th and 1st composite number?

5. Find the sum of all prime numbers between 90 and 100.

6. List all even numbers from 20 to 30.

7. List all prime numbers between 20 and 30.

Finding Prime Factorisation of any Number

Except prime numbers, all numbers can be reduced or expressed in terms of their prime factors.

Activity 2.7 Finding prime factorisation of any number (work in groups)

What you need: pen and note book

What to do:

1. Prime factorise 720 and write your answers in;

a) Multiplication form b) Power form c) Sub script form

2. Find the number whose prime factors are 2, 2, 2, 5, and 3,

3. Present your work to the whole class.

Exercise 2.8

1. Find the prime factors of the following numbers.

a) 75 b) 80 c) 60 e) 1440

2. Find the number whose prime factors are shown below.

a) (2 × 3 × 5 × 7} b) (2, 3, 2, 2,} d) 100 c) 52 x 32

3. Prime factorise 48 and write it as a product of its prime factors.

Relating Common Factors with HCF and Multiples with LCM

Supposing a school plans to tile the floor of a hall which measures 50 m by 30 m. They are looking for the biggest tile size to use such that they don’t break any of them. What will be the size of this tile? How many tiles will be required for the entire hall? Such problems and many others in real life can best be handled with the knowledge of HCF and LCM.

Factors, Common Factors and Highest Common Factor (HCF)

Activity 2.8 Finding factors (work in groups)

What you need: counters, pen, notebook

What to do:

1. Put the 12 counters in complete groups (without leaving a remainder).

2. List the complete groups you can form.

3. List the group numbers you have formed out of the 12 counters.

4. State the factors of 12.

5. Repeat 1 to 4 using different numbers of counters.

6. Present your work to the whole class.

Exercise 2.9

1. Make groups of 12 and 18 counters.

a) Identify the factors of 12 and 18.

b) Find the common factors of 12 and 18.

c) What is the highest common factor of 12 and 18?

2. Find the factors, common factors and the highest common factor of the following numbers.

a) 24 and 36 b) 15 and 45 c) 100 and 144 d) 720 and 400 e) 98 and 48 f) 45 and 50

Multiples, common multiples and lowest common multiples

Ivan exercises every 12 days and Jane every 8 days. Ivan and Jane both exercised today. How many days will it be until they exercise again together? You can solve this problem using common multiples.

Example

Find the multiples, common multiples and lowest common multiple of 30 and 40.

M30 = (30, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, …)

M40 = (40, 80, 120, 160, 200, 240, 280, 320, 360, 400, …)

Solutions

Common Multiples = (120,240, 360, …)

Lowest Common Multiple = 120

Exercise 2.10

1. What is the LCM of the following

a) 18 and 24 b) 40 and 50 c) 45 and 60

d) 12 and 20 e) 24 and 36 f) 30 and 36

2. In a hospital, a doctor checks on a patient after every 36 minutes while a nurse checks on the same patient after every 24 minutes. If they last checked on the patient together at 9:00 am, at what time will they next check the patient at the same time?

Working out and using Divisibility Tests of some Numbers

Activity 2.9 Working out and using divisibility tests of some numbers (work in groups)

What you need: notebook, pen.

What to do:

1. Study the table and complete the table below.

Exercise 2.11

1. Identify all the numbers divisible by 2 in the list below: 23177, 6089, 4588, 96110

2. Is 7,368 divisible by 3? Prove without using a calculator.

3. Which number in the list below is divisible by 4? 7368, 29472, 38463, 12441

4. Test whether 28,039 is divisible by 11.

5. Mulindwa has UGX 191,232,000. He wants to divide it equally amongst his 8 children. Without dividing, show whether the money is equally divisible.

6. A class captain has 539 oranges and he intends to share them with either 3, 4 or 7 of his friends, as long as he gets no remainder. Carry out a divisibility test to establish how many friends he will share the oranges with.