MATHEMATICS

PROTOTYPE

MATHEMATICS

LEARNER’S BOOK

SENIOR ONE

PROTOTYPE

LOWER SECONDARY

CURRICULUM

PROTOTYPE

PROTOTYPE

MATHEMATICS

LEARNER’S BOOK

SENIOR ONE

LOWER SECONDARY

CURRICULUM

DISCLAIMER!!

This material has been developed strictly for training purposes. Content and images

have been adapted from several sources which we might not fully acknowledge. This

document is therefore restricted from being reproduced for any commercial purposes

National Curriculum Development Centre

P.O. Box 7002,

Kampala- Uganda

www.ncdc.co.ug

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Contents

Acknowledgements ........................................................................................... viii

Topic 1 .................................................................................................................. 1

NUMBER BASES .................................................................................................... 1

Sub-topic 1. 1 Identifying numbers of different bases on an abacus .............. 2

Sub-topic 1. 2: Place Values Using the Abacus ............................................... 4

1.3 Converting Numbers ................................................................................ 5

1.4: Operation on Numbers in Various Bases .................................................. 6

Topic 2: ............................................................................................................... 11

WORKING WITH INTEGERS ................................................................................ 11

Introduction ................................................................................................... 12

Subtopic 2.1 Natural Numbers ...................................................................... 12

Sub topic 2.2: Differentiate between natural numbers and whole

numbers/integers. ......................................................................................... 14

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

Situations ....................................................................................................... 14

Sub-topic 2.4: Use the Hierarchy of Operations to Carry Out the Four

Mathematical Operations on Integers ........................................................... 17

Sub-topic 2.5: Identify Even, Odd, Prime and Composite Numbers .............. 24

Sub-topic 2.6: Find the Prime Factors of any Number ................................... 25

Sub-topic 2.7: Relate Common Factors with HCF and Multiples with LCM ... 27

Sub-topic 2.8: Work Out and Use Divisibility Tests of Some Numbers .......... 27

Sub-topic 2.9: Least Common Multiple (LCM) ............................................... 28

Topic 3: ............................................................................................................... 30

FRACTIONS, PERCENTAGES AND DECIMALS ...................................................... 30

Sub-topic 3.1: Describe Different Types of Fractions .................................... 31

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Sub-topic 3.2: Convert Improper Fractions to Mixed Numbers and Vice

Versa .............................................................................................................. 32

Sub-topic 3.3: Operations on Fractions . ....................................................... 33

3.33 Subtraction of Fractions with Same Denominators ............................... 37

3.34 Subtraction of Fractions with Different Denominators ......................... 38

3.35 Addition of Mixed Fractions .................................................................. 39

3.36 Subtraction of Mixed Fractions .............................................................. 40

3.37 Subtraction of Fractions with Different Denominators ......................... 44

3.38 Multiplication of Fractions ..................................................................... 45

3.39 Multiplying Mixed Fractions .................................................................. 47

3.310 Division of Mixed Fractions Flip And Multiply ..................................... 49

Sub-topic 3.5: Convert Fractions to Decimals and Vice Versa ....................... 52

Sub-topic 3.6: Identify and Classify Decimals as Terminating, Nonterminating

and Recurring Decimals ............................................................. 52

Sub-topic 3.7: Convert Recurring Decimals into Fractions ............................ 53

Sub-topic 3.8: Convert Fractions and Decimals into Percentages and Vice

Versa .............................................................................................................. 54

Sub –topic 3.9 Calculate a Percentage of a Given Quantity .......................... 55

Sub-topic 3.10: Works out Real-life Problems Involving Percentages ........... 56

Topic 4: ............................................................................................................... 58

RECTANGULAR CARTESIAN COORDINATES IN 2 DIMENSIONS .......................... 58

Sub-topic 4.2: Plotting Polygons (shapes) ...................................................... 60

Topic 5: ............................................................................................................... 64

GEOMETRIC CONSTRUCTION SKILLS .................................................................. 64

Sub-topic 5.2: Construction of Perpendicular lines ...................................... 66

Sub –topic 5.2: Using a Ruler, Pencil and Pair of Compasses, Construct

Special Angles ................................................................................................ 67

Sub-topic 5.3: Describing Locus Question...................................................... 68

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5.31: Relating Lines and Angles to Loci .......................................................... 68

Sub-topic 5.4: Construction of Geometric Figure .......................................... 69

Topic 6: ............................................................................................................... 71

SEQUENCE AND PATTERNS ................................................................................ 71

Sub-topic 6.2: Describing the general rule ..................................................... 79

Sub-topic 6.3: Generating Number Sequence ............................................... 80

Sub-topic 6.4 : Formulae for General Terms .................................................. 83

Topic 7: ............................................................................................................... 86

BEARINGS ........................................................................................................... 86

Sub-topic 7.1: Angles and Turns .................................................................... 87

Sub-topic 7.2: Bearings .................................................................................. 89

Topic 8: ............................................................................................................... 91

GENERAL AND ANGLE PROPERTIES OF GEOMETRIC FIGURES ........................... 91

8.1: Identify Different Angles ........................................................................ 93

Sub- topic 8.2 : Angles on a Line and Angles at a Point ................................. 94

Topic 9: ............................................................................................................... 97

DATA COLLECTION AND PRESENTATION ........................................................... 97

Sub-topic 9.1: Types of Data .......................................................................... 97

Sub-topic 9.2: Collecting Data ........................................................................ 99

Topic 10: ........................................................................................................... 105

REFLECTION ...................................................................................................... 105

Sub-topic 10.2: Reflection in the Cartesian Plane........................................ 107

Topic 11: Equation of Lines and Curves ........................................................... 108

Fundamental Algebraic Skills ....................................................................... 108

Subtopic 11.1 Function Machines ................................................................ 110

Sub-topic 11.2: Linear Equations ................................................................. 112

Topic 14: Time and Time Tables....................................................................... 116

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Sub-topic 14.1: Telling the Time .................................................................. 116

Sub-topic 14.2:12-hour and 24-hour Clocks ................................................ 118

Sub-topic 14.4: Timetables .......................................................................... 121

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Preface

This Learner’s Textbook has been written in line with the revised

Mathematics syllabus. The knowledge and skills which have been

incorporated are what is partly required to produce a learner who has

the competences that are required in the 21st century.

This h as b een d one b y p roviding a r ange o f a ctivities which will be

conducted both within and outside the classroom setting. The learner

is e xpected to be able to work a s an individual, in pairs and g roups

according to the nature of the activities.

The teacher as a facilitator will prepare what the learners are to learn

and this learner’s book is one of the materials which are to be used to

support the teaching and learning process.

Associate Professor Betty Ezati

Chairperson, NCDC Governing Council

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Acknowledgements

National Curriculum Development Centre (NCDC) would like to express

its appreciation to all those who worked tirelessly towards the

production of the Learner’s Textbook.

Our gratitude goes to the various institutions which provided staff who

worked as a panel, the Subject Specialist who initiated the work and

the Production Unit at NCDC which ensured that the work produced

meets the required standards. Our thanks go to Enabel which provided

technical support in textbook development.

The Centre is indebted to the learners and teachers who worked with

the NCDC Specialist and consultants from Cambridge Education and

Curriculum Foundation.

Last but not least, NCDC would l ike to acknowledge all those behind

the scenes who formed part of the team that worked hard to finalise

the work on this Learner’s Book.

NCDC is committed to uphold the ethics and values of publishing. In

developing this material, several sources have been referred to which

we might not fully acknowledge.

We welcome any suggestions for improvement to continue making our

service delivery better. Please get to us through P. O. Box 7002

Kampala or email us through admin@ncdc.go.ug.

Grace K. Baguma

Director, National Curriculum Development Centre

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Topic 1

NUMBER BASES

Key Words

base

binary

decimal

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

i) identify numerals in base(s) up to base 16.

ii) identify place values of different bases using abacus.

iii) convert numbers from one base to another.

iv) manipulate numbers in different bases with respect to all four

operations.

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Introduction

I Am an ordinary person, how many fingers do I have on:

i) one hand?

ii) two hands?

If you have heaps of oranges of ten, twelve and fifteen, how many

groups of tens, fives and fours do you get in each? And how many are

remaining in each heap?

In order to answer the above questions, you can use your

knowledge of decimal place value to develop your understanding

of numbers written in other bases.

Sub-topic 1. 1: Identifying numbers of different

bases on an abacus

In your primary education, you studied number bases such as bases

five, two and ten (decimal base). Remember the numerals for all the

various number bases you studied by doing the following activity:

Activity 1. 1: Getting familiar with number bases

In your groups, identify situations in which you have ever used

number bases in your life.

Real life situation Base Reason for the base chosen

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Question: Which possible base does each abacus below represent?

a) b)

c)

d)

Activity 1.2: List the numerals for the following bases

In your groups, list the numerals for the following bases:

i) Two (Binary) ii) Three. iii) five iv) seven v) eight. vi) nine vii)

eleven viii) twelve ix) sixteen

Now study the table below and fill in the gaps.

Base Numerals

Two 0, 1

Three 0, 1, 2

Four 0, 1, -, 3

Five 0, - , 2, - , 4

Nine 0, 1, 2, - , 4, - , 6, - , 8

Twelve 0, - , 2, - , 4, - , - , 7, - , 9, - , e

Sixteen 0,1,2,3,4,5,-,-,-,9,t,e,-,-,-,-

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Compare your answers and note what happens to the base number

when writing the numerals used in a particular base. Give reasons.

Sub-topic 1. 2: Place Values Using the Abacus

You have already learnt how to represent numbers on an abacus. The

representation of numbers on an abacus helps you to identify the

place value of digits in any base.

Activity 1.3: Making abaci

In groups work in pairs to make different abaci, in different bases.

Compare your work with other members of the group .

Activity 1.4: Reading and stating the value of digits in bases

In groups, represent the following numbers on an abacus:

a. 123four

b. 274ten

c. 1312five

Read and state what each digit in the numbers above represents on an

abacus using the stated bases.

Exercise

State the place value of each numeral in the following numbers:

a) 321four b) 354six c) 247eight

State the value of each numeral in the following numbers:

b) 567nine b) 381twelve c) 11010two

Represent the following numbers on the abacus:

(a) 1101two (b) 2102three (c) 2021four (d) 5645seven (e) 8756nine

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1.3 Converting Numbers

Numbers can be converted from one base to another, and when you

do this, you get the same numbers written in different bases.

You learnt how to convert from base ten to any other base.

Activity 1.5: Converting numbers from base ten to any other base

In groups, convert the following numbers in base ten to bases

indicated: 456, 1321, 5693, 56 and 436.

(a) Five (b) Nine (c) Eight

You can also convert numbers from any base to base ten (decimal).

Example: Convert (a) 101two (b) 324five (c) 756eight to base ten.

Solution:

(a) 101two = (1x 22) + (0x21) + (1 x 20) = 1x4 + 0x1 + 1x1= 4+0+1 = 5

(b) 324five = (3x52) + (2x51) + (4x50) = 3x25 + 2x5 + 4x1 = 75+10+4 = 89

(c) 756eight= (7x82) +(5x81) +(6x80) = 7x64 + 5x8 + 6x1 = 448+40+6 = 494

Activity 1.6: Converting numbers in a given base to another base

In pairs, discuss how to convert numbers in different bases to various

bases in the exercise below.

Exercise

Convert the following numbers to the bases indicated: (a) 762eight to

base seven; (b) 234five to base six; (c) 561seven to base nine; (d) 654six to

base four; (e) 5432six to twelve.

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1.4: Operation on Numbers in Various Bases

James had two jackfruit trees in his compound. At one time one tree

had 8 fruits ready and the other 7 fruits. He harvested them at the

same time. He decided to put them in heaps of nine fruits. How many

heaps of nine did he get and how many remained?

When you put the fruits in heaps of 9, you are adding in base 9.

Addition

The two jack fruit trees above had a total of 15 (that is 8 +7) ready

fruits.

You can add numbers in various bases. For example, add the following

numbers:

(a) 234five to 23five (b) 153seven to 453seven

Solution

(a) (b)

Exercise: Add the following numbers:

(a)321four to 122four. (b) 456seven to 342seven

(c) 764eight to 361eight. (d) 210three to 211three

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Subtraction

Subtraction in other bases is done in the same way it is done in base

ten.

Examples: Subtract (a) 342eight from 567eight (b) 432six from 514six

Solution:

(a) (b)

Exercise

Subtract the following numbers in the given bases:

(a) 351six from 510six(b) 672nine from 854nine

(c) 845twelve from t23twelve(d) 231five from 421five

Multiplication

Multiplication is done in the way it is done in base ten.

Example: Multiply 423five by 12five

Solution

Exercise:

Multiply the following:

(a) 241five by 13five. (b) 345six by 24six

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(c) 534seven by 123seven. (c) 156eleven by 534eleven

Division

The most common method of dividing numbers in different bases is by

converting the numbers to base ten first and after division, you can

convert the answer to the given base.

Example: Divide 1111two by 101two

Solution: Convert 1111two and 101two to base ten

1111two = (1x23) + (1x22) + (1x21) + (1x20)

= 8 + 4 + 2 + 1

= 15.

101two = (1x22) + (0x21) + (1x20)

= 4 + 1 = 5ten

Therefore, 1111two divided 101two is the same as 15 divided 5.

15÷5 = 3

3ten = 3÷2 = 1 remainder 1 = 11two

Therefore, 1111two÷ 101two = 11two

Exercise:

1. Add: (a) 654seven to 514seven (b) 278nine to 756nine

2. Subtract: (a) 412six from 554six (b) 435eight from 764eight

3. Multiply: (a) 1121three by 212three (b) 312four by 122four

4. Divide: (a) 100011two by 111two (b) 150nine by 20nine

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Activity 1.6: Operations on numbers with mixed bases

In your groups work in pairs discuss how you would carry out the four

mathematical operations on numbers with mixed bases by getting

your own examples. Compare your answers with other members of the

group.

Number Game: You are given four boxes containing numbers in base

ten. The boxes are labelled Box 1, Box 2, Box 3 and Box 4.

9 1 15 7

Box 1

6 14 2 7 15

Box 2

15 14 6 12 4 7

Box 3

15 14 9 12

Box 4

Task: Working in groups, select one number from any of the boxes

given. Your mathematics teacher will ask you whether the number you

selected appears in Box 1, Box 2, Box 3 and Box 4. From the responses

you give, the teacher will tell you the number you selected. Discuss

how the teacher was able to tell you the number you had selected.

Situation of Integration

A community is hit by famine and the government decides to give each

member in the household a potato to solve their problem of hunger.

Support: Each package contains an equal number of potatoes of five.

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There are 10 households in the community with 3, 5, 7, 4, 6, 5,8,12, 13

members respectively.

Resources: Knowledge of Bases, knowledge of mathematical

operations

Task: Determine the number of packages of potatoes the government

will take to that community. In case there are remaining potatoes,

discuss what the government should do with them.

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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.

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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?

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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

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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.

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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 r emove t he h ot s tones a nd t he

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

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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?

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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

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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

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

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

Exercise

1. Work out the following in degrees:

Note - x- = +, + x + = +

-x + = - , - ÷ - = +

- ÷ + = -

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2. Work out the following:

a) 8 + -6

b) 61 + + 7

c) 49 - - 30

d) 77 - + 50

e) -15 + + 20

f) -3 - - 13

25

5

x

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3. 2.Using a number line work out:

a) -2 ++ 3

b) +5 + - 6

c) – 8 - -5

4. A national park guide on o ne o f t he m ountains i n E ast A frica

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

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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.

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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

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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.

No Odd Prime Even Composite

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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

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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:

Prime Factor Number

2 36

2 18

3 9

3 3

1

36 = 2× 2 × 3 × 3 = 22 × 32

This approach of determining prime factors is called prime

factorisation.

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

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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

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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

2 8 12

2 4 6

2 2 3

3 1 3

1 1

2 × 2 × 2 × 3 = 24

The LCM of 8 and 12 is 24.

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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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Topic 3:

FRACTIONS, PERCENTAGES AND

DECIMALS

Key Words

recurring, numerator, denominator, terminating,

non-terminating, reciprocal, whole

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

i) describe different types of fractions.

ii) convert improper fractions to mixed numbers and vice versa.

iii) work out problems from real-life situations.

iv) add, subtract, divide and multiply decimals.

v) convert fractions to decimals and vice versa.

vi) identify and classify decimals as terminating, non-terminating

and recurring decimals.

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vii) convert recurring decimals into fractions.

viii) convert fractions and decimals into percentages and vice versa.

ix) calculate a percentage of a given quantity.

x) work out real-life problems involving percentages.

Introduction

In Chapter Two you studied place values in number bases. In this topic,

you will use knowledge of place values to manipulate fractions,

decimals and percentages. You will convert fractions to decimals,

decimals to percentages and vice versa.

Sub-topic 3.1: Describe Different Types of

Fractions

Activity 3.1

Create a park of different cards and label them with different types of

fractions, decimals and percentages.

From the park of the cards, you pick a card and place it in the most

appropriate play area.

Observe the fractions in each play area by looking at the denominators

and numerators.

In your groups explore and explain the common of the classification

made in the different play areas.

Exercise

1. Sarah shades 3/7 of a shape. What fraction of the shape is left

unshaded?

2. A cake is divided into 12 equal parts. John eats 3/12 of the cake

and Kate eats another 1/12. What fraction of the cake is left?

3. A car park contains 20 spaces. There are 17 cars parked in the car

park.

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a) What fraction of the car park is full?

b) What fraction of the car park is empty?

4. Ali eats 3/10 of the sweets in a packet.

Tariq eats another 4/10 of the sweets.

a) What fraction of the sweets has been eaten?

b) What fraction of the sweets is left?

5.

a) Draw a square with its four lines of symmetry.

b) Shade 3/8 of the shape.

c) Shade another 2/8 of the shape.

d) What is the total fraction now shaded?

e) How much is left unshaded?

Sub-topic 3.2: Convert Improper Fractions to

Mixed Numbers and Vice Versa

Mixed Numbers and improper Fractions

So far you have worked with fractions of the form a/b where a < b, e.g.

¾, 2/7, 5/6 …

You also need to work with what are sometimes called improper

fractions, e.g. 5/4, 7/2, which are of the form a/b when a and b are

whole numbers and a > b.

Example

Convert 13/4 into an improper fraction.

Solution

13 ÷ 4 = 3 remainder 1

This is written as 3 ¼.

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Exercise

1. Draw diagrams to show these improper fractions:

(a) 7/2 (b) 8/3 (c) 18/5

Write each improper fraction as a mixed number.

2. Convert these mixed numbers to improper fractions.

(a) 1 3/5 (b) 7 1/3 (c) 3 4/5 (d) 6 1/9

3. Write these fractions in order of increasing size.

6 ½ , 18/5 , 3 ¼ , 5 1/3 , 17/3

4. In an office there are 2 ½ packets of paper. There are 500 sheets of

paper in each full packet. How many sheets of paper are there in the

office?

5. A young child is 44 months old. Find the age of the baby in years as

a mixed number in the simplest form.

Sub-topic 3.3: Operations on Fractions

In the previous sub-topic, you studied how to find equivalent fractions.

In this sub-topic you are going to use the knowledge of equivalent

fractions to add and subtract fractions.

3.3.1: Work out problems from real-life situations

Now we start to use fractions in a practical way.

Example

(a) Find 1/5 of UGX. 10000

(b) Find 4/5 of UGX. 100,000

You can, do this practically, but it is much easier to work out.

(a) 1/5 of 10000 = 1/5 x 10000 = 2000

(b) 4/5 of 100000 = 4/5 x 100000 = 400000/5 = 80,000

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Exercise

1. Find:

(a) ½ of 12 (b) 1/8 of 40 (c) ¼ of 32

2. Find:

(a) 2/9 of 18 (b) 7/9 of 45 (c) 7/8 of 56

3. In a test, there are 30 marks. Nasim gets 3/5 of the marks. How many

marks does she get?

4. In a certain school there are 550 pupils. If 3/50 of the pupils are lefthanded,

how many left-handed pupils are there in the school?

Activity 3.3: Addition of Fractions

In your groups, use a sheet of paper to work out

5

3

5

1 + . Fold the paper

into five equal parts shade off one part of the five equal parts

Shade the three parts of the five equal parts

How many parts have been shaded?

Represent the shaded parts in a fraction form. Show the working.

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Activity 3.4: Addition of Fractions with the Same Denominators

Slice a hexagon into 6 pieces:

Each piece is of the hexagon. Right?

And is of the hexagon.

So, what if we wanted to add

Hmm... that would be

Count them up

So

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In your groups, use the same method to work out the following:

a)

7

2

7

3 +

b)

9

4

9

5 +

3.3.2: Adding Fractions with the different Denominators

In the previous topic you studied about lowest common multiple. In

this section, you will apply the knowledge of LCM.

Change the using the knowledge of equivalent fractions

Change the using the knowledge of equivalent fractions

The main rule of this game is that we cannot add the fractions until the

denominators are the same!

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We need to find something called the least common denominator

(LCD)..which is the LCM of our denominators, 2 and 3.

The LCM of 2 and 3 is 6. So, our LCD 6.

We need to make this our new denominator

3.3.3: Subtraction of Fractions with Same Denominators

Let's try

Look at a Chapatti in a conical shape cut into 8 pieces. Each piece

is of the Chapatti.

Take away (that's 3 pieces):

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We're left with 4 pieces - that’s.

So

But, look what we really did!

We just subtracted the numerators!

which is

3.3.4: Subtraction of Fractions with Different Denominators

Subtraction works the same way.

The LCM of 11 and 22 is 22... So, the LCD is 22.

We just need to change the .

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Done!

3.3.5: Addition of Mixed Fractions

What if we need to add

?

Hey, remember, that's just .

Done!

That was easy, but, what about mixed numbers?

How about this?

All we have to do is change these to improper fractions... Then we can

add them!

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3.3.6: Subtraction of Mixed Fractions

?

Well, we can't just stick it together like we would if it was addition.

We need to get a common denominator... But, the 5 does not even

have a denominator!

That's OK... Just think of a Chapatti cut into 8 pieces...

How many pieces would there be in 5 chapattis? Yep!

pieces

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So

Check it: is the same as which is . Yep!

Back to the problem:

What's ?

Well, that's of 6. Think about it:

You have 6 chapattis.

and you get to eat of them.

This is like splitting up the chapatti between 3 people:

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You get 2 chapattis

Your friend gets

2 chapattis

And your dog gets

2 chapattis

You get 2 chapattis

Your friend gets

2 chapattis

And your dog gets

2 chapattis

So of 6 is 2.

But, how do we do this with just math? EASY!

We know how to multiply two fractions... Right?

So, just make both things be fractions. Check it out:

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is already a fraction...

But, what about the 6?

Guess what? We can write 6 as .

Let's try

Look at a chapatti cut into 8 pieces. Each piece is of the

Chapatti.

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Take away (that's 3 pieces):

We're left with 4 pieces, that's.

So

But, look at what we really did!

We just subtracted the numerators!

which is

3.3.7: Subtraction of Fractions with Different Denominators

Subtraction works the same way.

The LCM of 11 and 22 is 22... So the LCD is 22.

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We just need to change the .

Done!

3.3.8: Multiplication of Fractions

What’s ?

Well, that's of 6. Think about it…

You have 6 chapattis.

and you eat of them.

This is like splitting up the pizza between 3 people:

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You get 2

chapattis

Your friend gets

2 chapattis

And your dog gets

2 chapattis

So of 6 is 2.

But, how do we do this with just math? EASY!

We know how to multiply two fractions... Right?

So, just make both things be fractions. Check it out:

is already a fraction...

But, what about the 6?

Guess what? We can write 6 as .

Think about it:

is the same as ... which is 6!

(You can do this with any number!)

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Back to the problem:

Just what we figured!

3.3.9: Multiplying Mixed Fractions

What about this?

Yikes! I am sure I don't want to try to think about pizza for this one!

Let's stick to the math:

Again, let's change these into improper fractions and go for it!

This is super easy!

Let's just do one:

We just multiply straight across...

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Now, think about it...

Cut a pizza into 10 pieces like

and look at 9 of the pieces:

We want of these

That would be 3 pieces. Right?

That's !

Doing math is cooooool!

Now that we understand what to do, we can just go for it.

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3.3.10: Division of Mixed Fractions Flip And Multiply

Check it out:

That's it -- then GO FOR IT!

Done!

Look at another one:

Use the same trick you do when multiplying by changing everything to

fractions and then go for it!

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Check it out:

How about another one?

Use the same trick you do when multiplying by changing everything

into fractions and then go for it!

Sub-topic 3.4: Add, Subtract, Divide and Multiply

Decimals

Activity 3.5: Fractions and decimals

In groups, copy and complete the table, by explaining how you have

obtained the answer. The first three have been done for you

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Tens Ones Tenth

(

10

1 )

Hundredth

(

100

1 )

Thousandth

(

1000

1 )

Fraction Percentage

5

2

1

50

1 2 4 12

5

2

1240

2 5

4

1

25

1 5 2

5

80

20

17

64

0 0 4

10

3

4 0 3

The column headings

will help you

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Sub-topic 3.5: Convert Fractions to Decimals

and Vice Versa

A fraction like ¾ means three quarters

or three parts out of four

or three divided by four

3 divided by 4 equals 0.75

So, the fraction ¾ is equal to 0.75 in decimal.

Activity 3.6: In pairs, convert the following fractions into decimals

a) 2/5

b) (b) 1/20 (c) 5/8 (d) 2/9 (e) 1/11

c) What do you notice about (d) and (e)?

Sub-topic 3.6: Identify and Classify Decimals as

Terminating, Non-terminating and Recurring

Decimals

Fractions like 3/5, 1/2, 3/8 can be converted into decimals and they end or

terminate: 3/5 = 0.6, ½ = 0.5 and 3/8 = 0.375.

Fractions like 2/3, 2/15, 1/11 do not end or terminate when converted

into decimals, 2/3 = 0.66666…, 2/15 = 0.133333… and

1/11 = 0.090909…

These decimals are referred to as recurring decimals

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Exercise

1. Write the following fractions as decimals:

(a) 3/8 (b) 7/10 (c) 17/50 (d) 13/25

2. Write the following as fractions in their lowest terms:

(a) 0.25 (b) 0.08 (c) 0.35 (d) 0.125

3. Write the following fractions as recurring decimals:

(a)a 2/11 (b) 1/3 (c) 1/6 (d) 7/9

Sub-topic 3.7: Convert Recurring Decimals into

Fractions

Recurring decimals can be converted into fractions.

Example: Convert this recurring decimal into a fraction: 0.333…

Note that the decimal repeats itself after one decimal place.

Let r = 0.333… (1)

Multiply both sides of the equation by 10 i.e. 10 x r = 10 x 0.333

10r = 3.333 (2)

Subtract equation (1) from equation (2):

That is, 10r = 3.333

- (r = 0.333)

9r = 3

r = 3/9 = 1/3.

Exercise

1. Convert the following recurring decimals into fractions

a) 0.77---, b) 0.133--- , c)1.25656 ---, d) 0.2727 ---, e) 0.01313

2. Convert the following numbers into recurring decimals

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a)

3

1

, b)

9

1 , c)

6

2

Sub-topic 3.8: Convert Fractions and Decimals

into Percentages and Vice Versa

Activity 3.7: Fraction percentage game

I am

20

7

Who is

67%?

I am

100

67

Who is

13%?

I am

100

13

Who is

22%?

I am

11

Who is

5%?

I am

1

Who

is72%?

I am

18

Who is

87%?

I am

87

Who is

4%?

I am

1

Who is

34%?

I am

8

Who is

42%?

I am

21

Who is

52%?

I am

13

Who is

45%?

I am

9

Who is

58%?

I am

29

Who is

64%?

I am

16

Who

is32%?

I am

17

Who

is2%?

I am

1

Who is

92%?

I am

23

Who is

98%?

I am

49

Who is

44%?

I am

11

Who is

82%?

I am

41

Who is

65%?

I am

13

Who is

14%?

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From the fraction percentage game, identify the equivalent

percentage for each fraction.

In your groups, use percentage to identify the smallest and largest

fractions from the fraction percentage game.

Sub –topic 3.9 Calculate a Percentage of a Given

Quantity

The percentage of a quantity can always be calculated in terms of

percentage increase or percentage decrease.

Example 1: Find the 10⁒ of 50,000

Solution: 10/100 x 50,000 = 5,000.

Example 2: Opio had 60 goats. Now he has 63 goats. What is the

percentage increase?

Solution: The increase in the number of goats is 63 – 60 = 3.

Percentage increase is 3/60 x 100 = 5⁒.

Activity 3.8: The table below shows students’ marks in two

mathematics tests. For each one, calculate the percentage

difference. Say if it is an increase or a decrease.

Student First Test Second Test

(a) Marion 50 45

(b) James 40 52

(c) Christina 20 35

(d) Sarah 60 50

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Sub-topic 3.10: Works out Real-life Problems

Involving Percentages

Exercise

1. In a closing-down sale, a shop offers 50% cut of the original

prices. What fraction is taken off the prices?

2. In a survey one in five people said they preferred a particular

brand of Coca Cola. What is this figure as a percentage?

3. Peter pays tax at the rate of 25% of his income. What fraction of

Peter’s income is this?

4. When Carol was buying a house, she had to make a deposit of

10

1

of the value of the house. What percentage was this?

5. I bought a coat in the January sales with

3

1 price cut of the selling

price. What percentage was taken off the price of the coat?

6. Adikinyi bought some fabric that was 1.75 metres long. How

could this be written as a fraction?

7. A car park contains 20 spaces. There are 17 cars parked in the car

park.

a. What fraction of the car park is full?

b. What fraction of the car park is empty?

Sub-topic 3.11: Identifying and classifying decimal

as terminating, non-terminating and recurring

decimals

Activity 3.6: Decimal as terminating, non-terminating and

recurring decimals

In groups list some terminating, none terminating and

recurring decimals. In pairs prove them. Compare your

answers with the members of the group.

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Situation of Integration

A primary school has two sections, that is, lower primary (P1-P4) and

upper primary (P5-P7). The head teacher of the primary school needs

to draw a timetable for both sections. The sections should start and

end their morning lessons at the same time before break time, start

and end their break time at the same time. The after break lessons

should start at the same time. The lunchtime for both sections should

start at the same time.

Support: The time to start lessons for the two sections is 8.00am. The

duration of the lesson for the lower section is 30 minutes and

that of the upper section is 40minutes.

Resources: Knowledge of fractions, percentages, natural numbers,

factors, multiples, lowest common multiples, and the

subjects taught in all classes and of time.

Task: Help the head teacher by drawing the timetable up to lunchtime

for the two sections. How many lessons does each section have

up to lunchtime?

Express the total number of lessons for the lower primary as a fraction

of the total number of lessons for the whole School. (Consider lessons

up to lunch time.)

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Topic 4:

RECTANGULAR CARTESIAN

COORDINATES IN 2 DIMENSIONS

Key words: coordinates, axes, plot, scale

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

i) identify the y-axis and x-axis.

ii) draw and label the Cartesian plane.

iii) read and plot points on the Cartesian plane.

iv) choose and use appropriate scale for a given data set.

v) identify places on a map using coordinates (apply coordinates

in real-life situations).

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Introduction

This topic is key in building the concept of location. The knowledge

achieved from this topic can be used in locating places. In order to

locate places you need a starting point (reference point).

Sub-topic 4.1: Identify the X-axis and Y-axis

Activity 4.1: Plotting Points

Now, plot the following points on a graph, (6,4), (5,9), (11,3), (5,6) and

(3, 4).

The x number comes first then the y number: (X, Y). These numbers are

called coordinates.

Exercise

1. Use a graph paper to:

a) Join the points with coordinates (0, 3), (5,6), and (5,0) to draw a

triangle.

b) On the same diagram join the points with coordinates (2, 0), (2,

6) and (7, 3) to draw a second triangle.

c) Describe the shape you have now drawn.

2. On the same graph paper join these points in order.

a) (4, 6), (5, 7), (6, 6), (4, 6).

b) (5, 8), (4, 8), (4, 7), (5, 8), (6, 8), (6,7), (5, 8).

c) (4, 5), (5, 4), (6, 5), (5, 3), (4, 5).

d) (5, 2), (3, 4), (3, 5), (2, 5), (2, 8), (3, 8), (3, 9), (7, 9), (7, 8), (8, 8), (8,

5), (7, 5), (7, 4), (5, 2).

We can also use negative numbers in coordinates. We can bring in

coordinate axes with positive and negative numbers.

Exercise

1. (a) Draw a set of axes and mark the points with coordinates (4, 0), (-

4, 0), (0, 4),

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(0, -4), (1, 2), (1, -2), (3, 3), (3, -3), (2, 1), (2, -1), (-1, 2), (-1, -2), (-3, 3), (-3, -

3),(-2, 1), (-2, -1)

(b) Join the points to form an 8 pointed star.

2. (a) On a graph paper, draw the rectangles with corners at the

following points with coordinates:

a) (-6, 6), (-5, 6), (-5, 4), (-6, 4)

b) (-2, 1), (-3, 1), (-3, 3), (-2, 3)

c) (3, 1), (3,3), (4, 3), (4, 1).

d) (10, 1), (10, 3), (9, 3), (9, 1)

e) (12, 4), (13, 4), (13, 6), (12, 6)

(b) Join the points with coordinates:

(1, -5), (1, -1), (2, 0), (5, 0), (6, -1), (6, -5)

Sub-topic 4.2: Plotting Polygons (shapes)

Here we look at polygons plotted on coordinate axes, but first, recall

the names of polygons.

Names of polygons

Number of sides Name

3

4

5

6

7

8

9

10

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

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Note:

In a regular polygon:

(a) all the sides are the same.

(b) all the angles are of the same size.

Activity 4.2: The line AB is one side of a square

What are the possible coordinates of the corners of the square?

Exercise

1. In each case the coordinates of 3 corners of a square are given.

Find the coordinates of the other corner.

(a) (2, -2), (2, 3) and (-3, 3)

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(b) (2, 3), (3, 4) and (1, 4)

(c ) (2, 2), (4, 4) and (4, 0)

(d) (-6, 2), (-5, -5) and (1, 3)

(e) (-5, -2), (-2, -1), and (-1, -4)

2. The coordinates of 3 corners of a rectangle are given below. Find

the coordinates of the other corner of each rectangle.

(a) (-4, 2), (-4, 1) and (6, 1)

(b) (0, 2), (-2, 0) and (4, -6)

(c ) (-4, 5), (-2, -1) and (1, 0)

(d) (-5, 1), (-2, 5) and (6, -1)

3. (a) The coordinates of 2 corners of a square are (-4, 4) and (1, -1).

Explain why it is possible to draw three different squares using

these two points.

(b) Draw the three different squares.

(c ) If the coordinates of the corners had been (-5, 1) and (1, 3)

would it still be possible to draw 3 squares? Draw the possible

squares.

4. Half of a heptagon with one line of symmetry can be drawn by

joining the points with coordinates: (2, 4), (-2, 1), (-2, -1), (0, -3) and

(2, -3). Join the coordinates. You have drawn one half of the

heptagon. Complete the heptagon. Write down the coordinates.

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Sub-topic 4.3: Use of Appropriate Scale for Given

Data

Activity4.3: Plot the following points on the axes: (5, 50), (10,100),

(15,150), (20,200), (35, 350)

Do you realise that on the horizontal axis there are 5 units for each

space?

On the vertical axis there are 50 units for each space. So, what is the

scale for the axes?

Exercise

1. For each part, draw a pair of axes with suitable scales and plot the

points:

(a) (1, 15); (4, 35); (8, 45)

(b) (15, 100); (35, 500); (40, 700)

2. Plot the points (2, 60); (4, 50); (0, 70); (7, 60)

Situation of Integration

A Senior One learner has reported in her class and has settled at her

desk.

Support: The classroom is arranged in rows and columns. It is big a big

class with each learner having his/ her own desk.

Resources: Knowledge of horizontal and vertical lines i.e. rows and

columns, coordinates

Knowledge: counting numbers

Task: The mathematics teacher has asked her to explain how she can

access her seat, starting from the entrance of the class. Discuss

whether there are other ways of reaching her seat.

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Topic 5:

GEOMETRIC CONSTRUCTION SKILLS

Key Words: perpendicular lines, parallel lines, circumcircle, arcs

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

i) draw perpendicular and parallel lines.

ii) construct perpendiculars, angle bisectors, mediators and

parallel lines.

iii) use compass and a ruler to construct special angles (600, 450).

iv) describe a locus.

v) relate parallel lines, perpendicular bisector, angle bisector,

straight line and a circle as loci.

vi) draw polygons.

vii) measure lines and angles.

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viii) construct geometrical figures such as triangle, square, rectangle,

rhombus, parallelogram.

Introduction

In this topic you will learn how to construct lines, angles and geometric

figures. Skills developed from this topic can be applied in day-to-day

life.

Sub-topic 5.1: Draw perpendicular and parallel

lines

Activity 5.1: Drawing perpendicular and parallel lines

(a) In your groups, list objects in real-life situations that can be used to

draw lines.

(b) Use the objects in (a) above to draw perpendicular lines, parallel

lines and intersecting lines.

Activity 5.2: Identifying lines

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In your groups, take a sheet of paper; divide it into half, then into half

again in the same way. Now fold your paper again. What kind of lines

do you see?

Next, fold the same paper into half in the opposite direction. Unfold

your paper now.

How is the new line you have created, related to the previous lines?

In real-life situations, where do we come across perpendicular lines

and parallel lines?

Which letters in the alphabet have the above lines?

In this sub-topic, you will have more hands-on work on perpendicular

and parallel lines

Sub-topic 5.2: Construction of Perpendicular

Lines

Activity 5.3: Construction of perpendicular line from an external

point to a given line

In your groups, work in pairs.

Given line segment AB and point C outside the line, construct a

perpendicular line from point C to line AB.

Taking the centre as C and any radius, draw two arcs on line AB at x

and y.

Now taking x as the centre and any radius, draw an arc below or above

the line opposite point C without changing the radius. Taking y as the

centre, draw an arc to intersect the previous arc. Join the intersection

of the arcs to point C .Compare your answers and make notes.

Activity 5.4: Construction of a Perpendicular line to a given point

on a given line segment

In your groups, work in pairs.

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Given line PQ and point Z on PQ. Taking Z as the centre and any

radius, draw two arcs on either side of Z name the arcs x and y . Now

taking x as the centre and any radius draw an arc either above or below

the line, without changing the radius now taking y as the centre draw

an arc to meet the previous arc join the intersection of the arcs to

point Z . Compare your answers with other group members.

Activity 5.5: Construction of a Perpendicular Bisector

In your groups work as an individual.

Given line segment AB. Taking A as centre and AB as the radius, draw

two arcs below and above the line, then now taking B as the centre and

without changing the radius, draw arcs to meet the previous arcs. Join

the intersection of the arc. What do you notice? Compare your work

with your group members.

Activity 5.6: Construction of parallel lines

In your groups, work in pairs.

Given line AB and point C outside the line. Take C as the centre, draw

an arc at point A taking AB as radius and A as the centre, draw an arc at

point B. Now take radius AC and taking B as the centre, draw an arc

above B, then taking radius AB and C as the centre, draw an arc to

meet the previous arc at D. Join the intersection of the arcs (D) to point

C. What do you notice. Name and describe shape ABCD. Compare your

answer with members of the your group.

Sub –topic 5.2: Using a Ruler, Pencil and Pair of

Compasses, Construct Special Angles

Activity5.7: Construction of special angles

In pairs, construct the following angles: 90o, 45o, 15o, 30o, 60o, 120o, 75o.

In your groups, compare your answers.

Using a protractor, measure your angles.

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Sub-topic 5.3: Describing Locus Question

What is the path traced out by the tip of the seconds-hand of a clock in

the course of each minute?

Activity 5.8: Discovering what Locus is

In your groups, discuss what happens if a goat is tied to a rope of

length 4 metres and around the place where the goat is, there are

gardens at a distance of 5 metres.

In your groups, draw sketches of the area where the goat can feed

from.

In real-life situations, where are such scenarios applied?

Activity5.8: Sketching and Describing Loci

In your groups, sketch and describe what happens about the following:

a) A mark on the floor as the door opens and closes.

b) The centre of a bicycle wheel as the bicycle travels along a

straight line.

c) A man is walking and keeping the same distance from two trees

P and Q.

d) A student is walking in the assembly hall keeping the same

distance from two opposite walls.

e) Compare your answers with other groups.

5.3.1: Relating Lines and Angles to Loci

According to the activities above, Locus is a trace of a point under

some conditions.

Activity5.9: Demonstration of some simple Loci

a) In your groups, demonstrate how one can walk the same distance

from a given point.

b) How one can walk the same distance from two fixed points.

c) How one can walk the same distance from a line.

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d) How one can walk the same distance from two intersecting lines.

In your different groups compare your answers.

Exercise

1. Construct the locus of a point equidistant from a fixed point.

2. Construct a locus of a point equidistant from a given line.

3. Construct the locus of a point equidistant from two intersecting

lines.

4. Construct a triangle ABC where AB = 12cm, AC=9cm and Angle BAC

= 60o. Find the point with the triangle where the distance from that

point to all the vertices of the triangle is equal taking that point as

the centre and the distance from the centre to the vertices as the

radius draw a circle. (vi, vii are implied.)

Sub-topic 5.4: Construction of Geometric Figure

Construction of geometric figures most of the time is application of

locus.

Activity5.10: Construction of geometrical figures

In pairs, construct a perpendicular bisector of any line segment.

Measure the distance from the perpendicular line to any of the points

on either side of the perpendicular bisector. What have you found out?

In your groups, construct an equilateral triangle with length 6cm.

Construct a circumcircle of the triangle. What type of locus is applied

here?

Exercise

1. Construct a triangle ABC in which AB = 8.5, BC = 6cm and angle B =

30o.

Construct a circle through the vertices of the triangle. Work out the

area of the circle.

2. Construct triangle PQR with PQ = QR= 7cm angle Q = 45o. Construct

a circumcircle of the triangle.

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3. Construct a parallelogram ABCD in which AB=5cm, BC=4cm and

angle B is 120o.

4. Construct an equilateral triangle ABC of sides 7cm.Bisect AB and BC

and let the bisectors intersect at X. With X as the centre and radius

XA, draw a circle.

Situation of Integration

In a village, there is an old man who wants to construct a rectangular

small house of wattle and mud.

Support: A string, sticks, panga, tape measure and human resource.

Resources: Knowledge of horizontal and vertical lines i.e. rows and

columns, knowledge of construction of geometric figures.

Task: The community asks you to accurately construct the foundation

plan for this old man’s house.

Explain to the class how you have accurately constructed the

foundation plan. Discuss whether there are other ways of constructing

an accurate foundation plan.

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Topic 6:

SEQUENCE AND PATTERNS

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

i) draw and identify the patterns.

ii) describe a general rule of a given pattern.

iii) describe a sequence.

iv) determine a term in a sequence.

v) find the missing numbers in a given sequence.

Introduction

In this topic you will learn how to identify and describe general rules

for patterns. You will be able to determine a term in the sequence and

find the missing numbers in the sequence.

Sub-topic 6.1: Draw and Identify the Patterns

Activity 6.1: Identifying number pattens

In groups, work in pairs.

Look at the following sequences, how can you get the next number?

Compare your answers with other members.

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i) 3, 6, 9, 12, 15, …

ii) 2, 4, 6, 8, 10, 12, …

In (i), in order to get the next number, you add 2 to the previous

number. The numbers in this sequence are multiples of 3.

Sequence (ii), represents the multiples of 2.

Exercise

State the multiples of 3 found in this table:

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

This square shows multiples of a number. What is this number?

Write down the numbers that should go in each of these boxes. The

number square will help you with some of them.

a) The fifth multiple of … is …

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b) The …th multiple of … is 36

c) The 12th multiple of … is …

d) The 20th multiple of … is …

e) The …th multiple of … is 96.

f) The 100th multiple of … is …

Solution

a) the 5th multiple of 4 is 20

b) the 9th multiple of 4 is 36

c) the 12th multiple of 4 is 48

d) the 20th multiple of 4 is 80

e) the 24th multiple of 4 is 96

f) the 100th multiple of 4 is 400

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Exercise

1. On a number square like this one, shade all the multiples of 6. Then

answer the questions after the table.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

a) What is the 4th multiple of 6?

b) What is the 10th multiple of 6?

c) What is the 12th multiple of 6?

d) What is the 100th multiple of 6?

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2. The multiples of a number have been shaded on this square. What

is the number?

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Copy each statement about these multiples and write down the

numbers that go in the spaces.

a) The 3rd multiple of … is …

b) The 9th multiple of … is …

c) The 200th multiple of … is …

d) The …th multiple of … is 66

e) The … th multiple of … is 330.

3.

a) Write down the first 8 multiples of 8.

b) Write down the first 8 multiples of 6.

c) What is the smallest number that is a multiple of both 6 and 8?

d) What are the next two numbers that are multiples of both 6 and

8?

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4. a) Write down the first 6 multiples of 12.

b) What is the 10th multiple of 12?

c) What is the 100th multiple of 12?

d) What is the 500th multiple of 12?

e) If 48 is the nth multiple of 12, what is n?

f) If 96 is the nth multiple of 12, what is n ?

5. a) What multiples have been shaded in this number square?

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

b) What is the first multiple not shown in the number square?

6. a) Explain why 12 is a multiple of 6 and 4.

b) Is 12 a multiple of any other numbers?

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7. The number 24 is a multiple of 2 and a multiple of 3. What other

numbers is it a multiple of?

8. Two multiples of a number have been shaded on this number

square. What is the number?

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

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9. Two multiples of a number have been shaded on this number

square

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

a) What is the number?

b) What is the 19th multiple of this number?

10. Three multiples of a number are 34, 170 and 255. What is the

number?

11. Three multiples of a number are 38, 95 and 133. What is the

number?

12. Four multiples of a number are 49, 77, 133 and 203. What is the

number?

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Sub-topic 6.2: Describing the General Rule

Activity 6.2: Finding the Next Term

In your, groups work in pairs.

Can you use the given numbers of the sequence to deduce the pattern

and hence find the next term?

Example: What are the next 3 numbers in the sequence:

a) 12, 17, 22 …?

b) 50, 47, 44, 41, 38, …?

Compare your answers with other group members

Solution

a) To find the pattern, it is usually helpful to first find the differences

between each term i.e. the difference between 12 and 17 is 5; the

difference between 17 and 22 is 5.

So the next term is found by adding 5 to the previous term. This gives

you 27, 32, 37.

b) Again you find the difference between:

i) 50 and 47 is -3.

ii) 47 and 44 is -3.

iii) 44 and 41 is -3.

iv) 41 and 38 is -3.

So, the next term is found by taking away 3 from the previous term,

giving you 35, 32, 29.

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Exercise

1. Copy the following exercise and find the sequence in each case,

giving the next three numbers.

a) 18, 30, 42, 54, 66, …

b) 4.1, 4.7, 5.3, 5.9, 6.5, …

c) 8, 14, 20,…, 32, …

d) 3, 11, …, 27, 35, …

e) 3.42, 3.56, 3.70, 3.84, 3.98, …

f) 10, 9.5, 9, 8.5, 8, 7.5, …

2. Copy each sequence and fill in the missing numbers.

a) 2, 4, …, 16, 32, …

b) 100, 81, 64, …, 36, …

c) 6, 9, …, 21, 30, 30, …

d) 0, 1.5, 4, …, 12, …

e) 1, 7, 17, …, 49, …

Sub-topic 6.3: Generating Number Sequence

Activity 6.3: Generating a sequence

In your groups work in pairs.

You can use formulae to generate sequences. For example, the formula

5n, with n = 1, 2, 3, 4, … generates the sequence 5x1, 5x2, 5x3, 5x4, …

The sequence generated is 5, 10, 15, 20, …

Example: What sequence do you generate by using the following

formula?

a) 5n – 1

b) 6n + 2

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Solution

a) putting n = 1, 2, 3, 4, … gives 4, 9, 14, 19, …

b) putting n = 1, 2, 3, 4, … gives 8, 14, 20, 26, …

You can find the formula for this sequence, 11, 21, 31, 41, 51, 61, …

How you can find the sequence. The sequence begins with 11, and 11 =

10 + 1. Continue to add 10 each time the formula is 10n + 1.

Compare your answers with other members in the group.

Exercise

1. What number comes out of each of these number machines?

2. The sequence 1, 2, 3, 4, 5, … is put into each number machine. What

does each machine do?

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3. Write down the first 5 terms of the sequence given by each of these

formulae:

a) 9n b) 12n c) 2n + 4 d) 3n – 1 e) 3n - 2

4. a) What is the 10th term of the sequence 2n + 1?

b) What is the 8th term of the sequence 3n + 6?

c) What is the 5th term of the sequence 4n + 1?

d) What is the 7th term of the sequence 5n – 1 ?

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5. Draw double machines that could be used to get each of these

sequences from 1, 2, 3, 4, 5 …

Also write down the formula for each sequence of the following:

a) 5, 9, 13, 17, 21, …

b) 2, 5, 8, 11, 14, …

c) 6, 11, 16, 21, 26, …

d) 4, 9, 14, 19, 24, …

e) 102, 202, 302, 402, 502, …

Sub-topic 6.4: Formulae for General Terms

Activity 6.4 : Identifying the nth term

In your groups work in pairs.

Note: It is very helpful not only to be able to write down the next few

terms in a sequence, but also to be able to write down, for example,

the 100th or even the 1000th term.

Example: For the sequence 3, 7, 11, 15, …, …

Find:

a) the next three terms.

b) the 100th term.

c) the 1000th term.

Answer

a) You can see that 4 is added each time to get the next

term.

So you obtain 19, 23, 27.

b) To find the 100th term, starting at 3, you add 3 to 4 times ninety

nine times giving

3 + 4 x 99 = 3 + 396 = 399

c) Similarly, the 1000th term is

3 + 4 x 999 = 3 + 3996 = 3999

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I can go one step further and write down the formula for a general

term, i.e. the nth term.

This is 3 + 4 x (n – 1) = 3 + 4n - 4

= 4n – 1.

Compare your answers with other members of the group and the

examples given.

Exercise

1. For each sequence, write down the difference between each term

and formula for the nth term.

a) 3, 5, 7, 9, 11, …

b) 5, 11, 17, 23, 29, …

c) 4, 7, 10, 13, 16, …

d) 2, 5, 8, 11, 14, …

e) 6, 10, 14, 18, 22, …

2. a) Write down the first 6 multiples of 11.

b) What is the formula for the nth term of the sequence of the

multiples of 11?

c) What is the formula for the nth term of this sequence?

3. The formula for the nth term of this sequence is n2.

1, 4, 9, 16, 25, …

What is the formula for the nth term of the following sequences?

a) 0, 3, 8, 15, 24, …

b) 10, 13, 18, 25, 34,

c) 2, 8, 18, 32, 50, …

d) 1, 8, 27, 64, 125, …

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Situation of Integration

There is a family in the neighbourhood of your school. The family has a

rectangular compound on which they want to put up a hedge around.

The hedge shall be made up of plants of different colours.

Support: Physical instruments like hoes, machetes, tape measure

Resources: Knowledge of construction of figures like rectangles,

patterns, sequences

Task: The family requests you to plant the hedge around their

rectangular compound so that it looks beautiful.

Explain how you will plant the hedge, making sure that the plants at

the corners of the compound are the same in terms of colour.

Discuss whether there are other ways of planting the hedge.

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Topic 7:

BEARINGS

The diagram below shows the bearing of Kabale from where the lady is

standing.

Key words: angle, direction, bearing, scale, line, turn

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

i) review the compass.

ii) describe the direction of a place from a given point using

cardinal points.

iii) describe the bearing of a place from a given point.

iv) draw suitable sketches from the given information.

v) choose and use appropriate scale to draw an accurate drawing.

v i) differentiate between a sketch and a scale drawing.

vii) apply bearings in real life situations.

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Introduction

In this topic you will learn how to tell the bearing of a point from a

given point. You will determine accurately the distance between two

points.

Sub-topic 7.1: Angles and Turns

You will need to understand clearly, what the terms such as turn, halfturn,

etc. mean in terms of angles. There are 360o in one complete turn,

so the following are true.

You also need to refer to compass points: north (N), south(S), east(E),

west(W), northeast (NE), southeast (SE), southwest (SW) and northwest

(NW)

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Activity 7.1: Identifying the angles in relation to the compass

direction

Work in pairs

Do the following turns and in each case state the size of the angle you

have turned through.

i) Turn from N to S clockwise or anticlockwise

ii) Turn from NE to SE clockwise

iii) Turning clockwise from NE to E

Example

What angle do you turn through if you turn:

(a) from NE to NW anticlockwise?

(b) from E to N clockwise?

Compare your answers with the rest of the members in class.

Solution

(a) 90o (or ¼ turn)

(b) 270o (¾ turn)

Exercise

1. What angle do you turn through if you turn clockwise from:

(a) N to E? (b) W to NW? (c) SE to NW? (d) NE to N? (e) W to NE?

(f) S to SW? (g) S to SE? (h) SE to SW? (i) E to SW?

2. In what direction will you be facing if you turn:

(a) 180o clockwise from NE?

(b) 180o anticlockwise from SE?

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(c ) 90o clockwise from SW?

(d) 45o clockwise from N?

(e) 225o clockwise from SW?

(f) 135o anticlockwise from N?

(g) 315o clockwise from SW?

3. The sails of a windmill complete one full turn every 40 seconds.

(a) How long does it take the sails to turn through:

(i) 180o (ii) 90o (iii) 45o?

(b) What angle do the sails turn through in:

(i) 30 seconds? (ii) 15seconds? (iii) 25 seconds?

Sub-topic 7.2: Bearings

The bearing of a point is the number of degrees in the angle measured

in a clockwise direction from North line to the line joining the centre of

the compass with the point. A bearing is used to present the direction

of one-point relative to another point.

Activity7.2: Estimating bearings of some places within the school

compound

In groups, work in pairs and outside the classroom.

From your school flag post, estimate the bearings of each building

found in the School.

Note: Three figures are used to give bearings.

All bearings are measured in a horizontal plane.

Compare your answers with the other members of the group.

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Exercise

1. Find the bearing of each of the following directions:

(a) S (b) NE (c) N (d) NW

2. Find the bearing of each of the following directions:

(a) N600E (b) N350E (c) N900W (d) S400E

3. Draw a scale diagram to show the position of a ship which is 270 km

away from a port on a bearing of 110o.

Situation of Integration

Ajok is in Kampala City and has been told to use a car to move to Lira

town. She has never gone to Lira. She has been given the map of

Uganda showing routes through which she can access Lira town.

Support: Mathematical instruments, pencil, paper, pens, tracing paper

and map of Uganda

Resources: Knowledge of construction of figures like triangles, lengths

of sides of triangles, operations on numbers.

Task: Ajok wants to use the short distance from Kampala to Lira.

Explain how Ajok can determine the shortest distance. Using the map

given to her is it possible for Ajok to use the shortest distance she has

determined. Explain your answer.

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Topic 8:

GENERAL AND ANGLE PROPERTIES OF

GEOMETRIC FIGURES

Key words: line segment, transversal, parallel

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

a. identify different angles.

b. solve problems involving angles on a straight line, angles on

transversal and parallel lines.

c. state and use angle properties of polygons in solving problems.

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Introduction

In bearings you studied angle turns, and in this topic you will study

angles on the straight line, parallel lines and angle properties of

polygons. Equipped with the knowledge from this topic, you will be

able to solve problems related with angle properties.

You will need to understand clearly what the terms such as turn, halfturn,

etc. mean in terms of angles. There are 360o in one complete turn,

so the following are true.

i) Turning from N to S is 180o clockwise or anticlockwise.

ii) Turning from NE to SE is 90o clockwise (or 270o anticlockwise).

iii) Turning clockwise from NE to E is 45o (or 315o anticlockwise).

Example

What angle do you turn through if you turn:

a) from NE to NW anticlockwise?

b) from E to N clockwise?

Solution

c) 90o (or ¼ turn)

d) 270o (¾ turn)

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Sub-topic 8.1: Identify Different Angles

Activity8.1: Identifying objects that form angles

In your groups, work in pairs.

Identify objects in you class, which make 900, 1800, 3600

A protractor can be used to measure angles.

Note:

The angle around the circle is 360o.

The angle around a point on a line is 180o.

A right angle is 90o

Compare your answers with other members of the group and

classify them

Exercise

1. For each of the following angles, first estimate the angles and then

measure the angle to see how good your estimate was.

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2. Draw the following angles

(a) 20o (b) 42o (c ) 80o (d) 105o (e) 170o (f) 200o (g) 275o (h) 305o

3. Immaculate finds out the favourite sports for members of her class.

She works out the angles in the list shown below for a pie chart. Draw

the pie chart.

Sport Angle

Football

Swimming

Tennis

Rugby

Hockey

Badminton

Other

1100

70o

80o

40o

30o

10o

20o

Exercise

1. (a) Draw a triangle with one obtuse angle.

(b) Draw a triangle with no obtuse angles.

2. Draw a four-sided shape with:

a) one reflex angle.

b) two obtuse angles.

Sub- topic 8.2: Angles on a Line and Angles at a

Point

Remember that:

a) angles on a line add up to 180o

And:

b) angles at a point add up to 360o

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These are two important results, which help when finding the size of

unknown angles.

Activity 8.2: Identifying angles

Work as individuals

Draw two intersecting lines. Use your mathematical instruments to

measure the angles formed at the intersecting point.

i) How many angles have been formed at the point of

intersection?

ii) What is the size of each angle formed?

Compare your work with your friends and note your findings.

A polygon is a closed plane figure with straight sides.

Activity 8.3: Identifying the polygons

In pairs:

Find the number of sides of different polygons and their corresponding

names. Also determine the number and size of interior and exterior

angles of the regular polygons.

Compare your answers with other members’.

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Exercise

1. If the vertices of a regular hexagon are joined to the centre of

the hexagon, what is the size of each of the six angles at the

centre? Use your answer to construct a regular hexagon

ABCDEF of side 3cm. Start with a circle of radius 3cm. Measure

the length of the diagonal AC.

2. Find the sum of the interior angles of a polygon with 22 sides.

3. The interior angle of a regular polygon is 1620. How many sides

has the polygon?

Activity of Integration

A diagram of a table showing coffee production in Uganda from year

2015 to year 2019

Year 2015 2016 2017 2018 2019

Production

(tonnes)

20 23 18 30 49

Task: The chairperson of Karo Farmers Association was asked to

represent the information above on pie chart. As a senior one

learner help him solve the challenge.

Support: Mathematical set

Resource: Knowledge of angles

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Topic 9:

DATA COLLECTION AND

PRESENTATION

Key words: data, chart, pie, quantitative, qualitative, discrete,

continuous, hypothesis

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

a) understand the differences between types of data.

b) collect and represent simple data from local environment using

bar chart, pie chart and line graph.

Introduction

In this topic, you will learn different types of data, data collection,

presentation and analysis.

Sub-topic 9.1: Types of Data

Qualitative data is data that is not given numerically; e.g. favourite

colour, place of birth, favourite food, and type of car.

Quantitative data is numerical. There are two types of quantitative

data: discrete and continuous data. Discrete data can only take

specific numeric values e. g. shoe size, number of brothers, number of

cars in a car park. Continuous data can take any numerical value e.g.

height, mass, length.

Activity 9.1: Identifying types of data

In your groups identify which of the following terms best describes

each of the information listed (i) to (vii)?

Give reasons for your response.

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• Qualitative data

• Continuous Quantitative Data

• Discrete Quantitative Data

i) Age

ii) Birth place

iii) Height

iv) World Ranking

v) Aces

vi) First serve School

vii) School life

In your groups identify more examples.

Exercise

1. Mr Okot starts to make a database for his lesson.

Name Age Primary school Transport

to School

Height Reading Glasses

Alice 11 St. Johns Bus 145cm yes

Ben 12 St. Andrews Walk 160 cm no

Carol 12 Hilltop Car 161 cm no

David 12 Hilltop 152 cm no

Eddie 11 St. Andrews Walk 158 cm yes

Fredrick St. Andrews Bike 164 cm no

Graham 12 St. Johns Bus 166 cm yes

a) What is missing from Mr Okot’s database?

b) Which columns in the database contain quantitative data?

c) Which columns in the database contain qualitative data?

d) Write down what Mr Okot would put in his database if you joined

his class.

2. Which of the following would give:

(a) qualitative data

(b) discrète quantitative data

(c ) continuous quantitative data

(i) Mass (ii) Number of cars

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(iii) Favourite football team (iv) Colour of car

(v) Price of chocolate bars (vi) Amount of pocket money

(vii) Distance from home to school (viii) Number of pets

(ix) Number of sweets in a jar (x) Mass of crisps in a packet.

3. The table below shows a database that has no entries.

Name Age Favourite

food

Favourite

TV show

Favourite

pop group

Time spent

watching TV

yesterday

a) Collect data from 10 people to complete the data base.

b) State whether each column contains:

i) qualitative data.

ii) continuous quantitative data.

iii) or discrete quantitative data.

c) Answer the following questions:

i) What is the most popular TV show?

ii) Who is the oldest?

iii) What is the favourite pop group for the youngest person?

d) Write 3 more questions you could answer using your database and

write the answers to them.

Sub-topic 9.2: Collecting Data

In this section, you will see how data is collected, organized and

interpreted, using a tally chart and then displayed using:

i) Pictograms

ii) Bar charts

iii) Pie charts

Note:

A hypothesis is an idea that you want to investigate to see if it is true or

false. For example, you might think that most people in your school get

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there by bus. You could investigate this using a survey. A tally chart can

be used to record your data.

Activity 9.2: Collecting data

In groups identify the means of transport each learner use to come to

school. As a class identify how many of you use the same means of

transport.

i) Which means of transport is used by the majority?

ii) Which one is the least used means of transport?

Example

The learners in a class were asked how they got to school.

Method of Travel Tally Frequency

Walk ///// /// 9

Bike /// 3

Car ///// / 6

Bus ///// ///// // 12

TOTAL 30

Illustrate this data using:

a) a pictogram

b) a bar chart

c) a pie chart

What are the main conclusions that can be deduced from the data?

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Solution

(a) If (stick man) is taken to represent 2 people, then the pictogram

looks like:

i) Walk (4 and a half stick men)

ii) Bike (1 and a half stick men)

iii) Car (3 stick men)

iv) Bus (6 stick men)

(b) A bar chart for the data is illustrated below:

(c) To illustrate the data with a pie chart, you need to find out what

angle is equivalent to one pupil. Since there are 360o in a circle and

30 pupils, then angle per pupil is 360 ÷ 30 = 12o.

To find the angle for walk, when there are 9 pupils, it is simply:

9 x 12 = 1080

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The complete calculations are shown below:

Method of

travel

Frequency Calculation Angle

Walk 9 9 x (360 ÷ 30) 108o

Bike 3 3 x(360 ÷ 30) 36o

Car 6 6 x (360 ÷ 30) 72o

Bus 12 12 x (360 ÷ 30) 144o

TOTAL 360o

The corresponding pie chart is shown below:

From the data we can see that:

• the most common way of getting to school is by bus. (This is

called the modal class or the mode.)

• the least popular way of getting to school is by bike.

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Exercise

1. The children in a class were asked to state their favourite crisps.

The results are given in the tally chart below:

Flavour Tally Frequency

Ready Salted /////

Salt and Vinegar ///// ////

Cheese and onion ///// //

Prawn Cocktail ///

Smokey Bacon ///// /

TOTAL

(a) Copy and complete the table by filling the frequencies.

(b) Represent the data on a bar chart.

(c) Draw a pictogram for this data.

(d) Copy and complete the following table and draw a pie chart.

Flavour Frequency Calculation Angle

Ready Salted 5 5 x ( 360o ÷ 30) 60o

TOTAL

(e) What flavour is the mode?

2. (a) Do you think salt and vinegar crisps will be most popular crisps in

your class?

(b) Carry out a favourite crisps survey for your class. Present the

results in a bar chart and state which flavour is the mode.

(c) Was your hypothesis in (a) correct?

3. “Most children in my class are 1.3m tall.”

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(a) Collect data to test this hypothesis.

(b) Present your data in a suitable diagram.

(c) Was the original hypothesis correct?

4. Is the music group that is most popular with the boys in your class

the same as the music group that is most popular with girls?

(a) Write down a hypothesis that will enable you to answer this

question.

(b) Collect suitable data from your class.

(c) Present your data using a suitable diagram.

(d) Was the hypothesis correct?

Situation of Integration

The Games Master at your school wants to buy football boots for the

three teams in the school. The three teams are the under 18 years,

under 16 years and the under 14 years. The Games Master does not

know the foot size for each of the players.

Support: pens, paper, tape measure, team members

Resources: Knowledge of tabulation, of tallying, of approximation, of

central measures and of collection of suitable data.

Task: The total number of players for the three teams is 54. The Games

Master wants to know the size of the boots for each player and

the number of pairs for each size.

Explain how the Games Master will get the required data and how to

determine the total cost for buying the football boots for the 54

players.

Is there another way of getting the required data other than what you

have explained above?

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Topic 10:

REFLECTION

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

i) identify lines and planes of symmetry for different figures.

ii) state and use properties of reflection as a transformation.

iii) make geometrical deductions using reflection (distinguish

between direct and opposite congruence).

iv) apply reflection in the Cartesian plane.

Introduction

In this topic, you will learn how to identify the lines of symmetry, state

the properties of reflection as a transformation, make geometrical

deductions and apply reflection in Cartesian plane.

The image of a figure by reflection is its mirror image in the axis or

plane of reflection. For example the mirror image of the letter p for

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reflection with respect to a vertical axis would look like q. Its image by

reflection in a horizontal axis would look like b.

Sub-topic 10.1: Identify Lines of Symmetry for

Different Figures

Activity 10.1: Identifying lines of symmetry

In pairs:

1. Fold a piece of paper in half

2. Open the paper and put in one drop of ink on the fold

3. Close the paper over the ink and press down hard on the paper.

4. When the ink has dried, open up your paper.

(a) Look at both sides of the fold line. Are they the same size and

shape?

(b) Look at any two corresponding points on the ink blot, one on

either sides of the fold.

(i) What can you say about the distance from one point to

the fold line and the distance from the corresponding

point to the fold line?

(ii) If a line joins two corresponding points, what is the angle

between the line and the fold?

Exercise 1

1. Draw a rectangle on a tracing paper. Fold it to find the lines of

symmetry. How many lines of symmetry does a rectangle have?

2. Find the number of lines of symmetry of (a) a square (b) an

equilateral triangle (c) a rhombus

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Sub-topic 10.2: Reflection in the Cartesian Plane

Activity 10.2: Reflecting in a Cartesian plane

In your groups, work as pairs.

Plot the points P (5, 4), Q (-1, 3) and R (0, -2) on squared paper.

a) A mirror is placed on the x axis. Where would the images of the

tree points be?

What are the coordinates of the image points P’, Q’ and R’?

b) Draw another pair of axes. Plot the same points again. Take the

line y = 2 as the mirror line. Where would the images of the

three points be? What are the coordinates of the new image

points P’, Q’ and R’?

c) Draw another pair of axes. Draw the line x = 4. Plot the points (1,

-3). Using the line x = 4 as the mirror line, find the image of the

point (1, -3).

Compare your answers with other members in your group.

Exercise 2

1. Find the image of the point (2, 5) under reflection in the y axis.

2. After a point has been reflected in the x axis, its image is at (3,

2). Find the coordinates of the object point.

3. The points A(4, 2) , B(1, 3) and C(1,-2)are reflected in the line y =

x. Find the coordinates of A’ , B’ and C’, the images of A and B.

Situation of Integration

One of your relatives wants to make a barbershop /hairdresser. He

approaches you for help.

As a senior one graduate draw a plan of how you can help your relative

make his /her barber shop be up to date.

Support: Interior plan of the shop

Task: Advice the barber to make sure the customers can view

themselves with their images not distorted.

Resource: knowledge of reflection

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Topic 11: Equation of Lines and

Curves

Key words: variable, curve, substitution

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

i) form linear equations with given points.

ii) draw the graph of a line given its equation.

iii) differentiate between a line and a curve.

Introduction

In this topic you will tell the difference between a line and a curve, how

to form linear equations and draw graphs for the given linear

equations.

Sub-topic 11.1: Fundamental Algebraic Skills

In this section, you will look at some fundamental algebraic skills by

examining codes and how to use formulae.

Example

If a = 4, b = 7 and c = 3, calculate:

(a) 6 + b (b) 2a + b (c) ab (d) a (b – c) (d) a (b – c)

Solution

(a) 6 + b = 6 + 7 = 13

(b) 2a + b = 2x4 + 7 = 8 + 7 = 15

(c) ab = 4x7 = 28

(d) a (b – c) = 4 x (7 – 3) = 4 x 4 = 16

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Example

Simplify where possible:

(a) 2x + 4x (b) 5p + 7q – 3p + 2q

(c) y + 8y – 5y (d) 3t + 4s

Solution

(a) 2x + 4x = 6x

(b) 5p + 7q – 3p + 2q = 5p -3p + 7q+2q = 2p – 9q

(c) y + 8y – 5y = 9y – 5y = 4y

(d) 3t + 4s = 3t + 4s

Exercise

1. If a = 2; b = 6; c = 10 and d = 3, calculate:

(a) a + b (b) c – b (c) d + 7

(d) 3a + d (e) 4a (f) ad

(g) 3b (h) 2c (i) 3c - b

(j) 6a + b (k) 3a + 2b (l) 4a – d

2. If a = 3; b = -1; c = 2 and d = -4, calculate:

(a) a – b (b) a + d (c) b + d

(d) b – d (e) 3d (f) 5(d – c)

(g) a (b + c) (h) d(b + a) (i) c(b – a)

(j) a (2b – c) (k) d(2a – 3b) (l) c(d – 2)

3. Simplify, where possible:

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(a) 2a + 3a (b) 5b + 8b

(c) 6c – 4c (d) 5d + 4d + 7d

(e) 6e + 9e – 5e (f) 8f + 6f – 13f

(g) 9g + 7g – 8g - 2g -6g (h) 5p + 2h

(i) 3a + 4b – 2a (j) 6x + 3y – 2x –y

(k) 8t – 6t + 7s – 2s

(l) 11m +3n – 5p + 2q -2n +9q -8m + 14p

4. Sam asks his friend to think of a number, multiply it with 2 and then

add 5. If the number his friend starts with is x, write down a formula for

the number her friend gets.

Subtopic 11.2: Function Machines

In this section you will look at how to find the input and output of

function machines.

INPUT → FUNCTION MACHINE→ OUTPUT

Activity 11.1: Function machine activity

In pairs try out the numbers the first one is done for you.

Calculate the output of each of these function machines:

(a) 4 →x5 →?

(b) 5 →x2 →-1 →?

(c) -3 →+8→x7→?

(d) 8 →+6→x9→?

(e) -5→+3→x7→?

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Compare your answers with members of the group.

Solution

(a) The input is simply multiplied by 5 to give 20:

4 → x5 → 20

Exercise

1. What is the output of each of these function machines:

(a) 4→+6 →?

(b) 3 →x10→?

(c) 10→-7→?

(d) 14→÷2→?

(e) 21→÷3→?

(f) 100→×5→?

2. What is the output of each of these function machines:

(a) 3→×4→-7→?

(b) 10→-8→×7→?

(c) 8→-5→×5→?

(d) -2→×6→+20→?

(e) 7→+2→÷3→?

(f) -5→+8→×9→?

3. What is the input of each of these function machines:

(a)? →×5→30 (b)? → +8→ 12

(c)? → -9→ 11 (d)? → +4→ 5

(e)? → +12→ 21 (f)? → ×7→ 42

4. A number is multiplied by 10, and then 6 is added to get 36. What is

the number?

5. Karen asked her teacher, Maria, how old she was. The teacher

replied that if she double her age, added 7 and then divided by 3, she

would get 21. How old is Karen’s teacher?

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6. A bus has a maximum number of passengers when it leaves the bus

station. At first stop, half of passengers alighted. At the next stop 7

people alighted and at the next stop 16 people alighted. There are

now 17 people on the bus. How many passengers were on the bus

when it left the bus station?

Sub-topic 11.3: Linear Equations

An equation is a statement, such as 3x + 2 = 17, which contains an

unknown number. In this case, it is x. The aim of this section is to show

how to find the unknown number, x.

All equations contain an ‘‘equals” sign.

To solve the equation, you need to reorganize it so that the unknown

value is by itself on one side of the equation. This is done by

performing operations on the equation. When you do this, in order to

keep the equality of the sides, you must remember that “Whatever

you do to one side of an equation, you must also do the same to the

other side”.

Example

Solve these equations:

(a) x + 2 = 8 (b) x- 4 = 3 (c ) 3x = 12

(d) 2x + 5 = 11 (e) 3 – 2x = 7

Solution

(a) To solve this equation, subtract 2 from each side of the equation:

X + 2 = 8

X + 2 -2 = 8 – 2

X = 6

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(b) To solve this equation, add 4 to both sides of the equation:

X – 4 = 3

X – 4 + 4 = 3 + 4

X = 7

(c) To solve this equation, divide both sides of the equation by3:

3x = 12

3x ÷ 3 = 12 ÷ 3

X = 4

(d) This equation must be solved in 2 stages.

First, subtract 5 from both sides:

2x + 5 = 11

2x + 5 -5 = 11 – 5

2x = 6

Then, divide both sides of the equation by 2:

2x ÷ 2 = 6 ÷ 2

X = 3.

(e) First, subtract 3 from both sides:

3 – 2x = 7

3 – 3 – 2x = 7 – 3

-2x = 4

Then divide both sides by (-2);

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-2x ÷ -2 = 4 ÷ -2

X = -2.

Example 3

You ask a friend to think of a number. He then multiplies it by 5 and

subtracts 7. He gets the answer 43

a) Use this information to write down an equation for x, the unknown

number.

b) Solve your equation for x

Solution

a) As x = number your friend thought of, then

5x

X 5x So 5x -7 = 43

b) First, add 7 to both sides of the equation to give

5x = 50

Then divide both sides by 5 to give

X = 10

And this is the number that your friend thought of.

Exercises

1. Solve these equations:

a) x +2 = 8 b) x +5 = 11 c) x – 6 = 2

d) x – 4 = 3 e) 2x = 18 f) 3x = 24

x 5 -7

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g) 4

6

x = h) 9

5

x = i) 6x = 54

j) x + 12 = 10 k) x + 5 =3 l) x -22 = -4

m) -2

7

x =

n) 10x = 0

o) 5

2

x =

2. Solve these equations

a) 2x + 4 =14 b) 3x + 7 =25 c) 4x +2 = 22

d) 6x – 4 = 26 e) 5x – 3 = 32 f) 11x – 4 =29

g) 3x + = 4 = 25 h) 5x – 8 = 37 i) 6x + 7 = 31

j) 3x + 11 = 5 k) 6x + 2 = -10 l) 7x + 44 +2

3. Solve these equations, giving your answers as fractions or

mixed numbers

a) 3x = 4 b) 5x = 7 c) 2x + 8 = 13

d) 8x + 2 = 5 e) 2x +6 =9 f) 4x = 7 = 10

4. Solve these equations:

a) x + 2 =2x -1 b) 8x – 1 = 4x + 11 c) 5x + 2 = 6x - 4

d) 11x – 4 = 2x = 23 e) 5x +1 = 6x -8 f) 3x + 2 +5x + x =44

g) 6x + 2 – 2x = x + 23 h) 2X – 3 = 6x + x -58 i) 3x + 2 = x -8

j) 4x – 2= 2x - 8 k) 3x + 82 = 10 x + 12 l) 6x – 10 = 2x - 14

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Topic 14: Time and Time Tables

Key words: quarter, timetable, half, departure, arrival

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

i) identify and use units of time.

ii) use and interpret different representations of time.

iii) apply the understanding of time in a range of relevant real life

contexts.

Introduction

In this topic, you will learn various units of time, such as minutes,

seconds, hours, day, week, month, year. You will be able to understand

and apply time in a range of relevant real-life contexts.

Sub-topic 14.1: Telling the Time

In this section we look at different ways of writing times; for example,

7:45 is the same time as quarter to eight.

On a clock face, this can be represented as shown below.

Also remember that

One hour = 60 minutes

So that

Half an hour = 30 minutes

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Quarter of an hour = 15 minutes

Three quarters of an hour = 45 minutes

Example

Write each time using digits and show the position of the hands on a

clock face:

(a) twenty-five past eight.

(b) quarter to ten.

Solution

(a) Twenty-five past eight using digits is 8:25

(b) Quarter to ten can be thought of as:

15 minutes to 10 o’clock

Or

45 minutes past 9 o’clock

So, using digits, quarter to ten is 9:45

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Exercise

1. Draw the time given below on clock faces:

(a) ten past five (b) ten minutes to nine (c ) quarter to seven

(d) quarter past twelve (e) half past ten (f) twenty nine minutes to five

(g) ten minutes to two (h) twenty five minutes to six (i) twenty past

four

2. Draw the following time on clock faces:

(a) 4:00 (b) 5:30 (c ) 7:15 (d) 8:20 (e) 2:45 (f) 3:50

(g) 1:55 (h) 6:05 (i) 11:35

3. Write the following time in words:

(a) 9:30 (b) 4:00 (c ) 4:25 (d) 8:45 (e) 7:35 (f) 9:05

4. Write these times using digits:

(a) eight o’clock (b) quarter to seven (c ) ten past five

(d) half past six (e) ten to three (f) five to four

(g) twenty five to nine (h) twenty to three

Sub-topic 14.2: 12-hour and 24-hour Clocks

The 24-hour clock system can be used to tell if time is in the morning or

the afternoon. Alternatively, time can be given as am or pm.

Activity 14.1: Converting from 12 hour to 24 hour and vice versa

In pairs:

i) Write these times in 24-hour clock time:

(a) 3:06 am (b) 8:14 pm (c) 9:45am (d) 3:06pm

ii) Write these times in 12-hour clock time:

(a) 03:00 (b) 09:45 (c) 13:07 (d) 22:15

Solution

(a) As this is a.m. the time remains the same except you add a zero in

front of 3, so the time becomes 0306 in a 24-hour clock.

(b) As this is pm, you add 12 to the hours to give you 2014 in a 24-hour

clock.

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Example

Write these times using am or pm in a 12-hour clock.

(a) 14:28 (b) 07:42

Solution

(a) As the hours figure, 14, is greater than 12, subtract 12 and write as a

pm time. The answer is2:28pm.

(b) As the hours figure, 07, is less than 12, simply remove the zero and

then write the time as am. The answer is 7:42 am.

Exercise

1. Convert the following time to the 24-hour clock:

(a) 9:24am (b) 11:28pm (c ) 11:14a.m (d) 7:13pm

2. Write the following time in the 24-hour clock:

(a) quarter to eight o’clock in the morning

(b) ten minutes to midnight

(c) ten past nine o’clock in the morning

(d) half past two o’clock in the afternoon

3. Write the following24-hour clock in words

(a) 14 :30 (b) 15:55 (c) 07:45

4. Sarah leaves home at 09:00 and returns 7 hours later. Write the time

that Sarah gets home in the 24-hour clock and in the twelve-hour

clock using am or pm.

Sub-topic 14.3: Units of Time

In this section we explore the different units of time.

1 minute = 60 seconds

1 hour = 60 minutes

1 day = 24 hours

1 week = 7 days

1 year = 365 0r 366 days

Example

1. How many hours are there in May?

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Solution

Number of hours in May = 31 x 24 = 744 hours

Activity 14.2

In pairs find out if 25 February is a Friday. What will be the date on the

next Friday?

(a) If it is not a leap year.

(b) If it is a leap year?

Compare your answers with members of the group before you check

the solution.

Solution

(a) You could write out the 7 days like this:

Friday 25

Saturday 26

Sunday 27

Monday 28

Tuesday 1

Wednesday 2

Thursday 3

Friday 4

Or

25 + 7 = 32

32 – 28 = 4

So the next Friday will be 4th March.

(b) Using the addition method:

25 + 7 = 32

32 – 29 = 3

So, in a leap year, the next Friday will be 3rd March.

Exercise

1. How many hours are there in a week?

2. How many hours are there in:

(a) September?

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MATHEMATICS

PROTOTYPE

121

(c) February?

(d) one year?

3. 3. Rupert goes on holiday on Monday 20th June. He returns 14

days later. On what date does he return from his holiday?

4. 4. If 3rd October is a Monday:

(a) What day of the week will 1st November be?

(b) What will be the date of the first Monday in November?

5. Immaculate goes to the bank every Tuesday. The last time she

went was on Tuesday 20th October.

(a) What will be the dates of her next 2 visits to the bank?

(b) On the second Tuesday in November she is ill and goes to the

bank on Wednesday instead. What is the date of that

Wednesday?

Sub-topic 14.4: Timetables

In this section we consider how to extract information from timetables.

Exercise

1. The table below gives the timetable for a Bus that runs from Mbale

to Kampala.

Mbale depart 08:57

Iganga depart 10:06

Jinja arrive 16:57

Mukono arrive 17:23

Kampala arrive 17:42

(a) At what time does the bus leave Mbale?

(b) At what time does the bus arrive at Kampala?

(c) Where does the bus arrive at 16:57?

(d) Mr Okot arrives in Mbale at five past nine. Can he catch the bus?

2. Mike is in Brussels and wants to return to Ashford. He looks at this

train timetable:

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SENIOR ONE

122

Brussels to Waterloo

Brussels Midi

Lille Europe

Ashford

International

Waterloo

International

0856

0937

0938

1047

1102

1142

1141

1247

1302

1342

1341

1447

1456

1536

1536

1639

1702

1742

1741

1843

1756

1836

1837

1939

1856

1936

1938

203

2102

2142

….

2239

a) At what time should he catch a train if he wants to arrive in

Ashford at 17:41?

b) Which train should he avoid if he wants to go to Ashford?

c) If he catches the 14:56 train, at what time does he arrive in

Ashford?

d) He catches the 14:56, but falls asleep and does not get off at

Ashford. At what time does he get to Waterloo?

3. The Journey from Kabale (Uganda) to Kigali (Rwanda) takes 2 ½

hours. The time in Uganda is 1 hour ahead of Rwanda.

a) If you leave Kabale at 10:00, what will be the local time when

you arrive in Kigali?

b) If you leave Kigali at 17:45, what will be the local time when you

arrive in Kabale?

4. Jean earns UGX 4,000 per hour on weekdays, UGX 4,500 per hour on

Saturdays and UGX 6,000 per hour on Sundays.

The table below lists the hours she worked on each day for one week:

Day No. hours worked

Monday 4

Tuesday 2

Wednesday 8

Thursday 10

Friday 3

Saturday 5

Sunday 2

How much money did Jean earn that week?

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MATHEMATICS

PROTOTYPE

123

Situation of Integration: A primary school has two sections, that is,

lower primary (P1-P4) and upper primary (P5-P7). The head teacher of

primary school needs to draw a timetable for both sections. The

sections should start and end their morning lessons at the same time

before break time, start and end their break time at the same time. The

after break lessons should start at the same time. The lunchtime for

both sections should start at the same time.

Support: The time to start lessons for the two sections is 8.00am. The

duration of the lesson for the lower section is 30 minutes and that of

the upper section is 40 minutes.

Resources: Knowledge of fractions, percentages, natural numbers,

factors, multiples, lowest common multiples and of time.

Task: Help the head teacher by drawing the timetable up to lunch

break for the two sections. How many lessons does each section have

up to lunch break?

Express the total number of lessons for the lower primary as a fraction

of the total number of lessons for the whole School. (Consider lessons

up to lunch break)

National Curriculum

Development Centre,

P.O. Box 7002, Kampala.

www.ncdc.go.ug