Easyread Mathematics for Junior Secondary Schools 1: Ube Edition

By I. Y. Hali

EasyRead Mathematics for Junior Secondary Schools is a doing it yourself mathematics series that is written for pupils with learning difficulties in mathematics. This series of books is easytoread and easytounderstand; as the style used in writing the texts is a stepbystep approach and the explanations used in presenting those steps are extremely easy to follow.
This series of books is written not just to add to the number of the existing mathematics textbooks on the shelf, but to be a candle that will lighten the paths of millions of pupils whose paths have been darkened by fear of mathematics as a subject. The author makes this series of books user friendly to pupils who never gave mathematics a show of love, and so appealing to pupils with little or no prior knowledge of mathematics before now.
In the beginning, God created man with a sense of numbers, said Hali. This sentence is meant to mean more than a quotation that is rightly said by the author in this series of books. This supplies the key which opens pupils understanding to mathematics as a whole and pupils passion for mathematics as a subject. Taking the pupils on this wondrous journey through making the unknown known-man is born with an inbuilt knowledge of mathematics-Hali teaches pupils in the course of this series of books, how to use their minds and imaginations in improving their personal knowledge of mathematics and in preparing themselves toward achieving personal excellent grades on Junior High School Mathematics.
In this charming volume (EasyRead Mathematics for Junior Secondary Schools I), the author features hundreds carefully selected examples and imaginative exercises with solutions to all the carefully selected examples, and answers to all the imaginative exercises.

• Language: English
• Publisher: AuthorHouse UK
• Release date: Mar 30, 2016
• ISBN: 9781481769532

I. Y. Hali

I.Y. Hali is a University student of Mathematics and Statistics with the University of Maiduguri. He is a single author of ten other Mathematics textbooks; of which three are on the University series and one on the High School Mathematics and six on the Primary School Mathematics. Hali’s books are easy reading, and the style used in presenting the subject has helped pupils/students, who were once failures in Mathematics, in achieving personal excellent grades on examinations set by standard institutes. Hali was the recipient of the University-wide Bode Amoa Award for excellence in learning with a first-class first year GPA and other National Merit Awards both from the University of Jos. He was also celebrated on a National Scale as an unsung hero.

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2016 by I. Y. Hali. All rights reserved.

No part of this book may be reproduced, stored in a retrieval system, or transmitted by any means without the written permission of the author.

Published by AuthorHouse   03/30/2016

ISBN: 978-1-4817-6952-5 (sc)

ISBN: 978-1-4817-6953-2 (e)

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Because of the dynamic nature of the Internet, any web addresses or links contained in this book may have changed since publication and may no longer be valid. The views expressed in this work are solely those of the author and do not necessarily reflect the views of the publisher, and the publisher hereby disclaims any responsibility for them.

CONTENTS

Preface

What is Special about this Book

How to use this Book

CHAPTER 1 Whole Numbers

Man is Born with the Knowledge of Mathematics

The Right Approach to Mathematics

Early Counting

Counting, Writing and Reading

Counting, Writing and Reading in Units, Tens, Hundreds

Counting, Writing and Reading in Thousands

Counting, Writing and Reading in Millions

Counting, Writing and Reading in Billions

Counting, Writing and Reading in Trillions

CHAPTER 2 Basic Operations on Directed Numbers

Integers

Directed Numbers

Laws of Signs

Addition and Subtraction of Directed Numbers

Multiplication of Directed Numbers

Division of Directed Numbers

Applications of Directed Numbers on Everyday Activities

CHAPTER 3 Factors and Multiples

Factors

Prime Numbers

Composite Numbers

Prime Factorization

Common Factors

Highest Common Factor (HCF)

Multiples

Common Multiples

Least Common Multiple (LCM)

CHAPTER 4 Common Fractions and Percentages

Definitions and Examples of Fractions

Table 1: Names of some Common Fractions

Forms of Fractions

Laws of Fractions

Equivalent Fractions

Reducing Fractions

Arranging Fractions

Decimal Fractions

CHAPTER 5 Basic Operations on Fractions

Introducing Fractions

Converting Mixed Numbers into Improper Fractions

Converting Improper Fractions into Mixed Numbers

Addition and Subtraction of Fractions

Multiplication of Fractions

Division of Fractions

CHAPTER 6 Basic Operations on Decimals

Place Value System

Addition and Subtraction of Decimals

Multiplication of Decimals by Powers of 10

Division of Decimals by Powers of 10

Multiplication of Decimals

Division of Decimals

CHAPTER 7 Approximation and Estimation

Significant Figures

Decimal Places

Rounding Numbers

Estimation

CHAPTER 8 Binary System

The Denary (Base-Ten) System

Denary Arithmetic

The Binary (Base-Two) System

Transforming Binary Numerals into Denary Numerals

Transforming Denary Numbers into Binary Numbers

Binary Arithmetic

CHAPTER 9 Use of Symbols

Open Sentences

Use of Letters for Numbers

Word Proble ms

CHAPTER 10 Algebraic Terms

Introducing Terms in Algebra

Algebraic Expressions

Like and Unlike Terms

Adding and Subtracting Like Terms

Inserting and Removing Grouping Symbols

CHAPTER 11 Simple Equations

Equations

Solving Equations

Word Problems

CHAPTER 12 Points and Lines

Points

Lines

Vertical Lines

Horizontal Lines

Oblique Lines

Intersecting Lines

Rays

Opposite Rays

Line Segments

Perpendicular Lines

Parallel Lines

Traversal Lines

Angles

Acute Angles

Right Angles

Obtuse Angles

Reflex Angles

Complete Angles

Complementary Angles

Supplementary Angles

Angles at a Point

Adjacent Angles

Vertical Angles

Alternate Interior Angles

Alternate Exterior Angles

Corresponding Angles

CHAPTER 13 Construction of Parallel and Perpendicular Lines

Introduction and Definitions

Constructions of Parallel Lines

Constructions of Perpendicular Lines

CHAPTER 14 Angles

Complementary Angles (Second Version)

Supplementary Angles (Second Version)

Angles at a Point (Second Version)

Adjacent Angles (Second Version)

Vertical Angles (Second Version)

Alternate Interior Angles (Second Version)

Interior Angles on a Transversal

Alternate Exterior Angles (Second Version)

Corresponding Angles (Second Version)

CHAPTER 15 Properties of Plane Shapes

Introducing Triangles

Acute-angled Triangles

Right-angled Triangles

Obtuse-angled Triangles

Equiangular Triangles

Scalene Triangles

Isosceles Triangles

Equilateral Triangles

Some Properties of Triangles

Introducing Quadrilaterals

Parallelogram

Rhombus

Rectangle

Square

Kite

Trapezium

Circles

CHAPTER 16 Perimeter of Plane Shapes

Measuring Perimeters of Plane Shapes

Calculating Perimeters of Plane Shapes

CHAPTER 17 Areas of Plane Shapes

Area Units

Area of Triangle

Area of Square

Area of Rhombus

Area of Rectangle

Area of Parallelogram

Area of Kite

Area of Trapezium

Area of Circle

CHAPTER 18 Solid Mensuration

Volume Units

Cubes

Cuboids

Capacity of Solids

Computing Capacity of Solids in Kilolitres

Computing Capacity of Solids in Millilitres

CHAPTER 19 Introducing Statistics

Need for Statistics

Data

Organizing Data

Frequency Table

CHAPTER 20 Data Presentation

Introduction

Pictogram

Bar Charts

Pie Charts

CHAPTER 21 Mean, Median and Mode

Mean

Median

Mode

Answers

Chapter One

Chapter Two

Chapter Three

Chapter Four

Chapter Five

Chapter Six

Chapter Seven

Chapter Eight

Chapter Nine

Chapter Ten

Chapter Eleven

Chapter Twelve

Chapter Fourteen

Chapter Sixteen

Chapter Seventeen

Chapter Eighteen

Chapter Nineteen

Chapter Twenty

Chapter Twenty-One

Preface

EasyRead Mathematics for Junior Secondary Schools is a doing it yourself mathematics series that is written for pupils with learning difficulties in mathematics. This series of books is easy-to-read and easy-to-understand; as the style used in writing the texts is a step-by-step approach and the explanations used in presenting those steps are extremely easy to follow.

This series of books is written not just to add to the number of the existing mathematics textbooks on the shelf, but to be a candle that will lighten the paths of millions of pupils whose paths have been darkened by fear of mathematics as a subject. The author makes this series of books user friendly to pupils who never gave mathematics a show of love, and so appealing to pupils with little or no prior knowledge of mathematics before now.

In the beginning, God created man with a sense of numbers, said Hali. This sentence is meant to mean more than a quotation that is rightly said by the author in this series of books. This supplies the key which opens pupils’ understanding to mathematics as a whole and pupils’ passion for mathematics as a subject. Taking the pupils on this wondrous journey through making the unknown known-man is born with an inbuilt knowledge of mathematics-Hali teaches pupils in the course of this series of books, how to use their minds and imaginations in improving their personal knowledge of mathematics and in preparing themselves toward achieving personal excellent grades on junior secondary school mathematics.

This series of books has literally been prepared on our knees. In this spirit, we must confess, We don’t have the right to claim that we have done anything on our own in writing and sponsorship of this series of books. God gives us, freely, what it takes to do all that we do. My daily prayer for each reader is that he/she will always pray to the Most High God for better understanding before learning based on personal discovery.

University of Maiduguri

What is Special about this Book

Many writers of mathematics textbooks direct themselves to someone who studies, knows, teaches or an expert in mathematics. Only on rare occasions do mathematics textbooks appeal to pupils with learning difficulties in mathematics. EasyRead Mathematics for Junior Secondary Schools have been written not for someone who studies, knows, teaches or an expert in mathematics, neither is it written for pupils of excellent backgrounds and foundations in mathematics but for pupils of poor backgrounds and foundations in mathematics, who claim not to know anything about mathematics as well as arithmetic.

The style used in writing this series of books is easy reading for pupils with learning difficulties in mathematics; the author made this possible by giving out step-by-step rules that guide the pupils on how to go about understanding and solving each problem to the finale, and the author as well led the pupils through the paths of recovery to discovery.

How to use this Book

Even if you have the knowledge of mathematics before now, we beg you to suppose the contrary and follow our advice on how to use this book.

Advice 1 Begin by begging the Almighty God to aid your understanding.

Advice 2 Then throw away any preconceived notions that you may have about what mathematics is and any notion that you are not good in mathematics as well as mathematics is difficult.

Advice 3 Make a commitment to learn the material—not just a good intention, but a genuine commitment.

Advice 4 Study this book. Notice that we said study not read. Reading is a part of study, study involves much more.

Advice 5 Use paper, pen (or pencil) and calculator when you study. These are your basic tools, and you cannot study effectively without them.

Advice 6 Find a study partner not a reading partner if available.

Advice 7 Work out the solution of each given exercise patiently and check out. Make every effort not to find behind each answer given at the answers pages.

In the beginning, God created man with the sense of numbers.

I. Y. Hali

CHAPTER 1

Whole Numbers

Man is Born with the Knowledge of Mathematics

In the beginning, God created man with an inbuilt or a natural knowledge of mathematics. Man started working-out mathematics when they began to count things in small groups, and when they began to recognize an increase or a decrease of things in small groups, and when they began to understand the difference between an increase and a decrease of things in small groups.

In those earliest days, man was able to recognize the number of his wives, children, houses, slaves, farmlands, chickens, ducks, doves, goats, cows and donkeys.

Man continued to put into effect his knowledge of mathematics when he understood addition (+) as one of his wives, chickens, ducks, doves, goats, sheep, cows and donkeys gave birth to a young one. This will always lead to an increase in count of his children, chickens, ducks, doves, goats, sheep, cows and donkeys.

Man was able to understand subtraction (-) as he freed one of his male or female slaves. Also, man was able to recognize subtraction (-) as a lion or bear took a lamb out of his flock of sheep. This will always lead to a decrease in the number of his flock of sheep.

Man was able to understand multiplication (×) as any two or more of his wives, chickens, ducks, doves, goats, sheep, cows and donkeys gave birth to young ones at the same time. This count will always double (two times), triple (three times), quadruple (four times), quintuple (five times), sextuple (six times) the number of the pregnant female animals.

Man was able to understand division (÷) as he shared his wealth to his children in parts. Also, man was able to recognize division (÷) as he shared his hunted or slaughtered meat to his wives and children in portions.

Following this reminder, one has come to understand that every pupil is born with knowledge of numbers and that every pupil is created to be exceptionally good at working-out mathematics. Consequently, mathematics as a subject had been engraved into the brain of every pupil by God at birth; so every pupil is called to love and practice mathematics since it is part of us.

The Right Approach to Mathematics

‘Man is born with an inbuilt Knowledge of mathematics.’ This sentence is meant to mean more than a topic in this series of books. This supplies the key which opens one’s understanding to mathematics as a whole and one’s love for mathematics as a subject.

Many pupils in schools visualize mathematics comfortably on a distant as an uninteresting subject. Such a view is wholly false. Many pupils have been told that mathematics is difficult to know; much more of passing it with an excellent grade. Such a notion is by no means true. History reveals to us that God created man and animals with a natural knowledge of mathematics. This event occurred long before man started going to schools. God took the initiative of inculcating an inbuilt knowledge of mathematics in man; because He knew that a time is coming when man will reject this gift of His, called mathematics, and will find himself lost in the darkness of fear and failure.

Pupils are called to carry a survey of their grandmas and grandpas in their respectively villages who never went to schools. Do they know how to count? Do they know that the combination of two apples and five apples is seven apples? Can they differentiate between a man with five fingers of a single hand and a man with six fingers of a single hand?

Furthermore, pupils are called to survey the market men and women who never went to schools. Do they know that the combination of a twenty and a five is same as a twenty-five? Can they add a fifty and a five? Can they remove (subtract) two pieces of candies from a pack containing fifty pieces of candies?

Pupils who hate mathematics are in reality rejecting the gift of the knowledge of mathematics that God had inculcated in them, and will soon find themselves lost in the darkness of fear and failure. In the same way, pupils who hate their mathematics teachers are in reality rejecting the gift of the knowledge of mathematics that God had inculcated in them and will soon find themselves hating, fearing and failing mathematics.

Much of hatred and controversy that people have concerned mathematics as a subject has arisen through a failure to appreciate and embrace the gift of the knowledge of mathematics that God has inculcated in them at birth while the rest of hatred and controversy that people have concerned mathematics as a subject has arisen through a failure to love mathematics as a subject and a failure to show love for the mathematics teachers.

What is the right approach to mathematics? The following steps will give one the right approach to mathematics as a subject.

Step 1 Every pupil must come to understand that he/she is created by God with a natural knowledge of mathematics at birth.

Step 2 Each pupil must come to accept that mathematics is a part of him/her.

Step 3 Pupils must come to stop hating mathematics and the mathematics teachers.

Step 4 Pupils must come to love mathematics as a subject.

Step 5 Pupils must come to love their mathematics teachers.

Step 6 Pupils must come to understand that the gift of the knowledge of mathematics needs to be recharged (practiced) every day after mathematics lessons.

Step 7 Pupils must come to understand that if only one’s forefathers, foremothers, market men, market women, some animals and birds that never went to schools lived mathematics, they could do it better.

Early Counting

The early man started counting when he used his fingers, sticks, small stones, tally, cut marks on wood, stones or house. This counting was done in groups called the grouping systems. These grouping systems appear to come from counting five fingers of a single hand called base five, ten fingers of both hands called base ten, twenty fingers of both hands and toes called base twenty. More grouping systems appear to come from counting the spaces between five fingers of a single hand called base four, ten fingers of both hands called base eight, twenty fingers of both hands and toes called base sixteen. There are other grouping systems. Such grouping systems are base two-counting in the group of 2s, base seven-counting in the group of 7s, base twelve-counting in the group of 12s, base twenty four-counting in the group of 24s.

Most languages all over the world prefer using base ten when counting. And this makes it the most acceptable base worldwide.

Counting, Writing and Reading

Counting is an act of finding the number of place a quantity is occupying. This number of place a quantity is occupying is called the place value system of that quantity and each quantity is called a digit. A digit is any of the familiar symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Every digit in the place value system tells the numbers of times Ones, Tens, Hundreds, Thousands, Ten-Thousands, Hundred-Thousands, Millions, Ten-Millions, Hundred-Millions, Billions, Ten-Billions, Hundred-Billions or Trillions is taken.

Table 1

Counting, Writing and Reading in Units, Tens, Hundreds

A unit is a place value system with one digit only. Units are also called ones, and they are used to start up counting from zero (0) through nine (9). In the place value system, units occupy a single column named the units column as demonstrated in Table 2:

Table 2

A ten is a place value system with two digits. Tens are used to start up counting from ten (10) to ninety-nine (99). In the place value system, they occupy the second column to the left (of the units’ column).

When one’s counting system is more than the units counting system, one is expected to form another column-called the tens’ column for the new system of counting and to write down the digit for tens in the ten’s column and the digit for ones in the units’ column. For example, the number 25 is represented as 2 Tens and 5 Ones as shown in Table 3:

Table 3

The number 25 could as well be written in the expanded form. The expanded form of 25 is 2×10+5×1.

A hundred is a place value system with three digits. Hundreds are used to start up counting from one hundred (100) through nine hundred and ninety-nine (999). In the place value system, they occupy the third column to the left (of the tens’ column).

When one’s counting system is more than the tens counting system, one is expected to form another column-called the hundreds’ column for the new system of counting and to write down the digit for hundreds in the hundred’s column, the digit for tens in the ten’s column and the digit for ones in the units’ column. For example, the number 703 is represented as 7 Hundreds, 0 Tens and 3 Ones as revealed in Table 4:

Table 4

The expanded form of 703 is represented as 7×100+0×10+3×1.

Rule 1: To write hundreds, tens and ones in the place value system:

Step 1 Identify the hundreds, tens and ones. Beginning one’s counting with the rightmost digit, we know that the hundreds is the third digit, the tens is the second digit and the ones is the rightmost digit.

Step 2 Place the digits for the hundreds, tens and ones in their related columns.

Example 4

Write 984 in the place value system.

Solution

Step 1 Identify the hundreds, tens and ones:

984 is represented as 9 Hundreds, 8 Tens and 4 Ones

Step 2 Place the digits for the hundreds, tens and ones in their related columns:

Table 5

Example 5

Place 760 in the place value system.

Solution

Step 1 Identify the hundreds, tens and ones:

760 is represented as 7 Hundreds, 6 Tens and 0 Ones

Step 2 Place the digits for the hundreds, tens and ones in their related columns:

Table 6

Example 6

Express 352 in the place value system.

Solution

Step 1 Identify the hundreds, tens and ones:

352 is represented as 3 Hundreds, 5 Tens and 2 Ones

Step 2 Place the digits for the hundreds, tens and ones in their related columns:

Table 7

Rule 2: To express a quantity in the expanded form:

Step 1 Identify the hundreds, tens and ones. Beginning one’s counting with the rightmost digit, we know that the hundreds is the third digit; the tens is the second digit and the ones is the rightmost digit.

Step 2 Multiply each digit of Step 1 by its respective Hundreds, Tens and Ones.

Step 3 Sum the obtained result of Step 2 to get the expanded form.

Example 7

What is the expanded form of 123?

Solution

Step 1 Identify the hundreds, tens and ones:

123 is represented as 1 Hundreds, 2 Tens and 3 Ones

Step 2 Multiply each digit of Step 1 by its respective Hundreds, Tens and Ones:

1×100, 2×10, and 3×1

Step 3 Sum the obtained result of Step 2 to get the expanded form.

1×100+2×10+3×1

Example 8

Write 468 in the expanded form.

Solution

Step 1 Identify the hundreds, tens and ones:

468 is represented as 4 Hundreds, 6 Tens and 8 Ones

Step 2 Multiply each digit of Step 1 by its respective Hundreds, Tens and Ones:

4×100, 6×10, and 8×1

Step 3 Sum the obtained result of Step 2 to get the expanded form.

4×100+6×10+8×1

Example 9

Express 502 in the expanded form:

Solution

Step 1 Identify the hundreds, tens and ones:

502 is represented as 5 Hundreds, 0 Tens and 2 Ones

Step 2 Multiply each digit of Step 1 by its respective Hundreds, Tens and Ones:

5 × 100, 0 × 10, and 2 × 1

Step 3 Sum the obtained result of Step 2 to get the expanded form.

5 × 100 + 0 × 10 + 2 × 1

Rule 3: To write a quantity in words:

Step 1 Identify the hundreds, tens and ones. Beginning one’s counting with the rightmost digit, we know that the hundreds is the third digit, the tens is the second digit and the ones is the rightmost digit.

Step 2 Write the obtained digits of Step 1 in words.

Step 3 Transform the wordings of Step 2 into modern English.

Example 10

Write 52 in words.

Solution

Step 1 Identify the hundreds, tens and ones:

5 Tens and 2 Ones

Step 2 Write the obtained digits of Step 1 in words:

five Tens and two Ones

Step 3 Transform the wordings of Step 2 into modern English:

five Tens in modern English mean fifty two Ones in modern English mean two So, five Tens and two Ones is fifty and two. Traditionally, 52 is written in words as fifty-two.

Example 11

Write 363 in words.

Solution

Step 1 Identify the hundreds, tens and ones:

3 Hundreds, 6 Tens and 3 Ones

Step 2 Write the obtained digits of Step 1 in words:

three Hundreds, six Tens and three Ones

Step 3 Transform the wordings of Step 2 into modern English:

three Hundreds in modern English mean three hundred

six Tens in modern English means sixty

three Ones in modern English mean three

Therefore, three Hundreds, six Tens and three Ones mean three hundred and sixty and three.

Traditionally, 363 is written in words is three hundred and sixty-three.

Example 12

Write 780 in words.

Solution

Step 1 Identify the hundreds, tens and ones:

7 Hundreds, 8 Tens and 0 Ones

Step 2 Write the obtained digits of Step 1 in words:

seven Hundreds, eight Tens and zero Ones

Step 3 Transform the wordings of Step 2 into modern English:

seven Hundreds in modern English mean seven hundred.

eight Tens in modern English mean eighty.

zero Ones in modern English mean zero.

Therefore, seven Hundreds, eight Tens and zero Ones mean seven hundred and eighty and zero.

Traditionally, 780 is written in words as seven hundred and eighty.

Rule 4: To write a written quantity in digits:

Step 1 Identify the hundreds, tens and ones. Beginning one’s counting with the rightmost digit, we know that the hundreds is the third digit; the tens is the second digit and the ones is the rightmost digit.

Step 2 Multiply each digit of Step 1 by its respective Hundreds, Tens and Ones and sum the obtained result to get the expanded form.

Step 3 Add the obtained result of Step 2 to get the desired answer.

Example 13

Write nine hundred and fifty-six in digits.

Solution

Step 1 Identify the