Get Ahead Mathematics

Bilingual Teaching Guide andAnswer Key for Book 5

Parveen Arif Ali

Contents

Unit I: Numbers

Roman numerals ............................................................ 1Divisibility ....................................................................... 1Factors ...........................................................................7Multiples ......................................................................... 9Prime and Composite Numbers ................................... 11Least Common Multiple ............................................... 11Highest Common Factor .............................................. 15

Unit II: Algebra

Common Fractions ..................................................... 21 Decimal Fractions ...................................................... 23

Unit III: Unitary Method ...................................................31

Unit IV: Average ...............................................................33

Unit V: Geometry

Lines ............................................................................ 35Angles .......................................................................... 39Perimeter ..................................................................... 39Area ............................................................................. 41Volume ......................................................................... 43

Unit VI: Graphs

Line Graphs ................................................................. 45Pie Charts .................................................................... 47

Answers to Exercises in Mathematics Book 5 .................. 49

iii

Introduction

Get Ahead Mathematics is a series of eight books from level one to eight. The accompanying Teacher’s Guides contain guidelines for the teachers. The answer keys at the end of the Teacher’s Guides, for Books 1 to 5, contain answers to the mathematical problems in the books.The teacher should devise means and ways of reaching out to the stu-dents so that they have a thorough knowledge of the subject without getting bored. The teachers must use their discretion in teaching a topic in such a way as they find appropriate, depending on the intelligence level as well as the academic standard of the class. Encourage the students to relate examples to real things. Don’t rush. Allow time to respond to questions and discuss particular concepts. Come well prepared to the class. Read the introduction to the topic to be taught in the pupils’ book. Prepare charts if necessary. Practice diagrams to be drawn on the blackboard. Collect material relevant to the topic. Prepare short questions, homework, tests and assignments.Before starting the lesson make a quick survey of the previous knowl-edge of the students, by asking them questions pertaining to the topic. Explain the concepts with worked examples on the board. The students should be encouraged to work independently, with useful suggestions from the teacher. Exercises at the end of each lesson should be divided between class work and homework. The lesson should conclude with a review of the concept that has been developed or with the work that has been discussed or accomplished. Blackboard work is an important aspect of teaching mathematics. However, too much time should not be spent on it as the students lose interest. Charts can also be used to explain some concepts, as visual material helps students make mental pictures which are learnt quickly and can be recalled instantly.Most of the work will be done in the exercise books. These should be carefully and neatly set out so that the processes can easily be seen. Careful setting out leads to better understanding.The above guidelines for teachers will enable them to teach effectively and develop in the students an interest in the subject. These suggestions can only supplement and support the professional judgement of the teacher. In no way can they serve as a substitute for it. It is hoped that your interest in the subject together with the features of the book will provide students with more zest to learn mathematics and excel in the subject.

iv

v

Unit I: Numbers Pages 1-40

Roman Numerals Pages 1-5

Objectives:

To learn about the Roman system of numbers.

Write a sequence of numbers from 1 to 9 on the board. Explain that the numbers that we use today are called the Arabic numbers as Arabs were the first to introduce them.The Romans invented a system of numbers long before the Arabs. They used symbols such as l to represent 1 and ll to represent 2 and V to represent 5.Write the Roman numbers on the board and their Arabic equivalents next to them.Explain that some numbers such as 2, 4, 6, 8, 9 had no symbols. Symbols such as l or V or X were joined with each other in various combinations to represent these numbers and larger numbers.Help the students to read and write Roman numerals in sequence.Write an addition sum in Roman numerals on the board, XIV+XII. Ask the students to solve it.Explain that this system of numbers could not be adopted for Mathematics because it is limited to natural numbers and is difficult to read. It cannot represent very large numbers. It does not contain a zero and it cannot represent negative numbers, fractions, etc.Now this system is only used for indexing numbers for classes, watches and clock faces, chapter numbers of books, etc.

Divisibility Pages 6-13

Objectives:

To know what a divisor is.To know even and odd numbers.To know the tests for divisibility by 2, 3, 4, 6, 9, 5, 10.

1

2

Divisors Page 6Write 6 on the board. Ask how many numbers can completely divide 6. Write 1, 2, 3, 6 on the board. The number which can exactly divide a number is called a ‘divisor’ of that number. Give more examples to explain the meaning of divisor.

Even and odd numbers Page 7Write the numbers 0 to 20 on the board.Ask the children to point out the numbers that are completely divisible by 2. Cross them out and write the crossed out numbers, i.e.0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, separately on the board.Draw a circle around the numbers that are left, i.e. 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.Explain that the numbers which are exactly divisible by 2 are called ‘even numbers’. Numbers that are not exactly divisible by 2 are called ‘odd numbers’. Explain that any number whose last digit is a zero or an even number which is exactly divisible by 2 and so on is an even number. Make the students practice finding even and odd numbers before doing the exercises on page 7 in the book.

Tests of divisibility Pages 8-13If we want to find the divisors of large numbers, there are some simple and quick tests by which we can find out whether a number is divisible by a certain number or not.

Test for divisibility by 2Check whether the number is even or odd. If the number is even, it is divisible by 2. If it is odd, it is not divisible by 2, for example, 234 is divisible by 2, 335 is not.

3

4

Test for divisibility by 3Add all the digits of the number. If the sum is a number that is exactly divisible by 3, then the number is divisible by 3, for example, 342 is divisible by 3 because the sum of 3 + 4 + 2 is 9 which is exactly divisible by 3. 134 is not divisible by 3 as 1+3+4= 8 which is not exactly divisible by 3.

Test for divisibility by 4If in a number, the digits in the tens and units place are 00 or are exactly divisible by 4, then that number is exactly divisible by 4, for example, 2,300 is divisible by 4. 2,348 is divisible by 4 as the numbers in the tens and units place are completely divisible by 4. 3,450 and 1,234 are not divisible by 4 as the last two digits are not 00.

Test for divisibility by 6If the sum of the digits of a number is exactly divisible by 2 and 3, then it is exactly divisible by 6, for example 24 is divisible by 2 as well as 3, therefore, 24 is divisible by 6. 38 is divisible by 2 but not by 3 so it is not divisible by 6. 45 is divisible by 3 but not by 2 so it is not divisible by 6.

Test for divisibility by 9Add all the digits of the number. If the sum is a number that is divisible by 9 then that number is divisible by 9. For example 693 is divisible by 9 because the sum of the digits 6+9+3 is 18 which is divisible by 9. 1,234 is not divisible by 9 because the sum of its digits 1+2+3+4 is 10 which is not divisible by 9.

5

6

Test for divisibility by 5If the digit in the units place of a number is 5 or 0 then the number is divisible by 5, for example 25 and 250 are divisible by 5. 234 is not divisible by 5.

Test for divisibility by 10Any number which has a 0 in the unit’s place, for example 380, 230, 400, is divisible by 10. 2,345 is not divisible by 10.

Factors Pages 14-21

Objectives:

To know what factors are and how to find factors of numbers.To know what prime factors are.

Pages 14-16Draw eight circles on the board. Ask the students to arrange them in different groups so that nothing is left.On the board draw,groups of one: O,O,O,O,O,O,O,O (1 x 8) eight groupsgroups of two: OO,OO,OO,OO (2 x 4) four groupsgroups of four: OOOO,OOOO (4 x 2) two groupsgroup of eight: OOOOOOOO (8 x 1) one groupExplain that every time the circles were arranged, there were none left. So, 1, 2, 4 and 8 are called ‘factors’ of 8. We can find the factors of numbers by arranging them in groups. Explain with more examples and show that 1 is a factor of every number. It is interesting to note that every number is a factor of itself.

7

8

(1 x 8) O,O,O,O,O,O,O,O

(2 x 4) OO,OO,OO,OO

(4 x 2) OOOO,OOOO

(8 x 1) OOOOOOOO

Prime Factors Pages 17-21Write 12 on the board. Find two factors of 12, which are 3 x 4. 4 again has two factors 2 x 2. So the factors of 12 are 3 x 2 x 2. Since two and three are prime numbers, therefore, they are called the ‘prime factors’ of 12. Explain that we cannot make any more factors of the prime factors. The method by which we make prime factors is called ‘prime factorization’.Ask the students to make factor trees as shown in the book. Explain that we can make prime factors in many ways, for example the factors of 12 can be 2 x 6, where the factors of 6 are 2 x 3. So the prime factors of 12 are 2 x 2 x 3. This shows that although we can make factors in different ways the prime factors will always be the same.

Multiples Pages 22-23

Objectives:

To know what multiples are and how to find multiples.To know what common multiples are.

Ask the students to revise their multiplication tables before they come to class.

Write the multiplication table of 6 on the board. Circle the products, which are: 6,12,18, 24, 30, 36,…. Explain that when you multiply 1, 2, 3, 4, 5, 6 by 6 you get the ‘multiples’ of 6, i.e. 6,12,18, 24, 30…. Write a few more examples on the board to explain the concept of ‘multiples’.

Common multiples Page 24Write the multiples of 6 and 9 on the board. Draw a circle around the common multiples 18 and 36. Explain that 18 and 36 are the multiples that are common to both 6 and 9. These numbers are called the ‘common multiples’ of 6 and 9. Teach the students to pick out the common multiples of two or more numbers by giving more examples on the board.

9

10

Prime and Composite Numbers Pages 25-27Write 5 and 12 on the board and make their factors. 5 has two factors 1 and 5. 12 has six factors, 1, 2, 3, 4, 6, 12. Explain that numbers that have only two factors are called ‘prime numbers’. 2, 3, 5, 7, 11, 13, 17, 19, etc., are prime numbers. Numbers that have more than two factors are called ‘composite numbers’. 4, 6, 8, 9, 10, 12, 14, 15, 16, etc., are composite numbers. An easy way of finding whether a number is prime or composite is to use the tests of divisibility. For example, to find whether 35 is a prime number, see which numbers are its divisors. It is divisible by 1, 5 and 7 so it cannot be a prime number as it has more than two factors. Therefore, it is a composite number. To find whether 19 is a prime number, see if it is divisible by 2. It is not divisible by 2 because it is an odd number. Is it divisible by 3? No, because 1 + 9 =10. So it cannot have more than 2 factors, i.e. 1 and 19, therefore, it is a prime number.Discuss the ‘true and false statements’ exercise on page 27 of the book before the students proceed to do it.

Least Common Multiple (LCM) Pages 28-33Write the multiples of 3 and 5 on the board. Draw a circle around the common multiples, 15 and 30. The ‘least common multiple’ of 3 and 5 is 15, because 15 is less than 30. Explain with more examples before the students attempt the exercise on page 29 of the book.

11

12

LCM of two numbers Pages 30-31Explain that we can find the LCM of two or more numbers by prime factorization as shown in examples 1 and 2 on pages 30 and 31 of the book. To further elaborate, demonstrate how to find the LCM of 20 and 30. Explain that first we find the prime factors as follows,

2 20 2 302 10 3 155 5 5 5

1 1

Hence, the prime factors of 20 are 2 x 2 x 5 and the prime factors of 30 are 2 x 3 x 5. Here, 2 and 5 are the common factors and 2 and 3 are the uncommon factors. By multiplying the common and uncommon factors we get, 2 x 5 x 2 x 3 = 60. Therefore, LCM = 60.We can also find the LCM by dividing 20 and 30 together by their common multiples,

2 20, 302 10, 155 5, 153 1, 3

1, 1

then by multiplying all the factors, 2 x 2 x 5 x 3, we get the LCM which is 60.

LCM of three numbers Pages 32-33The LCM of three numbers can also be found by the same method of factorization as well as division. Explain using several examples, including examples one and two on pages 32 and 33 of the book before the students proceed to do the exercises on page 33.

13

14

2 20 2 302 10 3 155 5 5 5

1 1

2 20, 302 10, 155 5, 153 1, 3

1, 1

Explain that the least common multiple is the smallest number that is a multiple of two or more numbers.

Highest Common Factor (HCF) Pages 34-40Write the factors of 12 and 18 on the board. Factors of 12 are 1, 2, 3, 4, 6, 12. Factors of 18 are 1, 2, 3, 6, 9, 18. The common factors for 12 and 18 are 1, 2, 3 and 6. Of these 6 is the highest common factor, therefore, HCF = 6.Explain that the highest common factor means the highest or the biggest number that can divide two or more numbers exactly.

HCF of two numbers Pages 35-37HCF of two or more numbers can also be found by the prime factorization method as for LCM. For example, to find the HCF of 30 and 45

2 30 3 453 15 3 155 5 5 5

1 1

Therefore, the prime factors of 30 are 2 x 3 x 5 and the prime factors of 45 are 3 x 3 x 5. In this case, 5 and 3 are the common factors. Therefore, HCF=5 x 3=15.Explain the method of finding HCF by prime factorization using several examples, including the example on page 35 of the book, before the students are asked to do the exercise on page 37.

15

16

2 30 3 453 15 3 155 5 5 5

1 1

Sometimes, when we want to find the HCF of very large numbers, we can use the ‘long division method’. To elaborate further, demonstrate how to find the HCF of 334 and 314.Tell the students to first divide the greater number by the smaller number.

334-314

20

314 1

(remainder)

Make the remainder the divisor of the first divisor.

314 - 20 114- 100

14

20 15

(remainder)

Make the remainder the divisor of the second divisor.

20-14

6

14 1

(remainder)

Make the remainder the divisor of the third divisor.

14-12

2

6 2

Make the remainder the divisor of the fourth divisor.

6- 6

0

2 3

17

(remainder)

18

334-314

20

314 1

20-14

6

14 1

14-12

2

6 2

6-60

2 3

314 - 20 114- 100

14

20 15

The last divisor is 2, so the HCF of 334 and 314 is 2.

Explain the method by several examples, including examples one and two on pages 35-37 of the book, before the students are asked to solve the exercise on page 37.

HCF of three numbers Pages 38-40The HCF of three or more numbers can also be found by applying the methods of prime factorization or long division. To find the HCF of three numbers by long division, first we find the HCF of two numbers by long division and then we divide the third number by the HCF of the first two numbers, for example, to find the HCF of 128,112 and 80 as explained in example one on page 35 of the book, first find the HCF of 128 and 112 by long division. The HCF is 16. Then divide 80 by 16, which gives the quotient 5 and there is no remainder left. Since 16 is the last divisor, therefore, 16 is the HCF of 128, 112 and 80.

Unit II: Algebra Pages 41-62

Objectives:

To know how to reduce fractions to their simplest form, multiply common fractions by whole numbers, solve word problems involving fractions.To know how to change common fractions into decimal fractions and decimal fractions into their simplest common fraction.To know how to multiply decimal fractions by whole numbers and by decimal fractions.To know how to divide decimal fractions by whole numbers and by decimal fractions.To know how to solve word problems involving decimal fractions.

19

20

Common Fractions Pages 41-48Revision Page 41Revise the concept of common fractions learnt in the previous class from the review exercises on page 41.

Changing fractions to their simplest form Pages 42-44Explain that we have learnt how to write equivalent fractions in the previous class.

Now we are going to learn how to reduce fractions to their simplest form

by the ‘cancellation method’.

First, we find a common factor for the numerator and the denominator of the

fraction. Then we divide them both by the common factor by crossing them

out or by cancelling them. For example, to reduce 2040 , the common factor

of 20 and 40 is 5, therefore, divide 2040 by 5 which gives us 4

8 . The common

factor of 4 and 8 is 2, dividing 48 by 2 we get 2

4. The common factor of 2

and 4 is 2, dividing 24 by 2 we get 1

2. Hence, the simplest form of 2040 is 1

2.

Multiplying common fractions by whole numbers Pages 44-45

To multiply common fractions by whole numbers, for example, 35 x 3, we multiply the numerator of the fraction by the whole number,

i.e. 35

x 3 = 95

.

If the answer is an improper fraction we have to change it into a

compound fraction. Therefore, in this case 95

is changed to 1 45 .

Multiplying compound fractions by whole numbers Pages 46-47

To multiply a compound fraction by a whole number, first we have to

change the compound fraction into an improper fraction and then follow

the same steps as in the previous example.

21

22

2040

2040

2040

48

48

24

12

24

12

35

x 335

x 3 = 95

45

95

For example, for 3 23 x 5, first change the compound fraction into an

improper fraction, i.e. 3 23 = 11

3 . Then, multiply the numerator by the

whole number 5 x 113 = 55

3 . Since 553 is an improper fraction, therefore,

change it to a compound fraction, i.e. 18 13 .

Another method is to write the compound fraction in its expanded form

and then multiply by the whole number. For example:

3 x 435

= 3 (4 + 35)

= (3 x 4) + 3 (35)

= 12 + 95

= 12 + 145

= 13 45

Note: Word problems can be solved after working out which operation has to be used.

Decimal Fractions Pages 49-62Changing common fractions into decimal fractions.

Revision Pages 49-50Revise the concept of decimal fractions, learnt in the previous class, from the review exercises on pages 49-50.

Changing common fractions with different denominators into decimal fractions Pages 50-52

To change common fractions with different denominators into decimal

fractions, we have to change the common fraction into an equivalent

fraction so that the denominator is a multiple of 10, for example, to

change 15 into a decimal fraction, change the denominator 5 into a

multiple of 10, by multiplying it by 20.

23

24

3 x 4 35

= 3 (4 + 35 )

= (3 x 4) + 3 ( 35 )

= 12 + 95

= 12 + 1 45

= 13 45

3 23 x 5

5 x 113 = 55

33 =2

3113

18 13

553

15

We also have to multiply the numerator by 20 so the fraction becomes1 x 205 x 20 = 20

100 = 0.2

Note: A compound fraction can also be changed to a decimal fraction in

the same way, for example to change 3 34 to a decimal fraction:

3 + 34 = 3 + 3 x 25

4 x 25 = 3 + 75100 = 3 + 0.75 = 3.75.

Explain the method of changing common fractions into decimal fractions

by several examples.

Changing common fractions into decimal fractions by division Pages 52-54

Explain that some fractions which are in their simplest form and

have denominators that cannot be changed into multiples of 10, can

be changed into decimal fractions by dividing the numerator by the

denominator as explained in the following example.

To change 23 into a decimal fraction, write the fraction as for division )3 2.

As 2 cannot be divided by 3, we add a decimal point in the quotient above 2 and write a zero in the units column of the dividend.

3 (add a zero)

(add a zero)

(add a zero)

(remainder)

)0.666

20-1820

-1820

-182

Explain that in the above example, each time we have 2 as a remainder, we make it a 20 by adding a zero. This process can go on and on, but in the above case, we divide three times to get the nearest possible decimal fraction.

25

26

3)0.666

20-1820

-1820

-182

3 + 34 = 3 + 3 x 25

4 x 25 = 3 + 75100 = 3 + 0.75 = 3. 75

1 x 205 x 20 = 20

100 = 0.2

3 34

23

Changing decimal fractions to their simplest common fractionPage 55

To reduce a decimal fraction to its simplest form, we have to change

it to a common fraction and then divide it. For example, to change 0.4

into its simplest form, first change it into a common fraction, ie. 0.4 = 410

. Cancelling by the common factor 2, we can reduce 410 to its simplest

form, which is 25 .

Multiplying decimal fractions by whole numbers Pages 56-57To multiply a decimal fraction by a whole number, we multiply in the same way as we multiply two whole numbers. But, we must remember to put the decimal point in its correct place. For example, to multiply 4.7 by 9, write the numbers in vertical form 4.7 (one decimal place ) x 9

42.3 (one decimal place)

We put the decimal point in the answer by counting the number of decimal places in the decimal fraction.

Multiplying two decimal fractions Pages 57-58We multiply two decimal fractions in the same way as we multiply two whole numbers. But we count the number of decimal places in both the fractions and then add them to put the decimal point that many number of places in the answer counting from the right. For example, to multiply 9.7 by 8.6, write the numbers in vertical form: 9.7 (one decimal place) x 8.6 (one decimal place) 582 7760 83.42 (two decimal places from right)

27

28

0.4 = 410

4.7

x 942.3

9.7x 8.6

5827760

83.42

410

Dividing decimal fractions by whole numbers Page 59When dividing a decimal fraction by a whole number we divide in the same way as for whole numbers. The decimal point is placed directly above the decimal point of the decimal fraction in the dividend. For example, to divide 8.92 by 4

) 2.23

4 8.92- 8

9 8 12- 12 0

Dividing decimal fractions by decimal fractions Pages 60-62To divide decimal fractions, change the decimal fraction into a whole number by shifting the decimal point or by multiplying it by a power of 10.For example, to divide 7.5 by 2.5, shift the decimal one place to the right so that 2.5 becomes 25 and 7.5 becomes 75. We can now divide it in the usual way as for whole numbers, i.e.

)3

25 75-75

0

We must remember that the decimal in the divisor and the dividend must be shifted an equal number of places, for example, to divide 3.55 by 0.5, 0.5 can be made into a whole number by shifting the decimal point one place to the right.

29

30

)3

25 75-75

0

) 2.23

4 8.92- 8

9 8 12- 12 0

Therefore, we can only shift the decimal point of 3.55 one place to the right. So 3.55 becomes 35.5. Now we can divide in the way we have learnt to divide a decimal fraction by a whole number.

Note: Word problems should be solved by first determining which operation has to be performed.

Unit III: Unitary Method Pages 63-67

Objectives:

To know what the unitary method is and to apply the operations of multiplication and division to solve problems.

Ask the students a simple question involving the unitary method. For example, if one ball costs Rs 5, how much will 10 balls cost?Write the statement:1 ball costs Rs 510 balls cost Rs 5 x10 = Rs 50Explain that unitary means ‘one’. It is a method which we apply when we want to find the value of more than one thing when we are given the value of only one. For finding the values we multiply.Write, if 10 balls cost Rs 40, how much will 1 ball cost?Write the statement: 10 balls cost Rs 40 1 ball costs Rs 40 ÷ 10 = Rs 4Explain that when we want to find the value of one quantity when we are given the value of more things, we have to divide.Explain the remaining examples on pages 63-66 of the book. Discuss the word problems with the students before they proceed to solve them.

31

32

Unit IV: Average Pages 68-70

Objectives:

To know the meaning of ‘average’ and be able to find the average.

Draw five boxes on the board. Draw 2 circles in the first box, 1 in the second, 3 in the third, 5 in the fourth and 4 in the fifth.Ask the students to count the total number of balls in the boxes, which is 15.Now ask them to put an equal number of balls in the five boxes, which will be 3 for each box.Explain that there is an average of 3 balls per box.To be able to find an average we use an easy method.First, we add up all the given numbers, then we divide their sum by the total number of given quantities.In the above example, we first add 2 + 1 + 3 + 5 + 4 = 15. We, then, divide 15 balls by the total number of boxes, which is 5, i.e. 15 ÷ 5 = 3. Hence, 3 is the average number of balls per box.Discuss the word problems with the students before they proceed to solve them.

Unit V: Geometry Pages 71-103

Objectives:

To know what rays, straight lines, parallel lines, horizontal and vertical lines are.To know what an angle, a right angle, a right-angled triangle is.To know what perimeter is and how to find the perimeter of a rectangle and a quadrilateral.To know what area is and how to find the area of a rectangle and a square.To know what volume is and how to find the volume of a cube and cuboid.

33

34

Lines Pages 71-76Draw a plane on the board. Draw some dots on its boundary. Draw lines in different directions from the dots.Explain that the flat figure is called a ‘plane’ and the dots are called ‘points’. When the points are joined, they form lines. Lines can extend in all directions from one point to the other points on the plane. Also, explain that a line has no ‘end points’. It can go on forever in any direction.

Ray Pages 71-72Draw a ray with points ‘A’ and ‘B’ on the board with an arrowhead at one end only. Explain that a ‘ray’ has only one end point. The arrow-head tells us the direction in which it is extending. It is indicated by a short line with an arrowhead at one end only, for example it can be written as AB or BA.orAB or BA..Draw a point ‘A’ on the board and draw rays extending from it in different directions as shown on page 72 of the book. Mark these rays as B, C, D, E, F. Explain that all these rays have a common end point, but they are extending in different directions.

Straight line Page 72Draw a line on the board. Mark its ends with arrowheads. Mark two points ‘A’ and ‘B’ on it. Explain that the two points ‘A’ and ‘B’ are joined together by a line. The arrowheads tell us that the line has no end points.

35

36

AB or BA.AB or BA.

In fact, a ‘straight line’ is a line that extends in opposite directions, or we can say that two rays are joined to make a straight line. We can indicate a straight line as a line with arrowheads at both ends, e.g. in the given example it can be written as AB or BA.AB or BA.AB or BA

Parallel lines Page 73Draw two parallel lines on the board, and two lines that are at an angle from each other. Extend both pairs of lines. Ask the students what they notice. Explain that the first two lines will never meet no matter how far they are extended. The other pair of lines meet each other after a while because the distance between them is not the same throughout.The first pair of lines are called ‘parallel lines’. The distance between these lines always remains the same. The railway line is an example of parallel lines. All the lines on the pages of your exercise book are parallel lines.

Vertical and horizontal lines Pages 74-76Draw an upright ladder standing against a brick wall on the board. Explain that the poles of the ladder are standing vertically against the wall. These are called ‘vertical lines’. The rungs of the ladder are in the same direction as the ground. In fact, they are parallel to the ground. Such lines are called ‘horizontal lines’. Point to the door and the window of the classroom. Indicate the horizontal and vertical lines. Ask the students to look for other horizontal and vertical lines in the room. Draw the four directions on the board and show the students that a vertical line always lies in the north-south direction whereas a horizontal line lies in the east-west direction.

37

38

AB or BA.AB or BA.AB BA

Angles Pages 77-80Angle Page 77Draw two rays starting from a common end point to make an angle. Explain that two rays starting from a common end point but extending in different directions make an ‘angle’. Draw different angles on the board.

Right Angle Page 78Draw a point ‘A’ on the board. From ‘A’ draw a horizontal and a vertical ray. Explain that the angle, which is formed by joining the horizontal and vertical ray, is called a ‘right angle’. Show the students examples of right angles in the classroom.

Right-angled Triangle Pages 79-80Draw a triangle on the board. Ask the students what it is. Mark the angles of the triangle. Now draw a triangle with a right angle as one of its angles. Explain that any triangle which has a right angle as one of its angles is called a ‘right-angled triangle’.

Perimeter Pages 81-86We have already learnt in the previous class what ‘perimeter’ means. Now we are going to learn how to find the perimeter of plane figures by actual measurement.Draw a quadrilateral with a ruler on the board. Write the measurements of its sides in centimetres (cms). Now, to find the perimeter, we will add up the measurements of all its sides.

39

40

Perimeter of a rectangle Pages 82-84To find the perimeter of a rectangle we must know that two of its opposite sides are equal. The longer side is called its ‘length’ and the wider side is called its ‘breadth’. To find the perimeter of a rectangle, we can either add up the measurements of all its sides, or we can add the measurements of its length and breadth and then multiply the sum by 2 as it has two equal sides of lengths and two equal sides of breadths, i.e.Perimeter = (length + breadth) 2, orP=(l+b) 2

Perimeter of a square Pages 83-84Since a square has four equal sides, we can find its perimeter either by adding up the measurements of its sides or by simply multiplying the measurement of one side by 4, i.e. Perimeter = side + side + side + side, or P = side x 4

Perimeter of a quadrilateral Pages 85-86 To find the perimeter of a quadrilateral with all its sides different in measurement, we have to add the measurements of its four sides.

Area Pages 87-97Draw the outline of a book or a pencil box on the board. Explain that the outline of the object is its perimeter. The region inside the perimeter is called ‘area’. Ask the students to draw the area of their books, pencil boxes, etc.

41

42

Take a squared paper and draw the outline of a box on it. Count the number of squares the box covers. Then write on the board that the area of the box is in squares.Draw a square on the board with a ruler and write its measurements in cms. Explain that we find the area of the square in square units, for example, square centimetres or square metres.

Area of a rectangle Pages 92-93We can find the area of a rectangle by multiplying the measurement of its length by the measurement of its breadth. The answer will be in square units.

Area of a square Pages 93-95We can find the area of a square by multiplying a side by a side, because all its sides are equal.

Volume Pages 98-103Volume of a cube Pages 98-99Volume of a cuboid Pages 99-100We have learnt about the shape of a cube in the previous class and we know that a cube has equal length, breadth and height. A cuboid on the other hand does not have equal length, breadth and height.

43

44

Volume is the total space that a three-dimensional figure, i.e. a figure with length, breadth and height, occupies. We can calculate the volume of a figure by multiplying length, breadth and height. Volume = length x breadth x height V = l x b x hVolume is measured in cubic units like cubic cm, or cubic m, etc.

Unit VI: Graphs Pages 104-110

Objectives:

To know how to read line graphs and pie charts.

We have learnt how to read line graphs in the previous class. Graphs are a quick way of finding information about different things.The unchanging quantities are written along the horizontal axis, and the changing quantities are written along the vertical axis.

Line Graphs Pages 104-108Draw the line graph from the example in the book on page 104 on the board. Teach the students how to find information from the graph. Explain that when the line graph goes up it shows that the quantity is increasing and when it goes down the quantity is decreasing. When the line becomes horizontal the values are the same.

45

46

V = l x b x h

Pie Charts Pages 109-110Draw a circle on the board and divide it into four equal parts. Explain that each part represents a quarter, i.e. each part represents one-fourth part of a whole. We can divide a circle into as many parts as we like to represent different fractions.A pie chart is used to show how many parts of a whole are used to represent different things. For example, if a man spends 1

4 part of his income on food, 1

4 on clothes, 14 on his house, and saves 1

4 , we can represent his expenditure in the form of a pie chart. Draw a circle and divide it into four equal parts. Write food in one part, clothes in the second, house in the third and savings in the fourth. When we read the pie chart we know exactly how much of his income he spends on each item.Draw various pie charts to explain how to read them.

47

48

14

14

14

14

14

Answers to Exercises in Mathematics Book 5

Unit I: Numbers

Pages 4-51. I, V, X, XX, L, C, D, M2. II, IV, VI, VIII, XI, XV, XVIII, XIX3. 3, 5, 7, 9, 14, 16, 18, 20, 50, 1000, 500, 1004. F F T F T T F T T F T F F F5. II, III, V, VI, VIII, IX, XII, XIV, XVI, XVII, XVIII, X6. I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII7. I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XIX, XX

Page 61, 2, 41, 2, 4, 81, 2, 5, 101, 2, 3, 4, 6, 1211, 51, 3, 9

Page 71. 2, 4, 6, 8, 10, 12, 14, 16, 182. 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 983. 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 234. 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175,

177, 179, 181, 183, 185, 187, 189, 199, 191, 193, 195, 197, 199

49

5. ✔, ✔, —, ✔, —, —, ✔, ✔, —, —, ✔, — —, —, ✔, —, ✔, ✔, —, —, ✔, ✔, —, ✔

—, —, ✔, —, —, —, —, ✔, —, —, —, ✔, ✔, ✔, —, ✔, ✔, ✔, ✔, —, ✔, ✔, ✔, —,

Page 81. 100, 124, 426, 1002, 5002. 115, 119, 123, 1947, 12315, 14417

Page 91. Numbers divisible by 3. 801, 336, 2304, 1008, 3612, 3540.

2. Numbers greater than 100 and less than 150 divisible by 3. 102, 105, 108, 111, 117, 120, 123, 126, 129, 132, 135, 138, 141,

144, 147

Page 101. Numbers divisible by 4. 348, 412, 1200, 555, 3112, 128, 5016, 14392, 24900, 1204

2. Number below 25 divisible by 4. 4, 8, 12, 16, 20, 24

Page 101. Numbers divisible by 6. 72, 108, 144, 216, 300

Page 11 ✔, —, ✔, —, ✔, —, ✔, —, ✔, ✔, ✔ —, ✔, —, ✔, —, ✔, —, ✔, —, —, —

Pages 11-121. Numbers divisible by 9. 99, 693, 846, 1800, 9720, 486, 4509

2. Numbers not divisible by 9. 541, 273, 1378, 8269, 7952, 4609, 23564, 45260

50

Page 131. Numbers below 25 divisible by 5. 5, 10, 15, 20

2. Numbers above 10 below 100 divisible by 10. 20, 30, 40, 50, 60, 70, 80, 90

3. ✔, ✔, ✔, ✔, ✔, ✔, ✔, ✔, ✔ ✔, —, ✔, ✔, ✔, —, ✔, —, ✔

✔, ✔, ✔, ✔, ✔, ✔, ✔, ✔, ✔ ✔, ✔, ✔, ✔, ✔, ✔, ✔, ✔, ✔

Page 161. 1, 3, 9 1, 2, 3, 4, 6, 12 1, 2, 4, 8, 16 1, 2, 3, 6, 9, 18 1, 2, 3, 4, 6, 8, 12, 24 1, 2, 13, 26 1, 2, 4, 5, 8, 10, 20, 40 1, 3, 5, 9, 15, 45 1, 2, 5, 10, 25, 50 1, 3, 5, 15, 25, 75

2. 1, 3, 5, 15 3. T, F, T, T, T, F, F, T, T, F 1, 2, 4, 6, 8 1, 2, 4, 5, 10, 20 1, 5, 25 1, 2, 3, 4, 6, 9, 12, 18, 36

Pages 18-191 (a) 2 (b) 2 x 2 x 3 x 3

(c) 5 x 10

2 x 55 x 2 x 5

(d) 2 x 3 2 x 4

6 x 8

48

2 x 22 x 3 x 2 2 2

51

Page 20

(e)

9 x 2

18

3 x 33 x 3 x 2 (f)

7 x 4

28

2 x 27 x 2 x 2

(g) 2 x 3 2 x 5

6 x 10

60

2 x 3 x 2 x 5

Page 21

(h)

3 x 25

75

5 x 53 x 5 x 5 (i)

7 x 10

70

2 x 57 x 2 x 5

Page 212. 2 x 5, 2 x 2 x 2 x 2 x 2 x 2 3 x 2 x 2 x 7 2 x 2 x 5 x 5 5 x 5 x 5 2 x 2 x 2 x 3 x 3 2 x 2 x 3 x 3 x 3 2 x 2 x 2 x 2 x 3 x 3 2 x 2 x 2 x 2 x 2 x 3 2 x 2 x 5 x 7

Page 22 24, 32, 40 9, 18, 27, 36, 45 5, 10, 15, 20, 25 4, 8, 12, 16, 20 7, 14, 21, 28, 35

52

Page 231. 6, 12, 18, 24, 30, 36, 42, 48 7, 14, 21, 28, 35, 42, 49, 56 9, 18, 27, 26, 45, 54, 63, 72 6, 12, 18, 24, 30, 36, 42, 48

8, 16, 24, 32, 40, 48, 56, 64 5, 10, 15, 20, 25, 30, 35, 40 9, 18, 27, 36, 45, 54, 63, 72 7, 14, 21, 28, 35, 42, 49, 56

4, 8, 12, 16, 20, 24, 28, 32 6, 12, 18, 24, 30, 36, 42, 48 8, 16, 24, 32, 40, 48, 56, 64 6, 12, 18, 24, 30, 36, 42, 48

9, 18, 27, 36, 45, 54, 63, 72 10, 20, 30, 40, 50, 60, 70, 80 3, 6, 9, 12, 15, 18, 21, 24 8, 16, 24, 32, 40, 48, 56, 64

5, 10, 15, 20, 25, 30, 35, 40 10, 20, 30, 40, 50, 60, 70, 80

2. 25, 40, 65, 100, 135, 180

3. 14, 42, 63, 70, 105, 56, 28, 112

Page 24 Multiples Common Multiples 3, 6, 12, 15, 18 5, 10, 15, 20, 25 15

2, 4, 6, 8, 10 6, 12, 18, 24, 30 6

5, 10, 15, 20, 25 10, 20, 30, 40, 50 10, 20

2, 4, 6, 8, 10 8, 16, 24, 32, 40 8

4, 8, 12, 16, 20 6, 12, 18, 24, 30 12

Pages 25-271. 19, 37, 47 3,5, 7, 11, 13, 17, 19, 23 29, 31, 37 11, 13, 29 11, 29, 43 53, 59

53

2. 2, 4, 6, 8, 10, 12, 15 5, 10, 15, 20, 21, 25, 27, 30

51, 52, 54, 55, 56, 57, 58 81, 82, 84, 85, 87, 92, 94

102, 112, 117, 132 9, 154, 255, 369, 680

3. P, P, —, P, —, P, —, P, —, —, —, P, —, — —, —, C, —, C, —, C, —C, C, C, —, C, C, C

P, —, —, —, —, —, P, P, —, —, — —, C, C, C, C, C, —, —, C, C, C, C, C, C, C

4. F, F, T, T, T, T, T, F

Page 29 Multiples CM LCM 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 6, 12, 18 6

3, 6, 9, 12, 15, 18, 21, 24, 27, 30 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 12, 24 12

2, 4, 6, 8, 10, 12, 14, 16, 18, 20 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 10, 20 10

6, 12, 18, 24, 30, 36, 42, 48, 54, 60 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 18, 36, 54 18

5, 10, 15, 20, 25, 30, 35, 40, 45, 50 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 10, 20, 30 40, 50 10

5, 10, 15, 20, 25, 30, 35, 40, 45, 50 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 40 40

54

Page 31 60, 160, 20 96, 225, 16 48, 90, 72

Page 331. 12, 24, 36 2. 8, 30, 175 12, 20, 24 32, 24, 150 40, 64, 28 24, 60, 48

Page 341. Factor CF HCF 1, 2, 4, 8 1, 2, 3, 4, 6, 12 1, 2, 4 4

1, 2, 5, 10 1, 2, 4, 5, 10, 20 1, 2, 5, 10 10 1, 3, 5, 15 1, 5, 25 1, 5 5

1, 2, 4, 5, 10, 20 1, 2, 3, 5, 10, 15, 30 1, 5, 10 10

Page 37 Page 381. 15, 12, 12 2. 18, 2, 16 14, 4, 10 14, 12, 12 22, 96, 92 6, 24, 10

Page 4019, 23, 44, 2, 25, 9, 17

55

Unit II: Algebra

Page 41 Revision1. 20, 49, 7 8, 16, 52. 4 14 , 8 34, 10 45, 8 46, 3 18, 5 37

3. 166 , 26

12 , 184 , 76

5 , 726 , 62

15

Pages 42-43

1. 23263845

15253459

Pages 44

59

35

58

14

7157

107

183

1014

12

49

2027

518

15

34

Page 45

4 12 , 2, 2

4

1 45 , 3 1

3 , 9

4, 15, 7

Page 47 13 4

5 , 33 34 , 39

5 57 , 22 3

3 , 15 35

11, 34 16 , 23 4

5

56

Page 48 (Word problems)

1. 12 12 minutes 2. 36 marbles

3. Rs 250 4. 75 kg

5 Rs 62

Page 49 (Revision)1. 0.5, 0.7, 0.41, 0.130, 0.245, 0.69, 0.47, 0.138

2. 910 ,

18100 ,

7100 ,

23100 ,

10100 ,

5100 ,

310 ,

131000 ,

91000

213

1000 , 21

100

Page 503.1 10.3 16.114.03 6.7 12.1118.99 23.009 36.028

Page 511. 0.5, 0.4, 0.1, 0.6, 0.82. 0.75, 0.15, 0.50, 0.30, 0.60, 0.80, 0.50, 0.34

Page 523.75, 2.6, 1.28.7, 3.9, 2.354.24, 5.04, 6.25

Page 540.888, 0.666, 0.875, 0.375, 0.916,0.25, 0.166, 0.833, 0.571, 0.777

Page 55

320

25

14

1950

34

125

425

1150

49200

35000

57

Pages 56-57 15.6 35.2 5.4 6.9 57.72 21.14 77.85 3.92 111 6.402 137.656 457.002

Page 581. 6.465 168.72 8.994 7.5375 12.01674 4.5552

2. 2.58 4.608 13.6332 7.488 131.625 18.6 11.75757 0.475625 24.10758

Page 592.23 14.2 2.10.31 2.9 4.390.93 5.98 8.54

Page 618 21.2 237.1 182 6.32.7 2.3 2.72.3 0.34 0.29

Page 61-62 (Word problems)1. 7.5 metres 2. 1.49 metres3. Rs 4.30 4. Rs 48.215. 5 metres

Page 62 (More word problems)1. 7 metres 2. 37.5 cm3. 0.575 kg 4. 3.95 m5. Rs 8.1 6. 70 cm

58

Unit III: Unitary method

Pages 64-65 (Word problems)1. Rs 42 2. Rs 19443. 625 match sticks 4. 4.2 km5. Rs 20 6. 15 stamps7. 172 soldiers 8. 15 stamps9. Rs 5820 10. 108 metres.

Pages 66-67 (Word problems)1. 24 kg 2. Rs 256 3. 128 litres4. 10 kg 5. 75 metres 6. 700 eggs7. 180 books 8. Rs 189 9. 1311 km10. 70 students

Unit IV: Average

Page 6924 ÷ 4 659 ÷ 4 14.75180 ÷ 5 364.5 ÷ 3 1.5 m11.1 ÷ 3 3.7 kg14.5 ÷ 4 3.625 l

Pages 69-70 (Word problems)1. 540 km 2. 71.25 kg3. 31.75 runs 4. 73.33 marks5. 81.66 birds 6. 119.8 cm7. 6.57 kg 8. 52.5 cm

59

Unit V: Geometry

Page 75Straight lines AB, CD

Straight line XYray AB. parallel lines AB, XYray CD. straight line XY

Page 80 Page 83X X ✔ 21 cm✔ X X 31 cm✔ ✔ ✔ 16 cm✔ ✔ ✔ 17 cm✔ ✔ ✔ 25.2 cm

Page 84 Page 85 Page 8634.4 cm 10 cm 2. 15 cm31.2 cm 12.5 cm 32 cm59.2 cm 14.5 cm 18 cm38 cm 39 cm 10 cm7.2 cm 28 cm 16 cm 19 cm

Page 90(a) 4 (b) 9 (c) 6 (d) 12(e) 6 (f) 7 (g) 16 (h) 8

Page 914 square centimetres6 square centimetres6 square centimetres

60

Page 9326 sq cm20 sq cm73.5 sq cm7.14 sq cm32.3 sq cm

Page 94225 sq cm4.41 sq cm110.25 sq cm31.36 sq cm302.76 sq cm

Page 951. Area Perimeter 54 sq cm 30 cm 106 sq cm 110 cm 3.77 sq cm 8.4 cm 13.05 sq cm 14.8 cm 136.53 cq cm 46.8 cm

2. Area Perimeter 100 sq cm 40 cm 5.76 sq cm 9.6 cm 65.61 sq cm 32.4 cm 22.09 sq cm 18.8 cm 320.41 sq cm 71.6 cm

61

Pages 96-971. 250 sq cm 70 m2. 1860 m3. 44 cm

Page 97 (word problems)4. 15 sq m 5. 12 m6 200 cm 7. 900 sq cm, 120 m8. 625 sq cm, 100 cm. Yes

Pages 101-1021. 8 cubic cm 64 cubic cm 30 cubic cm 30 cubic cm 27 cubic cm 54.872 cubic cm 27 cubic cm 30 cubic cm 15.625 cubic cm

Page 1032. 432 cubic cm 112 cubic cm 102.486 cubic cm 378 cubic cm 750 cubic cm

Word problems1. 1,120 cubic cm2. 1,200 cubic cm3. 60 cubic m4. 3,600 cubic cm5. It is a cube. Volume is 1 cubic metre.

62

Unit VI: Graphs

Page 105-1081. Sunday, Saturday, Monday, Thursday, Friday2. 5 hrs, 3 hrs, 2 hrs, 1 hrs3. Matches, Science, English, English and Islamiat4. 5, 10, 15, 25, 555. Bina, Lubna, Aimen and Bela

Pages 109-110

1 12 2 1

2 3

8 1

4 1

8 1

4

3. 14 4. 1

5 1

8 4. 15

12 4. 1

5 1

8 4. 15

63