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2 Section 1: Additional material

Section 1: Additional material

Section 1 provides you with optional chapters that you can use with your class.

Preliminary chapters are intended for use as revision and to reinforce foundation knowledge that is needed in the year. You may wish to assign work from this chapter as remedial activities for students who struggle with the main course work.

Enrichment chapters are optional chapters that extend the scope of the work done in the year. If you have time at the end of the year, you can work through this additional chapter with your students. You may wish to assign work from this chapter to stronger students.

3Section 1: Additional materialSection 1: Additional material 3

Review of previous courseworkPreliminary chapter

Objectives

By the end of the chapter, you will be able to recall, from Book 1 of New General Mathematics, the facts and methods that relate to these themes:• number and numeration• basic operations• algebraic processes• mensuration and geometry• everyday statistics.

Teaching and learning materialsTeacher: • addition and multiplication wall charts • 1–100 number square • metre rule • measuring tape • chalk board instruments (ruler, set-square,

compasses) • solid shapes.

Students: Mathematics set.

To make the best use of Book 2 of New General Mathematics, you should be familiar with the contents of Book 1. This chapter contains themes and topics from Book 1 that are necessary to understand Book 2.

P–1 Number and numeration

hundreds tens units decimal point tenths hundredths thousandths

3 7 6 . 2 9 5

Figure P1

1 We usually write numbers in the decimal place value system (Figure P1). The symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are called digits.

2 We use the words thousand, million, billion, trillion for large numbers: 1 thousand = + 000 1 million = 1 thousand × 1 000 1 billion = 1 million × 1 000 1 trillion = 1 billion × 1 000Similarly, we use millionth, billionth, …, for small decimal fractions: 1 millionth = 0.000 001 1 billionth = 0.000 000 001When writing large and small numbers, group the digits in threes from the decimal point, for example, 9 560 872 143 and 0.067 482.

3 28 ÷ 7 = 4.Number 7 is a whole number that divides exactly into another whole number, 28.This means that 7 is a factor of 28, and 28 is a multiple of 7.

4 A prime number has only two factors, itself and 1. The number 1 is not a prime number. The numbers 2, 3, 5, 7, 11, 13, 17, … are prime numbers. They continue without end.The prime factors of a number are those factors that are prime. For example, 2 and 5 are the prime factors of 40.We can write 40 as a product of prime factors:either 2 × 2 × 2 × 5 = 40, or,in index form, 23 × 5 = 40.

5 The numbers 18, 24 and 30 all have 3 as a factor. Number 3 is a common factor of the numbers.The highest common factor (HCF) is the largest of the common factors of a given set of numbers. For example, 2, 3 and 6 are the common factors of 18, 24 and 30; 6 is the HCF.The number 48 is a multiple of 4 and a multiple of 6. Number 48 is a common multiple of 4 and 6.The lowest common multiple (LCM) is the smallest of the common multiples of a given set of numbers. For example, 12 is the LCM of 4 and 6.


6 A fraction is the number obtained when one number (the numerator) is divided by another number (the denominator). The fraction 5 _ 8 means 5 ÷ 8 (Figure P2).

numerator

dividing line

denominator

5–8

Figure P2

We use fractions to describe parts of quantities (Figure P3).

Figure P3 5 _ 8 of the circle is shaded

The fractions 5 _ 8 , 10 __ 16 ,

15 __ 24 all represent the same amount; they are equivalent fractions. 5 _ 8 is the simplest form of 15 __ 24 , that is, 15 __ 24 in its lowest terms is 5 _ 8 .

P–2 Basic operations

To add or subtract fractions, change them to equivalent fractions with a common denominator. For example:

5 _ 8 + 2 _ 3 = 15 __ 24 + 16 __ 24 = (15 + 16) ______ 24 = 31 __ 24 = 1 7 __ 24

13 __ 16 – 10 __ 16 = (13 – 10) ______ 16 = 3 __ 16

To multiply fractions, multiply numerator by numerator and denominator by denominator.For example:

5 _ 8 × 2 _ 3 = (5 × 2) ____ (8 × 3) = 10 __ 24 = 5 __ 12 in simplest form

12 × 5 _ 8 = 12 __ 1 × 5 _ 8 = (12 × 5) _____ (1 × 8) = 60 __ 8 = 15 __ 2 = 7 1 _ 2

To divide by a fraction, multiply by the reciprocal of the fraction. For example:

35 ÷ 5 _ 8 = 35 __ 1 × 8 _ 5 = (35 × 8) _____ (1 × 5) = (7 × 8)

____ 1 = 56

5 _ 8 ÷ 3 3 _ 4 = 5 _ 8 ÷ 15 __ 4 = 5 _ 8 × 4 __ 15 = (5 × 4) _____ (8 × 15) = 20 ___ 120 = 1 _ 6

1 x% is short for x ___ 100 . 64% means 64 ___ 100 .To change a fraction to an equivalent percentage, multiply the fraction by 100. For example: 5 _ 8 as a percentage = 5 _ 8 × 100% = 125 ___ 8 % = 62 1 _ 2 %.

2 To change a fraction to a decimal fraction, divide the numerator by the denominator. For example:

5 _ 8 = 0.6250.625

8 5.0004 8 20 16 40 40

When adding or subtracting decimals, write the numbers in a column with the decimal points exactly under each other. For example: Add 2.29, 0.084 and 4.3, then subtract the result from 11.06. 2.29 0.084 11.06 + 4.3 – 6.674 6.674 4.386To multiply decimals:a Ignore the decimal points and multiply the

numbers as if they are whole numbers.b Then place the decimal point so that the

answer has as many digits after the point as there are in the question.

For example, to multiply 0.08 × 0.3: 8 × 3 = 24There are three digits after the decimal points in the question, so: 0.08 × 0.3 = 0.024.To divide by a decimal, change the division so that the divisor becomes a whole number. For example, to divide 5.6 ÷ 0.07: 5.6 ÷ 0.07 = 5.6 ___ 0.07 = (5.6 × 100)

________ (0.07 × 100) = 560 ___ 7 = 80 3 Numbers may be positive or negative.

Positive and negative numbers are called directed numbers. Directed numbers can be shown on a number line (Figure P4).

–3 –2 –1 0 +1 +2 +3

Figure P4

These examples show how to add and subtract directed numbers.

5Section 1: Additional material

Addition Subtraction (+8) + (+3) = +11 (+9) – (+4) = +5

(+8) + (–3) = +5 (+9) – (–4) = +13

(–8) + (+3) = –5 (–9) – (+4) = –13

(–8) + (–3) = –11 (–9) – (–4) = –5

An integer is any positive or negative whole number as shown in Figure P4.

4 When rounding numbers, round down the digits 1, 2, 3, 4 and round up the digits 5, 6, 7, 8, 9. For example: 3 425 = 3 430 to 3 significant figures = 3 400 to the nearest hundred 7.283 = 7.28 to 2 decimal places = 7.3 to 1 decimal place = 7 to the nearest whole number

5 The everyday system of numeration uses ten digits and is called a base ten, or denary, system.The base two, or binary, system of numeration uses only two digits, 0 and 1.To convert between bases ten and two, express the given numbers in powers of two: 43ten = 32 + 8 + 2 + 1 = 25 + 23+ 21 + 20

= 1 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20

= 101 011two

10 110two = 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20

= 16 + 0 + 4 + 2 + 0 = 22tenUse these identities when adding, subtracting and multiplying binary numbers:Addition: 0 + 0 = 0 1 + 0 = 1 0 + 1 = 1 1 + 1 = 10Multiplication: 0 × 0 = 0 1 × 0 = 0 1 × 1 = 1Detailed coverage of number, numeration and basic operations is given in Book 1 Chapters 1, 2, 3, 4, 9, 12, 23 and 24.

Review test 1 (Number and numeration & basic operations

• Allow 25 minutes for this test.• If you do not understand why some of your

answers were incorrect, ask a friend or your teacher.

1 What is the value of:a The 7 in 3.75? b The 5 in 519.2?

2 Find the HCF of:a 6, 15, 36 b 84, 56, 70

3 Find the LCM of:a 4, 5 and 10 b 9, 10 and 12

4 Reduce these fractions to their lowest terms:a 45 __ 63 b 256 ___ 352

5 Simplify:a 5 __ 12 – 1 _ 8 b 5 3 _ 4 + 8 3 _ 5

c 2 2 _ 7 ÷ 8 __ 21 d 1 3 _ 4 × 3 _ 7 6 Express the first quantity as a percentage of

the second:a N20, N50 b 4 mm, 5 cmc 20 cm, 5 m d 360 g, 2 kg

7 Each row of this table contains equivalent expressions. Complete the table. (The first row has been done for you.)

Common fraction

Decimal fraction

Percentage

1 __ 2 0.5 50%

2 __ 5

0.28

5 __ 8

8 Simplify:a (–2) + (+6) b (+3) – (+4)c –5 + (–3) d 7 – (–1)

9 Round off these numbers to the nearest:i Thousand ii Hundred iii Tena 27 536 b 13 705

10 Round off these numbers to the nearest:i Whole number ii Tenthiii Hundredtha 0.865 b 7.009



11 Write this number in digits, grouping the digits in threes.

Sixty one billion, five hundred and seven million, nine hundred and seventy-two thousand, and forty-three

12 Convert:a 58ten to base twob 110 110two to base ten

Now try Review test 2.

Review test 2 (Number and numeration & basic operations)



1 What is the value of:a The 3 in 0.438? b The 9 in 29.7?

2 Find the HCF of:a 44, 88 and 55 b 144, 90 and 54

3 Find the LCM of:a 3, 4 and 5 b 5, 8 and 9

4 Reduce these fractions to their lowest terms:a 24 __ 30 b 270 ___ 378

5 Simplify:a 8 _ 9 + 1 _ 3 b 4 2 _ 5 – 5 5 _ 8

c 2 _ 5 of 1 2 _ 3 d 7 1 _ 8 ÷ 4 3 _ 4 6 Express the first quantity as a percentage of

the second:a N80, N150 b 140 m, 2 kmc 28 cm, 4 m d 26 g, 4 kg

7 Each row of this table contains equivalent expressions. Complete the table. (The first row has been done for you.)

Common fraction

Decimal fraction

Percentage

1 __ 2 0.5 50%

7 __ 10

0.55

92%

2 __ 3

0.525

8 Simplify:a (–8) + (+3) b (–1) – (+7)c 4 + (–10) d –2 – (–5)

9 Round off these numbers to the nearest:i Thousand ii Hundrediii Tena 5 977 b 20 066


11 Write these numbers in digits only:a N6 1 _ 4 billion b 650 millionths

12 Calculate these base two numbers. Give your answers as binary numbers. Check your results in base ten.a 11 011 + 101 b 11 011 – 101c 11 011 × 101

P–3 Algebraic processes

1 7x – 2x + 3y is an example of an algebraic expression. The letters x and y stand for numbers, and 7x, 2x, 3y are the terms of the expression.3y is short for 3 × y.3 is the coefficient of y.

2 Multiply and divide algebraic terms as follows:6a × 5b = 6 × a × 5 × b = 6 × 5 × a × b = 30ab3p × 8p = 3 × p × 8 × p = 3 × 8 × p × p = 24p2

12mn ÷ 3m = (12 × m × n) _______ (3 × m)

= 12n ___ 3 = 4n 3 Simplify algebraic expressions in these ways.

a By grouping like terms. For example: 7x – 2x + 3y = 5x + 3y

Notice that 7x and 2x are like terms; 5x and 3y are unlike terms.


b By removing brackets. For example: a + (2a – 7b) = a + 2a – 7b

= 3a – 7b If there is a positive sign before the

bracket, the signs of the terms inside the bracket stay the same when removing brackets. For example:

5x – (x – 9y) = 5x – x + 9y = 4x + 9y

If there is a negative sign before the bracket, the signs of the terms inside the bracket change when the bracket is removed.

4 x + 8 = 3 is an algebraic sentence containing an equals sign; it is an equation in x. x is the unknown of the equation. If we substitute a value for an unknown, the equation may be true or false. For example, 2y = 6 is true when y = 3 and is false when y = 5.To solve an equation means to find the value of the unknown that makes the equation true.Use the balance method to solve simple equations. For example:3x – 8 = 25Add 8 to both sides:3x – 8 + 8 = 25 + 8That is, 3x = 33.Divide both sides by 3:3x ÷ 3 = 33 ÷ 3That is, x = 11.In the balance method, always do the same to both sides of the equation.

Detailed overage of algebraic processes is given in Book 1 Chapters 5, 7, 10, 15 and 19.


Review test 3 (Algebraic processes)



1 Each sentence is true. Find the number that each letter stands for.a 13 + 9 = p b q – 14 = 14c 27 = r + 14 d 12 = s + s + s

2 Find the value of these expressions when x = 6:a 9 + x b 44 – xc x – 5 + x d 7 – x + x

3 Simplify these expressions:a 8p – 5p b 8q + 3qc 12r – 4 – 7r + 15d 6w – y + 5z + 9z + 5y – 4w

4 a i How many weeks are there in 28 days?

ii How many weeks are there in d days?

b i A student has five bananas. He eats one of them. How many does he have left?

ii Another student has b bananas. She eats two of them. How many does she have left?

5 Remove the brackets and then simplify:a 4 – (6a – 7) b 3a – (5 – a) c (8 – 3a) – (3 – 4a)d (15x + 6y) – (8x – 7y)

6 Solve these equations:a 4a + 13 = 21 b 5b + 9 = 44c 9c – 11 = 61 d 7 = 10 – 3k


Review test 4 (Algebraic processes)



1 Each sentence is true. Find the number that each letter stands for:a 13 – f = 8 b 24 = 15 + gc 30 = m – 6 d 18 – n = n


2 Find the value of these expressions when x = 9:a x + 13 b x – 10c x + 9 + x d x – 5 – x

3 Simplify these expressions:a 7f + 7f b 18g – 13gc 8m – n – 5m – 3nd 5x + 2y – 8 – 5x + 3

4 a i How many cm in 2 metres? ii How many cm in m metres? b i In Compound 1 there are four

goats and three sheep. How many animals are there altogether?

ii In Compound 2, there are six goats and n sheep. How many animals are there altogether?

5 Remove the brackets and then simplify:a 2a – (a + 8) b 4b – (7 – b)c (7x + 2y) – (3x + y)d (5y – 3) – (4 – 3y)

6 Solve these equations:a 4a + 3 = 31 b 3b – 14 = 10c 15 – 6c = 0 d 18 = 19d – 20

P–4 Mensuration and geometry

Figure P5 gives sketches and names of some common solids.

All solids have faces; most solids have edges and vertices (Figure P6).

sphere

cuboid

cone

cubecylinder

square-based

pyramidtriangular

prism

Figure P5

face

edge

vertex

Figure P6

The formulae for the surface area and volume of common solids are given in the table on page xxx.

An angle is a measure of rotation or turning.1 revolution = 360 degrees (1 rev = 360°)1 degree = 60 minutes (1° = 60´)

The names of angles change with their size. See Figure P7.

acute angle (between 0° and 90°)

right angle (90°)

obtuse angle (between 90° and 180°)

straight angle (180°)

reflex angles (between 180° and 360°)

Figure P7

Angles are measured and constructed using a protractor.

Figure P8 shows some properties of angles formed when straight lines meet.

a

ab

c

the sum of the angles on a straight line is 180°a + b + c = 180°


b

r

qs

p

vertically opposite angles are equalp = q and r = s

c

a + b + c + d + e = 360°

b

cd

e

a

d e

Figure P8

Figure P9 shows the names and properties of some common triangles.

scalene

isosceles

right-angled obtuse-angled

equilateral

60°

60° 60°

Figure P9

Figure P10 shows the names and properties of some common quadrilaterals.

square rectangle

parallelogram rhombus

Figure P10

Figure P11 gives the names of lines and regions in a circle.

circumference

diameter

radius

centre

chordarc

semi-circle

sector

segment

Figure P11

A polygon is a plane shape with three or more straight sides. A regular polygon has all its sides of equal length and all its angles of equal size.

Figure P12 gives the names of some common regular polygons.

equilateral triangle

square regular pentagon

regular hexagon

regular octagon

Figure P12

m = n (alternate angles) x = y (alternate angles)

m

ny

x

a = b (corresponding angles)p = q (corresponding angles)

q

b

a

p


The formulae for the perimeter and area of plane shapes are given in the table on page 130 of the Student’s Book.

The SI system of units is given in the tables on pages 229 and 230 of the Student’s Book.

Use a ruler and set-square to construct parallel lines (Figure P13) and a perpendicular from a point to a line (Figure P14).

Figure P13 Parallel lines (ruler and set-square)

P l

lP

Figure P14 Perpendicular from point P to line l (ruler and set-square)

Detailed coverage of geometry and mensuration is given in Book 1 Chapters 6, 8, 11, 13, 14, 16, 20 and 21.


Review test 5 (Mensuration and geometry)

• Allow 25 minutes for this test. • If you do not understand why some of your


1 a Add these capacities 1.885 ℓ, 325 mℓ and 2.04 ℓ. Give the answer in:

i litres ii millilitres.b How many minutes in:

i 210 seconds? ii 1 day?2 How many faces, edges and vertices does a

hexagonal prism have?3 In Figure P9, one of the triangles is marked

as scalene. Are there any other scalene triangles in Figure P9?

4 Copy this table. Complete the unshaded boxes.

Shape Length Breadth Perimeter Area

a Rectangle 5 cm 9 cm

b Rectangle 8 cm 30 cm

c Square 28 cm

d Square 64 km2

5 Copy and complete this table of circles. Use the value 3 for π.

Radius Diameter Circumference Area

a 7 cm

b 16 cm

c 15 cm

d 363 cm2

6 Copy and complete this table of triangles.Base Height Area

a 10 cm 7 cm

b 9 cm 3 cm

c 6 cm 12 cm2

d 9 m 54 m2


7 Copy this table. Complete the unshaded boxes.Shape Length Breadth Height Base

areaVolume

a Cuboid 8 cm 3 cm 11 cm

b Cuboid 2 cm 6 cm 60 cm3

c Cube 4 m

d Prism 5.5 cm 6 cm2

e Prism 28 cm 84 cm3

f Prism 4 m2 30 m3

8 Write down the sizes of the lettered angles in Figure P15.

a b

c d

Figure P15

a°

g°

c°

d ° e°

b°

f °

115°

i °

j °k °

h °

l °61°

n°

m° 82°140°

q°

p°63°


Review test 6 (Mensuration and geometry)



1 a Add these quantities 2.5 tonnes, 850 kg, 4.955 tonnes. Give the answer in:

i kg ii tonnes.b How many seconds in:

i 5 minutes? ii 2 hours?2 How many faces, edges and vertices does a

square-based pyramid have?

3 One of the angles of an isosceles triangle is 22°. What are the sizes of its other two angles? (Note: There are two possible answers; give both.)

4 Copy this table. Complete the unshaded boxes.

Shape Length Breadth Perimeter Area

a Rectangle 2 km 11 km

b Rectangle 12 cm 36 cm2

c Square 25 mm

d Square 34 m

5 Copy and complete this table of circles. Use the value 3 for π.

Radius Diameter Circumference Area

a 4.5 cm

b 10 cm

c 84 m

d 3 m2

6 Copy and complete this table of triangles.Base Height Area

a 11 cm 8 cm

b 5 cm 15 cm

c 2 m 20 m2

d 4 cm 22 cm2

7 Copy this table. Complete the unshaded boxes.Shape Length Breadth Height Base

areaVolume

a Cuboid 7 cm 4 cm 10 cm


c Cube 216 cm3

d Prism 8 cm 3.5 cm2

e Prism 12.5 cm2 100 cm3

f Prism 2 m 26 m3

8 Write down the sizes of the lettered angles in Figure P16.

a a °

b °

44° b

c ° d °

e °

f °

43°

146°


c dl °

h ° g °

j °k °i °

18°

70°

254°

n°m°

p°

Figure P16

P–5 Everyday statistics

1 Information in numerical form is called statistics.a Statistical data may be given in rank order

(that is, in order of size) such as these marks scored in a test out of 5:

0, 1, 1, 2, 2, 2, 3, 3, 5b Data may also be given in a frequency

table (Table P1).

Mark 0 1 2 3 4 5

Frequency 1 2 3 2 0 1

Table P1

The frequency is the number of times each piece of data occurs.

c Statistics can also be presented in graphical form. Figure P17 shows the data above in a pictogram, a pie chart and a bar chart.

Pictogram Pie chart

1

2

3

05

0 marks

1 mark

2 marks

3 marks

4 marks

5 marks

freq

uenc

y

marks

4

3

2

1

00 1 2 3 4 5

Figure P17

2 The average of a set of statistics is a number that represents the whole set.The three most common averages are the mean, the median and the mode. For the numbers given in rank order in point 1 above:a Mean = 0 + 1 + 1 + 2 + 2 + 2 + 3 + 3 + 5

= 2 1 _ 9 .b The median is the middle number when

the data are arranged in size order (= 2).c The mode is number with the greatest

frequency (also 2). If a set of data has two modes, we say it is bimodal.

Detailed coverage of statistics is given in Book 1 Chapters 17, 18 and 22.


Review test 7 (Everyday statistics)



1 The bar chart in Figure P18 shows the rainfall for each month in a year.

J F M A M J J A S O N D

50

40

30

20

10

0

Rai

nfal

l (cm

)

Figure P18

a Which month had most rainfall?b How many cm of rain fell in the

wettest month?


c Which months had no rainfall?d List the four wettest months in rank

order.e Find in cm the total rainfall for the year.

2 Calculate the mean for each of these sets of numbers:a 11, 13, 15 b 8, 9, 13c 2, 9, 7, 9, 8 d 5, 5, 1, 0, 9

3 Find the mode, median and mean for each of these sets of numbers:a 6, 6, 8, 11, 14b 1, 2, 5, 5, 6, 6, 6, 7, 7, 8c 8, 6, 3, 10, 6, 9d 6, 4, 10, 6, 11, 7, 5, 8, 6


Review test 8 (Everyday statistics)



1 The marks obtained in a test out of 10 are shown in this frequency table.

Mark 3 4 5 6 7 8

Frequency 4 9 6 9 5 2

a What was the highest mark?b How many students scored this mark?c Are the data bimodal?d How many students scored more than

5 marks?e How many students took the test?

2 Calculate the mean for each of these sets of numbers:a 17, 5, 7, 11 b 2, 10, 5, 7c 4, 11, 2, 8, 9, 2 d 7, 8, 10, 11, 14, 16

3 Find the mode, median and mean for each of these sets of numbers:a 3, 4, 4, 6, 7, 9 b 3, 7, 10, 10, 11, 11, 11c 2, 0, 16, 0, 7, 11d 5, 4, 2, 5, 2, 1, 3, 5, 3, 4, 5, 3

Answers to exercises

Review test 1 1 a 7 tenths b 5 hundreds 2 a 3 b 14 3 a 20 b 180 4 a 5 _ 7 b 8 __ 11

5 a 7 __ 24 b 14 7 __ 20 c 6 d 3 _ 4

6 a 40% b 8% c 4% d 18% 7 Common

fraction Decimal fraction

Percentage

1 _ 2 0.5 50%

2 _ 5 0.4 40%

7 __ 25

0.28 28%

5 _ 8 0.625 62.5%

8 a 4 b –1 c –8 d 8 9 a i 28 000 ii 27 500 iii 27 540 b i 14 000 ii 13 700 iii 13 710 10 a i 1 ii 0.9 iii 0.87 b i 7 ii 7.0 iii 7.01 11 61 507 972 043 12 a 111 010two b 54ten

Review test 2 1 a 3 hundredths b 9 units 2 a 11 b 18 3 a 60 b 360 4 a 4 _ 5 b 5 _ 7

5 a 11 __ 9 = 1 2 _ 9 b –1 9 __ 40

c 2 _ 3 d 1 1 _ 2 6 a 53.3% b 7% c 7% d 0.65%


7 Common fraction

Decimal fraction

Percentage

1 _ 2 0.5 50%

7 __ 10

0.7 70%

11 __

20 0.55 55%

23 __

25 0.92 92%

2 _ 3 0.666 7 66 2 _

3 %

21 __

40 0.525 52.5%

8 a –5 b –8 c –6 d 3 9 a i 6 000 ii 6 000 iii 5 980 b i 20 000 ii 20 100 iii 20 070 10 a i 2 ii 2.4 iii 2.39 b i 4 ii 4.3 iii 4.30 11 a N6 250 000 000 b 0.000 65 12 a 100 000 (27 + 5 = 32) b 10 110 (27 – 5 = 22) c 10 000 111 (27 × 5 = 135)

Review test 3 1 a p = 22 b q = 28 c r = 13 d s = 4 2 a 15 b 38 c 7 d 7 3 a 3p b 11q c 5r + 11 d 2w + 4y + 14z 4 a i 4 ii d _ 7 b i 4 ii b – 2 5 a 11 – 6a b 4a – 5 c 5 + a d 7x + 13y 6 a a = 2 b b = 7 c c = 8 d k = 1

Review test 4 1 a f = 5 b g = 9 c m = 36 d n = 9 2 a 22 b –1 c 27 d –5 3 a 14f b 5g c 3m – 4n d 2y – 5 4 a i 200 ii 100m b i 7 ii 6 + n 5 a a – 8 b 5b – 7 c 4x + y d 8y – 7 6 a a = 7 b b = 8 c c = 2 1 _ 2 d d = 2

Review test 5 1 a i 4.25 litres ii 4 250 ml

b i 3.5 ii 1 440 2 8 faces, 18 edges, 12 vertices 3 Yes: the right-angled triangle and obtuse-

angled triangle are also scalene (their three sides are of different lengths).

4Shape Length Breadth Perimeter Area

a Rectangle 5 cm 9 cm 28 cm 45 cm2

b Rectangle 8 cm 7 cm 30 cm 56 cm2

c Square 7 cm 28 cm 49 cm2

d Square 8 km 32 km 64 km2

5Radius Diameter Circumference Area

a 7 cm 14 cm 42 cm 147 cm2

b 8 cm 16 cm 48 cm 192 cm2

c 2.5 cm 5 cm 15 cm 18.75 cm2

d 11 cm 22 cm 66 cm 363 cm2

6Base Height Area

a 10 cm 7 cm 35 cm2

b 9 cm 3 cm 13.5 cm2

c 6 cm 4 cm 12 cm2

d 12 m 9 m 54 m2

7Shape Length Breadth Height Base

areaVolume

a Cuboid 8 cm 3 cm 11 cm 264 cm3


c Cube 4 m 64 m3

d Prism 5.5 cm 6 cm2 33 cm3

e Prism 28 cm 3 cm2 84 cm3

f Prism 7.5 m 4 m2 30 m3

8 a = 115°, b = 65°, c = 65°, d = 65°, e = 115°, f = 65°, g = 115°, h = 119°, i = 61°, j = 119°, k = 61°, l = 119°, m = 40°, n = 58°, p = 63°, q = 54°

Review test 6 1 a i 8 305 kg ii 8.305 tonnes b i 300 ii 7 200 2 5 faces, 8 edges, 5 vertices 3 Either 22° and 136° or 79° and 79°


4Shape Length Breadth Perimeter Area

a Rectangle 2 km 11 km 26 km 22 km2

b Rectangle 3 cm 12 cm 30 cm 36 cm2

c Square 25 mm 100 mm 625 mm2

d Square 8.5 m 34 m 72.25 m2

5Radius Diameter Circumference Area

a 4.5 cm 9 cm 27 cm 60.75 cm2

b 5 cm 10 cm 30 cm 75 cm2

c 14 m 28 m 84 m 588 m2

d 1 m 2 m 6 m 3 m2

6Base Height Area

a 11 cm 8 cm 44 cm2

b 5 cm 15 cm 37.5 cm2

c 20 m 2 m 20 m2

d 4 cm 11 cm 22 cm2

7Shape Length Breadth Height Base

areaVolume

a Cuboid 7 cm 4 cm 10 cm 280 cm3

b Cuboid 3 cm 6 cm 3 cm 54 cm3

c Cube 6 cm 216 cm3

d Prism 8 cm 3.5 cm2 28 cm3

e Prism 8 cm 12.5 cm2 100 cm3

f Prism 2 m 13 m2 26 m3

8 a = 136°, b = 44°; c = 34°, d = 137°, e = 43°, f = 34°, g = 110°, h = 52°, i = 18°, j = 110°, k = 52°, l = 110°, m = 106°, n = p = 37°

Review test 7 1 a May b 50 cm c January, October, November, December d May, April, June, July e 195 cm 2 a 13 b 10 c 7 d 4 3 a 6, 8, 9 b 6, 6, 5.3 c 6, 7, 7 d 6, 6, 7

Review test 8 1 a 8 b 2

c Yes (the marks 4 and 6 have a frequency of 9)

d 16 e 35 2 a 10 b 6 c 6 d 11 3 a 4, 5, 5.5 b 11, 10, 9 c 0, 4.5, 6 d 5, 3.5, 3.5

16 Section 1: Additional materialSection 1: Additional material16

Enrichment chapter Calculators and tables

Objectives

By the end of the chapter, you will be able to:• use a simple four-function calculator to

perform everyday calculations• use the memory and square-root functions

on a calculator• use tables to find squares and square roots

of numbers.

Teaching and learning materialsTeacher: Calculators (tables are provided in this chapter and at the end of the book).Students: Calculator (tables are provided).

This enrichment chapter provides help and guidance in using calculators and tables. Its aim is to:• develop efficient and effective use of calculating

aids• ensure that students become skillful in using

calculators so that they know how, when and when not to use them.

Calculator skills are not part of the national curriculum and their use is not currently allowed in JSCE examinations.

However, they can assist studies, for example, by helping to check whether answers are correct or not.

All JSS leavers should be able to use calculators sensibly, whether as an aid to further study or as a desirable life skill.

E–1 Know your calculator

There are various kinds of calculators:• hand-held calculators (such as that in Figure E1)• desk calculators used in homes, shops and offices• electronic calculators that you will find on all

computers and on most mobile phones.

Most calculators operate in much the same way. But you will find small differences. So learn about your calculator. Find out what it can do.

Figure E1 shows the main parts of a simple four-function calculator with percentage and square-root keys as well as a memory.

Other calculators, such as scientific calculators and programmable calculators, have many more keys and functions.

memory keysMC = memory

clearMR = memory

recallM– = take from

memoryM+ = add to

memory

percentage key

square root key

clear last entry

all clear

number keys and decimal point

equals key

solar power cell (if fitted)

display

arithmetical function keys

Figure E1

All calculators, however, have the basic functions shown in Figure E1. And these are the functions you will use most often.

Power

Most calculators get their power from a solar cell. This powers the calculator as long as light is available (daylight, electric bulb or even candle-light).

Display

The display shows the answers. The digits in the display are usually made of small line segments as shown in Figure E1.

Keyboard

The keyboard has four main sets of keys or buttons: 1 Number keys Press , , , , , , , , , and

the decimal point key (usually shown as a dot ) to enter numbers into the calculator.


2 Basic calculation keys Press , , , , , and to do

mathematical operations on the numbers you have entered, and to display answers.

3 Clearing keysThe key clears the last number you entered. Press if you enter a wrong number by mistake. On some calculators, you may see instead of .The key clears the whole calculation that you are working on.Use this key if you want to start from the beginning again. Often the key is linked to the calculator’s on key and appears as or just as , as in Figure E1.Press before starting any calculation.This gives 0 on the display.

4 Memory keysa Memory plus key Press to store the displayed number in

the memory of the calculator. If there is any previous number in the memory, it adds the displayed number to it.

b Memory minus key Press to subtract the displayed number

from the number in the memory. The result of adding or subtracting the number will be the new number in the memory.

c Memory recall key If there is a number in the memory, the

calculator usually shows a small M in one corner of the display.

Press to display the number in the memory.

d Memory clear key Press to clear the number stored in the

memory.

Exercise Ea (Class activity)

Copy and complete Table E1. For each key sequence:a First guess the outcome.b Then use your calculator to get a result.c If anything unexpected happens, make a

note in the right-hand column.

Key sequence Guess Result Notes

Table E1


Display on your calculator:a the highest possible numberb the lowest positive number.

To find what a snail lives in:a Calculate 5 × 31 × 499.b Turn your calculator upside down and read

the display. To find out what plants grow in:

a Calculate √ __________

50 481 025 .b Turn your calculator upside down and read

the display. a Use your calculator to complete Table E2.

Powers of 7 Value71 7

72 49

73 343

74

75

76

77

78

Table E2

b Look at the final digits of the values displayed in Table E2. Is there a pattern? If so, what is it?

c Is there a recognisable pattern in the final two digits?

d Try the above with a different starting number, For example, 3, 6, 11 or 13. Are there any patterns?

‘100 up’ is a game for one person. To start: Enter any 2-digit prime number into

your calculator. Aim: To get the calculator to display a number

in the form 100. *****, where * may be any digit.

Rule: You must multiply the number shown in the calculator display by any number of your choice.

Scoring: Record each multiplication as a trial. Try to achieve your aim in as few trials as possible, that is, you should keep your score as low as possible.

Here is a sample game:

Display Press keys

Trial no. (score)

Start 29 × 3 1

87 × 1.2 2

104.4 × 0.9 3

93.96 × 1.05 4

98.658 × 1.02 5

Finish 100.63116

The score for this game is 5. Starting with 29, can you do better? Try to beat 5, then play some games starting with other prime numbers.

E–2 The operations , , ,

Addition and subtractionUse the and keys to add and subtract numbers and to display the result.

Example 1Calculate 356 + 717.

Keystrokes:

Display:0 3 35 356 356 7 71 717 1 073

(answer)Rough check: 400 + 700 = 1 100

It is a good idea to start any new calculation by pressing the key. This clears any previous calculation or data that the calculator may contain.When using a calculator, it is possible to make keying-in mistakes. So make a habit of doing a rough check. Do the check mentally.

Example 2Calculate 89 – 54 – 17.

Keystrokes:

Display:0 8 89 89 5 54 35 1 17 18

(answer)Rough check: 90 – 50 – 20 = 20

In Example 2, notice the value 35 in the display.This is an intermediate result (89 – 54 = 35). It appears when the second operation is entered.


Example 3Calculate 9 – 16 + 18.

Keystrokes:

Display:0 9 9 1 16 −7 1 8 11

(answer)Rough check: 10 – 20 + 20 = 10

Notice that calculators give a negative outcome when subtracting a larger number from a smaller number. Thus 9 – 16 gives –7 as an intermediate result during the above calculation.

Exercise Eb

Do these calculations on your calculator. Write down what appears in the display when you press each key and underline the final answer.a 7 + 2 b 9 – 5c 57 – 29 d 38 + 48e 94 – 38 – 26 f 18 + 37 + 42g 123 + 456 – 543 h 38 – 82 + 71i 32.7 – 8.4 j 3.4 + 7.8 + 4.3

Look at these calculations. Six of them are incorrect.i 6 + 7 = 14 ii 48 + 19 = 912iii 22 – 12 = 10 iv 950 – 42 = 53v 235 + 680 = 3 015 vi 8.9 + 4.5 = 13.4vii 87 – 59 = 82 viii 36 + 48 = –12a Decide which ones appear to be incorrect.b Use your calculator to correct them.

Look at these calculations before doing them. What kind of answer do you expect?

Do the calculations.a 2 – 7 b 5 – 15c 16 – 49 d 36 – 73e 8 – 75 f 44 – 260g 256 – 911 h 56 – 46 – 66

An athlete buys some clothes. Figure E2 shows the bill. Check that the shop assistant has added up everything correctly.

Sports WorldTracksuit N 12 999Sweatshirt N 3 990Shorts N 2 490Running shoes N 8 645Total N 28 124

Figure E2

What will the items in Figure E3 cost in total?

Bottle orange juice N 429

Jar coffee N 582

Packet of tea N 195

Sugar N 180

Margarine N 299

Jar of peanut butter N 265

Oranges N 389

Meat N 850

Shampoo N 405

Bottle of apple juice N 376

Chicken N 1 450

Figure E3

Multiplication and divisionUse the and keys to multiply and divide numbers and to display the result.

Given a multiplication (or a division) in the form a × b (or a ÷ b), press the keys (or ) then either , , , or will display the answer.

Example 4Calculate 68 × 29.

Keystrokes:

Display: 0 6 68 68 2 29 1 972 (answer)Rough check: 70 × 30 = 2 100


Example 5Calculate 725 ÷ 25 × 14.

Keystrokes:

Display:0 7 72 725 725 2 25 29 1 14 406 (answer)Rough check: 700 ÷ 20 × 10 = 350

In Example 5, notice that 725 ÷ 25 = 29. appears in the display at an intermediate stage of the calculation.

Calculators have a limited number of spaces in the display (usually eight spaces). Therefore, there is a limit to the size of answer they can display. Try the calculation 68 000 × 29 000 on a calculator.

Figure E4 shows what happens on some eight-digit calculators.a b

c d

Figure E4

Note that 68 000 × 29 000 = 1 972 000 000 (requiring 10 digits).• The calculator displays in a and b show the

digits but are unable to display the full number properly. So they print a small (error) to warn the user.

• The calculator displays in c and d group the display into two parts: and . This is short for 1.972 × 1 000 000 000. Note the nine zeros in the second number. They correspond to the

. This type of calculator is called a scientific calculator. Scientific calculators can cope with very large numbers because they give the outcome in standard form: 68 000 × 29 000 = 1.972 × 109

Example 6How many seconds are there in a 31-day month?

Number of seconds = 60 × 60 × 24 × 31 = 2 678 400 (calculator)

Exercise Ec

Do these calculations on your calculator. Write down what appears in the display as you press each key. Underline your final answer.a 67 × 88 b 4 234 ÷ 58c 513 ÷ 19 d 462e 49 × 67 × 13 f 7 938 ÷ 81 ÷ 14g 102 × 104 ÷ 78 h 495 ÷ 33 × 41

Do these calculations. Give each answer:i As displayed on the calculator.ii Rounded off to 2 decimal places.a 6.74 × 9.08 b 51.73 × 24.79c 28 341 ÷ 85 d 74.184 ÷ 40.08e 4 ÷ 3 × 6 f 7 ÷ 11 × 44g 773 ÷ 4.17 × 5.308h 3.142 × 4.5 × 4.5

Look at these calculations. Six of them are incorrect.i 5 × 9 = 30ii 100 000 ÷ 100 = 1 000iii 67 × 84 = 3 216iv 690 ÷ 15 = 45v 1 000 ÷ 30 = 33vi 1.7 × 1.5 × 1.3 = 33.15vii 360 ÷ 18 ÷ 5 = 4viii 4 123 ÷ 814 = 5a Decide which ones appear to be incorrect.b Use your calculator to correct them.

Multiply 30 000 by 50 000 on your calculator. What is displayed?

5 a Calculate 10 000 000 ÷ 0.9.b Calculate 10 000 000 ÷ 0.9 ÷ 0.9.

How many seconds are there in a 365-day year?7 a Write your age to the nearest year.

b Calculate how many days you have lived (assume 365 days in a year).

c Calculate how many hours you have lived.d Calculate how many minutes you have lived.e Is it possible for your calculator to calculate

the number of seconds you have lived?


A health inspector gets a salary of N659 520 per month. How much does this represent:a per year? b per day?

Assume a 30-day month. Give answers to the nearest naira.

Twelve members of a club hired a bus to go to a match. If the bus company charged N31 500, how much did each member have to pay?

0 An aeroplane travels 550 km in 1 h.a How many km does it travel in 1 min?b How many metres does it travel in 1 min?c How many metres does it travel in 1 s?

E–3 Further calculator techniques

Mixed operations

Look again at Exercise Ea, Question 1, part g. In some cases, calculators appear to give two answers to the same problem. According to the rules of precedence in arithmetic:2 + 5 × 3 = 2 + 15 = 17

And5 × 3 + 2 = 15 + 2 = 17.

However, for the first expression, the calculator gives this result:

Keystrokes:

Display: 2 2 5 7 3 21 (answer)

For the second expression, the calculator gives this result:

Keystrokes:

Display: 3 3 5 15 2 17 (answer)

This is because the calculator follows the operations in the order that it receives them.

Example 7Calculate 34 + 8 × 52.

There are no brackets, but always do multiplication before addition.

Rearrange the numbers:34 + 8 × 52 = 8 × 52 + 34 = (8 × 52) + 34

= 450 (calculator)

Brackets

Example 8Calculate 2.3 × (8.9 – 2.1).

The brackets tell us to do subtraction first.

Rearrange the numbers, so that the subtraction comes before the multiplication:2.3 × (8.9 – 2.1) = (8.9 – 2.1) × 2.3 = 15.64 (calculator)

In the above examples, it is possible to ‘turn the calculation round’ because in general, a × b = b × a.

However, with division this is not possible.

Read Example 9 carefully.

Example 9Calculate 84 ÷ (37 – 23).

Do the subtraction in the brackets first.

Store the outcome in memory. Recall it when needed.

The sequence of working is: 6

Check the above sequence on your own calculator and note the changes in display.

It is important to enter numbers and operations in an order that will allow the calculator to give the correct results. This means doing calculations in brackets first and storing them if necessary. Then, do multiplications and divisions before additions and subtractions.

Exercise Ed

In six of these cases, calculators will give incorrect results if you enter the numbers and operations in the given order. In those cases rearrange the numbers so that calculators will give the correct results.a 89 × 6 – 231 b 45 + 68 ÷ 17c 22 + 42 ÷ 3 d 63 + 18 × 5e 18 × (17 – 15) f (19 + 9) ÷ 7g 487 × (6 + 3) h 100 × (31 – 14)

Do these calculations, rearranging the order where necessary.a 95 × 7 – 436 b 101 + 51 × 9


c 55 + 75 ÷ 5 d 666 ÷ 36 + 2.5e 49 × (19 – 3) f (434 – 343) ÷ 13g 8.438 + 36.2 ÷ 2.6h 8.8 × (6.12 – 3.47)

All of these calculations require part of the calculation to be stored (either on paper or in the calculator’s memory).a 68 – 14 × 3 b 216 ÷ (25 – 7)c 708 ÷ (28 + 31) d 444 – 261 ÷ 29e 46.7 ÷ (15.28 – 3.59)f 381.04 – 12.6 × 7.8

Squares and square roots

SquaresMany calculators have a ‘squares’ key, usually shown as . However, if your calculator does not have a ‘squares’ key, then it is usually possible to get squares by pressing the key twice.

For example, try ; this should give an answer of 81.

Square-rootsSimply enter a number and use the key. For example gives 1.4142136.

Exercise Ee

Use your calculator to find the value of these numbers. Round your numbers to 3 s.f.a 672 b 762 c 9.82

d 7832 e 6.772 f 48.722

Use your calculator to find the value of these numbers. Round your answers to 3 s.f.a √

__ 3 b √

__ 5 c √

____ 760

d √ ____

6.82 e √ ____

24.9 f √ _____

3 884 Calculator fun

a Find √ __________

53 787 556 . Turn the calculator upside down. Is the display on your hand or on your foot?

b Find 192 – 42. Turn the calculator upside down. Is the display male or female?

c Make up more like these.

E–4 Using tables

Table of squares

If calculators are not available, then use tables to do calculations. See page 231 of the Student’s Book.

Use the table on page 231 of the Student’s Book to convert 2-digit numbers to the squares of those numbers.

Example 10Use the table of squares to find 6.72.• 6 is the first digit. Look for ‘6’ in the left-hand

column of the squares table.• 7 is the second digit. Look for the column

headed ‘.7’ of the squares table.• Find the number across from ‘6’ and under

‘.7’. See Figure E5.

6

.7

Figure E5

The number is 44.89.Thus, 6.72 = 44.89.

Example 11Use the table of squares to find:a 192

b 1902.

a 19 = 1.9 × 10 192 = (1.9 × 10)2

= 1.92 × 102

From the table: 1.92 = 3.61 Thus, 192 = 3.61 × 100

= 361b 190 = 1.9 × 100 1902 = (1.9 × 100)2

= 1.92 × 1002

= 3.61 × 10 000 = 36 100


In Example 11, notice that:• 1.92 = 3.61• 192 = 361• 1902 = 36 100

When multiplying a number by increasing powers of 10, you multiply its square by increasing powers of 100.

Example 12Use the table of squares to find:a 0.82

b 0.252.

a 0.8 = 8.0 × 10–1

(0.8)2 = (8.0 × 10–1)2

= 82 × 10–2

From the table: 82 = 64.00 Thus, 0.82 = 64 × 1

___ 102

= 0.64b 0.25 = 2.5 × 10–1

(0.25)2 = (2.5 × 10–1)2

= 2.52 × 10–2

From the table: 2.52 = 6.25 Thus, 0.252 = 6.25 × 1

___ 102

= 0.0625

Examples 14, 15 and 16 on pages 24 and 25 explain how to use the square root table on pages 232 and 233 of the Student’s Book. Notice this about the square root table.1 There are no decimal points. Use inspection to

place them correctly.2 There are two sets of digits for each number.

Exercise Ef

Use the table of squares (on page 231 of the Student’s Book) for this exercise.

Find the value of these numbers:a 1.42 b 2.32 c 6.82 d 7.22

e 4.92 f 8.52 g 5.62 h 9.82

i 3.12 j 1.82 k 5.72 l 4.32

Find the value of these numbers:a 182 b 312 c 322 d 152 e 292 f 442 g 702 h 202 i 622 j 92 k 812 l 992

Round off these numbers to 2 s.f. Then find the approximate square of each number.a 1.732 b 2.882 c 78.62

d 52.12 e 9.6472 f 4.9752

g 63.622 h 80.532 i 36.032

Find the value of these numbers:a 1302 b 4102 c 8702

d 5002 e 2 7002 f 8 3002

Look at this pattern: 1.52 = 2.25 = 1 × 2 + 0.25 2.52 = 6.25 = 2 × 3 + 0.25 3.52 = 12.25 = 3 × 4 + 0.25

Does the pattern continue in the same way?6 Find the squares of:

a 12 and 21 b 13 and 31c What do you notice?

A square has a side of length 4.3 cm. Calculate its area in:a cm2 b m2.

A square plot has a side of length 240 m. Calculate its area in:a m2

b hectares. (1 hectare = 10 000 m2) A square has a perimeter of 30 cm. Find:

a the length of one of its sidesb its area.

Find out which of these statements are true:a 652 = 162 + 632 b 652 = 252 + 602

c 652 = 302 + 352 d 652 = 332 + 562

e 652 = 362 + 432 f 652 = 392 + 522

Find the values of these numbers:a 0.42 b 0.72 c 0.52 d 0.112

e 0.482 f 0.312 g 0.652 h 0.832

i 0.272 j 0.192 k 0.582 l 0.772

Approximate square rootsA perfect square is a whole number with an exact square root.52 = 5 × 5 = 25

Thus, √ ___

25 = 5.

We can find the approximate square root of a number by knowing the perfect squares immediately before and after it.


Example 13Between which whole numbers does the square root of these numbers lie?a 3.6 b 9.4 c 40 d 78

a √ ____

3.6 3.6 lies between 1 and 4.Thus, √

____ 3.6 lies between √

__ 1 and √

__ 4 .

That is, √ ____

3.6 lies between 1 and 2.b √

____ 9.4

9.4 lies between 9 and 16Thus, √


__ 9 and √

___ 16 .

That is, √ ____

9.4 lies between 3 and 4.c √

___ 40

40 lies between 36 and 49Thus, √

___ 40 lies between √

___ 36 and √

___ 49 .

That is, √ ___

40 lies between 6 and 7.d √

___ 78

78 lies between 64 and 81.Thus, √


___ 64 and √

___ 81 .

That is, √ ___

78 lies between 8 and 9.

Square root tablesUse the tables on pages 232 and 233 to find the square roots of 2-digit numbers. Examples 14 and 15 explain how to put in the decimal point and how to use approximations to choose the correct set of digits.

Example 14Use the square root table to find:a √

____ 5.7 b R57

a √ ____

5.7 5.7 lies between 4 and 9.Thus, √


__ 4 and √

__ 9 .

That is, √ ____

5.7 lies between 2 and 3.Thus, √

____ 5.7 = 2 point something.

In the square root table, look for ‘5’ in the left-hand column.

The next digit is ‘7’. Look for the column headed ‘7’.

Find the digits across from ‘5’ and under ‘7’. See Figure E6.

5

7

Figure E6

This gives 239 ___ 755 .

As the required result begins with a 2, the required digits are 239. You can ignore the 755.Thus, √

____ 5.7 = 2.39.

b √ ___

57 57 lies between 49 and 64.Thus, √


___ 49 and √

___ 57 .

That is, √ ___

57 lies between 7 and 8. Thus, √

___ 57 = 7 point something.

From the table:The required digits are 755. You can ignore the 239.Thus, √

___ 57 = 7.55

Notice that the square root table gives values rounded to 3 significant figures. Thus: √

____ 5.7 = 2.39 to 3 s.f.

On a calculator √ ____

5.7 = 2.387 467 3. However, 3 significant figures are accurate enough for most purposes.


____ 940 b √

______ 3 998

a 940 = 9.4 × 100 √ ____

940 = √ ____

9.4 × √ ____

100 = √

____ 9.4 × 10

√ ____

9.4 is just over 9.Thus, √

____ 9.4 is just over 3.

Thus, √ ____

9.4 = 3 point something.From the table: √

____ 9.4 = 3.07

Thus, 940 = 3.07 × 10 = 30.7 to 3 s.f.b We can only use the square root table for

2-digit numbers. So, first round off 3 998 to 2 s.f.3 998 = 4 000 to 2 s.f.Thus, √

______ 3 998 ≈ √

______ 4 000

≈ √ ___

40 × √ ____

100 ≈ √

___ 40 × 10


40 is between 36 and 49. Thus, √

___ 40 is between 6 and 7.

That is, √ ___

40 = 6 point something.From the table: √ ___

40 = 6.32Thus, √

______ 3 998 ≈ 6.32 × 10 ≈ 63.2

Notice again that the final answers are not exact. For example, √

______ 3 998 = 63.229 74 (calculator).


_____ 0.28 b √

_____ 0.47

a √ _____

0.28 √

_____ 0.28 = 28 × 10–2

= √ ___

28 × 10–2 = √

___ 28 × R10–2

= √ ___

28 × 10–1

√ ___

28 lies between √ ___

25 and √ ___

36 .Thus, √

___ 28 lies between 5 and 6.

From the table: √ ___

28 = 5.29Thus √

_____ 0.28 = 5.29 × 1 __

10 = 5.29

____ 10

= 0.529

b 0.47 = 47 × 10–2

√ _____

0.47 = √ ___

47 × 10–2

= √ ___

47 × √ ____

10–2

= √ ___

47 × 10–1

√ ___

47 lies between √ ___

36 and √ ___

49 .Thus, √

___ 47 lies between 6 and 7.

From the table: √ ___

47 = 6.86Thus √

_____ 0.47 = 6.86 × 1 __

10 = 6.86

____ 10

= 0.686

Exercise Eg

Use the tables at the back of the Student’s Book for this exercise.

Between which whole numbers does the square root of these numbers lie?a 7.6 b 9.6 c 17d 27 e 34 f 30g 48 h 60 i 75j 69 k 54 l 83

Find the square roots of these numbers:a 9 b 90 c 2.8d 28 e 4.7 f 47g 36 h 3.6 i 25j 2.5 k 6.3 l 63

Find the square roots of these numbers:a 7 b 70 c 700d 7 000 e 2.9 f 29g 290 h 2 900 i 29 000j 250 k 2 500 l 38m 380 n 3 800 o 38 000p 10 q 100 r 1 000s 2 t 430 u 500v 72 000 w 8 400 x 960

Round off these numbers to 2 s.f. Then find their approximate square roots.a 9.28 b 78.3 c 463d 8.45 e 61.3 f 613g 59.4 h 5.86 i 5.003j 500.3 k 6 394 l 1 982

Is √ ___

10 a good approximation for π?6 A square has an area of about 45 cm2.

a Find the length of one of its sides.b Hence find its perimeter to the nearest cm.

a Use the square root table to find m, if m = √

___ 41 .

b Using the value of m, use the squares table to find the value of m2.

c What do you notice? Find the square roots of these numbers:

a 0.21 b 0.34 c 0.43d 0.62 e 0.55 f 0.73g 0.86 h 0.92 i 0.69j 0.78 k 0.59 l 0.98

Summary

• In many everyday situations, people use calculators to do calculations.

• Where calculators are not available, it is possible to use tables to make calculations easier.

• Whether using calculators or tables, it is very useful to be able to estimate the size of the likely outcome of a calculation and where the decimal point is likely to be. This helps to detect errors (either from pressing the wrong key or from using a table incorrectly).

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