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2. mathematics
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Solutions for all
Mathematics
Grade 7
Learner’s Book
Schools Development Unit
SFA-Maths-Gr-7-LB-4980086
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Solutions for all Mathematics Grade 7 Learner's Book
© Schools Development Unit, 2013
© Illustrations and design Macmillan South Africa (Pty) Ltd, 2013
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First published 2013
13 15 17 16 14
0 2 4 6 8 10 9 7 5 3 1
Published by
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Typeset by MPS
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Contents
Term 1
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
Working with whole numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Multiples and factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Ratio, rate and finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Comparing and representing numbers in exponential form . . . . . . . . . 43
Calculations with numbers in exponential form . . . . . . . . . . . . . . . . . . . 51
Measuring and drawing angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Classifying triangles and quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . 78
Circles, congruent and similar shapes . . . . . . . . . . . . . . . . . . . . . . . . . 93
Revision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Term 2
Unit 11
Unit 12
Unit 13
Unit 14
Unit 15
Unit 16
Unit 17
Unit 18
Common fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Percentage and more financial mathematics . . . . . . . . . . . . . . . . . . . . 131
Decimal fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Calculations with decimal fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Functions and relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Area and perimeter of 2-D shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Surface area and volume of 3-D objects . . . . . . . . . . . . . . . . . . . . . . . . 194
Revision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Term 3
Unit 19
Unit 20
Unit 21
Unit 22
Unit 23
Unit 24
Unit 25
Unit 26
Unit 27
Unit 28
Patterns and relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Functions and relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Algebraic expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Algebraic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Transformation I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
Transformation II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Classifying 3-D objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Building 3-D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Revision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
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Term 4
Unit 29
Unit 30
Unit 31
Unit 32
Unit 33
Unit 34
Unit 35
Unit 36
Unit 37
Unit 38
Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
Patterns and relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
Functions and relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
Algebraic expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
Number sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Collecting, organising and summarising data . . . . . . . . . . . . . . . . . . . . 368
Representing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
Analysing, interpreting and reporting data . . . . . . . . . . . . . . . . . . . . . . 402
Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Revision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
Mental mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
Questions and activities purposed for enrichment are indicated
using this icon. Use these questions and activities to challenge
learners’ thinking about a concept being learnt.
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Term
1
Unit 1 Working with
whole numbers
In this unit you will:
•
•
recognise and use the properties of numbers in calculations
recognise and perform calculations with whole numbers, including multiple
operations using a variety of strategies.
Ordering and comparing
whole numbers
Getting started
1. Write the numbers in the place value columns. The first number has been
done for you.
a) 509 432 061
b) 469 322 954
c) 520 496 642
d) 986 052 004
e) 305 000 563
f) 582 052 039
Hundred
Ten
Hundred
Ten
Millions
Thousands Hundreds Tens Ones
millions millions
thousands thousands
5
0
9
4
3
2
0
2. Fill in <, = or > between the following:
247 898 000
b) 784 109 400
a) 247 889 000
6
1
785 190 400
3. Arrange the following numbers from the smallest to the biggest:
a) 456 734; 3 445 237; 7 645; 3 465 122; 389 456 332; 45 902
b) 2 385 703; 23 875 485; 67 383 586; 83 348; 487 483 458; 392 923
4. Arrange the following numbers from the biggest to the smallest:
a) 546 788; 234 577; 3 451 129; 3 451 139; 4 387 193; 4 387 196
b) 536 344 587; 429 224 943; 495 956 396; 206 496 105; 479 566 396
5. Round off each of the following numbers to the nearest:
i) 5
ii) 10
iii) 100
iv) 1 000
c) 5 555
a) 2 378
b) 14 324
6. Fill in the missing numbers in the sequences:
a) 456 735; 456 835; 456 935;
b) 756 221; 757 221;
; 457 135;
; 759 221;
;
;
; 761 221;
;
Term 1 • Unit 1
•
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7. Copy and complete the number grid:
3 979 997
3 989 999
3 999 999
4 010 001
8. Fill in the missing numbers on the number lines:
a)
27 497
27 499
249 998
b)
250 002
9. What whole number, if any, is halfway between the following pairs?
a) 4 253 and 4 257 b) 20 438 and 20 430 c) 471 340 and 471 350
Activity 1.1
Which operation comes first?
1. Nombeka tries to do the calculation 8 − 4 × 2.
a) Can both of Nombeka’s answers be correct?
I can’t decide which
method is correct.
When I group the 8 − 4, I get an answer of 8.
(8 − 4) × 2
4×2
=8
When I group the 4 × 2, I get an answer of 0.
8 − (4 × 2)
8−8
=0
b)
Which of Nombeka’s answers is correct? Explain why.
Both answers could
be correct, but that
would cause confusion.
So, we have an
important rule that
says:
Remember this
rule because you
need to use it when
you do all your
calculations!
•
c)
2
First do all the multiplication and
division. Work from left to right. Then
do all the addition and subtraction,
also working from left to right.
Using this rule, which answer is correct? Show your calculations.
Term 1 • Unit 1
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The teacher writes the sum again. This time she uses brackets: (8 − 4) × 2.
Which answer is correct now, 8 or 0?
We must do the
calculation in brackets
first. My answer is:
(8 − 4) × 2
4×2
=8
The teacher agrees.
Key ideas
•
Different answers are possible if we do not use brackets or rules. This would
cause too much confusion. To calculate answers correctly, we use the rules for
order of operations.
• When we use brackets to group 8 − 4, in (8 − 4) × 2, the answer can only be 8.
• Using brackets prevents having two correct answers to the same calculation.
• The rules for order are:
1. Brackets first.
2. Then multiplication and division operations. Work from left to right.
3. Then addition and subtraction. Work from left to right.
So, 8 + 4 × 2 = 8 + (4 × 2) NOT (8 + 4) × 2.
Exercise 1.1
Order of operations
1. Complete these calculations. Use the rules for order of operations.
a) 15 × 2 + 3 × 10
b) 15 × (2 + 3) × 10
c) (2664 ÷ 9) + 3 × (189 + 142 + 533) d) (1 + 9) × (6 × 6)
e) 199 − 255 ÷ 5
2. a) Rewrite: 27 + 16 × 5 − 2 so that your answer is either 129 or 105.
b) Calculate 27 + 16 × 5 − 2.
3. a) Manie buys eight avocado pears for R3 each. He also buys 16 loaves of
bread for R4 a loaf. Which expression shows how much he paid?
i) 8 + 16 × 3 + 4
ii) 8 × 3 + 16 × 4
b) Manie pays with a R100 note. Write an expression to show how much
change he gets.
Term 1 • Unit 1
•
3
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Activity 1.2
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Switching whole numbers
1. Use only one operation, like +. Can you swap the whole numbers and still get
the same answer?
a) Is 2 + 3 equal to 3 + 2?
b) Is 9 + 4 equal to 4 + 9?
Try this with other whole numbers. Does it work?
2. Can you swap these whole numbers and still get the same answer?
a) Is 2 × 3 equal to 3 × 2?
b) Is 9 × 4 equal to 4 × 9?
c)
Is 6 ÷ 3 equal to 3 ÷ 6?
e) Is 6 − 3 equal to 3 − 6?
Try this with other whole numbers.
d)
Is 10 ÷ 2 equal to 2 ÷ 10?
f)
Is 10 − 2 equal to 2 − 10?
Key ideas
•
You can swap whole numbers across addition (+) and multiplication (×)
without changing the answer. It does not matter in which order you add or
multiply the numbers.
These rules do not apply for subtraction (−) or division (÷).
•
Exercise 1.2
Commutative and associative properties
1. a) Calculate.
i) (15 + 7) + 8
iii) 3 × (4 × 2)
ii) 15 + (7 + 8)
iv) (3 × 4) × 2
b) What do you notice? Write a sentence.
2. Complete:
a) If 33 + 99 = 132, then 99 + 33 =
b) If 33 + 99 = 132, then 132 − 99 =
and 132 −
c) If 51 + (19 + 46) = 116, then (51 + 19) + 46 =
d) If 22 × 5 = 110, then 110 ÷ 22 =
and 110 ÷ 5 =
e) If 238 ÷ 14 = 17, then 14 × 17 =
and 238 ÷ 17 =
•
4
Term 1 • Unit 1
= 99
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Activity 1.3
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Writing sums in different ways
1.
Is 2 × (3 + 5) the same as (2 × 3) + (2 × 5)? Calculate to check.
2.
Look at the following examples. We have kept the same operations, × and +.
a) Is 3 × (20 + 7) the same as (3 × 20) + (3 × 7)?
b)
3.
Is 8 × (100 + 70 + 4) the same as (8 × 100) + (8 × 70) + (8 × 4)?
Will 2 + (3 × 5) be the same as (2 + 3) × (2 + 5)?
Key ideas
•
•
When the multiplication operation (×) is outside a bracket and the addition (+)
operation is between whole numbers inside the brackets, we can multiply each
of the numbers inside the brackets by the number outside the brackets first
and then add them.
Example: 5 × (1 + 2 + 3) is the same as (5 × 1) + (5 × 2) + (5 × 3).
This rule does not apply when the addition operation (+) is outside the bracket
and the multiplication operation (×) is between whole numbers inside the
brackets.
We can use this rule to work out multiplication problems, for example
5 × (426) = 5 × (400 + 20 + 6) = (5 × 400) + (5 × 20) + (5 × 6)
= 2 000 + 100 + 30 = 2 130
Exercise 1.3
The distributive property of multiplication
over addition and subtraction
1. Calculate the following multiplication sums. Use the above method.
a) 7 × 342
b) 3 × 945
c) 9 × 672
2. Can you rewrite 4 × (12 + 8) as (4 × 12) + (4 × 8)? Show how you check this.
3. Rewrite 4 × (12 − 8) so that your answer stays the same. Show your working.
4. Complete:
a) 4 × (12 + 9) = (4 ×
) + (4 ×
b) (9 × 64) + (9 × 36) = 9 × (
) = 48 +
+ 36) = 9 ×
=
=
Term 1 • Unit 1
•
5
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Activity 1.4
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One and zero
There are special rules that apply to the number zero and the number one.
1. What happens to numbers when you multiply or divide by 1? Solve these
problems. Check your answers using your calculator.
28 ÷ 1 =
8 344 × 1 =
a)
28 × 1 =
e)
What can you conclude about multiplying or dividing by 1?
b)
c)
8 344 ÷ 1 =
d)
2. What happens to numbers when you add or subtract 0? Solve these problems.
Check your answers using your calculator.
b) 429 − 0 =
c) 5 360 + 0 =
a) 429 + 0 =
d)
What can you conclude about adding or subtracting 0.
3. When we say that 6 × 4 = 24, we mean that 4 + 4 + 4 + 4 + 4 + 4 = 24.
What happens when we multiply by 0?
a) What does it mean when we say 4 × 0?
b)
What can you conclude?
8
8
We can write 8 ÷ 4 as . You know that = 2. So 4 × 2 = 8.
4
4
24
= as × 6 = 24. The answer is 4.
We can write
6
What happens when we divide 0 by a whole number?
a) Rewrite this division problem as a multiplication problem. Then, write the
132
answer:
=
11
b) Compare your answer to Chad’s answer:
4.
132
=
11
is the same as 11 ×
= 132
11 × 12 = 132. So 132 ÷ 11 = 12
c) Do these in the same way:
i) 0 ÷ 4
ii) 0 ÷ 8
iii)
0 ÷ 58
iv)
0 ÷ 347
d) Check your answers using your calculator.
e) What can you conclude?
5. What happens when we divide by zero? Use Chad’s method to find these
answers:
a)
i)
4÷0
ii)
8÷0
iii)
b) Check your answers using your calculator.
•
c) What can you conclude?
6
Term 1 • Unit 1
58 ÷ 0
iv)
347 ÷ 0
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Key ideas
•
•
•
•
•
We call the number 1 the identity element for multiplication and division.
When you multiply or divide by 1, the number stays the same.
Zero (0) is the identity element for addition and subtraction.
When you add or subtract 0 from any number, the number stays the same.
When you multiply any number by 0, your answer is always 0.
When you divide 0 by any number, your answer is always 0.
When you divide a number by 0, ask the question ‘What number do I need to
multiply 0 by to get my number?’ We cannot answer that. We say that the
answer is undefined.
Exercise 1.4
The properties of 0 and 1
1. Write the correct number in the space.
= 482
b) 629 ÷
a) 482 +
× 1 = 1 340
d) 45 ÷ 0 =
c)
e) 1 547 ×
=0
f) 365 −
= 629
=0
2. If a is any whole number, complete the equation. Choose an answer from the
box:
a
0
1
No solution
a) a × 1 =
d)
+0=a
Activity 1.5
b)
e)
÷a =0
÷1=a
c) a × 0 =
f) a ÷ 0 =
Addition and subtraction of whole numbers
Look at the population figures for four towns in the year 2013.
1. a) What is the total population of Orangeville and
Bollywood?
Town
Calculate your answer. Compare your answer
Orangeville
to Deepan’s answer.
Kanaladorp
Deepan:
Tinseltown
1 4 5 4 4
+
9 5 0 0
2 4 0 4 4
Bollywood
Population
14 544
22 905
188 044
9 500
Deepan arranged the numbers in columns. He made sure that the units or
ones were underneath each other. Deepan did the same for the other
columns.
Term 1 • Unit 1
•
7
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b)
What would have happened if the digits were not in their correct columns?
c)
Which column did Deepan add first?
d)
Deepan had 5 + 5 in the hundreds column. Yet, he wrote 0 as its answer.
Explain why Deepan did not write 10.
a)
Calculate the total population of the four towns.
b)
Calculate the difference in population between Tinseltown and Kanaladorp.
Key ideas
•
When you add in columns, first add the ones, then the tens, then the
hundreds, and so on.
Write the numbers in the correct columns, otherwise your answer will be wrong.
When you subtract in columns, first subtract the ones, then the tens, then the
hundreds, then the thousands, and so on.
You can use subtraction to check addition and vice versa. We call addition and
subtraction inverse operations.
•
•
•
Exercise 1.5
The column method for addition
and subtraction
1. Work out the answers. Use the column method. Do not use a calculator.
b)
2 1 4 6 2
a)
6 3 4 8
+
5 3 2 1
+ 1 5 4 9 9
+ 8 0 0 5 9
c) 15 431 + 5 806 + 1 949
e) 22 840 − 15 399 − 244 + 1 455
d) 2 844 + 43 + 8 649 + 520
f) 143 872 + 297 531 − 189 788
2. Do not use a calculator. Find:
a) 354 827 + 492 857 + 4 854
b) 768 354 + 54 731
3. Do not use a calculator. Find:
a) 784 566 − 723 944
b) 244 533 − 199 403
•
8
Term 1 • Unit 1
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Activity 1.6
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Multiplication of whole numbers
Mr Matabane gives his Grade 7 class the multiplication problem 159 × 27.
Elton writes:
27 × 159 = (20 × 159) + (7 × 159)
= 20 × (100 + 50 + 9) + 7 × (100 + 50 + 9)
= (20 × 100) + (20 × 50) + (20 × 9) + (7 × 100) + (7 × 50) + (7 × 9)
= 2 000 + 1 000 + 180 + 700 + 350 + 63
= 3 180 + 1 113
= 4 293
Grace writes:
1 5
×
2
1 1 1
+ 3 1 8
4 2 9
9
7
3
0
3
Grace arranged the numbers in columns. She made sure
that the units (or ones) were beneath each other. Grace
did the same for the other columns.
Remember that you can
swap whole numbers across
multiplication without
changing the answer.
1.
What would have happened if the digits were not
in their correct columns?
2.
What did Grace multiply first?
3.
Grace puts a 0 in the units column, before starting to multiply by
the 2 in the tens column. Why does Grace do this?
4.
Compare Grace’s and Elton’s methods. What is similar
about them?
Exercise 1.6
Practise multiplication
1. Calculate. Use either Grace’s or Elton’s method.
a) 2 364 × 37
b) 1 226 × 82
c) 3 437 × 24
2. Do not use a calculator. Find:
a) 236 × 35
b) 4 462 × 52
c) 5 349 × 21
Term 1 • Unit 1
•
9
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Activity 1.7
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Long division
Mr Matabane asks his Grade 7s to find the answer to: 5 184 ÷ 9.
Janine writes her answer on the board. Then Alison asks Mr Matabane if she may
write a different method on the board. Here are Janine’s and Alison’s answers:
Janine
5 184
500 9 4 500
684
70 9 630
54
69
54
5 184 9 500 70 6 576
576
9 5 184
4 500
684
630
54
54
5 184 9 576
Alison
1. Discuss and compare their methods.
2. Janine uses repeated multiplication and subtraction to solve the problem. Alison
uses long division. Which method do you prefer?
Key ideas
•
•
Long division involves repeated division and subtraction.
You can use multiplication to check division. We call division and multiplication
inverse operations.
Exercise 1.7
Practise long division
1. Use either Janine’s or Alison’s method to solve these:
a) 350 ÷ 14
b) 4 800 ÷ 15
c) 8 100 ÷ 25
d) 41 124 ÷ 6
e) 6 248 ÷ 11
f) 18 765 ÷ 15
2. Do not use a calculator. Find:
a) 768 ÷ 24
b) 850 ÷ 25
d) 1 512 ÷ 28
e) 5 168 ÷ 17
•
g) 9 936 ÷ 48
10
Term 1 • Unit 1
h) 6 900 ÷ 150
c) 286 ÷ 26
f) 2 106 ÷ 39
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Activity 1.8
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Rounding off and compensating as calculation
strategies
There are times when we perform mental or quick ‘pencil and paper’ calculations that
we choose to change the numbers in the calculation to easier ones; e.g. let’s say we
want to find the value of: 232 × 588
On a calculator, this calculation is no problem, and we get 136 416.
If we did a quick mental calculation, we can simplify the task by:
•
•
•
rounding down 232 to the nearest hundred: 200
rounding up 588 to the nearest hundred: 600
multiplying the ‘easier’ numbers: 200 × 600 = 120 000
So the strategy for multiplication is that when we simplify the numbers in the calculation, and make one bigger, then the other must be made smaller in order to compensate for the one we increased.
For division, the strategy is different: if one number is increased, the other is also
increased.
Keep in mind that these calculation strategies are estimations and do not give accurate
answers. Also, if you round up or down to numbers that are much bigger or smaller
than the original ones, your answers may also be further from the correct one.
1.
How does the correct answer to 232 × 588 compare with the estimated one
obtained from 200 × 600?
2.
If you made the following estimate based on rounding off to the nearest 10 and
compensating, how does the estimate compare with the correct value?
230 × 590
3.
Round up or down to an appropriate value in order to compensate for the
rounding in the following mental calculations (given in the table on the next
page). You may then use a calculator to check how close your estimate is to the
actual answer. The first and third one has been done for you.
4.
We can also use the idea of compensation as a strategy for finding the actual
answer to a sum – not only an estimate.
For example, the sum:
121 + 449
= (121 + 9) + (449 − 9) Note that we have added 9 and then subtracted 9
= 130 + 440
9−9=0
In effect, we have changed nothing!
= 570.
Now use the compensation strategy to calculate the answers to:
a) 548 + 762
b) 1 564 − 366
c) 847 − 353
d) 4 341 + 5 029
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Operation
Estimate
Checked against answer on
calculator (rounded to two decimal
places)
36 ÷ 17
40 ÷ 20 = 2
2,12
90 × 90 = 8 100
8 091
85 ÷ 25
87 × 93
89 + 57
103 − 23
76 ÷ 37
124 × 17
458 + 93
1 015 − 717
Key ideas
•
•
•
It is useful to round off in a way that reduces the magnitude of overestimation
or underestimation. This is called compensating for the rounding error.
Compensation can also be used by adding and then subtracting the same
amount (in other words adding zero) from the numbers in a sum in such a way
that number bonds are formed that then make it easier to calculate the answer.
In general, remember the following ideas for compensation. They are related
to the operation being used:
Operation
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12
Remember . . .
‘÷’
• If the dividend is rounded up, then round up
the divisor by an appropriate amount
• If the dividend is rounded down, then round down
the divisor by an appropriate amount
‘×’
• If rounding up, try to round down by an equal
amount in the other numbers
• Vice versa for rounding down
‘+’
• Same as for multiplication
‘–’
• Round all subtracted terms either up or down by the
appropriate amounts
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Practising estimation by rounding off
and compensation
Work with a friend to discuss and answer the following questions.
Mr Matabane has to order paper for his next lesson. The paper is packed in batches
of 30 sheets. There are 43 learners in the class. They each need two sheets of paper
for the lesson.
1.
Guess how many batches of paper Mr Matabane will order. Explain how you
reached your answer. Mr Matabane reasoned as follows:
I round off the number
of learners to 45. I need
1 1 batches of paper to
2
give each learner one
sheet. So if I order 3
batches, I will have enough
to give them two sheets
of paper each.
Did you get the same answer as
Mr Matabane?
2. Write an expression. Show how
many sheets of paper would be
left over after the lesson.
3. There are seven children
absent. Estimate whether
Mr Matabane can order fewer
batches of paper. Explain your
reasoning.
4. Mrs Sithole asks if Mr Matabane can
order for her three classes as well. There
are 107 learners in her three classes. Rewrite the sum 107 + 43 in such a way
that it can help you to add the two numbers more easily.
5. How many batches should Mr Matabane order now so that each learner will get
two sheets of paper?
Key ideas
•
•
It is useful to find an estimate when we work with large numbers. Estimating
helps us to judge whether our answer could possibly be right.
If you round the figures off so that they are higher than the actual figures, you
are overestimating. If you round the figures off so that they are lower than the
actual figures, you are underestimating.
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Whether it is best to overestimate or underestimate depends on the
circumstances.
For example, you would overestimate how much money you need for
groceries. This is so that you do not run out of money when you shop. You
would underestimate how much salt to put in food. This is because you can
always add more if you need to, but you cannot take any out if you put in too
much to begin with.
The strategy where you take away part of one number and add it to another to
make a calculation easier is called compensation. Compensation is another
strategy you can use for estimation.
•
•
Exercise 1.8
Working problems with estimation,
rounding off and compensating
1. Telkom is delivering new telephone directories. The delivery team has to
cover three suburbs today. The population of the three suburbs is: 14 789,
4 485, 6 916. Help the delivery team. Use Mr Matabane’s method.
a) Estimate how many directories the delivery team would take for the day.
b) Was your estimate enough? Check by writing out an expression.
Calculate how many directories the delivery team actually needed.
c)
i)
What do the words underestimate and overestimate mean?
ii)
Would it be better to underestimate or overestimate in this case?
Explain why you say so.
2. Ms Mandoza enters an agreement with Safetyfirst Protection Services
(SPS). SPS will protect her property for R97,00 per month. Ms Mandoza
wants to make a payment for the year. SPS told her the annual cost is
R1 800. Ms Mandoza immediately disagrees with their total.
a) Ms Mandoza realised very quickly that the figure was wrong. What would
her estimated total have been?
b) Use a calculator. Calculate the actual cost.
c) What was the difference between Ms Mandoza’s estimate and the actual
cost?
d) The joining fee for safety first is R223. Use the strategy of compensation
to estimate the first month’s payment.
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3. Estimate by rounding up or down in a way that compensates for the rounding:
a) Round off the numbers in the calculation to the nearest 10 and then
estimate (without using a calculator) the result of the following operations.
b) State whether it is an underestimate, or overestimate and then check the
actual answer by using a calculator.
4. Now use the compensation strategy to calculate the actual answers to:
a) 728 + 1 232
b) 3 654 − 3 246
c) 671 − 269
d) 12 413 + 8 247
Operation
Estimate
Underestimate/
Overestimate?
Answer on calculator
(2 dec. pl.):
719 ÷ 11
87 × 9
91 + 54
103 − 24
77 ÷ 37
123 × 19
348 + 33
1 003 − 798
Activity 1.10
Multiplication using doubling and halving
We take any number and double it. We then halve the answer.
What number do we get?
Let us try 15.
Choose another number.
Try it.
Double 15 is 30.
Half of 30 is 15.
Each time you × 2
and then ÷ 2, you
end up with the
number you started
with!
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We say that halving is the inverse of doubling. How can we use halving and doubling
to help us multiply two numbers?
Consider: 15 × 4 = 60. We know that if we × 2 and then ÷ 2, we do not change the
answer. But let us check: 15 × 4 × 2 ÷ 2 = 60
We can also write this multiplication sum as: (15 × 2) × (4 ÷ 2) = 30 × 2 = 60.
We have doubled the first number and halved the second number.
1.
Could we have also written (15 ÷ 2) × (4 × 2)? Check the answer.
2.
Why have we doubled the first number and halved the second number?
3.
Two even numbers multiplied, for example 26 × 4. You want to simplify the sum.
Which number will you halve? Which number will you double?
4. Does doubling and halving simplify a multiplication problem when both numbers
are odd numbers, for example 27 × 7?
Exercise 1.9
Simplifying multiplication sums
1. Rewrite these multiplication sums as simpler sums. Use doubling or halving.
a) 4 × 18 = 8 × 9 = 72 b) 5 × 16
c) 3 × 24
d) 4 × 32
e) 5 × 48
f) 22 × 4
g) 32 × 500
h) 24 × 50
i) 282 × 5 000
2. Which of these problems are made easier by using doubling and halving?
a) 2 × 24
b) 8 × 54
c) 21 × 7
d) 9 × 42
e) 8 × 25
f) 7 × 68
Activity 1.11
Solving problems using a calculator
Solve the following problems. Use a calculator. For each problem, first write down the
expression that you need to enter into your calculator.
1. A car odometer shows you how many kilometres the car has travelled. Before
leaving on holiday, the odometer reading in Vusi’s car was 115 427 km. On his
return from holiday, the odometer reading was 116 719 km.
a) How many kilometres did Vusi drive during his holiday?
b)
2.
The census shows that two towns have populations of 89 743 and 121 492.
a) What is the total population of the two towns?
•
16
Check your answer. Use the inverse operation from the one you used in a).
Show your working.
b)
What is the difference in their populations?
c)
Check your answers for a) and b).
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3.
A city has 1 287 644 people. 345 612 are men and 356 932 are women. How
many children are there?
4.
A dam has 3 400 000 m3 of water in it. Owing to a drought, no more rainfall is
expected for approximately 5 months.
a) Consumption of water from the dam is 35 400 m3 per day. How many days
will the water last at this rate of consumption? Give your answer rounded off
to the nearest day.
b)
The council appeals to everyone to cut down on their water usage. Water
usage drops to 23 600 m3 per day. How many days will the water last now?
c) Will the water still run out before the expected rainfall?
d) To make the water use sustainable over six months, what must the daily
consumption rate be?
Key ideas
•
•
The Ishango and Lebombo bones are part of the history and indigenous
knowledge systems of Africa and Southern Africa that contributed to the
advancement of knowledge all over the world – we cannot be sure but the
markings on the bones suggest possible understandings of number concepts,
basic numeral systems, basic lunar calendar recording or even basic
arithmetic.
By Indigenous Knowledge (IK) we mean the local or traditional knowledge of
the people of Southern Africa and Africa (or other societies) that has been
developed over many years (even thousands of years). It is often not recorded
in books, but passed down from one generation to the next via practical
learning or oral (verbal) communication and teaching.
Summary
•
The rules for solving a sum in the correct order are:
1. Brackets first.
2. Then multiplication and division operations. Work from left to right.
3. Then addition and subtraction. Work from left to right.
•
•
•
It does not matter in which order you add or multiply the numbers. But for
subtraction and division, the order is important.
We call the number 1 the identity element for multiplication and division.
Zero (0) is the identity element for addition and subtraction.
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When you multiply any number by 0, your answer is always 0.
When you divide 0 by any number, your answer is always 0.
When you divide a number by 0, the answer is undefined.
Addition and subtraction are inverse operations.
Division and multiplication are inverse operations.
Estimating answers when we are working with large numbers helps us judge
whether our answer could possibly be right.
Two estimation strategies are rounding off and compensation.
If you round the figures off so that the answers are higher than the actual
figures, you would be overestimating. If you round the figures off so that the
answers are lower than the actual figures, you would be underestimating.
•
•
•
•
•
•
•
Check what you know
1. Consider the expression 50 − 30 ÷ 5.
a) If you want your answer to be 4, where will you place brackets?
b) If you want the answer to be 44, where will you place brackets?
c) You use the rule that says ‘First multiply or divide, then add or subtract’.
Can you still get two answers?
d) Use the rule in c) to work out 348 + 15 × 20. Is your answer
(348 + 15) × 20 or 348 + (15 × 20)?
2. Calculate. Write whether each statement is true or false.
a) 10 + 20 + 50 = 50 + 20 + 10
b) 52 + 32 − 11 = 52 + 11 − 32
c) 80 ÷ 8 × 4 = 80 ÷ 4 × 8
d) 40 × (8 + 4) = 40 × 8 + 40 × 4
3. Are the following statements true or false? If false, give the correct answer.
a) 12 × 1 = 1
b) 11 × 0 = 0
c) 24 × 0 = 24
d) 1 × 9 = 0
4. Write the following expressions differently, but do not change the answer:
a) 36 × 2 × 10
b) 34 × (7 − 4)
c) (2 × 3) + (2 × 4)
5. Calculate:
a) 545 883 + 783 372
c) 344 856 − 121 008
6. Calculate:
a) 364 × 56
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Term 1 • Unit 1
b) 654 489 − 344 218
d) 745 563 − 674 333
b) 1 256 × 74
c) 2 347 × 74
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7. Calculate:
a) 5 002 ÷ 61
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b) 12 888 ÷ 24
c) 10 902 ÷ 46
8. Lynn sells socks to raise funds for a holiday. She sells 11 pairs for R17 per
pair. Lynn estimates that she has made R150.
a) What estimated figures did Lynn use?
b) Calculate how much money she has made. Use your calculator.
c) How close was Lynn’s estimate?
d) Use the strategy of compensation to estimate how much Lynn has sold
altogether if she earned R243 the day before.
9. Use doubling and halving to work out the answers:
a) 5 × 66
b) 5 × 14
c) 20 × 24
d) 14 × 17
e) 36 × 15
f) 420 × 50
10. Sibongi is very angry. She signed up for a self-defence course for three
months. The course cost R148 per month with a registration fee of R40,00.
Sibongi made her first payment, which consisted of the registration fee and
the first month’s payment. Then she received an account for R376,00.
Sibongi argued that she only owed them R296,00.
a) Write out the expression that shows how Sibongi calculated her
outstanding balance.
b) Write out the expression that shows how the self-defence company
calculated her outstanding balance.
c) Who is correct? Explain why.
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Term
1
Unit 2 Multiples and
factors
In this unit you will:
•
recognise, name and find factors, prime factors and the highest common
factor (HCF) of a group of numbers
recognise, name and find multiples and the lowest common multiple (LCM)
of a group of numbers.
•
Getting started
Multiples, factors and prime numbers
1. The picture shows the first 14 multiples of 6.
a) Write down the first ten multiples of 9.
b) Write down the numbers that are
multiples of both 6 and 9.
c) Explain what a multiple is.
2. This picture shows the factors of 16.
a) Write down the factors of 12.
b) Write down the factors that are
common to both 12 and 16.
c) Explain what a factor is.
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3. This picture shows prime numbers up to 30.
a) Write down all the factors
for the prime numbers in
the picture.
b) Write down all the prime
numbers between 30
and 50.
c) Explain what a prime
number is.
d) Which of the following
numbers are prime numbers?
34
29
76
57
99
4. List the factors of the following numbers:
a) 20
b) 36
c) 64
Activity 2.1
23
d) 98
Finding factors of 2-digit and 3-digit whole
numbers
Ntsiki and Thobeka want to find all the factors of 128. Which method do you prefer?
Ntsiki writes:
I want to find the factors of 128. I will look for all the pairs of numbers that multiply to
give 128.
1 × 128 = 128; 2 × 64 = 128; 4 × 32 = 128; 8 × 16 = 128
All the numbers in the pairs will be the factors.
Thobeka writes:
I want to find the factors of 128. I will find all the numbers that divide into 128 without
leaving a remainder.
I can divide 128 by 1: 128 ÷ 1 = 128 and 128 ÷ 128 = 1. So 1 and 128 are factors of 128.
I can divide 128 by 2: 128 ÷ 2 = 64 and 128 ÷ 64 = 2. So 2 and 64 are factors of 128.
I can divide 128 by 4: 128 ÷ 4 = 32. So 4 and 32 are factors.
I can divide 128 by 8: 128 ÷ 8 = 16. So 8 and 16 are factors.
Ntsiki and Thobeka list the factors of 128 as: 1; 2; 4; 8; 16; 32; 64; 128.
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Key ideas
•
A factor of a whole number divides into that number without leaving a
remainder. Example: The factors of 12 are: 1, 2, 3, 4, 6, 12.
When we arrange the factors from smallest to biggest, a pattern forms showing
the pairs of factors that are multiplied to give the number. Example:
•
1; 2; 4; 8; 16; 32; 64; 128
Exercise 2.1
Factors
1. Find all the factors of the following numbers.
a) 36
b) 21
c) 120
d) 200
e) 921
f) 60
2. Fill in the missing factors.
a) The factors of 24 are 1; 2;
b) The factors of 144 are
Activity 2.2
;
; 4;
;
;
; 8;
;
;
; 24
;
; 12; 16; 18; 24; 36; 48; 72; 144
Finding common factors and the HCF
Answer the questions.
1. You can see the factors and the common factors of 28 and 42.
Factors of 28: 1; 2; 4; 7; 14; 28
Factors of 42: 1; 2; 3; 4; 6; 7; 14; 21; 42
Common factors of 28 and 42: 1; 2; 7; 14.
a) What do we mean by ‘the common factors of 28 and 42’?
b) Which of the common factors of 28 and 42 is the biggest or highest?
2. a) Find the common factors of 180 and 675.
b) Which of the common factors is the highest common factor?
Key ideas
•
•
•
•
Common factors of two numbers are the factors that are the same for both.
A common factor of three whole numbers must be a factor of all three.
We write the highest common factor as HCF.
1 is a common factor for all numbers. We do not list it as a common factor.
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Worked example
1. Find the common factors of 8, 12 and 20.
2. Find the HCF of 8, 12 and 20.
SOLUTION
1. Factors of 8:
1; 2; 4; 8
Factors of 12:
1; 2; 3; 4; 6; 12
Factors of 20:
1; 2; 4; 5; 10; 20
The common factors of 8, 12 and 20 are 2 and 4.
2. The highest common factor of 8, 12 and 20 is 4.
Exercise 2.2
Factors and the HCF
1. Find the highest common factor (HCF) of the following:
a) 36 and 120
b) 120 and 200
c) 21, 75 and 921
2. a) What are the common factors of 156 and 64?
b) What is the highest common factor of 156 and 64?
3. Two whole numbers multiply to give 72. What whole numbers can they be?
List all possible answers.
Activity 2.3
1.
We know that:
•
•
2.
Multiples and the Lowest Common Multiple
Multiples of 200 are 200, 400, 600, 800, and so on.
Multiples of 150 are 150, 300, 450, 600, and so on.
a)
What will the next five multiples of 200 and 150 be? Write them down.
b)
Which multiples of 200 and 150 are common multiples? Common multiples
means multiples that are the same for both 200 and 150.
c)
Which one is the smallest or lowest common multiple?
a)
Write down the first six multiples of 6, 8 and 12.
b)
List the common multiples.
c)
Which one is the lowest common multiple?
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Key ideas
A multiple of a number is 1 × the number or 2 × the number or 3 × the number,
or . . . 97 × the number, etc. Example: 48 is a multiple of 6 because 48 is 8 × 6.
Common multiples of two or more numbers are the multiples that are the same
for the numbers. Example: A common multiple of 2, 4 and 5 is 20.
We write the lowest common multiple as LCM.
•
•
•
Worked example
1. Find the first five multiples of 120, 150 and 300.
2. Find the LCM of 120, 150 and 300.
SOLUTION
1. Multiples of 120: 120, 240, 360, 480, 600
Multiples of 150: 150, 300, 450, 600, 750
Multiples of 300: 300, 600, 900, 1 200, 1 500
2. The lowest common multiple of 120, 150 and 300 is 600.
Exercise 2.3
Multiples and the LCM
1. List the first 10 multiples of the following numbers:
a) 9
b) 10
c) 12
d) 15
e) 20
f) 25
2. Find the lowest common multiple (LCM) of the following groups of numbers:
a) 10 and 15
b) 3 and 12
c) 3 and 5
d) 2, 3 and 8
e) 2, 3 and 4
f) 3, 6 and 9
g) 15 and 30
h) 20 and 25
3. Is 104 a multiple of 13 or 4 or both? Explain your answer.
4. a) Find two whole numbers that will be multiples of both 11 and 15.
b) Find the LCM of 11 and 15.
Activity 2.4
Whole numbers as products of prime factors
Express 12 as a product of its prime factors.
The factors of 12 are 1; 2; 3; 4; 6; 12.
Of these factors, only 2 and 3 are prime numbers.
2 × 3 = 6 but if we multiply by 2 again we get 12.
We can write 12 = 2 × 2 × 3.
We say that 2 and 3 are the prime factors of 12.
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1.
Express 625 as a product of its prime factors.
2.
Compare your answer to Nandipha’s and Francina’s answers.
Nandipha’s answer:
625 ÷ 5 = 125
Francina’s answer:
5
625
125 ÷ 5 = 25
5
125
25 ÷ 5 = 5
5
25
5÷5=1
5
5
625 = 5
1
5 × 5 × 5 × 5 = 625
625 = 5 × 5 × 5 × 5
Whose method do you prefer?
Key ideas
•
•
•
A prime number has only two factors – the number itself and 1.
We can write any number as the product of its prime factors. Sometimes we
may need to use the factor more than once.
Numbers that have more than two factors are not prime numbers. We call them
composite numbers. For example, 4, 6, 8, 9, etc. are composite numbers.
Exercise 2.4
Prime factors
1. Draw a 10 by 10 grid and fill it with numbers 1 to 100. Circle all the prime
numbers. What digits do most prime numbers end in?
2. Which of the following are prime numbers?
a) 241
b) 295
c) 148
d) 133
e) 269
f) 121
3. Look at the table on the next page. It shows the factors of whole numbers
between 30 and 40.
a) Complete columns B and C of the table.
b) Can you write any whole number as a product of its prime factors?
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Number
A. Factors
B. Prime factors
C. Product of
prime factors
30
1; 2; 3; 5; 6; 10; 15; 30
2; 3; 5
30 = 2 × 3 × 5
31
1; 31
31
31 = 31
32
1; 2; 4; 8; 16; 32
2
32 = 2 × 2 × 2 × 2 × 2
33
1; 3; 11; 33
34
1; 2; 17; 34
35
1; 5; 7; 35
36
1; 2; 3; 4; 9; 6; 12; 18; 36
4. Write the following numbers as products of their prime factors.
a) 220
b) 100
c) 210
d) 126
e) 924
Activity 2.5
f) 105
Using the product of prime factors to find
the HCF
To find the highest common factor of two or more numbers, we list all their factors to
see which are common so that we can choose the highest one. For example, to find
the highest common factor of 120, 300 and 900:
Factors of 120
Factors of 300
1; 2; 3; 4; 5; 6; 8; 10; 12; 15; 20; 24; 30; 40; 60; 120
1; 2; 3; 4; 5; 6; 10; 12; 15; 20; 25; 30; 50; 60; 75; 100; 150; 300
We can see that 1; 2; 3; 4; 5; 6; 10; 12; 15; 20; 30 and 60 are all common factors. The
HCF is 60.
Now we will use the product of prime factors to find the HCF.
Step 1:
Step 2:
First write the numbers as products of their prime factors.
Circle the common prime factors.
•• ••
••••
••• •
120 = 2 × 2 × 2 × 3 × 5
300 = 2 × 2 × 3 × 5 × 5
900 = 2 × 2 × 3 × 3 × 5 × 5
Step 3:
Step 4:
•
26
Multiply the common prime factors of the three numbers to find the HCF.
The common prime factors are 2, 2, 3 and 5.
The HCF = 2 × 2 × 3 × 5 = 60.
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Finding the HCF through prime factors
1. Find the highest common factor (HCF) of the following groups of numbers:
a) 220 and 100
b) 210 and 220
c) 105 and 210
d) 126 and 924
e) 220 and 924
f) 126, 210 and 924
2. Priscilla is making up parcels of clothes for Aids orphans. She has 16 jerseys
and 24 shirts. She wants each of the bundles to have the same number of
jerseys and the same number of shirts. Using the HCF, find the maximum
number of parcels that Priscilla can make up.
Summary
Multiples and factors
•
•
•
•
•
•
•
•
•
The factors of a whole number are the numbers that can divide into it
without leaving a remainder.
A common factor of three whole numbers must be a factor of all three whole
numbers. We call the highest common factor the HCF.
A multiple of a number is 1 × the number, or 2 × the number, or 3 × the
number, etc.
Common multiples of two numbers are the multiples that are the same for
the two numbers. We call the lowest common multiple the LCM.
A prime number has only two factors – the number itself and 1.
We call numbers that have more than two factors composite numbers.
We do not consider the number 1 a prime number. This is because it only
has one factor, being 1.
We can write any number as the product of its prime factors. Sometimes we
may need to use a factor more than once.
We can use the product of prime factors to find the HCF.
Check what you know
1. Find all the factors of the following numbers:
a) 18
b) 24
c) 28
d) 104
e) 256
2. a) Find three common factors of 140, 350 and 105.
b) What is the highest common factor of these three numbers?
3. List the first ten multiples of the following numbers:
a) 18
b) 20
c) 25
d) 35
e) 40
f) 100
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4. Find the lowest common multiple (LCM) of the following groups of numbers:
a) 2 and 5
b) 20 and 100
c) 9 and 5
d) 3, 5 and 10
5. Write the following as products of their prime numbers:
a) 252
b) 350
c) 88
d) 264
e) 396
6. Find the highest common factor (HCF) of the following groups of numbers.
a) 252 and 350
b) 88 and 264
c) 264 and 396
d) 252 and 900
e) 396, 252 and 900
f) 88, 264 and 900
7. Fay-yaadh is using two different strings of flashing lights to decorate his
house. One string flashes every 9 seconds. The other string flashes every
15 seconds. When Fay-yaadh switches on both strings, they both flash at the
same time. Work out how long it will be before they flash at the same time
again. Use the LCM.
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Term
1
Unit 3 Ratio, rate and
finance
In this unit you will:
•
•
•
solve ratio problems where two or more quantities of the same kind are
compared
solve rate problems where two different kinds of quantities are compared
solve financial problems involving whole numbers, percentages and decimal
fractions.
Getting started
Concentrate
Read the label on the bottle of juice concentrate. It gives
you the instructions for mixing the juice. You want to mix
a jug of juice.
1. How many parts of the jug of juice will be
concentrate?
2. How many parts will be water?
3. How many parts are there all together?
4. What fraction of the juice will be concentrate?
2
If you said , you are right! We can write the fraction
5
2
as the ratio 2 : 5.
5
Juice
Concentrate
Mix two parts
syrup with three
parts water.
The ratio describes the relationship between the quantities of concentrate and
juice. For every 2 parts of concentrate there are 5 parts of juice.
5. What fraction of the juice is water? Write this fraction as a ratio.
6. What does the ratio 2 : 3 represent?
7. You want to make ten litres of juice for a party. How many litres of
concentrate will you need to buy?
8. A bottle of juice concentrate costs R17,50. How much will you need to
budget for juice?
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Activity 3.1
April 22, 2013
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Ratio
A teacher asks a class of 48 learners if they play sport. He records the results
in a table:
Do you play sport?
Yes
32
No
16
TOTAL
48
The learners write about the results:
Anele writes:
The number of learners playing sport compared
to the number of learners not playing sport is 32
to 16. I write this ratio as 32 : 16.
1.
Simplify the ratio 32 : 16.
2.
Find an equivalent fraction to
Grace writes:
The number of learners playing sport
compared to the total number of learners
in the class is 32 out of 48. I write this
32
ratio as 48 or 32 : 48.
32
48
32
in its simplest form.
48
4. Now write the ratio 32 : 48 in its simplest form.
5. Write a ratio to compare the number of learners not playing sport to the total
number of learners in the class.
3. Write the fraction
Key ideas
•
•
A ratio is a comparison of two similar quantities. We use the same units to
measure the quantities. We can write ratios without including the units.
We write ratios with the ‘ : ’ symbol. Example: In a class of 20 boys
and 30 girls, the ratio of boys to girls is 20 : 30. We can simplify this to 2 : 3.
We read this as ‘2 is to 3’.
We can also write ratios as fractions.
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Worked example
SuperFood strawberry jam is made with fruit, sugar and water. The table below shows
how much of each ingredient is in the jar.
Size of jar
Fruit
Sugar
Water
Small
200 g
120 g
20 g
60 g
1. What fraction of the contents of the jar is a) fruit, b) sugar and c) water? Write
the fraction in its simplest form.
2. Write the fraction of the contents of the jar that is a) fruit, b) sugar and c) water
as a ratio.
SOLUTION
1. a) Fruit:
120 g
3
=
200 g
5
b) Sugar:
2. a) 3 : 5
Exercise 3.1
b) 1 : 10
20 g
1
=
200 g
10
c) Water:
60 g
3
=
200 g
10
c) 3 : 10
Ratio
1. For your class, write the ratio of:
a) boys to girls
c) girls to boys
b) boys to the whole class
d) girls to the whole class.
2. Thabo kept a diary of his activities during the week. He then drew up a table
to show an average day. Look at his table:
Activity
Sleeping
At school
Doing homework
Playing sport
Watching TV
Eating
Time
8 hours
7 hours
1 hour
2 hours
4 hours
2 hours
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a) What is the ratio of time spent doing homework to the time spent at
school?
b) For what fraction of the day does Thabo sleep?
c) Which three activities have time in the ratio 1 : 4?
d) What is the relationship between the time Thabo spends at school and
the time he spends playing (excluding eating and sleeping)?
Activity 3.2
Rate
Ashley invites all her cousins to come to her birthday party at her house in Pinelands
in Cape Town. Use the table below to discuss and answer the questions that follow.
Cousin
Travel from
Distance travelled
Time
Melanie
Perth, Australia
12 000 km
24 hours
Tamson
Centurion, SA
1 200 km
4 hours
Amy
Plumstead, SA
12 km
12 minutes
Simeon
Pinelands, SA
2 km
20 minutes
1.
a)
How do you think Melanie travelled to Ashley’s party?
b)
How fast did Melanie travel, i.e. how many kilometres an hour?
2. Tamson also comes to the party by plane. How fast does her plane travel?
3.
4.
a)
Compare how long Melanie and Tamson take to get to the party. Use a ratio.
b)
Simplify the ratio as much as possible. (Hint: Simplify by dividing both
numbers by their highest common factor).
a)
How do think Amy and Simeon got to the party?
b)
At what rate does each of them travel? (Remember to include the units).
Key ideas
•
Melanie flies by aeroplane at 500 kilometres per hour to Ashley’s party. This
is the speed or rate at which she travelled. This tells us the relationship
between the distance Melanie covered and how long it took her. We write this
relationship as:
•
speed =
32
distance
time
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Fractions can represent the relationship between different things, e.g. between
distance and time. We call this relationship a rate.
A rate shows us the relationship between quantities that are not the same. For
example, distance and time or rands and kilograms. Rates must always
include units such as km/h or R/kg (say: rands per kilogram). The ‘/’ is a
symbol for saying ‘per’.
Worked example
Star Supermarket sells apricot jam in three different size tins: small, medium and large.
The table below shows the size and cost of each tin.
Small
Medium
Large
Size of tin
200 g
450 g
1 kg
Cost of tin
R8,00
R9,00
R25,00
Which tin is the best value for money? Explain your answer.
SOLUTION
Small tin: 200 g for R8
200
= 25 grams per rand
8
Medium tin: 450 g for R9
450
= 50 grams per rand
9
Large tin: 1 kg for R25
1 000 = 40 grams per rand
25
The medium-sized tin is the best value for money. You get the most grams per rand
from this tin.
Exercise 3.2
Rate
1. Complete the table:
Distance (km) =
a)
495
c)
450
Speed (km/h) ×
70
55
110
d)
Time (h)
2
b)
11
9
2. Cape Town is 1 200 kilometres from Johannesburg.
a) A taxi takes 15 hours to make the trip. How fast does it travel?
b) How fast is it supposed to travel to get back in 10 hours?
c) If the taxi only travels at 80 km/h for the first 10 hours of the journey, how
far does it still have to go? If the trip takes 14 hours altogether, how much
faster did the taxi travel for the last part of the journey?
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Activity 3.3
April 22, 2013
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Sharing a whole in a given ratio
A well-known ratio of the chemicals nitrogen (N), phosphorus (P) and potassium (K)
in fertiliser, is the ratio N : P : K = 3 : 2 : 1.
1. How many parts are there in total in the mixture?
Use the pie chart to help you.
2. What fraction of the mixture will be N?
3. You have 1 250 g of the N, P and K mixture.
How many grams of N are there?
4. How many grams of P and how many grams
of K are there in the 1 250 g mixture?
Potassium, K
1 part
Phosphorus, P
2 parts
Nitrogen, N
3 parts
Key ideas
The ratio 3 : 2 : 1 means that a total of 3 + 2 + 1 = 6 parts make up the whole
1 250 g.
3
In fraction form, this means that of 1 250 g will be N.
6
3 1 250
3 750
3
=
= 625 g of N.
Calculate: of 1 250 = ×
6
6
1
6
3
The fraction tells us that 3 parts out of a total of 6 parts is N (nitrogen).
6
When you share an amount in a given ratio, first write the ratio in fraction form.
Then multiply the amount by the fractions in the ratio.
•
•
•
•
•
Exercise 3.3
Sharing wholes
1. The girl : boy ratio in a class of 40 learners is 3 : 2.
a) Work out the number of boys and girls in the class.
b) Check your answer. Add the number of boys and the number of girls.
Do you get 40?
2. An old man died. He left 90 cows for his three daughters. The community
has to share the cows between the eldest daughter, the second eldest
daughter and the youngest daughter in the ratio 5 : 3 : 1. How many cows will
each daughter get?
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Activity 3.4
April 22, 2013
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Managing a school musical – Calculating
income
Jaylow High School musical a hit!
Jaylow High School staged a musical this week to
celebrate its tenth anniversary. For months before
the event, the learners and teachers worked hard
to create a perfect performance. They certainly
succeeded! The musical was staged for a full week,
with eight shows.
There are 1 450 tickets available for each show. The tickets cost R15 each.
RECORD OF TICKET SALES
Mon 27 May: 49 tickets not sold
Wed 29 May: sold out
Fri 31 May:
sold out
Sat. afternoon: sold out
1.
Tues 28 May:
Thurs 30 May:
Sat. morning:
Sat. evening:
13 tickets not sold
28 tickets not sold
26 tickets not sold
sold out
Mr Rezandt is the Head of Finance at Jaylow High. He writes an expression to
calculate the income from ticket sales on the Saturday:
(1 424 × 15) + 2 × (1 450 × 15)
Calculate the income from ticket sales on the Saturday. Use Mr Rezandt’s
expression.
2.
Two learners wrote different expressions to calculate the income from ticket
sales on the Saturday:
(3 × 1 450) − 26 × 15
(15 × 1 424) + (15 × 1 450)
+(15 × 1 450)
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a)
Do the learners and Mr Rezandt all get the same answer?
b)
Another learner wrote:
15 × (1 426) + 1 450 + 1 450
Is this expression correct?
3.
Donald wrote the following expression to calculate the income from ticket sales
on the Saturday:
15 × 1 450 × 3 − 26
4.
a)
Calculate Donald’s answer. Use the rules for multiple operations.
b)
Explain why Donald did not get the same answer as the others.
c)
Rewrite Donald’s expression so that he also gets an answer of R64 860.
a) Write an expression to calculate the total income from ticket sales for all
eight performances of the musical.
b)
Calculate the total income for all eight performances of the musical.
Activity 3.5
Calculating profit or loss
1. Mr Rezandt wants to calculate the school’s total profit. The profit will be the
amount left once the school has covered all the expenses from the total income.
Mr Rezandt has to deduct the total cost of hiring the auditorium. He also has to
deduct the cost of props from the ticket sales.
Money paid out:
Props – R5 250
Hire of auditorium
Mon to Thurs 5 R380 per night
Fri
5 R580
Sat
5 R1 400
a)
Write a suitable expression to calculate the total profit made by the school,
then calculate the total profit of the musical.
b) Compare your expression to your friend’s. Did you get the same answer?
2. If only 67 tickets were sold for each show:
a) What would the total income from ticket sales for all eight performances of
the musical have been?
b) By how much would the expenses be greater than the total income? This
amount will tell us what the school’s loss would have been.
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Key ideas
•
•
Profit means financial gain. In particular, profit means the difference between
the amount earned (income) and the amount spent in buying, operating or
producing something (expenses). We say that:
Profit = total income − expenses.
When our income is less than our expenses, we have made a financial loss
instead of a gain.
Exercise 3.4
Calculating income, profit and loss
Supa Supermarket ordered 750 two-litre bottles of milk for the weekend. They
paid the supplier R6 000 for the milk. They sell the milk at R14 per bottle.
1. Work out the profit. Use a calculator.
2. Mrs Moonsamy bought R238 worth of milk for the school’s feeding scheme.
How many litres did she buy? (Be careful – litres, not bottles!)
3. On the Sunday evening, Supa Supermarket reduced the price of the
remaining 23 bottles of milk. Mr Isaacs bought seven bottles of milk, a tub of
margarine at R11 and a piece of cheese for R15. He paid with a R100 note.
He got R39 change.
a) What was the total cost of Mr Isaacs’ groceries?
b) What was the reduced price of one bottle of milk?
c) What loss did the supermarket make on the reduced bottles of milk?
4. Calculate the total profit made by the supermarket selling milk, after reducing
the remaining stock.
Key ideas
A budget is a financial plan. It shows our expected income and expenses for a
period of time.
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Activity 3.6
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Budgeting
Sam wants to save up to travel home for the holidays. The table below shows the
budget Sam makes of his weekly expenses:
Expenses
Weekly salary
Income
R1 150
Rent
R385
Food
R420
Cellphone
R65
Electricity
R40
Transport
R130
TOTAL
1.
a)
b)
Calculate Sam’s total weekly expenses.
How much does Sam save each week?
2. It will cost Sam R880 to travel home for the holidays. Sam needs a further R3 000
for presents for his family. He also needs R2 500 to spend while he is at home.
a) How many weeks will it take for Sam to save up for his holiday?
b)
c)
Will Sam be able to afford to take a three-week holiday each year? Explain.
How much does Sam need to save each week in order to afford his holiday?
Round your answer off to the nearest rand.
Exercise 3.5
Working with budgets
Nokuthula is making biscuits to sell at a food market. Look at her budget in the
table alongside. Answer the questions.
1. Nokuthula makes a batch of
100 biscuits. What are her total
expenses?
2. If Nokuthula sells all the biscuits
at R3 each, what is her profit?
3. If Nokuthula only sells half of the
biscuits, what is her profit?
4. How many biscuits does Nokuthula
have to sell before she starts
making a profit?
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Term 1 • Unit 3
For 100 biscuits
Expenses
@ R3 each
Butter
R27
Flour
R9
Castor sugar
R6
Sweets to decorate
R20
Icing sugar
R7
TOTAL
Income
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Accounts, loans and simple interest
Zoliswa borrows R5 000 from the bank in the form of a loan. The bank says that Zoliswa
must pay the money back over a period of six years. She has to pay simple interest of
R750 per year.
Interest is the amount of money that you pay to borrow money. Simple interest is
interest calculated as a set amount to be added to the original amount borrowed or
loaned each year.
1.
2.
a) How much interest will Zoliswa have to pay over the period of the loan?
b)
c)
How much money will Zoliswa have to pay back altogether?
How many months are there in the time period of the loan?
d)
How much money will Zoliswa have to pay each month to pay back the
total amount of money she owes? Round off to the nearest cent.
Zoliswa buys some new clothes on her account. Read her statement and answer
the questions.
Best FASHION Bargains
STATEMENT
Jeans
Casual top
Hoodie
TOTAL
Payment within 30 days
Late payment
Payment plan – 3 months
Zoliswa Manqina
R199
R59
R79
R337
R337
R40 per month added
R145 per month
a)
If Zoliswa is late with her payment by two months, how much will she end up
paying for the clothes?
b)
If Zoliswa takes the three-month payment plan, how much will she end up
paying for the clothes?
c)
How much money can Zoliswa save by paying her account straight away
instead of taking the three-month payment plan?
d)
Zoliswa’s mother gives her a discount voucher for Best Fashion Bargains.
The voucher says that Zoliswa can get a R30 discount if she pays straight
away. How much would Zoliswa pay?
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Key ideas
•
•
When you work with money, round your answer off to the nearest cent.
Interest is the amount of money that you pay to borrow money.
Simple interest is interest calculated only on the original amount you borrowed.
A discount is an amount of money that is subtracted from the usual amount.
•
•
Exercise 3.6
Accounts, interest and savings
1. Cyril opens a savings account at the bank. He deposits R3 000. Each year
he earns simple interest of R360.
a) How much money will Cyril have in his account after five years?
b) How many years will it take for his money to double?
2. Charlene borrows R2 500. Her simple interest charge is R250 per year. She
has to pay back her loan over three years. What are Charlene’s monthly
instalments?
3. Devide’s yearly school fees are R1 428. The school offers a discount of R108
if fees are paid over three months.
a) What is the monthly amount if school fees are paid over a year?
b) What is the monthly amount if school fees are paid over three months?
Summary
•
•
•
•
•
•
•
We can write ratios and rates as fractions.
Ratios show the relationship between quantities with the same units.
Rates show the relationship between quantities with different units.
We can write ratios as two numbers separated by a colon without units.
Rates are a single value. We always write rates with units.
We always simplify ratios to the smallest whole numbers.
We make use of the following two equations for problems of distance, speed
and time:
distance
speed =
time
distance = speed × time
•
•
40
When you share an amount in a given ratio, first write the ratio in fraction
form. Then multiply the amount by the fractions in the ratio.
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A budget is a financial plan. It shows our expected income and expenses for
a period of time.
Profit means financial gain.
When our income is less than our expenses, we have made a financial loss.
Interest is the amount of money that you pay to borrow money.
Simple interest is interest calculated only on the original amount.
A discount is an amount of money that is subtracted from the given amount.
Check what you know
1. The ratio of women engineers to men engineers in a construction company
is 2 : 7.
a) There are six women engineers. How many men engineers are there in
the company?
b) How many engineers are there in the company altogether?
c) What fraction of the total number of engineers are women?
d) The company decides to improve their gender equality. The company
wants to change the ratio of women engineers to men engineers to 2 : 5.
The company cannot afford to employ more than 28 engineers in total.
When the company achieves this ratio, how many women engineers and
men engineers would they have?
2. The Cape Argus Pick ’n Pay Cycle Race is 110 kilometres long. How fast must
I cycle to complete the race in five hours?
a) I average 25 km/h. How long will I take to finish?
b) 1 000 people complete the race. They get cola to drink at the end. They
drink 250 litres of cola altogether. How much cola is that per person?
3. Together, the ages of a father, son and grandson add up to 100. Their ages
are in the ratio 13 : 6 : 1. How old are they?
4. Mrs Khumalo has a budget of R1 200 per week for groceries at a pre-school.
Each day she buys:
•
•
•
•
5 loaves of bread @ R9 each
10 litres of milk @ R14 per two-litre bottle
2 kilograms of cheese @ R41 per kilogram
1 pocket of oranges @ R26
a) Write an expression and calculate how much Mrs Khumalo’s groceries
costs each day.
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b) Write an expression and calculate how much money would be left over at
the end of each week.
5. Look at the table below. It shows the weekly budget for running the pre-school.
Weekly budget
Expenses
Pre-school fees
Food
R48 per child per week
R1 200
Materials
Two assistants
Income
R200
R1 500
TOTAL
a) What are the expenses involved in running the pre-school for one week?
b) Each of the 60 children at the pre-school pay R48 per week to attend.
What profit does Mrs Khumalo make running the pre-school?
c) Mrs Khumalo wants to pay herself a salary of R1 000 per week. She also
has to cover her expenses. How much does she have to charge per child
per week?
6. Jackson borrows R15 500. He pays simple interest of R1 860 per year. He has
to pay back the loan over ten years.
a) How much interest will Jackson have to pay over the period of the loan?
b) How much money will Jackson have to pay back altogether?
c) How many months are there in the time period of the loan?
d) Jackson wants to pay back the total amount of money he owes. How much
money will Jackson have to pay back each month?
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Term
1
Unit 4 Comparing and
representing
numbers in
exponential form
In this unit you will:
compare and represent whole numbers in exponential form.
Getting started
Squares and cubes
Look at these square numbers:
1
4
9
16
1. Why do we call the numbers 1; 4; 9 and 16 square numbers?
2. What are the next three square numbers?
3. The whole number 10 is not a square number, although the
picture drawn here looks ‘square’. Explain what is wrong with
this square picture of ten.
10
4. Look at these cubic numbers:
1
a) Why do we call the numbers 1, 8 and 27 cubes?
b) What are the next two cubic numbers?
5. How can you find the squares and cubes without using a
drawing?
6. We carry on the pattern of squares and cubes shown
above and alongside. What is the tenth square number
and the tenth cubic number?
Term 1 • Unit 4
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27
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Key ideas
•
Multiplying each whole number by itself makes a square number. For example,
1 × 1 = 1; 2 × 2 = 4; 3 × 3 = 9, etc. The seventh square number is 7 × 7 = 49.
We say that the number 49 is the square of 7.
For the cubic numbers the whole number is multiplied by itself twice. For
example, 1 × 1 × 1 = 1; 2 × 2 × 2 = 8; 3 × 3 × 3 = 27, etc. The cube of 5 is
5 × 5 × 5 or 125.
•
Exercise 4.1
Square numbers and cubic numbers
1. Copy and complete the table below.
Number
Squared
1
1×1=1
Number Squared Number
7
1
2
8
2
3
9
3
4
10
4
5
11
5
6
12
6
Cubed
2×2×2=8
2. a) What is a square number?
b) What is the square of 12?
c) 169 is the square of 13. Find the number that 625 is the square of. Use a
calculator.
d) Fill in the missing numbers: × = 289. The same number must go in
both blocks.
3. a) Show how you work out the cube of 9.
b) 27 is the cube of 3.What is 216 the cube of?
c) The same number goes in each block in × × = 64. What is the
number?
4. A palindrome is a number or word that reads the same forwards or
backwards, e.g. aha, 101, racecar.
Using the digits 1, 2, 3 and 4, find:
a) a three-digit square number that is a palindrome
•
b) a three-digit cubic number that is a palindrome.
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Activity 4.1
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Making sense of exponential notation
There is a shorter way of writing 2 × 2 × 2 × 2 and of writing 9 × 9. We call this the
exponential form:
2 × 2 × 2 × 2 = 24
and 9 × 9 = 92
We call 2 the base and 4 the exponent
We call 9 the base and 2 the exponent
1.
What does the exponent ‘4’ stand for in 24 ?
2.
What does the exponent ‘2’ stand for in 92 ?
3.
Write the following in a shorter way. Use the exponential form.
a) 5 × 5 × 5
b) 12 × 12 × 12 c) 24 × 24
d)
53 × 53
Key ideas
•
•
•
•
•
2 × 2 × 2 = 8. We get 8 by multiplying 2 by 2 by 2.
We can write this in a short way: 23 = 8. We say: ‘2 to the power of 3 equals 8’
or ‘2 cubed equals 8’. We do not say ‘two to the three equals 8’.
We can write 9 × 9 as 92 = 81. We say: 9 squared equals 81. We do not say
‘nine to the two equals 81’.
We say that 92 is written in exponential form. We call the number ‘9’ the base
and the number ‘2’ the exponent. The exponent tells us how many times the
base is multiplied by itself.
Any number to the power of one stays that number. 41 = 4.
Exercise 4.2
Writing in exponential form
1. Copy and complete the table below. The first row has been done for you.
Repeated
multiplication
6×6
Exponential
form
62
We say
Value
Six squared
36
Ten squared
100
122
Five cubed
1×1×1×1
14
One to the power of four
54
1
625
10 × 10 × 10 × 10
2. What does six to the power of one equal?
3. Is 32 + 42 equal to 72 ? Show your calculations.
Term 1 • Unit 4
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Activity 4.2
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Squares and square roots, cubes
and cube roots
I know that
25 × 25 = 252 = 625,
we say that ‘twenty-five
squared equals six
hundred and twenty-five’
But how can I reverse
the process?
Do you mean that if
I start with 625, how do
I reach an answer of 25?
That’s it!
We call 625 the square of 25, We call
25 the square root of 625, We write
it like this:
√625 = 25
25 × 25 = 252 = 625.
I get it. So if 23 = 8. is 2
the cube root of 8?
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Term 1 • Unit 4
Quite right, We write it like this;
3
√8 = 2
2 × 2 × 2 = 23 = 8.
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1.
Explain why 25 is the square root of 625.
2.
Find the square root of 121. Use the same reasoning.
3.
Find the cube root of 125 in the same way.
Key ideas
•
•
•
•
√
√
The sign
stands for the square root of a number. We read 64 = 8 as ‘the
square root
of 64 is equal to 8’. The square root of 64 is 8 because
8 × 8 = 64.
√
√
3
3
The sign
stands for the cube root of a number. We read 64 = 4 as ‘the
cube root of 64 is equal to √
4’. The cube root of 64 is 4 because 4 × 4 × 4 = 64.
With the square root sign,
, you expect a little 2, like the little 3 in the cube
root sign. The little 2 is usually left out, but you√must imagine
that it is there.
√
3
If you have a calculator, check to see if it has
and
keys. Use these to
work out square and cube roots.√On some
calculators you need to first put in
√
3
or
key. On other calculators you need to
the number and
press the
√ then√
3
first press the
or
key and then put in the number.
Exercise 4.3
Square and cube roots
Copy and complete the table below.
We say
The square root of 36
The square root of nine
The cube root of 27
The cube root of eight
We write
√
36
√
144
√
3
Repeated
Value
multiplication
√
= 6×6
=6
√
= 7×7
√
4×4
√
3
= 3×3×3
√
= 3 6×6×6
=
27
√
3
125
√
3
=
√
3
1×1×1
=3
=1
64
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Activity 4.3
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Inverse operations of squaring and cubing
Work with a friend. Discuss and answer the questions.
Set Y
1
Set X
1
3
9
5
6
B
25
A
64
9
C
D
Some
whole numbers
169
Matching
square numbers
1. Some whole numbers are shown in Set X. Their square numbers are shown in
Set Y.
a) Find the values of A, B, C and D.
2.
b)
Explain how you worked out the missing square numbers in Set Y.
c)
Explain how you worked out the missing whole numbers in Set X.
Some whole numbers are shown in Set X and their cubic numbers are shown in
Set Z.
Set X
Set Z
1
1
3
27
5
6
F
9
H
Some
whole numbers
a)
125
E
512
G
2 197
Matching
cubic numbers
Find the values of E, F, G and H.
b) Explain how you worked out the missing cubic numbers in Set Z.
c)
Explain how you worked out the missing whole numbers in Set X.
3. Compare and discuss your answers in class.
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Key ideas
•
•
•
•
•
You get the square number in Set Y by multiplying the whole number by itself.
So, the square of 6 is 36, because 6 × 6 = 36. We also write this as 62 = 36.
You get the cubic number in Set Z by multiplying the whole number by itself
twice. So, the cube of 6 is 6 × 6 × 6 = 216. We also write this as 63 = 216.
To work out the whole number in Set X, you have to answer the question,
‘What number multiplied by itself, once or twice, will give me the number in
Set Y or Set Z?’ Then you find, for example, that 82 = 64 and 83 = 512.
Square roots and cube roots are the inverse operations of squaring and cubing
numbers.
Any number to the power of one stays that number. Example: 41 = 4
Exercise 4.4
Mixed calculations
1. Find the values of each of the following sets of calculations. Do not use a
calculator.
Set 1
Set 2
82
42
a)
b) 32
c) 52
√
d) 4
√
e) 100
a)
b) 33
√
c) 36
√
3
d) 8
e) 102
Set 3
√
a) 1
√
b) 9
√
c) 64
√
3
d) 125
√
3
e) 8
Set 4
√
a) 144
√
3
b) 8
c) 102
d) 63
√
e) 81
2. Say which of the following are true or false. If the statement is false, make it
true.
b) 22 = 12 × 2
c) 501 = 50 × 50
a) 122 = 144
d) 12 × 2 = 24
e) 12 × 12 = 12 × 2
f) 1 × 1 × 1 × 1 = 4
√
4
i) 92 = 3
g) 1 = 1 × 1 × 1 × 1 h)
9 = 81
Summary
•
•
•
•
Multiplying each whole number by itself makes a square number.
A cubic number is formed when a whole number is multiplied by itself twice.
We can write 9 × 9 as 92 = 81.We say: 9 squared equals 81.
We say that 92 is written in exponential form. We call the number ‘9’ the
base and the number ‘2’ the exponent. The exponent tells us how many
times the base is multiplied by itself.
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1
Any number
√ to the power of one stays that number. Example: 4 = 4.
The sign √ stands for the square root of a number.
√
3
3
The sign
stands for the cube root of a number. Example: 64 = 4
Finding the square roots and cube roots of numbers are the inverse
operations of squaring and cubing numbers.
•
•
•
•
Check what you know
1. Write each of the following in exponential form.
a) 4 × 4 × 4 × 4 × 4 × 4 × 4
b) 11 × 11 × 11 × 11
c) 145 × 145 × 145
d) 17 × 17 × 17 × 17 × 17 × 17
e) 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6
2. We can write the number 84 as the product of prime numbers:
84 = 2 × 2 × 3 × 7. We can write this in exponential form: 84 = 22 × 3 × 7.
Write the following numbers as products of their prime numbers. Use the
exponential form.
a) 36
b) 40
c) 54
d) 198
e) 525
f) 48
3. Which is larger: 34 or 43 ? Explain your answer.
4. Evaluate the following:
b) 51
c) 72
d) 43
a) 32
5. Complete the table below. Fill in:
• each number squared
• the number cubed
x
3
Square
(x 2 )
32 = 3 × 3 = 9
Square root
of the square
√
x2
√
√
9 = 32 = 3
•
•
e) 101
your answer square rooted
your answer cube rooted.
Cube
(x 3 )
33 = 3 × 3 × 3 = 27
Cube root
of the cube
√
3 3
x
√
√
3
3
27 = 33 = 3
4
5
11
12
6. Solve the following. Do not use your calculator. You may have to use prime
factorisation in some cases.
√
√
√
196
b)
10 000
c)
225
a)
7. The number 100 is a perfect square. Its square root is a whole number.
•
a) Is 14 a perfect square? Explain why or why not.
b) List all the perfect squares from 0 to 100.
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Term
1
Unit 5 Calculations with
numbers in
exponential form
In this unit you will:
•
•
•
perform calculations with numbers in exponential form
explore the order of operations with numbers involving exponents, and
square and cube roots
solve problems involving numbers in exponential form.
Getting started
Order of operations
Complete these calculations. Use the rules for the order of operations.
1. 11 × 2 + 5 × 8
2. 11 × (2 + 5) × 8
3. 25 × 4 − 3
4. 25 × (4 − 3)
5. 88 − 24 ÷ 8
6. (88 − 24) ÷ 8
7. 12 ÷ 12 × 4 + 218 × 0
8. 12 ÷ 12 × (4 + 218) × 0
Key ideas
When expressions do not have exponents, the rules for the order of operations
are:
1. Simplify the operations inside the brackets first.
2. Then do the multiplication and division operations. Work from left to right.
3. Finally do the addition and subtraction. Work from left to right.
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Activity 5.1
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Order of calculations involving exponents
Is (8 − 3) 2 the same as 82 − 32 ?
Mthunzi and Gloria write down their solutions. Compare their solutions with your own.
I found that 82 – 32 = 55
I worked out the sum inside
the brackets first. So
(8 – 3)2 = 25
Mthunzi
(8 − 3) = (8 − 3) × (8 − 3)
=5×5
= 25
2
Gloria
8 − 3 = (8 × 8) − (3 × 3)
= 64 − 9
= 55
2
2
We know that 25 = 55 so (8 − 3) 2 = 82 − 32
Key ideas
We need to add a new rule:
1. Simplify the operations inside the brackets first.
2. Simplify all exponents. Work from left to right.
3. Then do the multiplication and division operations. Work from left to right.
4. Finally do the addition and subtraction. Work from left to right.
Worked example
Solve (8 − 6) 3 × 42 − 5
SOLUTION
(8 − 6) 3 × 42 − 5
= 2 3 × 42 − 5
= 8 × 16 − 5
= 128 − 5
= 123
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Exercise 5.1
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Calculations with exponents
1. Evaluate the following pairs of expressions:
a) i) (7 + 4) 2
ii) 72 + 42
b) i) (10 − 5) 2
ii) 102 − 52
c) Are any of the pairs equal? Explain.
2. Now evaluate the following pairs of expressions:
a) i) (3 × 2) 3
ii) 33 × 23
b) i) (9 ÷ 3) 2
ii) 92 ÷ 32
c) Are any of the pairs equal? Explain.
3. Evaluate the following expressions. Use your new rules.
b) (42 − 12 ) ÷ 3
a) (4 − 1) 2 ÷ 3
d) (8 + 2) × 5 × 34 ÷ 9
c) 8 + (2 × 5) × 34 ÷ 9
f) 75 − (9 − 4) 2 ÷ 5
e) (27 + 42) ÷ 3 − 5 × 22
Activity 5.2
Order of calculations involving roots
√
√
√
Is 16 + 9 the same as 16 + 9? Mthunzi and Shaheeda write down their solutions
as shown. Compare their solutions with your own.
I worked out each
root and added
them. So 16 + 9 = 7
Mthunzi
√
√
16 + 9 = 4 + 3
=7
We know that 7 = 5 so
I worked out the sum under
the root sign first. I found
that 16 + 9 = 5
√
√
√
16 + 9 = 16 + 9
Shaheeda
16 + 9 = 25
=5
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Key ideas
•
•
The root sign acts as a bracket to the expression under it.
We need to add to our first and second rules:
1. Simplify the operations inside the brackets and under any root signs first.
2. Simplify all exponents. Work from left to right.
3. Then do the multiplication and division operations. Work from left to right.
4. Finally do the addition and subtraction. Work from left to right.
Worked example
How do we solve a problem like
√
65 − 16 × 32 − (12 ÷ 4)?
SOLUTION
√
65 − 16 × 32 − (12 ÷ 4)
√
= 49 × 32 − 3
=7×9−3
= 63 − 3
= 60
Exercise 5.2
Roots and exponents
1. Evaluate the following pairs of expressions:
√
√
√
a) i) 144 + 25
ii) 144 + 25
√
√
√
b) i) 169 − 25
ii) 169 − 25
c) Are any of the pairs equal? Explain.
2. Evaluate the following pairs of expressions:
√
√
√
a) i) 9 × 4
ii) 9 × 4
√
√
√
3
3
b) i) 3 64 ÷ 8
ii) 64 ÷ 8
c) Are any of the pairs equal? Explain.
3. Use your new rules to evaluate the following expressions:
√
√
√
√
b) 10 × ( 16 − 4)
a) 10 × 16 − 4
√
√
c) (5 × 2) 2 − 61 + 3
d) (4 × 3) 2 − 9 × 3 59 + 5
√
√
3
3
125 × 32 + 10
e)
f) (24 − 12) × ( 27 + 7)
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4. For each of the following:
i) write the expression in exponential form
a) 3 × 3 × 3 + 2 × 2
ii) calculate the answers.
b) (4 × 4 + 3 × 3 × 3 × 3 + 3) ÷ (2 × 2 × 5)
c) 5 × 5 × 5 + 5 × 5 × 7 + 7 × 72
5. What is the difference between:
a) 103 and 3 × 10
6. Find the value:
a) 5 + 5 + 5 + 5
d) 5 × 5 × 5 × 5
b) 64 and 4 × 6
c) 35 and 5 × 3.
b) 54
c) 3 × 5 + 2 × 5
e) 5 × 4
Summary
√
√
3
If you have a calculator, check to see if it has
and
keys. Use these to
work out square and cube roots.
The order of operations when working with exponents and roots:
•
•
1.
2.
3.
4.
Simplify the operations inside the brackets and under any root signs first.
Simplify all exponents. Work from left to right.
Then do the multiplication and division operations. Work from left to right.
Finally do the addition and subtraction. Work from left to right.
Check what you know
√
√
16 − 4 is equal to:
√
b) 4
20
a)
√
2.
12 is equal to:
√
√
b) 6
6+ 6
a)
c)
3. Calculate:
a) 32 + 42
d) 53 − 92
c) 102 ÷ 22
f) 32 − 16 + 23
1.
b) 53 − 102
e) 82 ÷ 42
4. Evaluate:
√
√
49 + 9 + 42
a)
√
c) √25 + 22 × (52 ÷ 5)
3
e)
8 × 42 + 18
c)
√
12
d) 2
√
d) 4
3×4
√
√
b) ( 49 + 9) + 42
√
d) (6 × 5) 2 ÷ 32 × 3 25 + 2
√
3
f) (25 − 22) × ( 27 + 7)
Term 1 • Unit 5
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Term
1
Unit 6 Measuring and
drawing angles
In this unit you will:
•
•
use a protractor to accurately measure and draw angles
classify angles according to their size.
Getting started
Looking at angles
1. Match each of the following angles with their names.
A
B
C
D
E
a)
acute: less than a right angle
b)
a right angle
c)
obtuse: more than a right angle, but less than a straight angle
d)
a straight angle
e)
reflex: more than a straight angle, but less than a revolution.
2. Draw each of the following angles. Name each angle. Show the size of the
angle with the curved line:
a) less than a straight angle, but more than a right angle
3
b) more than a straight angle, but less than a turn angle
4
c) less than a right angle
3
d) more than a turn angle, but less than a revolution.
4
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3. Name the angles A, B, and C.
a)
b)
c)
908
A
1808
908
3608
B
1808
2708
Activity 6.1
908
3608
1808
2708
C
3608
2708
What is an angle?
Drawing an angle
The line joining point A to
point B is called
the line segment AB.
Step 1:
Step 2:
Step 3:
A
B
Draw a line segment on a piece of paper. Name it AB.
Place your pencil on point A. Hold your ruler along the line AB.
Rotate your ruler a short distance anti-clockwise around the pencil. Draw a
new line segment from A to C. Your drawing should look similar to this:
C
ˆ or
We name the angle you have drawn, BAC
ˆ We can also call it A. This is because
CAB.
there are no other angles at the vertex A.
arm
A
Vertex
Now place your pencil onto point B. Rotate your ruler clockwise around the
pencil. Draw in a new line segment from B to D.
1. What letter marks the vertex of your new angle?
2. Name your new angle.
C
4
5
6
D
7
4
3
A
8
9
10
11
12
1
3
2
5
Step 4:
arrow shows rotation
B
arm
13
B
14
15
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Key ideas
•
We call the opening between two connected lines an angle. The angle
measures the amount you rotated your ruler from the line AB to the line AC.
We measure the size of an angle in degrees. The symbol for degrees is a
tiny circle that we write after the number of units. For example, we write 60
degrees as 60◦ .
We show rotation by drawing an arrow from the starting position to the end
position around a point of rotation.
We call the point A, where you placed your pencil and around which you
rotated your ruler, the vertex of the angle. The lines AB and AC are the arms
of the angle.
We call the 180 angle units on the protractor degrees.
We use three letters to name an angle. We use the letters that name the two
arms, with the letter of the vertex in the middle. We call the angle you have
ˆ or CAB.
ˆ We call it A if there are no other angles at the vertex.
drawn BAC
•
•
•
•
•
Exercise 6.1
Naming angles
1. Name the angle formed between the following line segments. Use three
letters.
A
G
a) GA and AE
E
M
b) KA and AG
K
c) EA and AM
d) MA and AG
e) GA and AK
2. Two crossing lines are shown. Write down the three-letter names for the
angles marked 1 to 4:
Y
H
2
K
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Term 1 • Unit 6
1
N
3
4
M
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Activity 6.2
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Using a protractor to measure angles
90
100 1
10
12
80 7
0
0
60 1 3
0
50
0
10
2
180 170 1 0
60
30
15
0 40
14
0
80
70
0
60 110 10
0
50 12
0
13
170 180
160 10 0
0
15
20
0
0
14 0 3
4
This is a semi-circular protractor. It is a half circle.
It has been divided into 180 equal angle units. We
call these units degrees.
Why is the protractor marked in degrees from 0◦ to 180◦ in two rows?
Measuring an angle
Step 1: Find the centre mark in the middle of the
straight edge of the protractor.
100 1
10
12
80 7
0
0
60 13
0
50
0
10
2
180 170 1 0
60
30
15
0 40
14
0
90
170 180
160 10 0
0
15
20
0
0
14 0 3
4
80
70
0
60 110 10
0
50 12
0
3
1
Line up the zero on the straight edge of
the protractor with one of the arms of the
angle.
Step 4:
Find the point where the second arm of
the angle intersects the curved edge of
the protractor. You may have to make
the arms longer so that they reach
past the protractor’s edge.
90
100 1
10
12
80 7
0
0
60 1 3
0
50
80
70
0
60 110 10
0
2
0
1
5
0
13
90
100 1
10
12
80 7
0
0
60 13
0
50
80
70
0
60 110 10
0
2
0
1
5
0
13
90
100 1
10
12
80 7
0
0
60 1 3
0
50
0
10
2
180 170 1 0
60
30
15
0 40
14
0
Step 3:
80
70
0
60 110 10
0
2
0
1
5
0
13
30
15
0 40
14
0
0
10
2
180 170 1 0
60
30
15
0 40
14
0
170 180
160 10 0
0
15
20
0
0
14 0 3
4
Term 1 • Unit 6
170 180
160 10 0
0
15
20
0
0
14 0 3
4
Place the centre mark at the vertex of the
angle you wish to measure. Like this:
170 180
160 10 0
0
15
20
0
0
14 0 3
4
Step 2:
0
10
2
180 170 1 0
60
1.
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60
90
100 1
10
12
80 7
0
0
60 13
0
50
80
70
0
60 110 10
0
2
0
1
5
0
13
90
100 1
10
12
80 7
0
0
60 13
0
50
2. Measure the following angles. Use the steps on pages 59–60.
a)
b)
c)
d)
e)
f)
•
60
Term 1 • Unit 6
170 180
160 10 0
0
15
20
0
0
14 0 3
4
Now go to the outer circle of markings.
Read the single degrees. You should
get 5 degrees. The size of the angle is
40◦ + 5◦ = 45◦ . This is the measure of
the angle in degrees.
80
70
0
60 110 10
0
50 1 2
0
3
1
170 180
160 10 0
0
15
20
0
0
14 0 3
4
Step 5: Read the number that is written on the
protractor at the point of intersection.
Always start at 0◦ . Count the tens on
the inner circle of numbers: 0◦ ; 10◦ ;
20◦ ; 30◦ ; 40◦ . . .
0
10
2
180 170 1 0 3
0
60
15
0 40
14
0
book
0
10
2
180 170 1 0 3
0
60
15
0 40
14
0
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g)
h)
Key ideas
•
A protractor has degree units. They are marked from right to left and from left
to right.
You must be sure about the direction of rotation you are measuring for a
particular angle when you use the protractor. For example, the angle below
is rotated anti-clockwise from 0◦ on the right. This means that the angle size
is 50◦ . It is not 130◦ .
•
Exercise 6.2
50
1
80
70
0
60 110 10
0
2
1
30
90
100 1
10
12
80 7
0
0
60 13
0
50
170 180
160 10 0
0
15
20
0
30
14 0
4
You can measure any angle, as long as you place
the vertex of the angle at the centre of your
protractor.
0 10
20
180 170
160 30
15
0 40
14
0
•
Measuring angles
Measure the size of each angle below. Use your protractor. Remember to place
the vertex of the angle at the centre mark on the protractor.
1.
2.
3.
4.
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Sorting angles according to size
Work with a friend. Answer the questions.
1. Study the cards. They show right angles and angles that are not right angles.
Right angles
Not right angles
Not right angles
Mixed angles
1
2
3
4
a)
What is the difference between the right angles and the angles on Card 2?
b) What is the difference between right angles and the angles on Card 3?
c)
How many right angles do you see on Card 4?
d) Draw the angles on Card 4 that you can group with the angles on Card 2.
e)
Draw the angles on Card 4 that you can group with the angles on Card 3.
2. Look at the cards above. Describe a right angle.
3. We call the angles on Card 2 obtuse angles.
a) Estimate the sizes of these angles.
b)
Describe obtuse angles. Use your own words.
4. We call the angles on Card 3 acute angles.
a) Estimate the sizes of these angles.
b) Describe acute angles. Use your own words.
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Key ideas
We can sort angles by comparing them to right angles or straight angles.
Angle type
Size
Right angle
equal to 90◦
Drawing
908
right angle
Straight angle
equal to 180◦
Acute angle
less than 90◦
Obtuse angle
greater than 90◦
but less than 180◦
Reflex angle
greater than 180◦
but less than 360◦
Exercise 6.3
1808 straight angle
less than
908
greater than 908
less than 1808
greater than 1808
Naming angles
1. Look at these two reflex angles. Answer the questions that follow.
ii)
i)
a) Estimate the sizes of these angles.
b) Describe the reflex angles above. Use your own words.
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2. For each of the angles below:
i) Identify the type of angle shown.
ii) Estimate the size of the angle.
iii) Measure the angle accurately. Use a protractor.
a)
b)
c)
d)
3. Write down the letters of an angle that is:
A
G
a) a right angle
E
M
b) an obtuse angle
K
c) an acute angle
4. Complete this table:
Angle type Angle size, in degrees
Acute angle
Between 0◦ and 90◦
Right angle
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Angle shape
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Obtuse angle Between 90◦ and ?
Straight angle
Reflex angle
Revolution
Activity 6.4
?
?
?
?
?
?
Drawing an angle of a particular size
We use a protractor to draw and measure angles.
ˆ = 50◦
Drawing the angle DEF
Step 1:
Step 2:
Draw a straight line. Use your ruler. This line forms one
E
side of your angle. Remember that you must name the
Vertex
vertex with the middle letter.
Find the centre mark in the middle of the
80 90 100 11
0
70
12
80 7
0
0
0
60 110 10
straight edge of the protractor.
60 13
20
0
0
1
50
0
10
2
180 170 1 0
60
13
Line up the zero on the straight edge of
the protractor with your line.
Step 5:
Always start at 0◦ . Use the inner circle.
Count in tens around the curved edge
of the protractor. Count from 0◦ to the
size of the angle you want to make.
Mark the point.
90
100 1
10
12
80 7
0
0
60 13
0
50
0
10
2
180 170 1 0
60
30
15
0 40
14
0
Step 4:
80
70
0
60 110 10
0
2
0
1
5
0
13
90
100 1
10
12
80 7
0
0
60 13
0
50
80
70
0
60 110 10
0
2
50 1
0
13
90
100 1
10
12
80 7
0
0
60 13
0
50
0 10
20
180 170
160
30
15
0 40
14
0
0
10
2
180 170 1 0
60
30
15
0 40
14
0
170 180
160 10 0
0
15
20
0
0
14 0 3
4
80
70
0
60 110 10
0
50 12
0
3
1
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170 180
160 10 0
0
15
20
0
30
14 0
4
Place the protractor so that its centre mark
is at the vertex of the angle you want to draw.
170 180
160 10 0
0
15
20
0
0
14 0 3
4
Step 3:
0
170 180
160 10 0
0
15
20
0
0
14 0 3
4
30
15
0 40
14
0
5
F
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Remove the protractor. Draw a line connecting your point
to the vertex. You have formed an angle.
E
Vertex
Exercise 6.4
Drawing angles
Draw the following angles. Follow the steps on pages 65–66.
ˆ = 43◦
ˆ = 176◦
2. HˆIJ = 65◦
3. KLM
1. DEF
ˆ = 97◦
ˆ = 125◦
ˆ = 14◦
4. NOP
5. QRS
6. TUV
Activity 6.5
Reflex angles
1. Measure this reflex angle. Follow the steps.
Step 1: Measure the remaining part of the revolution.
Step 2: Subtract this amount from 360◦ . You have
found the size of the reflex angle.
2. Draw the reflex angles below. Follow these steps.
Step 1: Subtract the reflex angle from 360◦ .
Step 2: Draw the new angle.
Step 3: Mark the rotation on your drawing. This shows the reflex angle.
ˆ = 315◦
b) ABC
a) HˆIJ = 195◦
ˆ = 343◦
ˆ = 276◦
c) KLM
d) DEF
Exercise 6.5
Drawing more angles
Draw the angles:
ˆ = 61◦
1. ABC
ˆ = 195◦
4. TUV
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ˆ = 217◦
2. DEF
ˆ = 348◦
5. XYZ
3. HˆIJ = 134◦
D
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Summary
Measuring and drawing angles
•
•
•
•
•
•
•
Angles are formed between any two lines that meet or cross.
The vertex of the angle is the point at which the two lines meet.
We can also call the two lines leading from the vertex the arms of the angle.
An angle measures in degrees the amount of rotation from one arm to the
other.
We use a mathematical instrument called a protractor to measure angles.
We call the units on the protractor degrees.
We write degrees with a tiny circle after the number of units.
We use three letters to name an angle. We use the letters that name the two
arms, with the letter of the vertex in the middle. If there are no other angles
at the vertex, we can also use the letter of the vertex to name the angle.
Check what you know
1. How many degrees make up each of the following?
a) a full rotation
b) a half rotation
c) a quarter rotation
d) a third of a full rotation
2. The angles shown here form a straight line. Do
not measure the angles. What is the sum of
these angles?
3. Angles are shown around point Q
alongside. What do you think the sum
of these angles will be? Explain.
Check by measuring with a protractor.
S
R
T
Q
V
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4. For each of the angles below:
i) Use the letters shown to name the angle.
ii) Identify the type of angle shown.
iii) Estimate the size of the angle.
iv) Measure the angle accurately with your protractor.
a)
A
B
b)
C
D
B
C
c)
d)
E
G
P
R
Q
F
5. Look at the sketch. Only the size of
angle FEH is given.
E
F
45°
a) What do you think the size of
angle FGH is?
b) Now measure angle FGH.
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Term
1
Unit 7 Constructions
In this unit you will:
•
•
accurately construct geometric figures including circles, parallel and
perpendicular lines
use a compass, ruler, set square and protractor appropriately to construct
geometric figures.
Getting started
•
•
•
Always use a sharp pencil.
If you need to mark a point, make it small. This is so
that it can mark a position accurately.
We use capital letters to label points in geometrical
drawings.
Activity 7.1
1.
2.
General rules
A
B
C
Recognising parallel and perpendicular lines
a) Copy the point and line alongside onto lunch wrap
or tracing paper.
b) Find the shortest distance from the point
to the line.
c) Draw in the line that shows the shortest
distance.
d) Use your protractor to measure the angle formed by the two lines.
e) Add a second point on the same side of the line and the same distance from
the line. Add four more such points.
f) What do you notice about these points?
Join the points.
a) What is the distance between the lines
alongside?
b) How did you measure this distance?
c) What do you notice about these lines?
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Key ideas
•
The shortest distance from a point to a line is the
direct and straight length to the line, as shown here:
•
These lines are perpendicular to each other. The
angle formed by the two lines is 90◦ . We use the
symbol ⊥ to show that two lines are perpendicular.
We call the lines in Question 2 parallel lines. The
perpendicular distance between two parallel lines is the same everywhere
along the lines. We use the symbol || to show that two lines are parallel.
•
Exercise 7.1
Perpendicular and parallel lines
Which of the following pairs of lines are:
1. perpendicular?
2. parallel?
a)
b)
e)
Activity 7.2
c)
f)
Step 1:
Draw a line CD on your page.
Mark point P somewhere on the line.
Step 2:
Construct a perpendicular line at P.
Place your set square in the position
shown. Draw line segment PQ ⊥ CD.
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g)
Constructing perpendicular lines using
a set square
Draw two perpendicular lines. Follow the steps.
•
d)
Q
C
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Exercise 7.2
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Angular spiral pattern
1. Copy the pattern. Start in the middle with a line
segment of 1 cm. Make each additional
perpendicular line segment 1 cm bigger than the
previous one. Use your set square to make the
perpendicular lines.
2. a) Construct your own pattern. Use perpendicular
lines.
b) A friend wants to do the same construction.
Write a list of clear instructions for them.
Activity 7.3
Constructing parallel lines using a set square
and a ruler
Draw two parallel lines. Follow the steps.
Step 1:
Draw line segment AB on your page.
Step 2:
Construct a parallel line to AB. Place the
set square with the base along line AB
as if you were going to draw a perpendicular line.
A
B
Step 3:
Place a ruler along the perpendicular side
of the set square. Hold the ruler firmly in
place. Slide the set square along the ruler
until the base is in the correct position for
the parallel line.
A
B
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Exercise 7.3
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Practising parallel lines
1. Draw an angle of 55◦ . Now make a pattern.
Fill your page with lines that are parallel to
the arms of the angle, similar to the pattern
alongside.
2. Now fill your page with your own pattern
of parallel and perpendicular lines.
Activity 7.4
558
Using a compass to construct a circle
The compass is a tool or instrument we use to construct accurate circles.
1. Draw a circle with a compass. Follow the steps.
Step 1: Put the two arms of the compass together. The tip of
the pencil should be just a little longer than the point
of the compass. Tighten the hold for the pencil.
Step 2: Open the arms of the compass to half the width
you want for your circle.
Step 3: Place the sharp point on the paper at the place
where you want the centre of your circle.
Step 4: Rotate the compass on its point, with the pencil
end drawing the circle.
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Use your compass to practise drawing circles. Find
your own best way to hold the compass down while
you draw a circle.
Make sure that the point of your compass is stuck in
firmly when you rotate it.
3.
Use your compass to construct big and small circles. Construct them inside and
outside one another, touching or crossing. Make the circles as big or as small as
you wish.
Key ideas
•
We use the point of the compass to mark the centre
of the circle. We use the pencil to draw the circumference.
ci
•
You can change the size of your circle by changing the width
of your compass. We call the distance from the centre of the
circle to the circumference the radius.
rc u
m fere nc
rad
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Activity 7.5
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Constructing circles of particular sizes
Construct a circle with a radius of 5 cm. Follow the steps.
Step 1:
Use a ruler to measure a distance of 5 cm
between your compass point and pencil.
Step 2:
Place your compass point where you want the centre of your circle to be.
Step 3:
Draw your circle by rotating the pencil around the compass point.
Exercise 7.4
Circle designs
1. Make this design. Follow the instructions.
Step 1: Draw a circle with a radius of
3 cm in the centre of your page.
Step 2: Keeping the radius 3 cm, place
the point of the compass
anywhere on the circumference
of the circle. Draw another circle.
Step 3: At every point where the second
circle cuts the circumference of the
first circle, place the point of the
compass and draw another circle.
2. The next design only uses parts of a circle. We call a
part of the circumference of a circle an arc. Use circles
with a radius of 5 cm to make this design.
arc
3. Construct a circle with a radius of 3 cm. Use your compass.
a) What is the longest line that we can draw from one point on the circle to
another?
•
b) How does the longest line compare in length to the radius of the circle?
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Activity 7.6
1.
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Following instructions for geometric figures
Follow the instructions to construct the figure. Draw a rough sketch from the
instructions to plan your construction.
Draw line segment CD = 6 cm.
ˆ = 115◦ . Make line BC = 6 cm.
Step 2: Construct angle BCD
Step 3: Construct BA parallel to CD with BA = 6 cm.
Step 4: Join AD.
Step 1:
2.
What do we call this quadrilateral?
Exercise 7.5
Constructions
1. a) Construct the figure. Follow the instructions. Draw a rough sketch to help.
Step 1: Draw line segment AB = 5 cm.
Step 2: Construct DA perpendicular to AB at point A. Construct CB
perpendicular to AB at point B. Make DA = 5 cm and BC = 5 cm.
Step 3: Join DC. Check that DC || AB and that DC = 5 cm.
Step 4: Draw in the lines DB and AC. Name the point where they
intersect E.
Step 5: Draw a circle using E as the centre. Use AE as the radius.
b) Describe the shape you have constructed.
2. a) Follow the instructions to construct the figure. Draw a rough sketch.
Step 1: Draw BD = 6 cm. Mark point E at the midpoint of BD.
Step 2: Draw AC perpendicular to BD and passing through point E.
Make line AE = 3,5 cm and line EC = 7,5 cm.
Step 3: Join ABCD.
b) What do we call this quadrilateral?
3. Follow the instructions to construct the figure as shown.
Step 1: Construct AB = 7 cm.
ˆ = 23◦ and AG = 8,5 cm.
Step 2: Construct BAG
E
Step 3: Construct CD ⊥ AG
with CD passing
through B. CD and
238
A
7 cm
AG intersect at point H.
Step 4: Construct EF || CD with EH = 2,3 cm.
G
C
H
F
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D
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Summary
Lines that are perpendicular to each other form an angle of 90◦ .
Lines that are parallel to each other have the same perpendicular distance
between them everywhere along the lines.
The compass is a tool or instrument we use to construct accurate circles.
We use the point of the compass to mark the centre of the circle. We use the
pencil to draw the circumference.
We call a part of the circumference of a circle an arc.
The distance from the centre of the circle to the circumference is the radius.
•
•
•
•
•
•
Check what you know
1. Draw a pair of lines that are:
a) perpendicular
b) parallel
c) not parallel or perpendicular
2. Look at the figure. Answer the questions.
Check your answers. Use your instruments.
A
G
E
F
a) Name a pair of parallel lines.
b) Name a pair of perpendicular lines.
ˆ
c) Measure FCD.
ˆ
d) Measure GFC.
ˆ
e) Measure EFC.
D
C
B
3. Draw any acute-angled triangle such as the one alongside.
A
a) Draw perpendicular lines from BC to A, from AB to
C and from AC to B. What do you notice?
b) Draw another triangle ABC. Check whether what
you observed in a) happens again.
B
4. a) Two circles are drawn through a single point. How many
more circles can you draw through the same point? Show
your answer. First draw freehand circles. Then draw
compass constructions.
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b) How many circles can we draw through two points?
c) How many circles can we draw through three points?
5. Construct the figure. Follow the instructions.
Step 1: Construct BC = 7 cm.
ˆ = 60◦ with AB = 7 cm.
Step 2: Construct ABC
ˆ = 60◦ with AC = 7 cm.
Step 3: Construct ACB
Step 4: Construct AD ⊥ BC. Construct BE ⊥ AC. Construct CF ⊥ AB.
Step 5: Name the point where AD, BE and CF intersect, G.
Step 6: Construct a circle with G as the centre and GE as the radius.
Describe the figure that you have constructed.
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Term
1
Unit 8 Classifying
triangles and
quadrilaterals
In this unit you will:
•
describe, sort, name and compare triangles according to their sides and
angles
describe, sort, name and compare quadrilaterals according to the properties
of their sides and angles
solve simple geometric problems with triangles and quadrilaterals.
•
•
Getting started
Different shapes
Complete the table below. Write the number and name of each shape in the
correct column.
4
1
3
8
2
7
9
12
11
16
17
14
19
18
Three-sided
shapes
20
Four-sided
shapes
4 – isosceles triangle 1 – rectangle
2 – parallelogram
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6
10
15
13
•
5
Term 1 • Unit 8
21
22
24
Five-sided Six-sided Eight-sided
shapes
shapes
shapes
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Activity 8.1
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Classifying triangles
These triangles have been classified into three groups:
Equilateral triangles
1.
2.
3.
Isosceles triangles
Scalene triangles
What reasons have been used to group these triangles?
In which group would you place each of the following triangles?
a) A triangle that has three equal sides.
b) A triangle in which each side has a different length.
c) A triangle with two sides of the same length.
Explain what the following are. Use the table and the questions above.
a) An equilateral triangle.
c) A scalene triangle.
b)
An isosceles triangle.
We can also classify the triangles above as follows:
Group 1
Group 2
Group 3
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