Math Smart - 7

CHAPTER 8.6 In this chapter

Pupils should be able to:
Fraction of a Quantity • multiply a fraction by an
integer
• calculate simple fractions
and percentages of
quantities
We can use fractions to find out a part of a group of items.
For example, we can work out what fraction of













the bikes are yellow, the shoes have laces and the people in the room
wear glasses.




Paul has 12 sweets. He wants to share them equally
with his 3 friends. How many sweets does each ^


Solution
Method 1 He can divide the sweets into 4 groups of equal size.
Each boy gets 3 sweets.

Method 2 ^ of 12 sweets = 12 sweets-4
= 3 sweets I I


Example 2


Peter has 12 sweets.
Find I of the 12 sweets.


Solution
Method 1 Method 2
He can divide the sweets ^ of 12 sweets = 12 sweets -r 3
into 3 groups of equal size. = 4 sweets

Each group consists of 4 sweets. I of 12 sweets =2x4 sweets
^ of 12 sweets = 4 sweets = 8 sweets.
So, I of 12 sweets =2x4 sweets
= 8 sweets.

Example 3

^ of a number is 60. What is the number?

Solution
We can use a bar diagram to help us.

1 part = 60
60
V. J 5 parts = 60 X 5 = 300
The number is 300.
Check My
Understanding


O Work out the following,
a) ^ of 20 b) ^ of 21 c) ^ of 20

d) iof24 e) ^ of 32 f) I of 45

O Find the value of the following,

a) ^ of 10 sweets b) I of 100 m c) i of $100

d) ^ of 24 m e) ^ of 1 h in minutes f) ^ of 1 h in minutes
g) ^ of 1 min in seconds h) "io of cm in millimetres i) y of 48 kg


O Work out the following,
a) I of 40 pencils b) I of 27 eggs c) I of 36 bananas
d) I of 45 sweets e) I of 48 apples f) I of $10

g) ^ of 20 plates h) ^ of $3 in cents 1) I of 2 weeks in days

j) I of 1 m in centimetres k) I of 2 m in centimetres I) ^ of 1 min In seconds

O There are 12 boys running in a race. Only half of them will receive prizes at the end of the race.
How many boys will receive prizes?
© One-third of the 36 balls in a bag are red. How many red balls are there?
© In a packet of mixed nuts, one-eighth of the nuts are almonds. If there are 48 nuts in a packet,
how many nuts in the packet are not almonds?
O Anna and Peter does the same job. In one week, Anna works for 12 hours and Peter works for 8 hours.
Together they are paid a total of $120 for the week's work. They are paid the same rate.
a) What fraction of the total number of hours does Anna work?
b) How much more pay does Anna receive than Peter?
© * Challenge! Mary has a tube of 12 sweets. She gives one-third of the sweets to a friend. She eats
one-quarter of the remaining sweets. How many sweets does Mary have left?








UNIT 8 I Fractions

o Journal Writing



28?
Give
reasons
of
of
or
24
for
O Which is larger? | | your answer.
O Let us look at the camel problem again.
The father left his sons 17 camels.
The eldest son got 2 camels.
The middle son got ^ of the 17 camels.
The youngest son got ^ of the 17 camels.


!r«rjr


fnrjTjr «rirjr


frrrfrfr + mrfr
=
!r

frff jr






«rfr«r




a) Calculate of 18 camels) + (^ of 18 camels)+ (^ of 18 camels).

b) Then calculate (^ of 17 camels) + {^ of 17 camels) + (^ of 17 camels).

c) Write down what you notice about the two answers.






Try and Apply


O A ball is dropped from a height of 125 cm. Each time it hits the ground, it bounces to |
of the height from which it fell previously. How high does it bounce after hitting the
ground the third time?

Q One-third of the animals in Scott's flock are goats and the rest are sheep. There are 12
more sheep than goats. How many animals are there altogether in Scott's flock?

Revision



O John says that J of this shape is coloured.
Give a reason why John is not correct.


0 What fraction of each shape is shaded? Leave your answers in the simplest form.

a) I f—1 ^ 1 b)








Q Fill in the missing numbers.
a) ^ - n b) 18 - ^
48 - t—I
DJ
□
O Write these fractions in simplest form.
a) ^ c) ^ d) ^ 40 e) ^ TJ
f) ^ 91
Q
24
a;
10 27
Q Which of these fractions does not have an equivalent in the list?
A. J. 6_ 6_ _4 18. 15. 7_
10' 12' 10' 15' 12' 24 25 28
Q Look at the number lines.

H
A



D IQ E 12
4
I I I I I I I I I I I I I I I I I I I I I I I I I
0 F2 3 456 78G10111213 1415 16 17 1819_ H 212^23 24
8 8888 88 8 8 8888 8 888 888 8






H 1 1 \ 1 F -I 1 1 1 1 1 h —I 1 1 h
1 2 5 4 5 6 7 K 9 10 11 L 13 14 15 16 17 18
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6


a) Write down the missing fractions at points A, B, C, D, E, F, G, H, I, J, K and L.
b) Which is greater, 2 ^ or2^?Why?


O Express these improper fractions as mixed numbers.
a) f b) f


0 Express these mixed numbers as improper fractions.
a) 4I b) si



UNIT 8 I Fractions

o Arrange the fractions ^ descending order.
In

<E) Which is more?
1 1
a) of 24 or of 25
4 5
7 1
b) of 45 or of 60
1 2
2 3
c) of 30 or of 40
5 8
d) Work out the following. Give your answers in simplest form.
a) |+^ b) :^ + ^
T2-1
® Mr Jones buys a television set that costs $6000. He wants to pay one-eighth
of the price in cash. He wants to pay the balance over a period of 12 months.
a) How much money will he pay in cash?
b) How much money will he pay each month?

® Are the following statements true or false?
If they are false, correct them so that they are true.
a) is less than one-half.
10
b) If the denominator and the numerator are equal, the fraction is 1.
c) If the denominator is twice the numerator, the fraction is equal to a half.

d) If you add 1 to the denominator, you make the frartion greater.
e) If the denominator is more than the numerator, the fraction is less than 1.







Mathematics Connect




BODY MASS INDEX






Ttf






Underweight Overweight
BMI<18.5 BMI 25-29.9

A person's body mass index (BMI) is a measure of body fat based on height and weight. It is calculated
by dividing the person's mass in kilograms by height in centimetres squared, rounded to one decimal
place. A normal BMI is less than 25. A person is overweight or obese if the BMI is between 25 and 29.9
and greater than 30 respectively. BMI measurements give doctors information about patients' health,
and they in turn suggest health advice for them.

Reading and writing fractions
.
^
and
• A proper fraction has a numerator that is smaller than the denominator, for example |
.
• An improper fraction has a numerator that is greater than the denominator, for example |, y and ^
• A mixed number is a number that is made up of a whole number and a proper fraction.
The whole number is 1 *"1 ^ —The proper fraction is ^


Parts of a whole
of the shape is shaded


Equivalent fractions

Equivalent fractions are fractions that are equal in value even though their numerators
or denominators are not the same.

3 1
3
6 '
Simplest form
We simplify fractions by dividing the numerator and denominator by their highest common factor (HCF).

Converting between improper fractions and mixed numbers
A mixed number can be converted to an improper fraction, for example 3 ^ ^.

An improper fraction can be converted to a mixed number, for example 13 y


Comparing and ordering fractions
We compare and order fractions by first converting them to fractions with the same denominator.

Add or subtract fractions

if the fractions do not have the same denominator, express them as equivalent fractions with the same
denominators by finding the lowest common multiple (LCM) of their denominators. Then add or subtract
the numerators.
3 .i_24^_7__31 5I _ 2I -3^ - 2
5 40 " 40 40 ~ 40 ^3 ^2 ^6 6
7_ 24 _7_ 17
40 40 40 40
=^1
Fraction of a quantity
We divide the quantity by the denominator, then multiply by the numerator.
I of 24 sweets = 24 sweets -r 8 x 3 = 9 sweets



i i i i









Fractions

We find mathematical
patterns all around
us. Understanding how
numerical and geometrical
patterns are formed helps us
to find solutions to a scenario
quickly, or allows us to make use of
patterns to create new algorithms in
technology.
Take for example if everyone in class
shakes hands once with everyone else, can
'//
you find out how many handshakes that
took place? What do you think this problem
has to do with connecting computers in a
Local Area Network (LAN)?


A
./

In this chapter
umber Sequences
Pupils should be able to:

• generate terms of an
integer sequence
• find a term given its 9.1.1 Number patterns and sequences
position in the sequence
• find simple term-to-term
Investigate!
rules


Micky drew the triangles shown to
form a pattern. How many lines
are there In Figure 13?

Figure 1 Figure? Figures
Understand the problem
Figure 1 has lines.
Figure 2 has lines.
Figure 3 has lines.
What about the number of lines in Figure 13?

Make a plan
Can you make a table to see the relationship between the figure
number and the number of lines?
Figure Number of lines


2
3
Look for a pattern in the number of lines in the figures.

3, 5, 7,...
Starting with 3 lines in Figure 1, we to find the number
of lines in the next figure.

Try it out
For Figure 13, how many lines do we need to add to the number of
lines in Figure 1?
So, the number of lines In Figure 13 = + x

Figure 13 has ines.

Look back
-»5&"
Is the pattern correct?
Did I multiply correctly?
Did I add the correct number to the number of lines in Figure 1?
How can i check that my answer for Figure 13 is correct?



Terms and Sequences
I

Example 1


Look at the figures below. Find the number of dots, sticks and triangles in
Figure 9.












Figure 1 Figure 2 Figure 3 Figure 4

Figure Number of dots Number of sticks Number of triangles
1 3 3 1
2 6 9 4
3 10 18 9
4 15 30 16

Solution
Understand the problem
The figures are formed using sticks and each figure has a certain number
of dots and triangles.
Two triangles are added to Figure 1 to form Figure 2, three triangles are
added to Figure 2 to form Figure 3, and so on.
We have to find the number of dots, sticks and triangles in Figure 9.

Make a plan
We look for number patterns from the table.
For total number of dots: 3, 6, 10, 15, ...
For total number of sticks: 3, 9, 18, 30, ...
For total number of triangles: 1, 4, 9, 16, ...


Try it out
-^5
We use the number patterns to find the information on Figure 9.
To find the number of dots, we add the next number.
+ 3 +4 +5 +6 +7 +8 +9 +10
^. , ,


The number of dots in Figure 9=15 + 6 + 7 + 8 + 9 + 10
= 55
To find the number of sticks, we add (3 X the next figure number). — 6 = 3X2, 9 = 3X3, 12 = 3X4
+ 6 +9 +12 +15 +18 ^21^^+j^^27^




The number of sticks in Figure 9 = 30 + 15 + 18 + 21 +24 + 27
= 135

To find the number of triangles, we square the figure number.
(1 XI), (2X2), (3X3), (4X4),...
The number of triangles in Figure 9 = 9x9
= 81
Look back
is the pattern for finding the number of dots correct?
Is the pattern for finding the number of sticks correct?
Is the pattern for finding the number of triangles correct?



A sequence is a list of numbers that follow a pattern or are governed by a rule.

3, 10, 17, 24, ... is a sequence of numbers.
Each number in the sequence above is 7 more than the number before.
The numbers in the sequence are called the terms of the sequence. We start at 3,
then add 7 to each term to get the next term.
1" term 2^*^ term 3"^ term 4'^ term 5'^ term
3 10 17 24

+ 7 + 7 + 7 + 7
The term-to-term rule is 'add 7 to each term to get the next term'. A term and the
term that comes immediately after are called consecutive terms.


Example 2


For each sequence,
a) find the term-to-term rule.
b) write down the next two terms.

O 1,3,9,27,81,...
O -21,-18,-15,-12,-9, ...
512, 256, 128, 64, 32, ...
O 1.-1,1,-1,1,-1,...
Solution

o 1, 3, 9, 27, 8 1,

x3 x3 X3 x3

a) The term-to-term rule is 'multiply each term by 3 to get the next term'.
b) The next two terms are 81 x 3 = 243 and 243 x 3 = 729.
Q -21, -18, -15, -12, -9,

+ 3 + 3 + 3 +3

a) The term-to-term rule is 'add 3 to each term to get the next term'.
b) The next two terms are -9 -h 3 = -6 and -6 + 3 = -3.





UNIT 9 I Terms and Sequences

Q 512, 256, 128, 64, 32, Think and Share

■r2 ^2 -rl -tI
Design your own number
a) The rule is 'divide each term by 2 to get the next term'. patterns, then ask your
b) The next two terms are 16 and ^ = 8. partner to find the
next three terms in the
o sequence.

X (-1) X (-1) X (-1) X (-1)
a) The rule is 'multiply each term by (-1) to get the next term'.
b) The next two terms are 1 and -1.





Check My

Understanding

O For each sequence, write down the term-to-term rule and the next two terms,
a) The powers of two: 1, 2, 4, 8, 16, ...
1.^ 2, ^4, ^8,^ 16^ _ ^ ' ■"


x2 x2 X2 x2 ...

b) The multiples of five: 5, 10, 15, 20, 25, ...
c) The powers of five: 1, 5, 25, 125, 625, ...
^ For each of these number sequences, write down the term-to-term rule and the next two terms,

a) 7,9,11,13, ... b) 9. 12, 15, 18, ...
c) 5,6,8,11,15, ... d) 100,99,96,95,92, ...
e) 30,43,56,69, ... f) 97,94,91,88, ...
g) 15,8,1,-6, ... h) -11,-15, -19, -23, ...
i) 2,7,12,17,22, ... j) 2, 0, -2, -4, -6, ...

k) 12,36, 108,324,972, ... I) 540, 320, 160, 80, 40,
O Write the next three terms In each of these number sequences,

a) 5,7,9, 11, ... b) 11, 16,21,26, ... c) 10,2, 10,4, 10,6, 10,8,
d) 98,90,82,76, ... e) 24,17,10,3, ... f) 4,9,16,25, ...
O Find the missing terms in each of these number sequences.

a) 18, 15, 1 I 9, 6, 1 to. I I

b) 32, 1 i 50, 1 l 68, 1 I
c) I i 109, 100, 91,1 11 i I I

©* Challenge! Write down the next two terms in each of these algebraic sequences.

a) X, 2x, 3a', 4x, ...
b) jr + 1, A + 3, a: + 5, a: + 7, ...
c) 2a:, Zy + 2, 2a: + 4, 2a: + 6, ...
d) 3x-1,6A-1,9.r-1, IZv-l, ...
e) xy, Axy, 9a:t, ^6xy. ...
f) ab, Sab, 27ab, 64ab, ...

O The rule in each sequence is to add the same number to each term to get the next term.
Find the missing numbers.
a) 10, on, □,22, ... b) 8,0000050, ... c) 3,000^^

O The rule of each sequence is to subtract the same number from each term to get the next term.

Find the missing numbers.
a) 40,00034, ... b) 90,ODD50. ...
0 i3,odddddd-3 d) -2,onn-26


© Look at the number sequence.
42, 22, 18, 22, ...
12 = 4 X 2 + 14

= 8+14
= 22
We say that the term-to-term rule in the sequence is to 'multiply the digits in each term and add 14' to
find the next term.
a) Find the next four terms in the sequence.
b) What is the 20'^ term in the sequence?


































Terms and Sequences
I

CHAPTER 9.2


In this chapter
Generating Number Pupils should be able to:


• describe the general
Sequences and Finding term in a number

sequence
• generate sequences
the General Term from spatial patterns





We can use a formula to describe the relationship between the numbers in a
sequence. We do this by comparing the terms in the number sequence to its
position in the sequence.
2, 4, 6, 8, 10, ... is a sequence of positive even numbers.

We can use Ti, T2, T3,... to denote each term in the sequence.

Ti (1"term) = 2
T2 (2"'^ term) = 4
T3 term) = 6
T4 (4^^ term) = 8
T5 {5^" term) = 10
So, Tp represents the term or general term.

We can present the sequence in a table.

Position of term in
1 2 3 4 5 ... n
the sequence, n
2 X 1 2X2 2X3 2X4 2X5 2X n
Term, In
= 2 = 4 = 6 = 8 = 10 = 2n

We see that the term is always twice the position of the term. We say the
general term in the sequence is 2n. So, In = 2/i. n is the variable representing
the position of the term.

So, to find the 110*^" term in the sequence, we use the formula In = 2n and get
Tiio = 2 X 110 = 220.

9.2.1 Simple number sequences






For each of the following sequences,
a) use a table to find a formula for the general term of the sequence.
b) find the 16*^ term, Tie-

O Multiples of three: 3, 6, 9, 12, 15,... in
@ Powers of four: 4, 16, 64, 256,... II
@ Perfect squares: 1,4, 9, 16, 25, ...
O Perfect cubes: 1, 8, 27, 64, 125, ...

Solution
O Multiples of three: 3, 6, 9, 12, 15, ...
a)
Position, n 1 2 3 4 5 ... n

1 X 3 2X3 3X3 4X3 5X3 « X 3
Term,
II
= 3 = 6 = 9 = 12 = 15 = 3/?
00
Hence, = 3n.
II
II
b) Tie = 3X16 = 48
@ Powers of four: 4, 16, 64, 256, ...
a)

Position, n 1 2 3 4 5 ... n
44
41 4^ 4^ 4/j
Term, Tp
= 4 = 16 = 256 = 1024
Hence, In = 4/i.
b) Tie = 4'^ = 4294 967 296

Q Perfect squares: 1, 4, 9, 16, 25, ...
a)
Position, n 1 2 3 4 5 ... n
,j2
V 2' 3^ 4^
Term, T^
1
= = 4 = 9 = 16
Hence, Tn = n^.
b) Tie = 16^ = 256
O Perfect cubes: 1, 8, 27, 64, 125, ...
a)
Position, n 1 2 3 4 5 ... n

1^ 43 5^ n'
Term, T^
= = 64 = 125
1
Hence, T^ =
b) Tie =16^ = 4096
210.) UNITS I Terms and Sequences

Example 2

The term of a sequence is In = Sn + 7.

a) Find the first five terms of the sequence.
b) Find the difference between the 2"'' and the 5^^ terms of the sequence.
Solution

a) Ti=6x 1+7 = 6 + 7 = 13
12 = 6x2 + 7 = 12 + 7 = 19
13 = 6x3 + 7= 18+ 7 = 25
14 = 6x4 + 7 = 24 + 7 = 31
Is = 6x5 + 7 = 30+ 7 = 37

b) Difference between 2"^ and 5^^ terms = 15 -T2
= 37-19
= 18

Check My
Understanding



O For each number sequence,
a) write down the rule.
b) find the value of the term, In, and
c) find the 75'''term.

i) 2,4,6,8,...
ii) 2,4,8, 16, ...
iii) 3, 9, 27, 82,...
iv) 8, 16, 24, 32, ...
v) 8,64,512,4 096,...
vi) 10, 100, 1 000, 10 000, ...

O The m"'term of a sequence is In = 3rt-5. Find
a) the 3"* term.
b) the sum of the 3'^^ term and the 10'" term.

O The term is given. Use the term to find the first five terms of the sequence.
a) Tn = n + 3

1
b) Tn = 2« -
c) Tn= 11rt-9
d) Tn = 3«-3
e) Tn = 2« + 7
f) Tn = 4«-10

9.2.2 Challenging number sequences


For number patterns that are not directly related to its position in the sequence,
we can compare it with a known number sequence, like the multiples of a
number, perfect squares or cubes.


Example 3

Find the formula for the general term (Tn) of each sequence.
O 4,7, 10, 13, 16,...
@ 1, 6, 11, 16, 21, ...

Solution
o
Find the differences between the terms.

Position n 1 2 3 4 5
TermTn 4, 7, 10, 13, 16,




We start at 4 and add 3 each time.
We say the common difference = 3.

Because the common difference is 3, we compare the given sequence to
the multiples of 3.

Position in the sequence, n 1 2 3 4 5
Terms 4 7 10 13 16
Multiples of 3 3 6 9 12 15

1
1
3 + 6 + 1 9 + 12+1 15 + 1
1
(Multiples of 3) +
= 4 = 7 = 10 = 13 = 16
So, each term in the given sequence = multiples of 3 + 1.
Ti = 4 = 3 + 1= 3x1 + 1
T2 = 7 = 6 + 1= 3x2 + 1
T3 = 10 = 9 + 1= 3x3 + 1
T4 = 13 = 12 + 1 =3 X 4 + 1
T5 = 16 = 15 + 1 =3 X 5 +
1
So, Tn = 3 X /! + 1 = 3/7 + 1.
©

Find the differences between the terms.
1
Position n 1 2 3 4 5
Term In 1, 6, 11, 16, 21, ...

+ 5 +5 +5 +5
We start at 1 and add 5 each time.
We say the common difference = 5.



UIMIT 9 I Terms and Sequences

Because the common difference is 5, we compare the given sequence to
-2^ the multiples of 5.


Position in the sequence, n 1 2 3 4 5
Terms 1 6 11 16 21
Multiples of 5 5 10 15 20 25

5-4 10-4 15-4 20-4 25-4
(Multiples of 5) -4
= 1 = 6 = 11 = 16 = 21

So, each term in the given sequence = multiples of 5 - 4.
Ti = 1= 5-4 = 5x1-4
12 = 6 = 10-4 = 5x2-4
T3 = 11 = 15-4 = 5 X 3-4

14=16 = 20-4 = 5x4-4
T5 = 21 =25-4 = 5 X 5-4
So, Tn = 5 X /? - 4 = 5// - 4.


Check My
Understanding


For each of the following sequences,
O "find a formula for the general term of the sequence, Tn.
© determine the 100'^ term.
a) 10, 12, 14,16,... b) 9,14,19,24,... c) 16,20,24,28,
d) 13,24,35,46,... e) 3,7,11,15,... f) 2,6,10,14,...
g) 5,9, 13, 17, ... h) 7, 12, 17,22, ... i) 2,8,14,20,...







Investigate!



Find out what palindrome numbers are on the Internet.
Then investigate the pattern found in palindromic numbers.

a) Use a calculator to find all the palindromic multiples of 13 that are less
than 1000.
b) Describe the pattern in these numbers.

9.2.3 Number seauences in
geometrical patterns



Geometrical or spatial patterns are number patterns represented by diagrams.
Recording the numbers in a table helps us spot the pattern and describe the
general rule for the pattern.



-• Think and Share

6 people can sit around one rectangular table
as shown.

If 10 tables are put together so that they join
at their breadths one after another, how many
people can sit around these 10 tables?
Try drawing a diagram to find out. Can you
spot a pattern? Can you write a formula to
represent the general term, 1^?




Example 4


These patterns of hexagons form a sequence.


t tt ttt tttt



Pattern 1 Pattern 2 Pattern 3 Pattern 4

O Write down the term-to-term rule.
@ Find a formula Tn for Pattern n.
O How many hexagons will be in Pattern 25?

Solution
The number of hexagons in each pattern form the sequence 3, 6, 9, 12, ...

O The term-to-term rule is "add 3 hexagons to each pattern to get the next
pattern". The common difference is 3.
@ Use a table to find Tn-


Pattern number, n 1 2 3 4
3 6 9 12
Number of hexagons
= 3x1 = 3X2 = 3X3 = 3X4









Terms and Sequences

Each term Is a multiple of 3.
Ti = 3 = 3 X 1
T2 = 6 = 3 X 2
Tb = 9 = 3 X 3
T4= 12 = 3 X 4
So, Tn = 3 X n.

O T25 = 3 X 25 = 75


Example 5


These patterns are made from a number of sticks.

G □□ □□□





Pattern 1 Pattern 2 Pattern 3

O Draw the next two patterns in the sequence,
o Write down the term-to-term rule.
O Write down a formula for Tp.
O Hovv many sticks are there in Pattern 75?


Solution
O The next two patterns are:
□□□□□□□CD






Pattern 4 Pattern 5
@ We can use a table to derive the term-to-term rule.


Pattern number, n 1 2 3 4 5
Number of sticks 4 7 10 13 16

The term-to-term rule is "start at 4 and add 3".
The common difference is 3, so we compare the number of sticks to multiples
of 3.
O We can find T^ by adding in the multiples of 3 and finding the difference.

Pattern number, n 1 2 3 4 5
Number of sticks 4 7 10 13 16
Multiples of 3 3 6 9 12 15

3-I-1 6 + ^ 9-t- 1 12 + 1 15 + 1
(Multiple of 3) + 1
= 4 = 1 = 10 = 13 = 16

So, each term in the given sequence = multiples of 3 + 1.
Ti=4 = 3 + 1=3xi + 1
1
T2 = 7 = 6 + 1= 3x2 +
1
T3 = 10 = 9 + 1= 3x3 +
1
T4 = 13 = 12 + 1 =3 X 4 +
1
T5 = 16 = 15 + 1 =3 X 5 +
So, = 3 X /; + 1 = 3« + 1.

O T75 = 3 X 75 + 1 = 225 + 1 = 226



Check My
Understanding


O Recall the Think and Share we did earlier. A function is being held in the school hall.
6 people can sit around one table, but if two tables are pushed together, then 10 people
can sit around the tables, as shown In the diagram below.
'ft











1 table: 6 people 2 tables: 10 people

a) Draw diagrams to work out the number of people that can sit around
0 3 tables, ii) 4 tables and lil) Stables.

b) Complete the table below.

Number of tables 1 2 3 4 5
Number of people that can sit around the tables

Write down the term-to-term rule. Use the table to help you.

Add two rows to the table as shown below to determine T-i, T2, T3, T4, T5 and T^, the formula for
finding the number of people who can sit around n tables.
Number of tables 1 2 3 4 5
Number of people that can sit around the tables
6 10
Multiples of ...
Difference between the number of people and
the multiples of...

d) How many people can sit around 75 tables?






216 UNIT 9 Terms and Sequences

O The fences around a park are made of 1-m pieces of wood as shown.
1 section of the fence 2 sections of the fence
5 pieces of wood 9 pieces of wood











Pattern 1 Pattern 2
a) Draw diagrams to show Pattern 3 (3 sections of the fence), Pattern 4 (4 sections of the fence)
and Pattern 5 {5 sections of the fence).
b) Complete the following table.

Pattern number 1 2 3 4 5
Number of pieces of wood 5 9

Write down the term-to-term rule. Use the table to help you.

Since the common difference is 4, let us compare the number of pieces of wood to the
multiples of 4.

1 2 3 4 5
Pattern number
Number of pieces of wood 5 9
Multiples of 4 4 8

1
4+1 8 +
1
(Multiples of 4) +
= 5 = 9
Use the table below to determine the general term. In, the formula we can use to find the
number of pieces of wood in n sections of the fence
d) Use In to calculate
i) how many pieces of wood are required for a fence made up of 50 sections (T50).
ii) how many sections of fence can be made with 823 pieces of wood.

O A white cross Is made up of square white tiles as shown below.

Length of each arm = 1 tile Length of each arm = 2 tiles
Total number of tiles = 5 Total number of tiles = 9
white tile
/







Pattern 1 Pattern 2

a) Draw diagrams to work out Pattern 3 and Pattern 4.

b) Complete the following table.

Pattern number 1 2 3 4 5
Total number of tiles
5 9
Write down the term-to-term rule. Use the table to help you.

Add two rows to the table as shown below to determine Ti, J2, T3, T4, T5 and T^, the formula for
finding the number of tiles needed in Pattern n.

Pattern number 1 2 3 4 5
Number of tiles
5 9
Multiples of...

Multiples of ... + ...

d) How many tiles would be needed for Pattern 52?




nevision


Write down the next two and the term-to-term rule for each of the following sequences.
a) -7.0,7,14,21,... b) 125, 121, 117, 113, 109, ...
c) 20,31,42,53,64, ... d) 2,9, 16,23,30, ...
e) 4,1,-2,-5,-8,... f) -23,-19,-15,-11,-7, ...
g) 88,79,70,61,52,... h) 8,4,2,1, 1, i, ...

i) 1,9,25,49,81,... j) -2,0,4, 10, 18, ...
O Use the given rules to find the first four terms in each of these number patterns.

a) Start with 0 and add 4 to each consecutive term.
b) Start with 243 and divide each consecutive term by 3.
c) Start with 11 and add ^ to each consecutive term.
d) Start with 1 and multiply each consecutive term by 0, 5.
For each of the following sequences, find a formula for the general term of the
sequence (Tn) and hence, find the 200^'^ term.

a) 6,9,12,15,... b) 5, 14, 23, 32, ...
c) 10, 12, 14, 16, ... d) 25, 26, 27, 28, ...

e) 1,4,9,16,... f) 2 2 1 3 3 1
O Look at this number sequence: 9, 16, 25, 36, 49, ...
a) Write down the next two terms of the sequence.
b) Find, in terms of n, the formula for the term (Tn) of the sequence.
c) Hence, find the 30'^ term.









JNIT9 Terms and Sequences

A
0 gardener uses pieces of wood to divide up a vegetable garden Into squares as shown below.








1 square 2 squares 3 squares

a) Draw diagrams to work out
i) the number of pieces of wood needed to make 4 squares.
ii) the number of pieces of wood needed to make 5 squares.

b) Complete the following table.
Number of squares 1 2 3 4 5
Number of pieces of wood


Use the table to write down the term-to-term rule.
Add two rows to the table as shown below to determine Ti, T2, T3, T4, T5 and Tp, the formula
for finding the number of pieces of wood needed to make n squares.

Number of squares 1 2 3 4 5
Number of pieces of wood
Multiples of...

Multiples of ... + ...
= number of pieces of wood

d) How many pieces of wood are needed to form:
i) 21 squares (T21)? ii) 40squares (T40)? iii) 111 squares am)?
0 lighting company advertises that it can make L-shaped advertising signs made up of light globes.
A
They make up some small L's as samples.
o
o o

o o o
o o o o
00 000 0000 00000
Pattern 1 Pattern 2 Pattern 3 Pattern 4
Height of L = 2 Height of L = 3 Height of L = 4 Height of L = 5

Total number of Total number of Total number of Total number of
globes = 3 globes = 5 globes = 7 globes = 9

a) Draw diagrams to work out Pattern 5 and Pattern 6.

b) Complete the table.

Pattern number 1 2 3 4 5
Height of L
Number of globes 3 5

Write down the term-to-term rule.

c) Add two rows to the table as shown below to determine Ti, T2, T3, T4, T5 and T^, the
formula for finding the number of globes needed in for a sign with a height of n globes.

Pattern number 1 2 3 4 5
Height of L
Number of globes
3 5
Multiples of...
Difference between the number of globes and
the multiples of...

d) How many globes will be needed to make a sign with a height of 120 globes?






Help
• A sequence is a list of numbers that are connected by a rule, called the term-to-term rule.
Sheet
• The numbers in the sequence are called the terms of the sequence
• The general term of the sequence, T^, can be represented by an algebraic expression, for
example Tn = -2n or Tn = 3h + 1. We can use a table to find Tp.






Mathematics Connect




By studying number patterns, we become more aware of how nature works.
Amazing
Observing patterns allows us to predict the behaviour of living things and other
Mathpm;
Mathematician
phenomena. Some examples are:
• Civil engineers use their observations of traffic patterns to construct safer cities.
• Meteorologists use patterns to predict thunderstorms, tornadoes, and
hurricanes.
• Seismologists use patterns to forecast earthquakes and landslides.

One famous number sequence that occur frequently in nature and is used to
Fibonacci was born in determine the golden ratio in beautiful photos is the Fibonacci sequence. The
1170. He was an Italian Fibonacci sequence does like this: is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21,...
mathematician from the
We find the next term in the sequence by adding the two numbers that come
Republic of Pisa and was
before it.
considered to be "the
1
most talented Western Ti =
mathematician of the T2 = 1 + 0 =
1
Middle Ages".
T3=Ti+T2 =1+1=2
O Find the next three terms in the sequence
T4 = T2 + T3 = 1 + 2 = 3 and share your working with your partner.
T5 = T3 + T4 = 2 + 3 = 5 @ Find out how Fibonacci sequence appears in
Tg = T4 + T5 = 3 + 5 = 8 nature and where we can see it. How is the
Fibonacci sequence used in modern day?




J I Terms and Sequences

UNIT 10





















You will learn about:
2D shapes in different orientations
Common solids and their properties

Side, angle and symmetry properties
of special quadrilaterals and triangles,
and regular polygons

Line and rotation symmetry in 2D
We find shapes and symmetry ail
shapes and patterns
around us. Can you see how the
Order of rotation symmetry Taj Mahal at Agra was planned by
architects using a mirror Image?

CHAPTER 10.1

In this chapter
hapes and Solids
Pupils should be able to:
• Identify, describe,
visualise and draw 2D
shapes in different ^ RECALL
orientations
• recognise and describe Look at the polygons below. Write their names in the space below.
common solids and
some of their properties J L
j
• name and identify side,
angle and symmetry 1 r
properties of special
quadrilaterals and
triangles, and regular
J " L
polygons with 5, 6 and
k
8 sides

n ,. r
















A 2D shape is a flat shape that has two dimensions, height and width, but no depth.




.c
U)
'a>
The orientation of a 2D
shape is a description of
how the shape is placed in width width
the space It occupies. The
A solid has three dimensions: A flat shape has two
properties of these shapes
height, width and depth. dimensions: height and width.
have not changed. They Hence, it is called a 3D shape. Hence, it is called a 2D shape.
are drawn in different
positions.
These shapes are closed shapes known
as polygons.
They have four straight sides.
So, they are also known as quadrilaterals.




UNIT 10 I Shapes and Symmetry

Investigate!



Orientate the 2D shapes so that they fit in the grid provided.
O You will receive a set of 2D shapes from your teacher.
Q Arrange and place the 2D shapes on your grid so that they all fit into
the grid, with no gaps and no overlapping of shapes. Try different
orientations of the 2D shapes.
Q Two 2D shapes have already been placed for you.

How would you name these shapes?



Polygons


A polygon is a two-dimensional closed shape with straight sides. It is a closed
figure.







All the sides are joined up. The lines cross each other.
They do not cross. This is a polygon. This is not a polygon.


Many polygons are named according to the number of sides they have. For
example, triangles have three sides, quadrilaterals have four sides, pentagons
have five sides, hexagons have six sides and octagons are eight sided polygons.

Each of the shapes below have 5 sides and 5 vertices. They are pentagons.



vertex



Each shape below has 6 sides and 6 vertices. They are hexagons.









Each shape below has 7 sides and 7 vertices. They are heptagons.








Each shape below has 8 sides and 8 vertices. They are octagons.

In a regular polygon all sides are of the same length and all angles are of the
same size.







equilateral square regular regular regular
triangle pentagon hexagon octagon
Let us look closely at each type of polygons.

Triangles

We can classify triangles in two ways. Let us recall some.

1. Grouping according to the lengths of their sides

• Triangle A has 3 equal sides.
It is an equilateral triangle.
• Triangles D and E each have 2 equal sides.
They are Isosceles triangles.
• Triangles B, C and F do not have any equal sides.
They are scalene triangles.

Grouping according to the types of angles
• The angles in triangles A and D are all acute angles.
They are acute-angled triangles.
• Observe that triangles C and E each have an obtuse angle.
They are obtuse-angled triangles.
• Observe that triangles B and F each have a right angle.
They are right-angled triangles.


Example 1


O Group the triangles according to:
a) the lengths of their sides and
b) the types of angles.



Solution
a)

Type of triangle Triangle Type of triangle Triangle
Equilateral A Acute-angled A, E, G

Isosceles E, F Obtuse-angled B, C
Scalene B, C, D, G Right-angled D, F







B UNIT 10 I Shapes and Symmetry

Quadrilaterals
You have learnt about rectangles and squares. Let us recall a few other
quadrilaterals such as the parallelogram, rhombus, kite and the trapezium.

The table below shows the basic properties of these quadrilaterals.

Quadrilateral Property
Rectangle a Two pairs of parallel sides
-\-w—
J L AB H DC, AD n BC
Opposite sides are equal.
AB = DC, AD = BC
n 1 h r

Square Two pairs of parallel sides
AB H DC, AD H BC

All sides are equal.
AB = BC, CD = AD



Parallelogram Two pairs of parallel sides
AB H DC, AD H BC
Opposite sides are equal.
AB = DC, AD = BC



Rhombus Two pairs of parallel sides
A. \* jD
AB II DC, AD a BC
All sides are equal.
AS = fiC = CD = AD


Kite No parallel sides.

Adjacent sides are equal.
AB = AD, CB = CD










Trapeziums One pair of parallel sides.

AD//SC

Example 2


Name each quadrilateral below. Give reasons for your choice.

a) b)














Solution

a) ABCDdoes not have any parallel sides. b) PQ//5/? and PS//Q/?
AB = AD and CB=CD. P0=5P and P5= QR.
ABCD is a kite. PQRS is a parallelogram.

c) WXHZY d) EF// HG and EH//FG.
l/l/XVZ is a trapezium. EF=FG=GH=EH
EFGH \s a rhombus.


Check My
Understanding


Q Draw each 2D shape in three different orientations on a dot grid paper.
Use a protractor, ruler and sharp pencil to make sure that you draw accurately.


Shape
a) b) c) d) e)
\








Q Name each quadrilateral below. Give reasons for your choice.
a) b) d)




















Shapes and Symmetry

Solids


A solid is a 3D object. The three dimensions are: height, width and depth. Many
objects around us are cuboids, spheres, cylinders, cubes and cones. Here are some
examples.









o

Cylinders
Spheres
Cuboids


A



JL


Cones
Cubes

Cubes and Cuboids

Solid Number of Faces Number of Vertices Number of Edges

Cuboid


12




Cube

12





The face of a solid is any flat surface in the solid.








Left side face Right side face Front face Back face Bottom face Top face

The edge of a solid is The vertex of a solid is the
the line formed where edges point where two or more vertices
edges meet. Vertices is the
two faces meet.
plural of vertex.

Check My
Understanding

^ Use the diagram of the solids and their nets to complete the table.


Properties of solid
Solid Net of solid Name of solid Number Number ! Number
of faces of edges of vertices

























Q


























Prisms
Prisms are named according to their parallel faces.
trapezoidal face

vertex

rectangular
face


edges
rectangular
triangular face
face
Triangular prism Trapezoidal prism
UNIT 10 I Shapes and Symmetry

If you look carefully at the prisms, you would notice that
O the faces of a prism are made up of 2 types of shapes. trapezoidal: in the shape
0 one of the shapes is the uniform cross-section of the prism. of a trapezium
0 the uniform cross-section of solid A is a triangle so it can be cut into many
triangles of the same shape and size.
0 the uniform cross-section of solid B is a trapezium so itcan be cut into many
trapeziums of the same shape and size.

A triangular prism has 5 faces. A trapezoidal prism has 6 faces.
3 faces are rectangles and 4 faces are rectangles and
2 faces are triangles. 2 faces are trapeziums.
It has a total of 6 vertices. It has a total of 8 vertices.
It has a total of 9 edges. It has a total of 12 edges.




In the prism on the right, the opposite faces opposite
are in the shape of a pentagon. faces
So, this prism is called a pentagonal prism.



This is another type of prism. Its opposite faces
opposite
are in the shape of a hexagon.
faces
So, it Is called a hexagonal prism.



Pyramids

The solids below are pyramids.
. vertex
vertex


triangular triangular
face
face
triangular
rectangular
base
base
V edges
edges
Rectangular pyramid Triangular pyramid

Notice that
a
0 pyramid does not have a uniform cross-section so it cannot be cut into
A triangular pyramid is
pieces of the same shape and size. also called a tetrahedron. ^
0 the slanted faces of a pyramid meet at a vertex at the top of the pyramid.
a
0 pyramid has a base.
a
0 pyramid is named after its base.
A rectangular pyramid has 5 faces. A triangular pyramid has4 faces.
1 face is a rectangle and 4 faces are triangles. All 4 faces are triangles.
There are 5 vertices. There are 4 vertices.
There are 8 edges. There are 5 edges.
The base of the pyramid is a rectangle. The base of the pyramid is a triangle.

Example 3














o




Each object above has the shape of a geometric solid. Name the geometric solid
and state the number of flat faces, vertices and the number of edges.


Solution
O The solid is a cube with 6 flat Q The solid is a cone with 1 flat
faces, 8 vertices and 12 edges. face, 1 vertex and 0 edges.
Q The solid is a sphere with 0 flat Q The solid is a cylinderwith 2 flat
faces, 0 vertices and 0 edges. faces, 0 vertices and 0 edges.
Is a cuboid a prism? What @ The solid is a cuboid with 6 flat @ The solid is a cube with 6 flat faces,
about a cube? faces, 8 vertices and 12 edges. 8 vertices and 12 edges.


Example 4













o








Each object above has the shape of a geometric solid. Name the geometric solid
and state the number of flat faces, vertices and the number of edges.


Solution
O The solid is a prism that has 5 faces, Q The solid is a pyramid with 5 faces,
6 vertices and 9 edges. 5 vertices and 8 edges.
Q The solid is a prism that has 8 faces, Q The solid is a pyramid with 6 faces,
12 vertices and 18 edges. 6 vertices and 10 edges.



UNIT 10 Shapes and Symmetry

What is the shape of the cross-section of each prism below?








Solution

b)





The cross-section The cross-section The cross-section is
is a triangle. is a rectangle. a parallelogram.


Example 6


What is the shape of the base of each pyramid below?
a) / , b) A 0










So ution
b)











The base is a square. The base is a triangle. The base is a rectangle.

Check My
Understanding




O a) b)












f)








i) Name the geometric solids.
ii) State the number of faces of each solid.
iii) State the number of vertices in each solid.
iv) State the number of edges in each solid.
Q Use the diagram of the solid and its net to complete the table.


Properties of solid
Solid Net of solid Name of solid Number Number Number
of faces of edges of vertices






































UNIT 10 Shapes and Symmetry

In this chapter

Pupils should be able to:
• recognise line and
rotational symmetry in
2D shapes and patterns
• identify the order of
rotational symmetry
Line of symmetry
• draw lines of symmetry
and complete patterns
with two lines of
i Investigate! symmetry
m

o Take a square piece of paper and fold it in half.

e Use a pair of scissors to cut out the shape as shown.
What shape will the hole in the middle be?

0 Draw a dotted line along the fold line. Is this line the Talk about the properties
line of symmetry? Does the shape have any other lines of symmetry? of each shape.
Discuss the shapes when
o Can you find the fold lines or lines of symmetry for the following
the paper was folded
shapes in order to cut them out?
in half and when it was
unfolded.
X Challenge yourself to
make other shapes by
folding a square piece of
paper in half and cutting
out the shapes.
What shapes do you think
are impossible to make?
^ RECALL


O Circle the shapes that have reflective symmetry.


* M i


Q Circle the shapes that do not have rotational symmetry.


4





A figure with at least one line of symmetry diagonal line
of symmetry
is a symmetric figure. When a figure is
folded along a line of symmetry, the two
horizontal line
parts of the figure fit exactly. A line of
of symmetry
symmetry is also called an axis of symmetry.
A shape can have more than one axis of
symmetry. The plural for axis is axes. diagonal line
of symmetry
vertical line of symmetry

When we cut a symmetric figure into two parts, one part is a mirror image of the
Spotlight
other part.

We find examples of
symmett7 in architecture,
pottery, mosaics, rugs,
fabrics and artwork.
A mandala is a symbol
used in Hinduism If you place a mirror along the line of If you fold the shape along the line of
and Buddhism. symmetry, the reflection is a mirror symmetry, one half folds perfectly over
Many mandalas are image of half of the shape. the other half.
symmetrical. Discuss the
lines of symmetry in this
mandala with a partner.











r






s Figure 10.1 Symmetry occurs all around us — in nature, buildings and in patterns.


Check My
Understanding


O How many axes of symmetry does each shape have? Draw all the axes of
symmetry.
a) A b)








Q State whether the dotted line in each figure Is an axis of symmetry of the
figure.

a) i b)













E Shapes and Symmetry
UNIT 10

Symmetry in patterns


Some patterns have lines of symmetry.
For example, this pattern has two lines
of symmetry.



Check My
Understanding

O This pattern has 2 lines of symmetry.
Colour in the unshaded parts of the
pattern. Use the lines of symmetry as
a guide.

















Q Complete the pattern. Use the lines of
symmetry as a guide. You may colour
the pattern to help you.















Q Complete the pattern. Use the lines of
symmetry as a guide.




^ X






Q Design your own pattern with two lines of symmetry. Swap with a partner.

J
Complete each other's patterns.

10.2.3 Rotation symmetry



A shape has rotation symmetry when it still iooks the same after a rotation (of less
than one full turn). The order of rotation symmetry, is the number of times a shape
fits onto itself in one full turn.


Example 1


Q Does this shape have rotation symmetry?
If yes, what is its order of rotation symmetry?
Solution
If this star is rotated about its centre point, it will
look the same in five different positions.




B D

D C D C
Position 1 Position 2 Position 3 Position 4 Position 5 Back to
position 1
This star shape can be rotated to five different positions and still look the same as the
figure in the original position. We say it has a rotation symmetry of order 5.

P Spotlight Q Does this shape have rotation symmetry?
if yes, what is the order of rotation symmetry?
This symbol for recycling Solution
has rotational symmetry
of order 3.







L,. This right-angled, scalene triangle has no rotation symmetry. The shape cannot
be rotated into a different position and still look the same. It looks the same only
once in one full turn.


Investigate!
m

Find an image of an airplane propeller. Trace its
shape on a piece of tracing paper. Place your tracing
directly over the propeller. Use a pin to pin down
Draw a shape with 2 lines the tracing in the centre of the propeller. Rotate it.
of symmetry and rotational e Make a 30^ 60°, 90°, 120° turn.
symmetry of order 2.
e The tracing fits onto itself after a turn.
How many such turns can you make
before the tracing moves back to its original position? What is the order of
rotational symmetry of the propeller about its centre?

UNIT 10 I Shapes and Symmetry

0 Write down the order of rotation symmetry, if any, of these letters,



MATHS



0 Write down the order of rotation symmetry, if any, of these shapes.

























a) Shade 1 more square b) Shade 1 more square so that c) Shade 1 more square so that
the figure has no lines of the figure has no lines of
so that the figure has
symmetry but has rotational symmetry and no rotation
1 line of symmetry.
symmetry of order 2. symmetry.
Then draw the line of
symmetry.












0*ChallGnge!
The figure has been torn in half. Complete the drawing so that the figure has a rotation symmetry
of order 2.

0 Complete the table. One has been done for you.

Number of lines of Order of rotational
Name of shape Draw lines of symmetry
symmetry symmetry


Scalene triangle




A
Isosceles triangle




Equilateral triangle




Square




Rectangle





Parallelogram




Rhombus





Trapezium




Kite




Regular pentagon




Regular hexagon





Regular octogon






Shapes and Symmetry

Revision

Q Complete the table by drawing each 2D shape in three different orientations.
Shape Orientation


a)






b)








Q Match each solid with its correct net.













c D






^ Complete the table.

oo ¥



equilateral triangle infinity symbol yen symbol


$




regular hexagon dollar symbol extinction symbol


Shapes Number of lines Order of rotation
of symmetry symmetry
equilateral triangle
infinity symbol
yen symbol
regular hexagon
dollar symbol
extinction symbol

Q Complete the pattern.
Use the lines of symmetry as a guide.




















O Journal Writing


Describe these 2D shapes and solids to someone who cannot see or feel them.
Talk about the number of sides, side lengths, number of angles, angle sizes, faces, vertices,
edges and orientation where relevant.
a) b)
J L


n r


d) e)











Mathematics Connect %,




Rotational symmetry in wind turbines

Wind turbines use the energy in moving air (wind) to
generate renewable energy. The blades of this wind
turbine have rotation symmetry. Without rotational
symmetry, the blades would 'wobble' in the strong wind
as they are not balanced or stable. This would make it
harder for the blades to rotate and less energy would be
generated.









Shapes and Symmetry
1

Help
A polygon is a 2D closed figure with straight sides.
Sheet
In a regular polygon, the sides are all the same length and all angles are the same size.







equilateral square regular regular regular
triangle pentagon hexagon octagon


. k
A triangle is a polygon
with three sides.


scalene triangle equilateral triangle isosceles triangle

A quadrilateral is a polygon with four sides.






square rectangle parallelogram rhombus trapezium kite


A solid is a 3D object. It has 3 dimensions: height, width and depth.

7^

§


cube (square cuboid triangular cylinder pentagonal square cone sphere
prism) (rectangular prism prism pyramid
prism)

Some solids have faces, edges and vertices. A sphere has no faces, edges or vertices.

9
vertex
edge




A line of symmetry cuts a shape Into two equal parts that mirror each other.

The dotted line is an axis of The dotted line is not an axis of
symmetry of the semicircle. symmetry of the quadrilateral.



The order of rotation symmetry is the number of times a shape fits onto itself in one full turn.
This shape has 6 lines of symmetry.
It will fit onto itself 6 times when rotated.
Its order of rotation symmetry about its centre is 6.

Number of lines Order of rotation
Name of shape Draw lines of symmetry
of symmetry symmetry


Scalene triangle



s
Isosceles triangle





Equilateral triangle




Square ••




Rectangle





Parallelogram




Rhombus





Trapezium




Kite




Regular pentagon





Regular hexagon




Regular octogon






Shapes and Symmetry

UNIT 11





Coordinate




Geometry






and Graphs Cape Town







You will learn about:
Reading and plotting coordinates in all four quadrants
Writing coordinate pairs that satisfy a linear equation
Plotting graphs corresponding to the linear equations

Recognising straight line graphs
Drawing and interpreting graphs in real life contexts









are exposed to a variety of data
jr everyday lives. A line graph '
I coordinate grid is a statistical Table Mountain
used In several Industries to ^ National Park
ent data effectively. Line graphs can ■xamptfl
:ate trends such as increase, decrease
0 change. For example, the changes in'
perature over a period of time. It Is useful ^
Camps Bay
»arn how to Interpret graphs accurately and
kly In order to understand data without
ng to process a large amount of numbers,
ket researchers, bankers, engineers, -f
■
itists and doctors use line graphs frequently
elp them Identify trends and formulating
erns. Graphs help them make sense of
data they have collected, and can also be
1 to predict future patterns and trends,
gie Earth uses coordinates from satellite
rmatlon to Identify places and landmarks
imagery
I as airports, places of interest and shops.
j could this information be used?

CHAPTER 11.1


In this chapter
Coordinate Grid
Pupils should be able to:
• read and plot coordinates
of points determined by
geometric information in ^ RECALL
all four quadrants
• generate coordinate
pairs that satisfy a linear Look at the grid reference. Can you
equation, where y is describe the position of an object using
given explicitly in terms
grid reference?
of jr queer





O Spotlight © Look at the coordinate grid.
a) Is the point (2, 5) the same as the
point (5, 2)7
Recall what an ordered
pair is and follow the b) Mark and label point A (1, 1), point
characters in the video B (2, 5) and point C (4, 1) on the
to find out what the grid
four quadrants in a c) Mark and label point P (0, 4), point
coordinate grid look like. Q (3, 1), point R (4, 4) and point
S (5, 3) on the grid.
Then draw the lines PQ and R5. 1 2 3 4 5






Grid references


We have learnt to give the position of an object on a grid by stating the column
first, then the row. For example, a square in Column D and Row 3 is D3.

Try and Apply!



Let's play Battleships
Battleships\% a game played on coordinate grids. Each player's
fleet of ships is marked on their grids. The location of the ships
are hidden from the other player. Players take turns calling
'shots' at the other player's ships. The objective of the game is to
destroy the opposing player's ships before your opponent sinks
all your ships.
► Figure 11.1 A battleship is a large, heavily armoured warship
equipped with many powerful guns.



Coordinate Geometry and Graphs

How to Play:

Each player needs two 10x10 grids — one on which they place their fleet (My Ships), and the other to
record their attack on their opponent (Their Ships).


10 10
9 9

8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
A B C D E F G H I
A B C D E F G H I
My Ships Their Ships


Use the given table to mark your aircraft carrier, battleship, submarine, Types of ships Size
cruiser and destroyer on My Ships grid.
1 X Carrier 5
• You can place the ships horizontally or vertically, but not diagonally. 1 X Battleship 4
• Each ship must be separated from other ships by at least one grid 1 X Cruiser 3
space. 2 X Submarine 2
• Do not show this marked grid to your opponent. 2 X Destroyer 1


Take turns to call out a coordinate position where you think the opponent has placed a ship.
For example, B6I
• If your opponent has a ship there, they call out "hit". Otherwise they call out "miss".
• Each hit destroys one part of the ship. When all the parts of a ship are hit, your opponent
must say "Hit and sunk."
The aim is to sink all of your opponent's ships.


You can record each coordinate position you have called out by shading it for a 'hit' and
5 drawing a cross x for a 'miss'. Draw a line through a ship when it is sunk.
• On your My Ships grid, record the locations that your opponent has called out.


A player wins the game when all the ships on their opponent's grid have been sunk.

'• Think and Share


In another Battleships qaxwQ, Rosa marked the hits and
misses on Nathan's ships as shown below. A hit is shown
by shading in the block. A miss is shown by drawing a
X
cross. A line is drawn through the blocks when a ship is
X X X X
sunk.
X X X
O Which one of Nathan's fleet has Rosa already sunk?
X X X X X
O Which squares around this sunk vessel can Rosa cross X X
out knowing that no ships are on these squares? How
X
does she know?
@ What type of ship might Nathan have at G10?

O Part of the cruiser is on G3. Give the grid references
for further hits on the cruiser.
X X
O If the hit at A2 was on an aircraft carrier, give the A B C D E F G H
possible grid references of the other hits on it. Nathan's Ships








11.1.2 Plotting coordinates



After recalling how to use a grid reference, let us now look at coordinate grids.
A grid reference gives the position of a square, but a coordinate point can give a
more specific position on a grid.

Coordinates are an ordered pair of numbers. They give the position of a point on
a set of axes. Look at the coordinate grid below.



We call this line the
y-axis. The numbers
on this line are
The arrows on the .v-axis
y-coordinates.
and the yaxis show that
secon 1 irst
the lines go on and do not qu idra It quadrant
stop. The point .We call this line
where the
the x-axis. The
A-axis and numbers on
2nd 1st y-axls meet 5 - 4 - 3 - 2 -1 0 5 . X
} this line are the
3rd 4th is called the x-coordinates.
origin. 7
1 Tird t. urti 1
qu^drant ■3, qu nt

-•4
-S

2461 UNIT 11 I Coordinate Geometry and Graphs