Math Smart - 7

We derive the formula for the volume of a cuboid asl x bx h.
Voiume of a cuboid = Length x Breadth x Height
= ixbxh








The cuboid is made up of 1-m cubes. Find the voiume of the cuboid.

















Solution
Volume of cuboid = Length x Breadth x Height
=6x4x5
= 120 m^

The volume of the cuboid is 120 m^.





The cake, in the shape of a cuboid, is 10 cm long, 6 cm wide and 7 cm high
Find the volume of the cake.
Solution
Voiume of cake = Length x Breadth x Height
= 10 X 6 X 7
7 cm
= 420 cm^
10 cm
The volume of the cake is 420 cm^.
6 cm




Calculate the volume of this cuboid.
Remember to write the
Solution
2 cm units. The dimensions are
V = lhh
in cm, so the volume is in
=6X3X2
cm*
= 36 cm^
6 cm
The volume of the cuboid is 36 cm^





<c^

This is a cube. Calculate the volume of this cube.
Ail the edges of a cube
Solution
are of the same length.
V=lbh
=4x4x4
= 64 m^
4 m
The volume of the cube is 64 m^.
Try and Apply!


Write down the dimensions
The cube has an edge of 12 cm. Find its volume.
of a cuboid that you can build
12 cm
using Solution
a) forty-eight Icm^ blocks. Volume of cube = Edge x Edge x Edge
b) twenty-three Icm^ blocks. = 12 X 12 X 12
= 1728 cm^
All the blocks must be used. Is
this possible? The volume of the cube is 1728 cm^.

We derive the formula for the volume of a cube is as / x / x /.
Volume of a cube = Edge x Edge x Edge
= / X / X / m
= /^
Check My
Understanding


O The solid is made up of 1-m cubes.

Find the volume of the solid.

© The cuboid has a length of 9 cm, a breadth of 5 cm and a height of 8 cm. 8 9 cm

Find the volume of the cuboid.
5 cm
@ The cubical block of cheese has an edge of 8 cm. 8 cm

Find the volume of the cheese block.


O The box, in the shape of a cuboid, measures 20 cm by 15 cm by 10 cm. 10 cm
Find the capacity of the box.
W 20 cm

© The cuboid is 11 cm long, 8 cm wide and 7 cm high. 7 cm
Find the volume of the cuboid.
8 cm
11 cm
O The block of clay has an edge of 14 cm.
Find the volume of the clay block.
14 cm



UNIT 15 I Measuring Time, Area, Perimeter and Volume

O What is the volume of the cereal box?
30 cm




20cm
O* Challenge!

Bern! wants to know the volume of the box once
7 cm
she has folded up the net shown.
Use the net to help her calculate the volume
of the box.
Hint: You could construct the net to help you 4 cm
find the missing dimensions.



O* Challenge!
a) Calculate the volume of this block of wood. 20 cm


12 m 12 m








5 m — 5 m



b) A hole has been cut from the centre of the block of wood.
Calculate the volume of the block of wood after the centre has been cut out.



24 cm
Challenge!

Calculate the volume of this cuboid.
48 cm''
72 cm


Challenge!
The two cuboids have the same volume.
Calculate the height of cuboid B.





2 cm


4 cm
6 cm
4 cm

Finding the missing edge given the volume of a cuboid






The cuboid has a volume of 125 cm^.
Calculate the height of this cuboid.

6 cm
7 cm
Solution
y=l bh yyg know the volume, length and width of this cuboid.
126 = 7 X 6 X \/\/e need to calculate the height.
126 = 42 X h
h = 3 cm Check that 7 x 6 x 3 = 126.

The height of the cuboid is 3 cm.



Check My
Understanding



O The cuboids have the same volume.
Calculate the length of the missing edge in each cuboid.
(All dimensions are in cm.)


a = cm cm cm d = cm








































Measuring Time, Area, Perimeter and Volume
1

CHAPTER 15.4


In this chapter
Surface Area of Cubes Pupils should be able to:

• calculate the surface
area of cubes and
and Cuboids cuboids from their nets



^ RECALL
Investigate!


Look at the net below.
Pizza boxes are made by folding the net of
a box. The surface area of the pizza box Is
the sum of the areas of all the faces of the
box.

Flatten a pizza box. Use a measuring tape
to measure the dimensions of the box
a) What are the
so as to calculate the surface area of the
common shapes of all
flattened box.
its faces?
b) What type of solid
The surface area of a cuboid is the total area of all the faces of the cuboid. has a net like this?
We can use a net to calculate the surface area of a cuboid.









/ X W IXw



/ X h ixh
Solid







Net of solid

All cuboids have six rectangular or square faces. To find the surface area of a
cuboid, we add the areas of the six faces.
Area of each face:
yellow face = l x h
• red face = hx w
^ green face = / x w
There are two of each type of faces in the cuboid.

Example 1


6 cm
Calculate the surface area of this cuboid.
3 cm



Solution
You can draw the net of the cuboid to help you. Use the
information given to label the sides of the net.
6 cm
6 cm
B
3 cm
6 cm 3 cm 3 cm

A C A C
Cuboid
B

6 cm
Net of the cuboid
Find the area of each face.
Area of face A: 6 x 3 = 18 cm^
Area of face B: 6 x 3 = 18 cm^
Area of face C: 3 x 3 = 9 cm^
Surface area of the cuboid = (Area of face A x 2) + (Area of face B x 2) +
(Area of face C x 2)
= (18 X 2)+ (18 x 2) + (9 X 2)
= 36 + 36 + 18
= 90 cm^
Try and Apply!



Draw the nets of the
4 cm
cuboids shown on a piece
of square grid paper to
find out if they have the
same surface area. (Each
square grid is 1 cm^.)

Hint: The net of cuboid A
has been drawn for you.
Draw the net of cuboid B.
2 cm





Check that the dimensions of your net of cuboid B is correct by cutting out the net you have drawn
and folding it into a solid. Measure the length, width and height of the cuboid you have made. Is it the
same as the dimensions given in the diagram?


UNIT 15 Measuring Time, Area, Perimeter and Volume

Find the total surface area of this cuboid.

Hint Note the number of faces the cuboid has.
10 cm
3 cm
Solution
4 cm

3 cm



10 cm
10 cm 10 cm
3 cm 3 cm I
4 cm 4 cm 4 cm

Area of one blue Area of one red Area of one green Total surface area
rectangle face rectangle face rectangle face = 24 + 60 + 80
= 3x4 = 3 X 10 = 4 X 10 = 164 cm^
= 12 cm^ = 30 cm^ = 40 cm^
Area of two blue faces Area of two red faces Area of two green faces
= 12 X 2 = 30 X 2 = 40 X 2
= 24 cm^ = 60 cm^ = 80 cm^


Check My
Understanding


O The following are the nets of some solids. Calculate the surface area of each
solid based on the dimensions given,
a) ■) b) 6 cm 30 mm

30 mm
3 cm

10 cm 2 cm



3 cm

3 cm




O Calculate the surface area of these solids. Start by drawing the net of each solid
a) ^ ^ b) ^ c)

6 cm


cm
4 cm
2cm |i- •'
8 cm
5 cm 40 mm

d) e) f)

6.5 cm
50 mm
1 m

2 cm
4 cm
3 cm
200 mm 3 cm
2 cm


Remember to convert all dimensions to the same unit.


@ Calculate the surface area of this cuboid
a) incm^and /i = 5 cm
b) in mm^.

If the width of the cuboid is halved, what
w = 6cm
is the surface area of the cuboid in cm^?
/ = 8 cm

O* Challenge!
This bench is made by combining two cuboids.
Calculate the surface area of the bench that
needs to be painted.

4 cm
7 cm






2 cm 6 cm
O* Challenge!
Calculate the surface area of this open cube.
Hint: There is no lid.
Kim wants to paint the inside and outside
of the open cube. Calculate the total surface
area to be painted.
2 m


















UNIT 15 Measuring Time, Area, Perimeter and Volume

Revision


O Write the times in 12-hour and 24-hour notation.
For times in 12-hour notation, use a.m. or p.m.
a) 3 h 22 min later than the b) 71 min earlier than the
shown time. shown time.


(This clock
shows a
time before
12 noon.) /■




12-h 12-h

24-h 24-h
Q Express the following time durations in the units stated.

a) 10h = mm b) 10 h 38 min = mm
c) 215 min = d) 1 a minute = .
e) 48 h = mm f) 17 min 48 s = .

g) 35 min 23 s = h) 210 min =
i) 615 min =
@ Look at the train timetable and answer the questions that follow.


Train station Departure timing
Badlands 07 10 07 25 07 40 07 55
Castle 07 30 07 45 08 00 08 10

Lake 07 45 08 00 08 15 08 25
Orchid Fields 08 05 08 20 08 35 08 45
River Bend 08 15 08 30 08 45 08 55
Canyon West 08 40 08 55 09 10 09 20

a) At what time does:
i) the 07 40 train from Badlands station arrive at Orchid Fields station?
11) the 08 30 train from River Bend station depart from Badlands station?

b) Trains depart from Badlands station every quarter of an hour.
What are the departure times for the next three trains?

c) Rashni arrives at Castle station at 07 35.
What time is the next train he could catch to Canyon West station.

d) How long should it take to travel from
i) Badlands station to Canyon West station if I catch the 07 25 train?
ii) Lake station to River Bend station if I catch the 08 25 train?
iii) Castle station to Orchid Fields station if I catch the 07 30 train?

e) Jackie arrives at Lake station at 07 45. She would like to spend as long as possible at Lake
station but she needs to be at Canyon West station by 09 10. How long can Jackie spend at
Lake station?
f) Jim catches the train from Castle station to work in River Bend. He should be at work by
09 00. It takes him 10 min to walk from home to Castle station. It takes Jim 10 min to walk to
work from River Bend station. What is the latest time Jim can leave home and get to work
on time?
O Express

a) 92 m^ in cm^ b) 7000 mm^ in cm^ 35 000 cm^ in
d) 150 000 mm^ in m^ and e) 1.05 cm^ in mm^.

O This watch face measures 3.2 cm by 4.1 cm.
a) Find the area in mm^.
b) Find the area in cm^.
c) Find the perimeter in mm.
d) Find the perimeter in cm.



O Neo has a garden with the dimensions 7 m by 5 m.
In the centre of the garden, there is a square fish
pond with a length of 3 m.

Calculate:
a) the perimeter of the garden.
b) the perimeter of the fishpond.
c) the area of the garden including the fishpond.
7 m
d) the area of the garden without the fishpond.
^ Challenge! The town council wants to tar the streets in an area as shown. The streets are 10 m
wide. The blocks (including the gray pavements) are 1 km long and 500 m wide.

a) Calculate the area that needs to be tarred.


1 1
block

street

1 1


b) The roads have dotted lines painted on them to indicate the lanes. If each white rectangle
that forms each dash on the road measures 80 cm by 30 cm, what is the area of the shaded
portion on this stretch of road?
18m




5 m





UNIT 15 I Measuring Time, Area, Perimeter and Volume

9 cm

O Find the length of the missing sides. Then calculate
the perimeter and area of this composite figure. 4 cm
5 cm
20 m
O Calculate the perimeter and 5 m
area of the compound shape. 7 cm

25 m






4 m
Calculate the volume of each solid.
a) 3 mm b) 4 cm



3 cm
9 m
3 mm
2 cm
3 mm



d) 4 in e) f)


7.5 cm
n-c-.-:';
14 in. 8 mm 4 mm
12 mm
4.5 cm^
11 in. 7.5 cm
g) h)
8.5 cm
4 cm


10 cm
15 cm

The two cuboids have the same volume.
Calculate the width of cuboid B.
4 cm 5 cm



w
5 cm
10 cm

A box of pencils measures 5 cm by 5 cm by 15 cm.
How many such boxes of pencils can fit In the box shown? 40 cm


35 cm
1.5 m

Calculate the volume and surface area of each of these cuboids.
a)

30 mm
3.2 cm
5 cm
20 mm
3.2 cm

Peter builds a cupboard 2 m high on a square base with sides measuring 600 mm.
a) Calculate the perimeter of the square base.
b) Calculate the area of one of the vertical faces of the cupboard.
c) Calculate the volume of the cupboard in m^.
d) Calculate the surface area of the cupboard in m^.
e) Peter wants to paint the outside of the cupboard, but not its base.
Calculate the surface area that needs to be painted.
f) Peter starts painting at 09 30 and finishes painting at 13 05.
How long did it take for Peter to paint the cupboard?







Mathematics Connect



In 1583. Italian physicist Galileo Galilei discovered that
a pendulum always takes the same amount of time to
swing back and forth. So, in the past, pendulums were
used to make clocks because a pendulum provides a
regular indicator of time passed. If the string of the
pendulum is about 25 cm, it makes a 1-second swing,
back and forth. This in turn moves the gears in the clock
every 1 second, causing the hour and minute hands
in the clock to move, to show time. But a problem is
that air resistance and friction uses up the pendulum's
kinetic energy, and the clock will stop working when
the pendulum stops moving. Pendulum clocks also
due to gravity given its tiny size, it varies its swing by
depends on gravity to move. If there is a change in
pressure and temperature. So. a quartz clock may lose
gravity, like deep under the sea or high up on the
time over the years.
mountains, the clock will tell time that is either too fast
or too slow. Recently, scientists built the most accurate clock that
loses only 1 second every 15 billion years. It is made of
To solve this problem, we found quartz. Inside a quartz
strontium, which makes use of the regular vibrations
clock or watch, the battery sends electricity to the
of an atom between two energy states. Accurate clocks
quartz crystal, causing the quartz crystal to move back
play an important role in determining the accuracy
and forth exactly 32768 times each second. The circuit
of GPS satellites in navigation and global positioning
then sends out electric pulses, one per second. These
technology. The Strontium clock is so accurate, and
pulses power the watch display to show digital time or
sensitive to minute changes in gravity, that physicists
the pulses turn the gears that move the clock's hands.
are thinking of using it to map out the shape of the
Although the quartz crystal is not affected greatly
Earth!
Measuring Time, Area, Perimeter and Volume

Help
Time
Sheet
Unit Equivalent to
1 min 60s r.-io 2-1
1 hour 60 min
t /[
1 day 24 h
1 week 7 days

1 mth 28-31 days
Digital clock Analogue clock
12 months or 365 days, but a
lyr We can convert the amount of time from one
leap year has 366 days.
unit to another.
1 decade lOyr
s
1 century 100 yr 1 h = 60 min 1 min = 60
A time interval is the duration of time between two events.
0 75
? ?
1-5
R - 00 45
10 30
GO 45 1045 11 15
24-hour clock 12-hour clock 24-hour clock 12-hour clock
00 00 12 00 midnight (start) 15 00 3 00 p.m.
01 00 1 00 a.m. 16 00 4 00 p.m.
02 00 2 00 a.m. 17 00 5 00 p.m.
03 00 3 00 a.m 18 00 6 00 p.m.
06 00 6 00 a.m. 19 00 7 00 p.m.
10 00 10 00 a.m. 20 00 8 00 p.m.
11 00 11 00 a.m. 21 00 9 00 p.m.
12 00 12 00 p.m. (noon) 22 00 10 00 p.m.
13 00 1 00 p.m. 23 00 11 00 p.m.
14 00 2 00 p.m. 24 00 12 00 midnight (end)

Perimeter Area
The perimeter of a figure is the total Area is a measure of the amount of space that a flat
length around the edge of the figure. surface covers. The area of a figure is the number of
square units contained within the figure.
Quadrilateral Perimeter
Quadrilateral Area
Rectangle
Rectangle
2 x(/ + b)
/ X b
_J
Square

Square
4 X /
/ IX I

Abbreviation
Unit of nieasurement for area (not drawn to scale)
square millimetre
1 mm
A square that measures Converting between units in area
1 mm by 1 mm has an area 1 mm mm
of 1 square millimetre. X 1 000 000

square centimetre
1 cm
A square that measures
cm
1 cm by 1 cm has an area of 1 cm m^ cm^ mm^
1 square centimetre.
^ 100^ -r 102
square metre
1 m
A square that measures ^ 1 000 000
m
1 m by 1 m has an area of
1 m
1 square metre.



Volume

Solid Number of flat faces Number of vertices Number of edgef^
Cuboid



12



Cube


12




Volume of a solid Surface area of a solid
Definition A measure of how much space a solid takes up. The total area of all the faces of a solid.

Units mm^ cm^ or m^ mm^ cm^ or m^
Formula Volume = Ix bx h Total surface area
^Ibh = (Area of face A x 2) + (Area of face B x 2)
+ (Area of face C x 2)

= 2Ui + 2hb + 21b
height h
h
'Breadth b
length I n Face 2
Face 1
Face 3






UNIT 15 I Measuring Time, Area, Perimeter and Volume

UNIT 16




rarlsTormation






You will learn about:
transformation of 2D points and shapes by:
• translation
• reflection In a given line
• rotation about a given point
objects and images which are congruent after
translation, reflection and rotation





















































Transformation is when a point or a collection of points
moves across space. Is reflection the same as symmetry?
Where is the line of symmetry in this photo?

CHAPTER 16.1

In this chapter

Pupils should be able to:
• know that shapes
remain congruent after
translation, reflection
Investigate!
and rotation

Look at the pictures below. What do you notice about the objects in each
picture?
^ RECALL


Ashley marks the
coordinates (-1, 1), (3, 1)
and (3, -1) on a coordinate
grid. What Is the last point
she has to mark if she
wants to draw a rectangle?
'CvD V : ■/ i

Can you describe the changes or movements the objects went through?



The objects have changed positions. We say the objects have undergone a

transformation.

Types of transformation
Translation Reflection Rotation
Translation is a Reflection is a Rotation is a turn about
movement. transformation that acts a centre point.
like a mirror.
Every point of the Each point of the object Each point of the
object is moved the is reflected in the object is rotated about
same distance and in mirror line, the same a fixed point to a new
the same direction. distance away from the position in a different
mirror line {or line of orientation.
reflection)
On the chess board, the In the photograph, the In the diagram, the
knight has translated 2 Taj Mahal is reflected in triangular shape
squares up and 1 square the Yamuna River. bounded by the spokes
to the right. of the wheel is rotated
about its centre to a
new position.






UNIT 16 Transformation

Object and image

Just like objects, shapes can also go through a change or a transformation. In the
transformation, the original shape is called the object, and the transformed shape
is called the image.

Translation Reflection Rotation
ine of r ifle :tio 1

ect LWfi
PSi

object image
W
imcige :en re (f f irec tior of
Otatlor 1'
)tat on










Investigate!



A B C


bbject 1
image
5



J
image

image



Q Identify each transformation above.
Q Measure the dimensions (sides and angles) of the shape and the image In
each transformation.
Label the dimensions on each shape.
Q Compare the dimensions of the object to the image for each
transformation. What do you see?



Notice that each object's shape and size did not change in each transformation. In
translation, reflection and rotation, the object and the image are of the same size
and shape. Only the orientation or position of the object has changed. We say the
object and the image are congruent.

Using coordinates to describe transformations

You can describe the position of the object and the transformed Image using
coordinates on a coordinate grid.





The object and the image are points on the coordinate grid.

O Write down the coordinates of the object and the image.
Q State whether each statement is true or false.
a) The image is a translation of the object.
b) The image is a reflection of the object.
c) The image is a rotation of the object.




object im ige
—1 )—








The distance from the Solution
object to the image is 8
O Coordinates of the object: (-4, 3)
units, not 4 units.
Coordinates of the image: (4, 3)
Distance is counted from
0 True.
a)
one point to the other
The object has been translated by 8 units to the right.
point, not from the y-axis
The image is a translation of the object.
or x-axis.
We can also work out the
distance by finding the Ob ect im< ige
difference between their
x-coordinates:
Distance = 4-(-4)
= 4 + 4 -6-5 -4-3 -2 -1 0
= 8 units
b) True.
The object is 4 units to the left of the y-axis.
The image is 4 units to the right of the y-axis.
The objert has been reflected about the y-axis.
The image is a reflection of the object.

In this example, the line
line of reflection
of reflection is the y-axis.
ob ect im ige
However, an object and
an image can be reflected
along any line on the grid.


-6 -5 -4 -3 -2 -1 0

UNIT 16 I Transformation
E I

c) True.
The object is rotated about the origin.
The origin is the centre of rotation.
The image is a rotation of the object.

Try and Apply!
1 object —4- l inage! 1
-y'
>2 ! Use your protractor to
—zr
measure the angle of
—h
> —
< 2 rotation.

A. _ _^ _.3 _"> _1 a 1 ^ : 1 '; f
centre of rotation

Check My

Understanding


Q Look at this shape. Which image shows a reflection?
















0 Look at this shape. Which Image shows a rotation?




n
m
-j.'W






0 Look at this shape. Which image shows a translation?

m



ji in



I! B

CHAPTER 16.2


In this chapter
Translation, Reflection
Pupils should be able to:

• transform 2D points and
shapes by: and Rotation
o translation
o reflection
o rotation
16.2.1 Translation

^ RECALL Translation is a movement. Every point of

the object is moved in the same direction M
The coordinates of a for the same distance.
N
polygon PQRS is (5, 6),
The image and the object are congruent.
(2, 5), (1,8) and (3, 8).
Let us use coordinates to describe
Translate the polygon 5
translation of shapes.
units down. What are the
coordinates of the vertices Look at the polygon on the right.
of its image? The coordinates of its vertices are:
D(0,-2) E(3,-2) F(4,-3)
G{3.-4) H{0.-4) 1 (1,-3)











Let us look at where the vertices of the
Image of polygon DEFGHI will be after
we translate the polygon 6 units up and
-5 -4 -3 -I -ns ■*-x
4 units to the left. I 1












The coordinates of the vertices of the
image are:
D'(-4, 4) E'(-1,4) F'(0, 3)
G'(-1,2) H'(-4,2) I'(-3, 3)

-5 -4 -3 -2 -1 0




UNIT 16 I Transformation

Q Draw and label AABC on a coordinate grid. Use these coordinates: A (-4, 5),
B(-5, 1)andC{-2, 2).
Q Translate AABC 6 units to the right and 1 unit down. Label the points in the
image A', B' and C.
Q Write down the coordinates of AA'B'C.

Solution
0 Plot the points on the grid. Label them A, B and C. Join the points with
straight lines to form the triangle.









c

B
->■ X
_
-<
)
' ) - 1 -! - : - 0 n t ,


Q Move each point 6 units to the right, then 1 unit down.Mark the new points
after the translation as A', B' and C.






A'.
•l! '
4
! We use (') to label the
c
M points on the image after
'r the translation.
'
B
B', '
' - ) i 1 - ! 1 - 0 1 Notice that one of the
-
-
vertex (B') is located on
the .r-axis.

Q AABC is the object that is translated.
AA'B'C is the image formed as a result of the translation.
The coordinates of AABC are: A' (3, 4), B' {2, 0) and C (5, 1).

Whink and Share


What do you notice about the changes in the value in the coordinates after the translation? Can you use
addition and subtraction to work out the coordinates of the Image from the original coordinates?











-6-5 -4-3 -2 -1 0 -6 -5 -4 -3 -2 -1 0 3 4 5 6 -6-5-4 -3 -2 -1 0 1 2 3 4 5 T-'




Check My
Understanding


O Translate triangle XY2 8 units to the left and
2 units down. Draw the image of the triangle
after translation and label its vertices as X', Y'
and Z' on the coordinate grid. Then write the
coordinates of X', Y' and Z'. Use a piece of
graph paper to help you.


X
>












Translate polygon WXYZ 7 units down and
4 units to the right. Draw the image of
the polygon after translation and label its V/
vertices as W', X', Y' and Z' on the coordinate
grid. Then write the coordinates of W', X', Y'
and Z'.

-5 -4 -3 -2 H














UNIT 16 I Transformation

Q Look at the figures below. Draw and label each image after the described translation on a piece of
graph paper. Then write the coordinates of each translated image.

a) 3 units down, 4 units to the b) 7 units to the right, 3 units c) 5 units up, 2 units to the
left. down. right.



ill
F F
G N




5 -4 -3 -I2 - 1 0 * 5 -4 -'3 - 2 - 1 -5 -4 -3 - 2 -

7 ? -2
7 ..3
(t



Q Look at the figures and their images on the coordinate grids. Describe each translation in words,
a) b) c)





K'
M \A

*'X
-5 -4 - 2 - 5
8'



N' M'



e L'M'N'O' is the image of rectangle LMNO after it
was translated 6 units to the left and 5 units up.
t' 1
a) Draw rectangles LMNO and L'M'N'O' on a piece .mi. ..
of graph paper using the same set of axes shown
as shown on the right.
b) Write the coordinates of rectangle LMNO.
-
5 4 3 - I I u 1
-
-
lQ-D) T
7
N(n. □>

oO'Q) -S


Q*Challenge!

Point Q {0, 5) is translated 5 units up and 3 units to the left. What are the coordinates of point Q'?

Try and Apply!



We have learnt simple tessellation patterns in Stage 5. A tessellation is a repeating pattern of one or
more shapes that has no gaps or overlaps between the unit shapes.

We can obtain a tessellation using translation. Try making a unit shape to tessellate using this method.

Unit shape add cut










Translation tessellation





















16.2.2 Reflection
O Spotlight

Reflection Is a another type of transformation that acts like
Watch this video to find a mirror. In a reflection, every part of the object is flipped
out how translation along the mirror line (line of reflection) to form an image.
helps you to form unit The image and the object are congruent.
shapes that are good for
Let us use coordinates to describe reflections of 2D shapes.
tessellation.

HiWH



m
Look at the triangle RST on the right.
Here are some tips on
The coordinates of its vertices are:
how to craft your own
tessellation using square R{-2, 4)S(-2, 1)TM, 3)
sticky notes! Let us reflect triangle RST along
the v-axis. \
mm nirror lin
->-X
_5 _4 _3 _2 -1 U 3 4







Transformation

y
1 k

R n R • R' RL
CO j u
J T n J'
\ \ 0 p
\
\ V
S ■V Y
J
->-.V
-
5 4 - 3 _2 -1 u 2 3 4 i _5 -4 -2 2 3 4 S
The reflected triangle R'ST is a mirror image of triangle RSI. It has the same size
and shape as the original triangle. The two triangles are congruent.


Example 2

0 Draw and label AABC on a coordinate grid. Use these coordinates: A (-4, 5), B

(-5, 1)andC(-2, 2).
0 Reflect AABC along the y-axis.
0 Label and state the coordinates of the reflected image (AA'B'C). To reflect in the y-axis
means to use the y-axis
Solution
as the mirror line.
0 Plot the points on the grid. Label the points A, B and C Join the points with In this example, the line
straight lines. of reflection is the y-axis.





^ •A
The distance between
c point A and the y-axis
should be the same as the
B
distance between point B
_7 _6 -5 ^ -3 -2 -1 0 1 2 3 4 5 6 7 and the y-axis.

0 Each point of AABC is reflected along the y-axis.










ygl C


sf e( tior
>■ X
7 - 5 - 5 --4 - 5 ' _ 0 } 4 .> 6 /

0 AABC is the object that is reflected.
AA'B'C is the image formed as a result of the reflection.
The coordinates of the translated image are: A' (4, 5), B' (5, 1), C (2, 2)

Example 3



Reflect the polygon TUVW in the line PQ.
What are the coordinates of the vertices of its image?




s
i 4 /
/
T V
-- ->N
\ /
2
\ / /
/ V
■♦•.V ■*-.v

5 - 4 - 3 - 2 - > -5 -4 - 5: rif
/
s.\
/
w-
/
/
/


The coordinates of the vertices of its image are:
T' (3,^) U'(4,-3) V'(3, 0) W'(0,-3)

Check My
Understanding



Q Look at the polygon on the right.
The coordinates of its vertices are:
D(-4,3) E{3,3) F(3, 1) G(-2, 1) H (-3, 0)
Where will the vertices of the image of polygon DEFGH
be after we reflect the polygon in the x-axis?






Q For each reflection, state the axis about which the object is reflected.
a) >' b)













> X




Reflection about the -axis Reflection about the -axis

UNIT 16 I Transformation

d)












-
6 5 • 4 -i -2 - 1 0



. s
f,
7





Reflection about the -axis Reflection about the -axis


Q In each diagram, the line of reflection is indicated with a dotted line.
Draw and label the images.

a) b)
The line of reflection]
f
is also called the
/ p 0
L mirror line.
V
/
K







d)* Challenge! Spotlight
o
n
\
\ > F
\
Watch this video to see
examples of reflection
'
\ 1-
s
on a coordinate grid.
B #
V
\
pas
V
s
i

Q Look at the objects on the coordinate grids.
a) Draw and label the images after reflection about the line of reflection shown by the dotted lines.

b) Write the coordinates of each image.


i) ii)
1
F F
N
\ G
i
-1

-
•> X
5 -4 -3 -2 - 1 0 1 >. -5 ■ 4 - ^ i . 1 0 \


mil
1
iii) Iv)









->■ X
•5 -4 -3 -2 - -5 -4 - 3 ^2 -









Q Look at the objects on the coordinate grids.
a) On each diagram, draw the line of reflection.
b) Describe each reflection. Use the words: horizontal, vertical, x-axIs and y-axis.

') V '■) V i'i)

L' K' K I
■
■ K'
M' N' h


5 4 -^ - i - \ ^ X ->-x
•
5 -I -3 -


4









UNIT 16 I Transformation

@ Rectangle L'M'N'O' is the position of the image of rectangle LMNO after it was reflected about
the >'-axis.
a) Draw the object on the coordinate grid.
b) Write the coordinates of the figure LMNO.
c) Rectangle L'M'N'O' is reflected about the x-axis to form rectangle L"M"N"0".
i) Draw rectangle L"M"N"0".
li) Write the coordinates of rectangle L"M"N"0".



L( ) L"(
L' M'
M( M" (
0' N'
N{. N" (.
X 0(. O" (.
5 -4 - ^ - i . 1






s

©*Challenge! If a point Q (2, 4) is reflected about the.v-axis, what will the coordinates of point Q' be?

For Questions 8 and 9, use a piece of graph paper to draw a set of suitable axes and plot the graphs.

@*Chal!enge! 0*Challenge!
a) Plot MNO on the coordinate grid where a) Plot triangle ABC on the coordinate grid
M M, 4), N (-3, 2) and O (-5, 2). where A (1, 0), B (2, -2) and C (-1, -2).
b) Draw the graph of >' = -1. b) Draw the graph of y = x+ 1.
c) Draw the image of triangle MNO when it c) Draw the image of triangle ABC when it
is reflected about the line _y = -1. is reflected about the line >? = x + 1.
i) Label the image. i) Label the image.
ii) Write the coordinates of the image. ii) Write the coordinates of the image.

d) Draw another image of triangle MNO
when it is reflected about the line x = 2.
i) Label the image.
ii) Write the coordinates of the image.


Spotlight


A kaleidoscope is a tube in
Find out how a
which mirrors are used to
kaleidoscope is made
reflect an object and create
by watching this video
multiple images.
When you look through Q
a kaleidoscope you see
i
patterns like this)

16.2.3 Rotation


Rotation is a turn around a centre point called the centre of rotation. In a
rotation, every part of the object is turned around a fixed centre point to a new
position in a different orientation.
When a shape is rotated, The image and the object are congruent.
all points in the shape The centre of rotation can be on the object or at a distance from the object.
rotate through the same
angle.

The distance from the
centre of rotation to a
point on the object is the centre of centre of
rotation -
rotation
same as the distance from
the centre of rotation to
the corresponding point
on the Image.





Investigate!


Aim: To find out the effects of rotation on coordinate points and shapes and to map out
the image.
You will need:
• graph paper, a compass or dividers or pin, glue and the drawn-and-cut-out shapes from
the activity below
Instructions:

Rotation A Rotation B
O Look at the L-shape below. O Look at the L-shape below.
Draw and cut out an identical L-shape Draw and cut out an identical L-shape
(3 units by 4 units) on a piece of graph {3 units by 4 units) on a piece of graph
paper. paper.











0







Q Place the L-shape as shown on another piece Q Place the L-shape as shown on another piece
of graph paper. Trace and shade it. of graph paper. Trace and shade it.



UNIT 16 I Transformation

Q Mark point 0 as shown as the centre of rotation. Q Mark point 0 as shown as the centre of rotation.
Place your compass point on point O. Place your compass point on point O.
Q Then, rotate the L-shape through half a turn Q Then, rotate the L-shape 90° clockwise.
(180°), clockwise. 0 Paste the rotated L-shape on the graph paper.
0 Paste the rotated L-shape on the graph paper. 0 Draw arrows to show the rotation. Label the
0 Draw arrows to show the rotation. Label the object (B) and the image (B').
object (A) and the image (A').


Look at the two rotations you have completed. Complete the statements and answer the questions.

0 rotation A, the centre of rotation is the object.
In
What do you notice about this point of the object after the rotation?
In
0 rotation B, the centre of rotation is from the object.
What do you notice about the points on object B after the rotation?



Check My

Understanding

0 Draw each figure below on a piece of graph paper. Draw the Image after each rotation.

_ \ A rtrtA _ ^ ! L A L%.\ AAO J"l A c) 180° about point C
ll
1 QH®
a) 180° about point A b) 90° anti-clockwise
about point B




B
A
■ y
S:
■■■i




0 Draw each figure below on graph paper. Rotate each given figure about point X. Draw the image,
a) 180° b) 90° anti-clockwise c) 90° clockwise





r
]
X x' ■
■



y

Let us use coordinate points to describe rotations of 2D shapes.
Example 4


Q Given A (0, 2), B (1, 3) and C (-1, 3), draw and label AABCon a coordinate grid.
Q Rotate AABC 90° clockwise about the origin.
Q Write the coordinates of the rotated image (AA'B'C).

Solution
O Start at any point to rotate AABC. In this
example, we start with point A.

Draw an orange dotted line from A
to the origin, which is the centre
of rotation.
*-x
Q Place the sharp point of your compass
on {0, 0) and the pencil tip on point A (0, 2).
Rotate the compass 90° clockwise to draw
an arc. The image of point A is where the
compass pencil has stopped. Mark that point
and label point A'.

Q Use the coordinate grid. Make r R
sure that the green line is the same % -*• ^ y"
length as the orange line.
A
In this example, both lines are 4 units long. N.
s
4 u lits
90° cloc ;wis€ N
Q The point at the end of the green line Jo\ t/ii rot2tion \
is the position of point A'. This is the rotation N
of point A, 90° clockwise about the origin. 2 *1 0 0 A'
A un ts


Next, look at point B. We can also
use a protractor and a ruler to rotate
the point 90° clockwise.
C B
@ Draw a dotted line from point B immr nmmr r
to the origin. Place the protractor on TS «0 ^ h
'
t N 100
the dotted line as show. Mark a dot at ;\\ \J • S"? \
///
the 0° marking on the right side of '// ^ r / N •?<'
the protractor. :\i !// S''.
Draw a dotted line the same length as 2
-
OB through the dot. The end of the line
"s-
would be the image of point B. Label this
B'
point B'.











UNIT 16 Transformation

@ Lastly, rotate point C We can use
vertical and horizontal P
construction lines to show a path
n
.
from point C to the origin. Look at
A
the orange construction lines.
C
0 Rotate the orange construction lines
90° clockwise about the origin. i
This gives the green construction ~ 2 - 1 0 A'
lines. Make sure the green lines
are as long as the orange lines. Label B'
o
the end of the longer green line C. Spotlight
(J) Join points A', B' and C with
Watch this video to see
straight lines to form AA'B'C.
how to use a protractor
AABC has been rotated 90° and a ruler to carry out
clockwise about the origin to rotations.
form the image AA'B'C.

The coordinates of AA'B'C are:
A' (2, 0)
iiSi
B' (3, -1)
C (3, 1)







Given A (1, 3), B (3, 7), C (6, 6) and D (3, 1), plot and draw quadrilateral ABCD on a
coordinate grid. Rotate quadrilateral ABCD 1 S0° anti-clockwise about the origin,
and then draw its image.
Solution
Use a protractor and a ruler to help you determine how many turns is an angle of
180° to rotate each point through the origin.
V





Spotlight


Watch this video to learn
how to describe rotation
given the object and the
image.


ilif


Eli:®®

Check My
Understanding


O a) On a piece of graph paper, draw and label each object and the Image formed after rotation,
b) Write the coordinates of each image.
i) 90° clockwise about the ii) 90° anti-clockwise about ill) 180° about the origin
origin the origin
y y y
-s






-h—t
5-4 -3 -2 -5 -4 -3 -2 - -5 -4 -3 - 2 -




-4-


o a) Given the coordinates U (1, -2), W (0, 2), K (3, 2) and G (3, -3), draw and label figure UWKG on a
piece of graph paper.
b) What type of polygon is UWKG?
c) Rotate polygon UWKG 90° clockwise about the origin. Label the image formed.
d) Write the coordinates of polygon U'W'K'G'.


Q*Challenge! Look at the coordinate grid.

Q' N'
1 1
N 1

m
E'
/e R'
X

5 -4 -3 -2 -1




S
a) State whether each statement is true or false.
i) Polygon N'R'E'Q' is a reflection of polygon NREQ.
ii) Polygon N'R'E'Q' is a rotation of polygon NREQ 180° about the origin.
iii) Polygon N'R'E'Q' Is a rotation of polygon NREQ 90° anti-clockwise
about the origin.
iv) Polygon N'R'E'Q' is a rotation of polygon NREQ 90° clockwise about the origin.
b) Draw polygon N'R'E'Q'on a piece of graph paper.
Rotate polygon N'R'E'Q' 180° about the origin.
Label the image N"R"E"Q".




Transformation

Try and Appiyi


Q The diagram below shows 8 congruent triangles. The triangles are numbered as 1, 2, 3, 4, 5, 6, 7 and 8.

H B
\
/ / \
\ / /
2 S / 1 8 1
\/ N/
b L
/\ \
\ \
3 4 5 6
/ \ \
>
/ \ \
F E I
a) Name the transformation that moves triangle 1 to fit exactly onto
i) triangle 4, ii) triangle 7 and iii) triangle 3.

b) Move triangle 1 to fit exactly onto triangle 5 in one transformation. If you are unable to do so,
describe how you would do it in two transformations. Hint: you can use a combination of different
transformations.
c) Describe how to move triangle 1 to fit exactly onto triangle 6 in two transformations, in two
different ways.
B
Q Play this game to try to get the white ball into the black hole!
You can only use translation, rotation and reflection to transform
the ball to the final position on the grid. B



Revision

Answer Questions 1 to 7 on graph paper. Start each question on a new piece of graph paper.
O Translate each polygon. Draw the images of the polygons after translation and label their vertices on
the coordinate grid. Then write the coordinates of the vertices of the images in the boxes below.

a) Translate 6 units to the left b) Translate 2 units up



>


\- A
Q
\
F
5 • 4 -3 -2 -1 U ;









?■(□_□) Q'(n.n) D'(n,n)

^ Draw and label each image after the object has been translated.
a) Translate 4 units up b) Translate 6 units down
and 2 units to the right. and 3 units to the left.








1
/\ ■>x
/ '\
5 • 4 - 3 - 2 -1 0 1 ; n
\
/
A L


. s



Q Draw and label the image after the object has Q a) Draw the following object and the set of
been reflected about the v-axis. axes. Label the image formed after the
object is rotated 90° clockwise about the
origin.
b) Write the coordinates of the image.


r-b-



-5 -4 - 3 2 Jnr
-


5 - 4 - 3 - 2 1 0 1 I
-
?
\ 1







0 Draw and label object VSG, given the coordinates of the points: V {2, 0). S (1, 3), G (5, 0).

a) What type of polygon is VSG?
b) Rotate polygon VSG 90° anti-clockwise about the origin. Label the image formed.
c) Write the coordinates of the image.













UNiT 16 I Transformation

^ This diagram shows triangles A, B, C, D and E, and points P, Q, R, S and T.


When describing a
rotation, you need
to give the following
information:
• degree of rotation
• direction of rotation
• centre of rotation


Describe the following rotations.
a) Rotate from triangle A to triangle B.
b) Rotate from triangle A to triangle E.
c) Rotate from triangle B to triangle C
d) Rotate from triangle C to triangle D.

O Plot polygon PQRS on a coordinate grid where the coordinates of the points are P {-3, 3), Q (-1, 3),
R{-2, 1)andS(-4, 1).
a) Draw the graph of V = .v-
b) Draw the image of polygon PQRS when it is reflected about the line >' = x
i) Label the image.
ii) Write the coordinates of the image.






Mathematics Connect




The Singapore Flyer is one of the world's largest observation
wheels. The wheel rotates around the centre point. As the
wheel turns, it transforms the position of the people in the
carriages so that they are able to view Singapore
from different angles. Have you sat in a similar
observation wheel before?
Ij/ 0'/ /












iiiiii
111^'

Help ^ Transformations

Sheet A transformation is a change in the position of an object.


Translation Reflection

A translation moves every point on a shape the A reflection produces a mirror image of a shape
same distance in the same direction. along a line of reflection.



/
\ H :
\ /
r
> \ llr e 0
\ A
\
s ^re lec ion
N /
->-JC
s -4 -3 - N
-
N \ \
/ \
|^H>
/
ig^W
/
/ s
Shape D is the reflection of shape C along the
Shape A has been translated -6 units along the
line y^x. Both shapes are the same perpendicular
jc-axis and 3 units up along they-axis to shape B.
distance away from the line of reflection.


A rotation turns a shape about a fixed point. To perform or describe a rotation, three details
are needed:
• centre of rotation
• angle of rotation

• direction of rotation (clockwise or anti-clockwise)
When a shape is rotated, ail points in the shape rotate through the same angle.
The distance from the centre of rotation to a point on the object is that same as the distance
from the centre of rotation to the corresponding point on the image.


r~^
c'i I.


y X

A < C
^x
5 4 - 3 -2 -1 * ;
n




In this example, the object triangle ABC is rotated 90° anti-clockwise about the origin (0, 0).
The image and the object remains the same (congruent) after translation, reflection or rotation.



UNIT 16 I Transformation

UNIT 17












You will learn about:
The mode (or modal class for grouped data), median and
range
Calculating the mean
Comparing two simple distributions using the range and the
mode, median or mean
Drawing and interpreting
• bar-line graphs
• bar charts
• frequency diagrams for grouped discrete data
• pictograms
• pie charts
Drawing conclusions based on the shape of graphs and
simple statistics
Drawing conclusions based on simple statistics.











us
UK
Netherlands
Germany
According to the
Australia
information shown,
Cartada
Switzerland which country has the
Sweden most universities in the
South Korea top 100 universities in
Singapore the world?
Japan
Explain your answer.
Hong Kong
France
China
Belgium
irt .r.i.r.**
Turkey in m-i
Italy

In this chapter
CHAPTER 17.1
Pupils should be able to:

• find the mode (or modal
class for grouped data),
median and range
• calculate the mean,
including from a simple
^firsT^ rr
frequency table
• compare two simple
distributions using the
range and the mode,
We have learnt in Stage 6 some ways to interpret a given set of data by finding an
median or mean
average of the given data set An 'average' is a 'typical' value of the data set. It is
usually a 'middle' value and tells us approximately what most of the values in the
^ RECALL data set are close to. We can also call this the 'central tendency'.

There are three types of average — mean, median and mode.
Find the mean, median,
mode and range of the set
Mode
of values.
29 32 21 45 9 41 17 32 Mode is the value that occurs the most often. Let us see how we can find the
mode of a given data set.
Mean Median
Finding mode in discrete or ungrouped data set
Mode Range
Discrete data contains values that can be counted and has a finite number of
possible values. For example, days of the week or shoe sizes.
We do not need to group |
these values, hence [
we call it discrete or j
ungrouped data. Paul owns a shoe shop. The data set below shows the shoe sizes he sold in the
last month.

10, 5, 4, 5, 8, 5, 5, 6, 7, 5, 6, 11, 5, 5, 5, 5, 3, 4, 5, 5, 7, 6

Which shoe size did most of the customers buy?
Solutioii
Arrange the data values in order from the smallest to the largest:
When data in a data set is
listed in order of size, we 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 10, 11
say that the data set has
We can see that the number '5' occurs the most number of times.
been ranked.
So, most of the customers buy shoe size 5. The mode of this data set is 5.
In statistics, the value that occurs most frequently in a set of data is called
the mode.
We can also record the data set in a frequency table.


The number '5' occurred Shoe size 3 4 5 6 7 8 9 10 11
11 times. Frequency
1 2 11 3 2 1 0 1 1



386 UNIT 17 Data Handling

Find the mode of the data set below.
14, 18, 16, 13, 16, 14, 15, 16, 14

Solutio
First, rearrange the data in ascending or descending order. In this example, the
data is arranged in ascending order:

13, 14, 14, 14, 15, 16, 16, 16, 18
Sometimes a set of data
Find the value that occurs most often.
does not have a mode.
In this data set, the values 14 and 16 each occur three times, which is also the This means that there is
highest frequency in the set. no value that occurs more
frequently than the other
We say this data set has two modes: 14 and 16. This data set is bimodal.
values.
.1a! class in grouped data






The manager of the Jurong Bird Park is planning the number of persons
needed to work in the ticket office during different times of the day.
She collected the data to find out when is the busiest period for the bird park.

Time interval No. of visitors
entering the park
This is the mode. 09 00-10 59 114 The most number
We say this is -► 11 00-12 59 340 < of people entering
the modal time 13 00-14 59 201 the park.
interval. 92
15 00-16 59
The most number of people entered the park between 11 00 to 12 59. The modal class of a set
This is the modal class of this set of grouped data. of grouped data is the
During which time period should the manager have the most staff working in class with the highest
frequency.
the ticket office?





A class of 40 pupils took their height measurements during their Physical
Education lesson. Their heights are recorded in the frequency table below. Find
the height interval that the majority of the pupils fall under.

Read this as Heiqht (cm) Number of dudIIs
"height more -► 130 ^ h < 134 1
than or equal to 135 £ h <139 0
130 cm but less 140 <h <144 2
than 134 cm". 145 <h <149 2
150^ h <154 7
154sh<159
Modal height . " 160 <h <164 7
interval 8 The greatest
165 <h <169 8 number of pupils in
170 ^h <174 10 4 this height interval.
175^ h<179 4
180 <h <184 0
185fih<189 1 387

Solution
In order to answer the question, you need to find the modal class. There are
10 pupils with heights ranging between 170 cm to 174 cm. The modal class of this
data is 170 cm < h < 174 cm.
The mode is a particularly useful measure of central tendency when there is a
The mode is a good large number of values in the data set.
average when there are
For example, a shopkeeper or manufacturer may need to know which is the most
many identical values in
popular product so that they can make the best decisions in order to maximise
the data set.
their profit.


17.1.2 Finding the median



Another type of average is the median. Let us recall how we can find the median
of a set of data.





The annual salaries of seven employees are shown. The manager wants to use
the 'middle' value to represent the annual salaries of her employees. What is
the middle value?
What is the middle value?
Employee Annual Salary ($)
Is it a good representation
A 27 000
of what most of her
B 21 000
employees are getting?
C 23 000
D 28 000
E 30 000
F 25 000
G 29 000
hink and Share Let us rank the data. In this example, we can order the salaries from the lowest
to the highest annual salary:
$21 000, $23 000, $25 000, $27^000, $28 000, $29 000, $30 000
What if you have an even
number of data values in
the set?
The middle value of this set of
How do you find the ranked data is the 4'^^ value out
'middle' value? Is there a
of 7 values, which is $27 000.
middle value?
The middle value of a ranked data set is called the median.
















388 UNIT 17 Data Handling
II

Find the median of this data set.
5, 7, 2, 8,11,6, 6, 0,1
Solution
First, rank the data values in ascending order:
middle
In this example, there is
0, 1, 2, 5,06, 7, 8, 11
an c dd number of values.
V > \ J
4 numbers 4 numbers
There are 9 values. The number in the middle is the 5*^ value.
The middle value is 6. So, the median of this data set is 6.

Find the median of this data set.
4, 7, 2, 8, 11,5, 6, 0, 1,9

Solution
First, rank the data values in ascending order:
middle

In this example, there is
0, 1, 2, 4,{|[6)7, 8, 9, 11 an even number of values.
V I \ J
4 numbers 4 numbers

There are 10 values. The median is the mean of the two middle values.

5 + 6
The median of this data set is —^
= 5.5
The mean Is calculated by adding the values together and dividing the sum by
the number of values.
Therefore, the median is 5.5.



I

The median is the middle |
value of a set of data |
Find the median of each data set.
when it is arranged in j
a) 2,9,3,7,4, 1,6,0,8
order. J
b) 12,9, 13,7,14,21,17,10,18,15
When there is an even
c) 5,6,11,6,4,3,9,1,10,8
number of values
A horticulturalist records the lengths of some leaves from a tree in the arranged in order, the
frequency table below. What is the median length of the leaves? median is the mean of the ;
two middle values. |
Length (mm) 24 32 41 50 I UWIWlWixf

Frequency 1 4 3 2


389

17.1.3 Finding the range



Another way to interpret a set of data is by looking at how the data in the set is
spread out. Let us look at a scenario.






This weather forecast shows the expected temperature at different times of a
day. Linda wants to know the difference between the highest and the lowest
expected temperatures.
Weather Forecast^^^


06 00 07 00 08 00 09 00 10 00 11 00 12 00 13 00 14 00

O O


28°C 27°C 27°C 28°C 29°C 29°C 30°C 30°C 31°C

Which is the highest or lowest temperature? How can you find the difference?
What is the range?



The difference between the highest and the lowest value in a data set is called
the range.





What is the range of these numbers?

14,12, 7,1,5, 9,15,11,19
Solution
Ordering the data from the highest to the lowest, we get:

19, 15, 14, 12, 11, 9, 7, 5, 1

highest value lowest value
The difference between the highest and the lowest value is 19 - 1 = 18.
Spotlight The range of the data set is 18.



An evolutionary I
biologist found that
sprinters with the most
Five people participated in a marathon race. What is the range of their times?
symmetrical knees have
2.9 h, 7.4 h, 3.6 h, 5.8 h, 4.9 h
the best times and this
was especially true of Find the range of these quiz scores.
the 100-m sprinters in 92,81,85, 87, 90, 79, 84
Jamaica.
Find the range of the following hourly wages.
$7.50, $8.75, $9.50, $8.25, $8.50

390 UNIT 17

.4 Finding the mean



The mean is the most common way of finding an average. In real life, when we
use the word 'average' (such as average salary or average height), we are most Ungrouped data refers
likely talking about the mean. to data values that are
discrete (fixed and can be
Calculating the mean for ungrouped data counted). Discrete data
values do not need to be
Let us look at the annual salaries of the seven employees in the earlier Investigate!
grouped into classes or
again. Previously, the manager used the median to represent the annual salaries
intervals.
of her employees. The median was $27 000.
The manager now wants to use the mean to represent their annual salaries.

Emolovee Annual Salarv($)
A 27 000
B 21 OOP What is the mean?
C 23 000
Is it a good representation
D 28 OOP of what most of her
E 30 OOP employees are getting?
F 25 000
How close is it to the
G 29 OOP median value?
We can find the mean by adding the values together, and then divide the sum by
the total number of annual salaries like this.

$27 OOP + $21 OOP + $23 OOP + $28 OOP + $30 000 + $25 OOP + $29 000
Mean = Spotlight
7 O
_$183 OOP
One of the surveys done
7
in 2017 found that the
= $26 142.86 (round to 2 d.p.) Netherlands people
have the tallest average
The mean of the annual salaries is about $26 142.85.
height. This was no
surprise as the Dutch
Mean for ungrouped data
are well known for their
The mean is a measure of central tendency. The mean is calculated by adding
lofty stature. The average
all the values together and dividing the sum by the number of values.
male height is 1.838 m.
sum of values The Nilotic people that
mean =
number of values live in regions near the
Nile Valley, are also
considered among the
tallest people on Earth
with an average male
height of 1.9 m.
A restaurant collects soft drink cans for recycling. The number of cans
Find out more using this
collected in 3 weeks are given below.
link.
81, 95, 75, 71, 33, 36, 84, 89, 61, 69, 58, 53, 74, 79, 71, 45, 49, 81, 71, 94, 48
Find the mean, median, mode and range. HSiB
Mean: Median:
i
Mode: Range:


391

A library has Mathematics books shelved on 10 racks. The number of books
on each rack are given below.
38, 51, 65, 34, 46, 58, 46, 73, 39, 22
Find the mean, median, mode and range.

Mean: Median:
Mode: Range:

The mean mass of the 7 players in a netball team is 59 kg.
a) What is the total mass of the team?
b) A player with a mass of 64 kg is replaced by a player with a mass of
56 kg. Find the new mean mass of the team.

The ages of the members of a choir are given below.

25, 24, 29, 31, 26, 28, 33, 21, 24, 28, 37
a) Find the median age of the choir members.
b) Calculate the mean age of the choir members.
c) If the conductor is 48 years old, calculate the mean age of the choir
members and the conductor.
d) If the composer Is 18 years old, calculate the mean age of the choir
members and the composer.
e) Compare your answers in parts (c) and (d). Explain the difference.


Match each data set to the correct criteria.
Criteria Data set
Mode = 21 and 23
• • 103, 104, 100, 101, 103, 102, 103

Range -
7
Median = 19
• • 4, 9,4,7, 6, 4, 10, 9, 7, 3, 8, 7,4,5,9

Range = 7
Mean = 102.29

• • 21,23, 26, 23, 22,21,27, 28
Range = 3
Mean = 6.4 • •

0.4, 0.6, 0.9, 0.5, 0.3, 0.8, 1.2, 2.3
Median = 7
Range = 2 • •

20, 14, 16, 21, 21, 21, 17, 18
Median = 0.7
)* Challenge!
A team plays 4 matches. The goals scored in the matches are 6, 7, 4 and
7. How many goals do they need to score in their 5''' match in order to
increase their mean score to 7 goals per match?















392 UNIT 17

Calculating the estimated mean for grouped data

A shopkeeper records the number of shirts he sold In a week. As there were a lot
of shirts sold, the number of shirts sold were grouped in classes in the frequency
table below.


Number of Frequency The frequency in
The number of shirts sold (/) this column shows
how many shirts
shirts sold has
50-54 7
been grouped were sold per
55-59 6 week in each class.
into classes.
60-64 8
65-69 10
70-74 15
75-79 9
80-84 5
Total 60

The shopkeeper is planning the number of shirts to order. She needs to know
the mean number of shirts she sells in a week. But this frequency table does not
provide discrete data values. So, she is unable to find the exact mean. She can,
however, try to find an estimate of the mean.


1
Draw a new frequency table with four columns.
Midpoint Frequency
Number of fx
shirts sold (X) if) 5 Multiply the midpoint by
Calculate the midpoint
—^ 52 7 364 ^
by calculating the - the frequency for each
mean of the lowest 55-59 57 6 342 row.
and highest value in
60-64 62 8 496 For example, in the first
each class.
row, 7 X 52 = 364
65-69 67 10 670
For example, the
midpoint of the class 70-74 72 15 1080 Do this for all the rows.
interval 50-54 is:
75-79 77 9 693
50 + 54 _ ^2
80-84 82 5 410
Add all the values in
Do this for all the class Total 60 4055-
the last column.
intervals.
I
' Add all the values in
the frequency column.


The estimated mean of this grouped data is calculated as shown:

Estimated mean
= 67.58
The shopkeeper sells an estimated mean of 67.58 shirts every week. Since
the shopkeeper cannot sell 0.58 of a shirt, we round the estimated mean
to 68 shirts.

393

Estimated mean for grouped data

Grouped data is represented in a frequency table. The estimated mean is
calculated by dividing the sum of f.\ by the sum of/in a frequency table.
estimated mean = sum of/v
sum of/





Calculate an estimate for the mean of the grouped data.


Mass Frequency
(kg) (/)

30 < m s40 15
40 < m £ 50 28
50 < m £ 60 12
60 < m £ 70 10
80 < m £ 90 8
90 < m £ 100 5

Solutioii
Mass (kg) Midpoint (m) Frequency (/) /m
The midpoint of the class
interval in the first row is 30 < m £ 40 ^ 15 525
35
calculated by: '4(r<rrr£ 50 45 28 1260
30 + 40 50 < m £ 60 55 12 660
= 35
2
60 < m £ 70 65 10 650
70 < m £ 80 75 8 600
80 < m £ 90 85 5 425
Total 78 4120

estimated mean = sum of/a-
sum of/
- 4120
78
= 52.82

The estimated mean for the grouped data is 52.82 kg.






The frequency table shows the number of strawberry
Number of
shrubs in a nursery with different numbers of 0-9 10-19 20-29 30-39
strawberries
strawberries growing on each shrub.
Frequency 6 8 4 2
a) How many shrubs are there in this nursery?
b) What is the modal class of the grouped data?
c) Calculate an estimate of the mean number of strawberries.



394 UNIT 17

Bella records the number of times she hears her favourite Number of times song is
Frequency
song each week. played per week
Calculate the estimate of the mean number of times her 0-4 2
favourite song plays each week. Round your answer to the 5-9 8
10-14 11
nearest whole number.
15-19 4
20-24 2


Dina works at a call centre. She records the duration of Call duration (min) Frequency
each telephone call for a month.
0£t< 10 28
a) How many calls did Dina receive during the month? 10<t<20 22
20£t<30 18
b) Which call duration was the most common?
30 £ t < 40 17
c) Calculate an estimate of the mean duration of one
40 5 t < 50 20
telephone call during the month.
50 £ t < 60 3
60£t<70 13
70 s t < 80 2

Elsa conducts a survey to find out how many millilitres Millilitres of water Frequency
of water teeneagers drink each day. She records the 0 ^ .V < 500 12
information in the frequency table to the right. 500 <.v< 1000 35

a) What is the modal class of the data? 1000 <v< 1500 26
1500 £ v < 2000 9
b) How many teenagers are surveyed?
2000 s r < 2500 3
c) Calculate an estimate of the mean number of
2500 £ A- < 3000 1
millilitres of water consumed per day.



17.15 When to use mean, median

or mode



You have learnt that the mean, median and mode are measures of central
tendency. The objective of central tendency is to get an idea of a 'typical' value
to represent the data set. Different measures of central tendency are useful for
different sets of data.







The scores for six basketball teams in a tournament are as follows:
68, 70, 72, 74, 78, 112 hink and Share
The basketball team with a score of 112, scored much higher than the other
teams. A value that is much higher or much lower than the other values in a Did many of the
data set, is called an outlier. Since there is a high outlier in this data set, the basketball teams score
closer to the mean, or the
median is the best measure to represent the scores.
median?
Would you choose the mean or the median to represent the central tendency of
this set of data?


395

The mean does not represent the scores well because the mean of
higher than every score except for one.
The median of the data set is .
68, 70, 72I74, 78, 112

T

The mean of the data set is

Mean Median Mode
In a data set that In a data set that The mode is usually not a
has no outliers, has outliers, the suitable representation of
the mean is the median is the central tendency.
most suitable most suitable
representation of representation of 41,45, 43, 42, 47, 46, 15, 49, 15
central tendency central tendency as In the data set above, the mode
because it is the is not a good average
the values of the value in the data because all of the other values
data set are used set. are much higher than the
to calculate the mode.
mean. The mode is usually used to
represent popularity or majority
sentiments.



6 Comparing distributions








A teacher recorded the scores of 10 pupils for their English and Maths tests.
The teacher wants to compare the overall performance of these pupils in
English and Maths. Should she use the range, mean, median or mode for this
comparison?

Subject Scores
Maths
79 79 76 76 74 79 81 79 77 80
English
67 74 ICQ 68 67 83 70 85 102 64
Work out the range, mean, median and mode for the scores of each test. Fill
in the table below.

Subject Mean Mode Median Range
Maths
English
What does the average or spread tell the teacher about the scores in the
English and Maths tests?





396 UNIT 17
Data Handling
m