FUN ADVENTURE
Tools/Materials used wrapping paper, coloured paper or patterned paper,
glue, scissors, manila card
Participants 2 pupils in a group
Task
1 Draw a square, a rectangle, a right-angled triangle, an isosceles triangle,
and an equilateral triangle of various measurement on the coloured paper
or patterned paper.
MY IDEAL HOME
2 Cut the polygons. Think of your
favourite theme, for example,
my ideal home. Create
composite shapes for the theme
by pasting the polygons on a
piece of manila card.
3 Calculate the perimeters and areas of each
composite shape.
4 State the composite shape which has the largest
and the smallest perimeter and area respectively.
Are there any two composite shapes that have
the same perimeter and area?
5 Present your work. Display it at the creative corner
or put it inside the mathematics folio.
144 Teacher’s Notes 8.2 (i)
Prepare a variety of coloured papers and patterned papers. Help pupils to carry
out the activity above.
Encourage pupils to create various composite shapes that are different from
each other. Pupils can use various polygons that they have learned to create
patterns and find their perimeters and areas.
SELF-PRACTICE 12 cm P 10 cm S 6 cm
Q T R
Solve the following problems.
16 cm
a The diagram shows a composite shape of two
right-angled triangles. The length of QR is 2 times
the length of ST. Calculate the perimeter, in cm, and
the area, in cm2, of the whole diagram.
b Elisa and her friends did a project of creating composite shapes with a
computer. Which composite shape had the smallest perimeter and area?
i. ii.
10 cm 10 cm 8 cm 10 cm
6 cm
7 cm 2 cm 4 cm 4 cm 2 cm
8 cm 5 cm 5 cm
8 cm
c The shaded region of the diagram shows the region to 12 m
be used to rear animals. The perimeter of the shaded
region is 50 m. A pond in the shape of a right-angled 10 cm 9m
triangle will be built on the unshaded region as shown. B
Calculate the area of the pond. C
d The plan of a recreational park is shown in the diagram. 8m
A 90 m
A jogging track ABCD is built around the region. The
length of BC is 2 of the length of AB. 75 m
3
i. What is the length of the jogging track? D 45 m
9m
ii. What is the area of the recreational park?
e Mrs Lee wants to plant grass in the rectangular front yard
of her house as shown in the diagram.
1 of the area of the front yard is a mini garden. 20 m
4
1 of the area of the front yard is a car park.
6
Calculate the area of the front yard that will be planted with grass.
8.2 (i) Teacher’s Notes 11455
Help pupils to draw diagrams for the questions above while solving them.
C Solve the Problems involving Surface Area and Volume
1 Nabila builds a composite three-dimensional shape for her 6 cm 6 cm
art project which consists of a cuboid and a cube as shown in
the diagram. She wants to paint the surfaces of the composite
shape. Find the total surface area, in cm2, to be painted.
20 cm
Solution
18 cm
Given The length of the sides of the composite shape
Asked for Total surface area to be painted Separate each surface of
the composite. Draw and
Operation Multiplication, addition, and subtraction
Solve Make simulation write the length of the
Total surface area of the cuboid to be painted: sides of each surface.
20 cm The area of 2 surfaces 18 cm The area of 2 surfaces
= 2 × (20 cm × 6 cm) = 2 × (18 cm × 6 cm)
= 2 × 120 cm2 = 2 × 108 cm2
= 240 cm2 6 cm = 216 cm2
6 cm
20 cm The area of 1 surface 6 cm20 cm The area of 1 surface
= 1 × (20 cm × 18 cm) 18 cm 6 cm
= 360 cm2 = 1 × (20 cm × 18 cm)
– (6 cm × 6 cm)
18 cm
= 360 cm2 – 36 cm2
The surface area of the cube
to be painted: = 324 cm2
The area of 5 surfaces
Total surface area:
6 cm = 5 × (6 cm × 6 cm) 240 cm2 + 216 cm2 + 360 cm2
6 cm = 5 × 36 cm2 + 324 cm2 + 180 cm2 = 1 320 cm2
= 180 cm2
Check Total surface area= Total surface area of the cuboid + Total surface
to be painted area of the cube – The area of overlapping surfaces
= 1 176 cm2 + 216 cm2 – 36 cm2 – 36 cm2
= 1 320 cm2
The total surface area to be painted is 1 320 cm2.
146 Teacher’s Notes 8.3 (i)
Prepare several examples of different composite three-dimensional shapes
and ask pupils, in groups, to calculate the total surface area of these composite
three dimensional shapes.
Surf https://www.youtube.com/watch?v=6Na7dErsljE and
http://www.mathvillage.info/node/11
2 Puan Dayang baked a cake as shown in the 30 cm 12 cm
diagram. 12 cm
A
a Calculate the total surface area of the cake.
30 cm
b What is the total volume of the cake, in cm3?
B 40 cm
Solution Draw a diagram 40 cm
a 1 surface 1 surface
30 cm
30 cm 40 cm 40 cm 4 surfaces 4 surfaces
40 cm 12 cm 12 cm
40 cm
40 cm 30 cm
The total surface area of the cake:
[2 × (40 cm × 40 cm)] + [4 × (40 cm × 12 cm)] + [4 × (30 cm × 12 cm)]
= (2 × 1 600 cm2) + (4 × 480 cm2) + (4 × 360 cm2)
= 3 200 cm2 + 1 920 cm2 + 1 440 cm2
= 6 560 cm2
The total surface area of the cake is 6 560 cm2.
b The total volume of the cake:
= The volume of Cake A + The volume of Cake B
= (30 cm × 30 cm × 12 cm) + (40 cm × 40 cm × 12 cm)
= 10 800 cm3 + 19 200 cm3
= 30 000 cm3
The total volume of the cake is 30 000 cm3.
What is the suitable box size for this cake?
BRAIN TEASERS
A total of 6 cubes of equal size are needed to build 3 steps.
How many cubes are needed to build 11 steps?
8.3 (i) Teacher’s Notes 147
Use plan and elevation diagrams to draw the top and side views to help pupils
get an overview of the situation.
3 The diagram shows a model built by Year 6 Inovasi pupils for the
Decorating School Yard Competition. The model is a combination of three
wood blocks A, B and C. The volume of A is 1 of the volume of B. The
2
volume of C is 3 times the volume
of A. What is the total volume, 1m
in m3, of the model? 30 cm
Solution Draw the model C B A 1m
1m Draw the model to represent the problem.
100 cm = 1 m. Therefore, 30 cm = 0.3 m.
0.3 m3
30 cm
A
Volume A: 1 m × 1 m × 0.3 m = 0.3 m3
1m
Volume B: 1 × B = A
2
0.3 m3 0.3 m 3
B 1 × 1 × B = A×2
2 = 0.3 m3
2
1
× 2
= 0.6 m3
0.3 m3 0.3 m3 0.3 m3 Volume C: 3 × A = 3 × 0.3 m3
C = 0.9 m3
The total volume of the 3 blocks is: 0.3 m3 + 0.6 m3 + 0.9 m3 = 1.8 m3
Check Volume A: 1 m × 1 m × 0.3 m = 0.3 m3
Volume B: 2 times the volume of A = 2 × A
Volume C: 3 times the volume of A = 3 × A
The total volume of the 3 blocks = A + B + C
= A + (2 × A) + (3 × A)
= 0.3 m3 + (2 × 0.3 m3) + (3 × 0.3 m3)
= 0.3 m3 + 0.6 m3 + 0.9 m3
= 1.8 m3
The total volume of the model is 1.8 m3.
Calculate the total surface area of the model to be painted and decorated.
148 Teacher’s Notes 8.3 (i)
Get pupils to design various composite three-dimensional shapes using cubes
and cuboids to enhance pupils’ understanding of calculating surface areas
and volumes.
SELF-PRACTICE
Solve the following problems.
a The diagram shows a chocolate cake in the
shape of a cuboid. The total volume of the cake is
320 cm3 and the area of the base is 40 cm2.
Calculate the height, in cm, of the cake.
b The diagram shows a structure built by 3 cm 11 cm 2 cm
a group of pupils for their Design and 8 cm 3 cm
Technology project. The structure consists 6 cm
of two cuboids and a cube. 9 cm 4 cm
3 cm
i. What is the surface area, in cm2, of the blue
plane?
ii. Calculate the total volume, in cm3, of the
structure.
c A stage is built from a combination of a cuboid 0.3 m
and two cubes of the same size as shown
in the diagram. The volume of the cuboid is
2 times the volume of the cube. What is the
volume, in m3, of the stage?
d The diagram shows the measurement of a 60 cm 20 cm
kitchen cabinet built by Mr Tan.
1m
i. What is the surface area of the 4 doors of
the cabinet? 2m 2m
ii. Calculate the total volume, in cm3, of the
cabinet.
e A total of 4 cubical gift boxes of the same size as 6 cm
shown in the diagram will be put into another box.
Calculate the possible surface areas, in cm2, for the
box needed.
8.3 (i) Teacher’s Notes 149
Pose similar questions as above.
SKILFUL MIND
1 Draw the following polygons on triangular grid paper. Name the polygons.
Measure each angle and label the angles.
ab c
2 Draw the following polygons on squared paper. Name the polygons.
Measure their angles and write their characteristics.
ab c
3 Solve the following problems. 6m
11 m
a The diagram shows a plot of vegetable farm. The shaded
region represents the plot to grow long beans. Calculate 12 m
the area of that region. What is the area, in m2, of the 3 m B
farm for other vegetables? 20 m
D 12 m C
b The diagram shows the plan of an orchard. EDC is a A
straight line and the length of ED is 2 times the length 200 cm
of DC. The perimeter of the whole orchard is 96 m.
Find the length, in m, of AE. Calculate the area, in m2, E 8.1
of the orchard. 8.2 (i)
8.3 (i)
c The diagram shows a composite three-dimensional
shape which consists of a number of cubes measuring
2 cm × 2 cm × 2 cm. There is a hole in the middle of the
shape. What is the volume of cubes, in cm3, needed to fill
the hole?
d An art piece in a restaurant is shown in the diagram.
The red area is 1 250 cm2, which is 1 of the total area A
8
of the art piece. If the red area is 1 of the green area,
2B
calculate the length, in cm, of AB. Calculate the area,
in cm2, of the non-coloured part.
150 Teacher’s Notes
Pose similar questions as above.
9
A Distance between Coordinates
Horizontal and vertical distances of objects
from the origin
The diagram shows the location of a few islands. Damia and
her family are on a holiday and they are staying at a resort
which is located at the origin, O (0, 0). Help Damia determine
the distance of each island from the origin.
y On both axes, the
6 Aman Island distance between
5 one number and
AMAZING FACTS the next is 1 unit.
y
Vertical distance 24 Sentosa Island
1 3
O 1 2 3x Harmony Island
Horizontal distance 2
Bahagia Island
1
O 1 2 3 4 5 6 7x
Origin
a State the distance of Bahagia Island from Damia.
Damia is at the origin (0, 0). Count the horizontal distance along the x-axis
from the origin, 0 to 5.
The horizontal distance of Bahagia Island from Damia is 5 units.
b The coordinates of Harmony Island are (2, 1). Therefore, the horizontal
distance of Harmony Island from Damia is 2 units and the vertical
distance is 1 unit .
c The coordinates of Sentosa Island are (7, 3). Therefore, the horizontal
distance of Sentosa Island from the origin is and the vertical
distance is .
9.1 (i) State the horizontal and vertical distances of Aman Island from Damia. 151
Teacher’s Notes
Use coloured chips on squared papers or checker boards which are labelled
as coordinate planes to state the horizontal and vertical distances.
Recall the positions of the x-axis and y-axis on the first quadrant with pupils.
Horizontal and vertical distances between coordinates
1 The plane below shows coordinates A and B. What are the horizontal and
vertical distances from Point A to Point B?
y Horizontal distance From Point A, move 2 units
5 to the right and move 3 units
upwards to Point B.
B (4, 4)
4
3 Horizontal distance:
2 Vertical distance 4 – 2 = 2 units
1 A (2, 1) Vertical distance:
4 – 1 = 3 units
x
O 1 2345
The distances from Point A to Point B are 2 units horizontally and 3 units vertically.
2 Coordinates for Point C, D, E, F and G are shown in the plane below. Calculate
the horizontal and vertical distances between:
a Point C and Point E. b Point F and Point E.
y The horizontal distance from Point C to Point E:
4 – 1 = 3 units
5
F (6, 4) The vertical distance from Point C to Point E:
3 – 2 = 1 unit
4
The horizontal distance from Point F to Point E:
3 E (4, 3) 6 – 4 = 2 units
C (1, 2)
2 The vertical distance from Point F to Point E:
D (3, 1) G x 4 – 3 = 1 unit
1
O 1 234567
The distances from Point C to Point E are 3 units horizontally and 1 unit vertically.
The distances from Point F to Point E are 2 units horizontally and 1 unit vertically.
Distance C to F D to F D to G G to E Complete the
Horizontal table.
Vertical
152 Teacher’s Notes 9.1 (i)
Emphasise the starting and the ending points. Start with the horizontal distance
followed by the vertical distance when calculating distances.
Surf http://www.mathopenref.com/coordpointdistvh.html
3 The diagram below shows the floor plan of the National Science Centre.
Sali and his friends are visiting the centre and they are at the Cafe.
Source: http://www.psn.gov.my/wp-content/map/tab.html#tab1
y Key:
AA Cafe
6 B Aquarium
5 E (4, 5) H (7, 5) CC Wonderspark
DD Eureka
4 C (2, 3) F (6, 3) E Pathways to Science
3
2 B (0, 3) F Science Show Stage
1 D (4, 1) G (6, 1) GG Kids Discovery Place
A
x
O 1 2 34 56 7 8
H Kidz World
Now, we are at A (0, 0).
a The distance between the Cafe to the Aquarium is 3 units vertically.
b The distance between Wonderspark and Pathways to Science are
2 units horizontally and 2 units vertically.
c The distance from Eureka to the is 2 units horizontally.
d Sali moves from Wonderspark to Eureka , and then to Kidz World .
At first, he covers a distance of units horizontally and units
vertically. Then, he moves 3 units and .
e Horizontal and vertical distances from C to E, to , to ,
and to are the same.
9.1 (i) State the horizontal and vertical distances from the Cafe to 153
the Kidz World . Is the distance the same as the distance
from the Kidz World to the Cafe ? Discuss.
Teacher’s Notes
Use floor plans of different locations to conduct similar activities.
FUN ADVENTURE
LUCKY COUNTER Player Dice
Tools/Materials recycled paper such (A and B)
as calendars or square
manila cards, ruler, x-axis y-axis
Player Turn 1 2 3 4 5 1 2 3 4 5
coloured pens, tables, First
variety of coloured Second
counters, 2 dice (A and B) Third
Participants 4 pupils in a group Fourth
Steps to prepare the game
1 Fold a piece of manila card horizontally
and vertically to get 8 × 8 squares of the
same size.
2 Draw horizontal and vertical lines on the folded markings.
Label numbers on the x-axis and y-axis. Mark the point of
origin (0, 0) and end point at coordinates (6, 6).
Rules of the game
1 Determine the turns. The game starts from the origin.
2 The first player throws 2 dice simultaneously. Record the value on dice A as x-axis and
the value on dice B as y-axis. For example: 4 units horizontally and 2 units vertically.
Player Dice
(A and B)
x-axis y-axis
Player Turn 1 2 3 4 5 1 2 3 4 5
First 4 2
3 Move the counter along the x-axis and y-axis according to the
numbers recorded in the table.
4 Change turns. The next player repeats steps 2 and 3. Repeat the
game for 5 rounds. If the counter is out of the grid, the player will
restart.
5 The winner is the first player to reach the coordinates (6, 6).
154 Teacher’s Notes 9.1 (i)
Prepare enough materials for the activity. A chessboard can be modified and
used as the game board.
SELF-PRACTICE
1 Write the horizontal and vertical distances between the points for the following
planes.
ay by
6 6 D
5 5
4 A B 4
3
3
2 2 C
1
1
O 1 23456 x O 1 23456 x
From to units From to unit
unit units
Horizontal distance: Horizontal distance:
Vertical distance: Vertical distance:
cy dy
6 6
5
4 F 5
3
2 E 4 G
1 1 23456 3
H
O to 2
From 1
x O 1 23456 x
From to
Horizontal distance: units Horizontal distance: units
Vertical distance: units Vertical distance: unit
2 Mark the Point P (0, 2) on the plane. State the coordinates of Point Q, given the
distances from Point P.
a Horizontal distance: 3 units b Horizontal distance: 5 units
Vertical distance: 0 unit Vertical distance: 1 unit
y y
66
55
44
33
22
1 1 x
O 1 23456 x O 1 23456
Teacher’s Notes
9.1 (i) Carry out activities of stating the distance between two points involving horizontal 155
distance to the left and vertical distance downward. For example, the distance
between Point A (4, 3) and Point B (1, 1).
SKILFUL MIND
Let’s help Akil and his friends play a game at the traditional game station.
y
H Key:
9D
8 A Hopscotch B Coconut
Shell Race
7 F C Top D Congkak
E Spinning
6
5 C G I E Five F Tating Lawi
Stones Ayam
4
3 A G Konda H Guli Garis
Kondi
2 B I Sepak Raga
1
O 1 23456789 x
1 Write the distance between the following stations.
a Top Spinning and Sepak Raga b Konda Kondi and Guli Garis
Horizontal: units Horizontal: unit
Vertical: unit Vertical: units
c Congkak and Coconut Shell Race d Tating Lawi Ayam and
Horizontal: units Five Stones
Vertical: units
Horizontal: unit
Vertical: units
2 Name two stations with the furthest horizontal distance.
The two stations are and .
3 Name two stations with the nearest vertical distance.
The two stations are and .
4 Which two stations have twice the horizontal distance between stations A
and C, and the vertical distance is the same as the vertical distance between
stations A and D?
156 Teacher’s Notes 9.1 (i)
Surf https://www.tes.co.uk/teaching-resource/coordinates-activities-6101284
10
A Ratio of Two Quantities
1 Let’s compare the number of
blue ball to red balls.
There are 3 red
balls and 1 blue
ball.
a The ratio of the number of blue ball to the number of red balls is one to three.
The ratio 1 to 3 is written as 1 : 3 .
b The ratio of the number of red balls to the Is 1 : 3 equal
number of blue ball is three to one to 3 : 1?
3:1
c The ratio of the number of blue ball to all the balls BOOST
is 1 : 4 . YOUR KNOWLEDGE
d The ratio of all the balls to the number of Ratio is the comparisan of two
red balls is 4 : 3 . quantities in the same unit of
measurement.
2
There are 2 blue cones,
4 yellow cones, and 3 red cones.
The total number of cones is 9.
Try to complete the table below.
Comparison of the Ratio
number of cones 2:4
All of cones What is the ratio of the
All of cones number of boys to the
number of girls?
10.1 (i) Teacher’s Notes 157
Provide examples of ratio in daily situations.
Surf http://www.mathsisfun.com/numbers/ratio.html
Emphasise that the objects or units of the measurement must be the same
in order to state the ratio.
3 Puan Sameen buys some ribbons to decorate a few wedding gifts. The length
of the pink ribbon and the green ribbon is 7 m and 3 m respectively.
Let’s state the ratio of the length of
the pink ribbon to the green ribbon.
7m 3m
a The ratio of the length of the pink ribbon to the green ribbon is 7 : 3.
b The ratio of the length of the green ribbon to the pink ribbon is 3 : 7.
c The ratio of the length of the pink ribbon to the total length of the ribbons
is 7 : 10. We can write 7 : 10 as 7 .
10
d The ratio of the total length of the ribbons to the green ribbon
is : or . Does ratio have units
of measurement?
Discuss.
SELF-PRACTICE
1 Write the ratio of:
a i. the number of yellow cars to the number of red cars.
ii. the number of red cars to the number of yellow cars.
iii. the number of yellow cars to the total number of cars.
b i. the number of orange hibiscus to the number of white hibiscus.
ii. the number of white hibiscus to all the flowers.
iii. the total number of flowers to the number of orange hibiscus.
2 Look at the picture. State the ratio of:
a the volume of green paint to the total volume of paint.
b the total volume of paint to the volume of blue paint.
3 The diagram shows a collection of three types of shapes. Write the ratio of:
a to . b to .
c
e to . d to all the shapes.
158 to the total number of shapes. f all the shapes to . 10.1 (i)
Teacher’s Notes
Conduct a simulation with various objects to enhance the concept of ratio.
Discuss with pupils the importance of ratio. It only consists of whole numbers and
it does not have units.
B Solve the Problems
1 Puan Sumitha prepared 12 cupcakes. 5 of the cupcakes are
chocolate flavoured and the rest are pandan flavoured. If
2 chocolate cupcakes and 5 pandan cupcakes are eaten
by her children, what is the ratio of the remaining chocolate
cupcakes to the remaining pandan cupcakes?
Solution
Given 12 cupcakes. 5 chocolate flavoured and the rest is pandan flavoured.
Asked for The ratio of the remaining chocolate cupcakes to the remaining
pandan cupcakes
Operation Subtraction
Solve Flavour Number Number Number Check
of of of the
The number : The number of
cupcakes cupcakes remaining of chocolate pandan cupcakes
eaten cupcakes cupcakes
Chocolate 5 2 5−2=3 3:2
Pandan 12 − 5 = 7 5 7−5=2
3+2:2+5
5:7
The ratio of the remaining chocolate cupcakes
to the remaining pandan cupcakes is 3 : 2.
. 2 The measurement of a room is shown in the diagram.
A 10 m long and 1 m wide cabinet is fixed to one side of
the wall in the room. What is the ratio of the length to the
width of the remaining space in the room?
8 m 10 m
Solution The length of the room is 10 m. The width of the room is 8 m.
➙Important The length of the cabinet is 10 m. The width of the cabinet is 1 m.
➙Draw a information Find the ratio of the length to the width of the remaining space
➙diagram.
in the room.
1 m The length of the room = 10 m
8 m 7 m The width of the remaining space in the room = 8 m – 1 m
=7m
10 m The ratio of the length to the width of the remaining
space in the room is 10 : 7.
10.1 (ii) Teacher’s Notes 159
Surf http://www.ixl.com/math/grade-6/ratios-word-problems to solve problems
online involving ratio.
3 8 kaya buns and 12 red bean buns are repacked into
4 packets with the same number of kaya and red bean
buns. What is the ratio of the number of kaya buns to the
number of red bean buns in each packet?
Solution
Given 8 kaya buns and 12 red bean buns are packed equally into 4 packets.
Asked for The ratio of the number of kaya buns to red bean buns in each packet
Solve The number of kaya buns in each packet is 8 ÷ 4 = 2.
The number of red bean buns in each packet is 12 ÷ 4 = 3.
The ratio of the number of kaya buns to the number of red bean buns is 2 : 3.
The ratio of the number of kaya buns to the number of
red bean buns in each packet is 2 : 3.
4 Arpita Kaur uses 7 cups of flour and 3 cups of milk
to make a dough. Her friend, Laili, wants to try the
same recipe but she reduces a cup of flour and adds
2 cups of milk compared to Arpita’s dough. What is
the ratio of the number of cups of flour to the number
of cups of milk used by Laili?
Solution Important information
➙ ➙
Arpita uses 7 cups of flour and Laili uses 1 cup of flour less and 2 cups
3 cups of milk. of milk more than Arpita.
Find the ratio of the number of cups of flour What will
to the number of cups of milk used by Laili. happen if Laili
does not follow
Flour used by Laili : 7 cups − 1 cup = 6 cups the correct ratio?
Milk used by Laili : 3 cups + 2 cups = 5 cups
The ratio of the number of cups of flour to the
number of cups of milk used by Laili is 6 : 5.
160 Teacher’s Notes 10.1 (ii)
Surf http://www.thinkingblocks.com/ThinkingBlocks_Ratios/TB_Ratio_Main.html
for exercises on solving problems involving ratio by drawing blocks.
Discuss different methods to obtain the answers.
SMART PROJECT
Tools/Materials manila card, notebook, pen, Participants 4 pupils in a
pencil, ruler, measuring tape group
Task
1 Prepare task cards for each group. Put them into envelopes. Example of the tasks:
1st envelope Task 2
Get the data on the number of pupils in each Year 6 class.
Task 1 Find the ratio of:
Measure the length and
width of the teacher’s table. a. the number of boys to the number of girls in each class.
State the ratio of the length b. the number of boys to the total number of Year 6 pupils.
to the width of the table. c. the number of all the Year 6 pupils to the number of Year 6 girls.
2nd envelope
Task 3 Task 4 Task 5
Measure the length, Polygon A has 7 sides. Polygon B has 3 sides
width, and height of less than polygon A. Find the ratio of the
your desk. Find the
number of sides of polygon B to polygon A. The ratio of the number of purple
ratio of the width to the crayons to all the crayons.
height of your desk.
2 Each group gets an envelope.
3 Each group completes the tasks within a certain time.
4 The leader of each group presents the answers. Teacher provides feedback.
5 Collect all the task cards along with the answers.
6 Display your work at the mathematics corner.
SELF-PRACTICE
Solve the following problems.
a Geok Kiaw has 6 storybooks. Lisa has 5 storybooks more than Geok Kiaw.
State the ratio of the number of Geok Kiaw’s storybooks to their total number
of storybooks.
b There are 20 balls in a basket. 9 of them are white and the rest are blue.
Calculate the ratio of the number of blue balls to the number of white balls.
c The distance from town A to town B is 29 km. The distance from town A to
town C is 60 km. If town B is located between town A and town C, what is
the ratio of the distance from town A to town B to the distance from town B to
town C?
10.1 (ii) Teacher’s Notes 161
Smart Project activity can evaluate the mathematics skills, analysing skills,
problem-solving skills, making research skills, and communication skills besides
inculcating cooperation.
Provide more similar questions as above.
SKILFUL MIND
1 Write the ratio of:
a the number of shaded parts to the number of unshaded parts.
b the number of unshaded parts to the number of shaded parts.
c the number of shaded parts to the whole.
d the whole to the number of unshaded parts.
2 A total of 8 pens and 9 pencils are still in Encik Rajen’s shop.
a What is the ratio of the number of pens to the number of pencils?
b State the ratio of:
i. the total number of pens and pencils to the number of pens.
ii. the number of pencils to the total number of pens and pencils.
3 Solve the following problems.
a Pak Amit prepares 48 packets of nasi lemak and D
40 packets of fried noodles. A total of 45 packets of
nasi lemak and 38 packets of fried noodles are sold.
What is the ratio of the remaining number of packets
of fried noodles to the remaining number of packets of
nasi lemak?
b The diagram shows a composite shape of two triangles. z
The ratio of the length of AB to the length of AD is 3 : 8. B
Calculate the value of z.
3 cm
c There are 25 ice creams in a box. If 24% of the A C E
ice creams are durian flavoured and the rest are
mango flavoured, what is the ratio of the number
of mango flavoured ice creams to the number of
durian flavoured ice creams?
d A and B are two cubical boxes. The length of each side B A
of box A is 2 times the length of each side of box B. If a
3 m length of string is used to tie both boxes, state the 10.1 (i)
possible ratio of the length of string to tie box B to the 10.1 (ii)
length of string to tie box A.
162 Teacher’s Notes
Discuss the application of ratio in daily situations.
Surf https://www.khanacademy.org/math/pre-algebra/rates-and-ratios/ratios_
and_proportions/e/ratio_word_problems
11
A Data Interpretation MATHEMATICS CORNER
Pictograph Steps to interpret data from a pictograph:
• What is the title of the pictograph?
1 Mass of Recycled Materials
Newspapers The title of the pictograph is Mass of
Aluminium cans Recycled Materials.
Plastic • What are the types of materials in
Others the data?
Key: represents 5 kg. The types of materials in the data are
newspapers, aluminium cans, plastic,
What will happen and others.
if the newspapers • What is the key?
are not recycled? One circle represents 5 kg.
Discuss. • What conclusion can be drawn from this
pictograph?
The conclusion is the mass of the
newspapers collected is the highest,
25 kg. The mass of the plastic is the
lowest, 2.5 kg. The difference between
the mass of aluminium cans and the
mass of other materials is 12.5 kg.
The total mass of the recycled materials
collected is 50 kg.
The data from the pictograph can also be shown as a bar chart and
a pie chart. Explain the differences of these three types of charts.
30 Mass of Recycled Materials Mass of Recycled Materials
Mass (kg) 25 Others Newspapers
Plastic
20
5% 10%
15
10
5 Aluminium 50%
cans
Ne0wspapersAlumcinainuTsmypes Plastic Others 35%
of Materials
BAR CHART PIE CHART
11.1 (i) Teacher’s Notes 163
Discuss the interpretation of data from other pictographs, bar charts, and pie charts.
Pose various questions such as to find the total or difference of the number of
recycled materials collected.
Discuss the materials that can be produced from recycled materials.
2 The horizontal pictograph shows the number Number of participants
of participants in the Traditional Dance in the Traditional Dance
Performance during the Birthday Celebration Makyung
of Duli Yang Maha Mulia Seri Paduka Baginda Singa
Yang di-Pertuan Agong. Bharatanatyam
Ngajat
Sumazau
Key: represents 2 persons.
a How many participants are there in each dance?
Makyung : 5.5 × 2 persons = 1 1 persons State the dances
that have equal
Singa : 5 × 2 persons = 10 persons number of
participants.
Bharatanatyam : 1 × 2 persons = 2 persons
Ngajat : 3.5 × 2 persons = 7 persons
Sumazau : 5 × 2 persons = 10 persons
b Calculate the fraction of Sumazau dancers.
NTuomtabl neur mofbSeurmoaf pzaaurtidciapnacnetsrs = 1 1 + 10 10 + 7 + 10
+2
= 4100
=1
4
The fraction of Sumazau dancers is 1 .
4
c Calculate the percentage of Makyung dancers.
Number of Makyung dancers × 10 0% == 14110140 %× 10 0%
The total number of participants
= 27.5%
The percentage of Makyung dancers is 27.5%.
164 Teacher’s Notes 11.1 (i)
Instil moral values such as self-esteem and love for culture.
Get pupils to research on Malaysian traditional dances.
Surf http://www.math-only-math.com/worksheet-on-interpreting-a-pictograph.html
Bar chart Cases of Dengue till 31.10.2014
1 The dengue cases reported States Perak
as of 31 October 2014 is Sarawak
shown in the bar chart.
Negeri
a Which states had more Sembilan
than 10 dengue cases
recorded? Johor
Based on the bar chart, 0 5 10 15 20 25 30
Perak and Johor had more Number of cases
than 10 cases recorded.
Source: http://idengue.remotesensing.gov.my/
idengue/index.php
b Which state had 4 times the number of dengue cases compared to
Negeri Sembilan?
5 cases were recorded in Negeri Sembilan. Therefore, 4 × 5 = 20.
20 cases were recorded in Johor. Therefore, Johor had 4 times the
number of dengue cases compared to Negeri Sembilan.
c The total number of dengue cases in and was the same as the
total number of dengue cases in and .
Selangor recorded the highest number of dengue cases
which were 100 cases more than the total cases in the four
states. Calculate the number of dengue cases in Selangor.
Discuss the steps to PREVENT
prevent dengue.
DENGUE
KILL AEDES
11.1 (i) Teacher’s Notes 165
Get pupils to collect data on diseases spread by animals such as JE virus,
Nipah virus, and Leptospira bacteria.
Construct the bar charts and then interpret the data.
2 The bar chart shows the Number of curry puffs Sales of Mak Jah’s Curry Puffs
number of curry puffs sold 140
by Mak Jah in four days. 120
100
We can interpret data
from this bar chart. 80
60
40
20
0
Monday Tuesday Wednesday Thursday
Day
a State the sales of curry puffs for each day.
Monday Tuesday Wednesday Thursday
120 pieces 90 pieces 105 pieces 105 pieces
Why do you think the sales on Monday was the highest? Give your reasons.
b What is the difference in sales between Tuesday and Wednesday?
105 pieces − 90 pieces = 15 pieces
The difference in sales between Tuesday and Wednesday is 15 pieces.
c Calculate the average number of curry puffs sold each day.
Average = Total sales On which day was the sales
Number of days equal to the average sales?
= 120 + 90 + 105 + 105 Calculate the total sales,
4 in RM, for four days if the
price of 5 pieces of curry
= 420 puff is RM2.
4
= 105
The average number of curry puffs
sold each day is 105 pieces.
166 Teacher’s Notes 11.1 (i)
Guide pupils to make interpretations related to total, maximum value, minimum
value, and average.
3 The number of visitors to Air Bayu Theme
Park during the school holidays is shown
in the bar chart below.
Visitors of Air Bayu Theme Park
Day Sunday
Saturday
Friday
Thursday
0 100 200 300 400 500 600 700 800 900 1 000
The number of visitors
a The total number of visitors b The fraction of visitors on Thursday
in four days is: of the total visitors from Thursday to
Saturday is:
1 Thursday 250 = 250
Friday 250 + 200 + 800 1 250
250 Saturday
200 Sunday =
800
+ 750
2 000
c The collection from the sale of tickets on Friday is RM5 000. Therefore,
the collection from the sale of tickets on Saturday is:
200 — RM5 000 Calculate the total sales of
×4 ×4 tickets on Thursday and Sunday.
800 — RM20 000
d The percentage of visitors on Saturday compared to the total visitors
from Thursday to Sunday is:
4 800 × 100% = 40%
1 2 000
11.1 (i) If the number of adults is 30% of the total 167
number of visitors, how many adults are there?
Teacher’s Notes
Surf http://www.superteacherworksheets.com/graphing/bar-graph-
simple-6_TWNBN.pdf
Pie chart Classification of Vertebrates
1 Sarah has 80 pieces of picture cards Amphibians
of vertebrates. The pie chart shows Reptiles
the classification of vertebrates. Fish 20% Mammals
15% 25%
a What is the percentage of the
amphibians? Birds
35%
Percentage of all the Percentage of amphibians:
group sectors in the
pie chart adds up to 100% – (25% + 20% + 15% + 35%) = 100% – 95%
= 5%
100% .
The percentage of amphibians is 5%.
b Calculate the difference between the number of mammal and bird picture
cards.
METHOD 1 METHOD 2
Percentage of mammal picture cards: 25% Calculate the number of picture
Percentage of bird picture cards: 35%
cards in each group:
Difference in percentage = 35% – 25% • Mammals = 25 × 80 pieces
= 10% 100
Difference = 20 pieces
between the = Difference in Total number • B irds = 35 × 80 pieces
number of percentage × of cards 100
picture cards = 28 pieces
= 10% × 80 pieces Difference between the number
= 11000 x 80 pieces
= 8 pieces of picture cards
= 28 pieces – 20 pieces
= 8 pieces
The difference between the number of
mammal and bird picture cards is 8 pieces.
168 List the amphibians that might be in Sarah’s 11.1 (i)
animal picture cards.
Teacher’s Notes
Ask pupils to interpret data from the pie chart. Pose questions involving fractions,
percentage, and average based on the data from the pie chart.
Surf http://www.superteacherworksheets.com/graphing/pie-graph-simple-2.pdf
2 The pie chart shows pupils’ participation Activities of the Uniform Units
in five activities carried out by the uniform Jungle
units. The total number of pupils involved Night Rampage
Venture
in the activities is 120.
15% Flying Fox
10% 25%
Cultural Night has the largest Cultural Night Group Activity
participation. Night Venture is 30% 20%
of least interest.
a State the combination of activities that has a total participation of 50%.
Cultural Night is 30% and Night Venture is 10%, Jungle Rampage
Group Activity is 20%. is 15%, and Flying Fox is 25%.
b Calculate the number of pupils for each activity.
Night Venture: Cultural Night:
10 × 12 0 pu pils = 12 pupils 100 × pupils = pupils
100
Jungle Rampage: Group Activity:
100 × 120 pupils = pupils × pupils = pupils
If you were given a chance to participate in the activities,
which activity would you choose? Give your reason.
11.1 (i) Teacher’s Notes 169
Ask pupils to get pie charts from websites or newspapers and create questions
for their classmates to answer.
Surf http://commoncoresheets.com/Piegraphs.php
SMART PROJECT
Tools/Materials A4 paper, coloured pencils Participants 4 pupils in a
or MS Excel group
Task
1 Each group collects data on various topics such as the way pupils go to
school, favourite television programmes, pets, and hobbies. Collect data
from 5, 8 or 10 pupils.
Example of data collected:
Ways of 5 pupils going to school Pets of 10 pupils
21 1 1 31 42
Favourite television programmes
Documentary Comedy Drama Entertainment
4 2 1 1
2 Present the data in a pictograph, bar chart or pie chart.
Ways of Going to School Number of pupils Pets Favourite Television Programmes
Bus 6 Entertainment
Bicycle 4 12.5%
Walk 2
Car 0 Drama Documentary
Key: represents 1 pupil 12.5% 50%
Cats
Fish Birds Rabbits Comedy
25%
Types of animals
3 Each group is required to get at least three pieces of information from their
pictograph, bar chart or pie chart.
Example: 1
5
a. The fraction of pupils going to school by car is .
b. The ratio of the number of pupils who like rabbits to cats is 2:3.
c. The number of pupils who like to watch documentary is 4.
50 × 8 pupils = 4 pupils
100
4 Compile your work into a scrapbook and display it at the mathematics
corner.
170 Teacher’s Notes 11.1 (i)
This project can evaluate thinking skills such as developing ideas and soft skills by
showing interest and enthusiasm to learn.
SELF-PRACTICE
1 Answer the questions based on the incomplete pictograph.
Sale of Face Masks in Bagus Pharmacy
January
February
March
April
May
Key: represents 20 boxes
a What is the difference between the sales of the face masks for January
and April?
b The number of face masks sold in March is 30 boxes less than February.
How many boxes of face masks were sold in March?
c The sales in May is 5% more than in April. Calculate:
i. the number of boxes sold in May. ii. the total sales from January to May.
d A box of face masks has 50 units and the cost price is RM30. The price of
a face mask is 80 sen. Calculate the profit gained in February.
e In your opinion, why does April record the highest sales of face masks?
Explain.
2 Answer these questions based on the bar chart.
Usage of Paint in the Living a State two colours with a total volume
Room and Bedroom of 1.4 ℓ.
Paint colour Green b What is the average volume of paint
Blue used?
White c Calculate the remaining volume of
Purple
paint if the volume of each tin is 1 ℓ.
0
0.5 1 d 3 of the remaining paint is used to
Volume of paint (ℓ) 5
paint a bookshelf. Calculate, in mℓ,
the volume of paint used.
11.1 (i) Teacher’s Notes 171
Surf http://www.edugain.com/sampleWorksheet/Grade6/Data-
Handling/Printed
3 Look at the bar chart and answer the following questions.
Number of Workers in Electronic Factories
Number of workers 500
400
300 QR S
200 Factory
100
0P
a Calculate the total number of workers in factories P, Q and R.
b The total number of workers in two factories is more than 800. State the
two factories.
c What is the fraction of workers in factory P from the total number of
workers in the four factories?
d 40% of the total number of workers are male. Calculate the number of
female workers in the four factories.
4 Look at the pie chart given and answer these questions.
Participation in a Jogathon a What is the percentage of the youth
participating in the jogathon?
25% Key: b A total of 148 male teenagers and
30% 152 female teenagers participated in
Teenage the jogathon. Calculate the number
(10 - 15 years old) of veteran participants.
Youth
(16 - 40 years old) c State the ratio of veterans to youths.
Veteran
(Above 40 years old) d In your opinion, what is the purpose
of this competition?
172 Teacher’s Notes 11.1 (i)
Surf http://www.superteacherworksheets.com/pictograph.html
http://www.superteacherworksheets.com/bar-graphs.html
http://www.superteacherworksheets.com/pie-graphs.html
B Solve the Problems
1 A group of 15 pupils received school shoes sponsored by the Parent-Teacher
Association (PTA) of the school. The number of pupils that received sizes
3 and 4 shoes are 4 pupils and 6 pupils respectively. Size 5 shoes were given to
3 pupils and the rest received size 6 shoes. Find the:
a mode. b range. c median. d mean.
Solution
Given 15 pupils received shoes Asked for Mode, range,
Size 3 4 pupils Size 4 6 pupils median, and mean
Size 5 3 pupils Size 6 ? pupils
Solve
The number of pupils who received size 6 shoes = 15 – (4 + 6 + 3)
= 15 – 13
Create a table. =2
Shoe size 3456
Number of pupils 4 6 3 2
a The highest frequency is 6. Therefore, the mode is size 4.
b Range = Maximum value – Minimum value
=6–3
=3 The range of the shoe size is 3.
c There are 15 data. Median is the value of the data that is in the middle.
3 , 3 , 3 , 3 , 4 , 4 , 4 , 4 , 4 , 4 , 5 , 5 , 5 , 6 , 6 The median is the
Therefore, the median is size 4. 8th datum.
d Mean = (4 × 3) + (6 × 4) + (3 × 5) + (2 × 6)
(4 + 6 + 3 + 2)
= 12 + 24 + 15 + 12
15
= 63 SMART CALCULATION
15 Mean = Total value of data
= 4.2 Total number of data
11.1 (ii) a The mean is size 4.2. 173
Teacher’s Notes
Recap the definition of mode, median, mean, and range learned in Year 5.
Ask pupils to use a calculator to check the answer.
Surf http://www.mathgoodies.com/lessons/vol8/range.html
2 The two bar charts show the scores of 10 boys and 10 girls in the athletic
events of the Malaysia Sports Camp.
Boys’ Athletic Events Girls’ Athletic Events
Frequency3 5
Frequency2 4
1 3
0 8 12 16 20 2 10 15 20
Score 1 Score
4 0
5
Find the mode, range, median, and mean for the boys’ scores.
Solution Create a table Score 4 8 12 16 20
Solve Frequency 1 2322
a The highest frequency is 3 boys.
Therefore, the mode is score 12.
b Range = Maximum value – Minimum value
= 20 – 4
= 16
The range of the score is 16.
c Arrange the data in ascending order. Median = 12 + 12
2
4, 8, 8, 12, 12, 12, 16, 16, 20, 20
24
middle = 2
The median is score 12. = 12
d M ea n = (1 × 4) + (2 × 8) + (3 × 12) + (2 × 16) + (2 × 20)
(1 + 2 + 3 + 2 + 2)
= 4 + 16 + 36 + 32 + 40
10
Use the same method to find
= 128 the mode, range, median,
10 and mean for the girls’ athletic
scores. Compare them to the
= 12.8
The mean is score 12.8. boys’ scores. What is your
conclusion?
174 Teacher’s Notes 11.1 (ii) b
Surf http://www.superteacherworksheets.com/mean-median-mode-range/
mean-median-mode-range-boxes.pdf
Emphasise that the mean, mode, median, and range refer to the values of the
data and not the frequencies.
3 The Malaysian Nature Society (MNS) is Contribution to MNS
actively involved in environmental issues.
The pictograph shows the contributions
received by MNS. Find the range, mode,
median, and mean.
Solution
Given The contribution and the Contoh
number of contributors
Asked for Range, mode, median, and Key: represents 1 person.
mean
Solve
a Range = Maximum value – Minimum value b Mode
= RM20 – RM1 The value that has highest
= RM19 frequencies represents the mode.
The range is RM19. Therefore, the mode is RM10.
c Median METHOD 2
METHOD 1 Contribution (RM) 20 10 5 1
The total number is 14. The The number of contributors 4 532
median is between the 7th and (persons)
8th. There are 14 data. The median is the
average of the 7th and 8th data.
RM 10 + RM 10 = RM20
2 2
Contoh Contoh = RM10
Therefore, the median is RM10.
d M ea n = (4 × RM20) + (5 × RM10) + (3 × RM5) + (2 × RM1)
(4 + 5 + 3 + 2)
= RM80 + RM50 + RM15 + RM2
14
= RM147 If there is one more
14 contributor who
contributes RM5,
= RM10.50 find the new mode,
The mean is RM10.50. mean, and median.
11.1 (ii) b Teacher’s Notes 175
Provide exercises to find the mode, median, mean, and range.
Instil moral values such as donating and loving the environment.
Surf http://www.mathworksheetsland.com/6/grade6-42b-5pack.pdf
4 The pie chart shows the percentage of Parcels of Different Mass Received
by Sri Maju Post Office
parcels of different mass received by
Sri Maju Post Office. The total number of
parcels is 12. 25% Key:
4 kg
Find: 50% 3 kg
25% 2 kg
a mode. b median.
c range. d mean.
Solution
a Mode METHOD 2 Create a table
METHOD 1 Mass (kg) Number of parcels
Look at the highest percentage.
The highest percentage = 50% 1 3
25
Therefore, the mode is 4 kg. 2 × 12 = 3
100
4
1
1 3
25
b There are 12 data. Therefore, the 3 100 × 12 = 3
median is the average of the 6th 4
1
and 7th mass.
1 6
50
2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4 4 100 × 12 = 6
Median = 3 kg + 4 kg 2
2
1
= 3.5 kg The highest frequency is 6.
The median is 3.5 kg. Therefore, the mode is 4 kg.
c Range = 4 kg – 2 kg
= kg
d M ea n = (3 × 2 kg) + (3 × 3 kg) + (6 × 4 kg)
(3 + 3 + 6)
What is the total amount of
= 6 kg + 9 kg + 24 kg money collected by the post
12 office if the parcel postage rates
are as shown in the table?
= 39 kg
12 Mass 2 kg 3 kg 4 kg
Price RM10 RM13 RM15
= 3.25 kg
The mean is 3.25 kg.
176 Teacher’s Notes 11.1 (ii) b
Surf http://www.melvister.com/2013/06/kadar-harga-terkini-penghantaran
BRAIN TEASERS Grades Achieved by
The incomplete bar chart shows the grades Number of pupils Year 6 Teguh Pupils
achieved by Year 6 Teguh pupils. If the mode 12
is grade B, state the minimum frequency for
grade B. Then, find the total number of pupils. 10
8
6
4
2
0 B CD
A
Grades
FUN ADVENTURE Participants 3 pupils in a group
Tools/Materials MS PowerPoint
Task
1 Show a pie chart of guessing activities such as animal’s name.
Example:
Letters in the Name of The pie chart shows the number of letters in
an Invertebrate Animal the name of an invertebrate animal.
Key: Count the number of the following letters:
Letter C
20% 40% Letter E C = E = H = L=
20% Letter H
Letter L Rearrange the letters to
20% get the name of the animal.
Letters in My Name Key:
The animal lives in the jungle. 10% 30% Letter A
Has a smart mind. 10% Letter D
Loves to cheat. Letter E
Has many friends. 10% Letter M
The animal is Letter O
. 10% 10% Letter R
10% 10% Letter S
Letter U
2 Get pupils to find the answers based on the information interpreted from the
pie chart given.
3 The fastest group to get the correct answer wins.
11.1 (ii) b Teacher’s Notes 177
Surf http://www.onlinemathlearning.com/mean-median-mode.html
Modify the question in Fun Adventure activity to problem-solving questions
involving mode, range, median, and mean.
SELF-PRACTICE
1 A total of 10 pupils in Year 6 Cerdas had collected aluminium cans for a
recycling campaign. The number of cans collected and the frequencies are
shown in the table.
Number of cans 18 22 26 30
Frequency 51 31
Find the mode, median, mean, and range of the data.
2 A total of 15 pupils took part in a poster competition during the month of
patriotism as shown in the pie chart.
Duration a What is the percentage of pupils who
completed the poster in 35 minutes?
Key:
20% 25 minutes b Calculate the number of pupils who took
20% 30 minutes 1
35 minutes more than 2 hour to complete the poster.
20% 40 minutes
?%
c Calculate the mode, median, mean, and
range.
3 Raj and his family of 4 want to go on a holiday to Penang for 3 days and
2 nights. Based on their budget, he checks a few hotels as shown in the table.
Type of hotel 2 3 4 5
Number of hotels 1 4 4 6
a Find:
i. mode. ii. median. iii. range.
b The accommodation budget Hotel rate per night
for Raj’s family is RM1 500. for twin sharing
In your opinion, which hotel
will Raj choose? Give your Type 2 3 4 5
reason. Rate RM120 RM190 RM360 RM750
178 Teacher’s Notes 11.1 (ii)
Pose more routine and non-routine questions to enhance pupils’ understanding.
Surf http://www.commoncoresheets.com/MMMR.php
http://www.helpteaching.com/questions/Range_Median_Mean_Mode/Grade_5
SKILFUL MIND
1 The pictograph shows the number of doughnuts sold by Kak Ji. The number
of doughnuts sold on Saturday is not shown.
Sale of Kak Ji’s Doughnuts
Tuesday
Wednesday
Thursday
Friday
Saturday
Key: represents 30 pieces.
a Calculate the number of doughnuts sold on Saturday if the sales on that
day is 2 times the sales on Friday.
b What is the average number of doughnuts sold each day?
c Calculate the percentage of sales on Tuesday from the total sales for 5 days.
d The profit on Thursday is RM84. Calculate the price of a doughnut if the cost
price is 30 sen each.
2 During the Physical Education period, Cikgu Farzi asked 4 pupils to practise
running on the track for 10 minutes. The distances of their run were recorded
as shown in the bar chart below.
Distance (km) Running Distances by 4 pupils
3
2
1
0
Akif Omeng Prem Chu
Name of pupils
a Who recorded the longest distance?
b The distance covered by and is the same, which is a distance of
km.
c Calculate the average distance, in km, covered by each pupil.
d Omeng’s running distance decreased by 0.3 km compared to last week.
Calculate his distance for last week. Give a reason for the poor achievement.
11.1 (i) Teacher’s Notes 179
11.1 (ii)
Surf http://www.superteacherworksheets.com/pictograph.html
http://www.superteacherworksheets.com/bar-graphs.html
3 The Earth’s atmosphere consists of several gases such Earth's
as nitrogen, oxygen, and other gases. The percentage Atmospheric Gases
of the gases is shown in the pie chart.
Nitrogen
a What is the percentage of other gases in the Earth’s 78%
atmosphere? Oxygen
21%
b The other gases consists of 0.03% of carbon dioxide
and rare gases. State the percentage of rare gases. Other gases 1%
c Oxygen is used in the process. On the other
hand, oxygen is released in the process.
4 The table shows the achievement of 10 Level of achievement 4 5 6
pupils in Year 6 Orchid for Mathematics. 2 6 2
Find the mode, median, mean, and range Frequency
from the data given.
5 The pictograph shows data of funds raised by Funds Raised
14 members of a finance club.
Contoh
a Determine the value of:
i. mode. ii. median. iii. mean. iv. range. Key: represents 1 person.
b What is the fraction of the members who
gave less than RM1 compared to the total
number of members?
6 Encik Sufi donates 15 packets of buns for his Number of packets Packets of Buns for
child’s school Canteen Day as shown in the School Canteen Day
bar chart. 6
a Find the value of: 5
i. mode. ii. median. iii. mean.
4
b 20% of the total number of buns are
chocolate flavoured and the rest are corn 3
flavoured. Calculate the number of corn
flavoured buns donated by Encik Sufi. 2
1
0 3 6 9 12
Number of buns
180 Teacher’s Notes 11.1 (i)
11.1 (ii)
Ask pupils to collect data from printed media or the Internet and find the
mode, range, median, and mean.
Surf http://map.mathshell.org/download.php?fileid=1619
12
A Recognise whether an Event is Likely or Unlikely
1 Certainly we can. The Earth moves
Can we expect day and night in its orbit around the Sun. That’s
how day and night occurs each day.
to occur tomorrow?
2 Oh, that’s impossible! When
100 is divided by 7, the answer
Can 100 be divided by 7 14 is 14 remainder 2.
without a remainder?
7 100
–7
30
– 28
2 remainder
An event in our daily lives is likely DISCMTIAORNTARY
or unlikely to occur. Can you
mention a few of these events? Likelihood means the
possibility of an event to
occur.
12.1 (i) Teacher’s Notes 181
Discuss several events in our daily lives that are likely or unlikely to occur.
3 A box has red, blue, and green marbles.
Is it likely that a red marble is Yes, it is likely! The red marbles
picked from the box? are in the box, teacher.
Teacher, the blue and
green marbles are also
likely to be picked.
Based on the situation above, are the It’s unlikely to be picked
white marbles likely to be picked from because there are no white
the box? marbles in the box.
What colours are likely or unlikely to be picked?
Discuss.
SELF-PRACTICE
State whether the following events are likely or unlikely to occur.
a There are 8 days in a week. b A combination of two triangles
make a rectangle.
c An even number can be
divided by 5 without remainder. d A penguin can be found on
Malaysian beaches.
182 Teacher’s Notes 12.1 (i)
Carry out simulation for the activity above. Subtitute the marbles with various
flavours of sweets or 3-D shapes.
Discuss events that are likely or unlikely to occur.
B Recognise the Five Likelihood of Events
It’s certain that
1 Today is Wednesday. tomorrow is
Tomorrow is Thursday and Thursday because
there is swimming class. today is Wednesday.
2 You will get 7 from the throw
of a dice.
That’s impossible! A dice has
only 1 to 6.
3 I got heads when I The event of getting heads or tails is
tossed the coin. equally likely because there are only
two possible outcomes.
AMAZING FACTS
The face of a coin that
shows an important/
prominent feature is heads
whilst the face that shows
the value is tails.
12.1 (ii) Teacher’s Notes 183
Carry out simulation of tossing of a coin to emphasise the meaning of
equally likely. Discuss other equally likely events.
4 Discuss reasonable
causes for the likelihood
of these events.
There are five likelihood of events which are certain, impossible,
more likely, less likely, and equally likely. Can you give other
examples of events and their likelihood?
BRAIN TEASERS
Three dice are rolled. What is the likelihood that the product of the three
numbers is 216?
SELF-PRACTICE
Match the events to their likelihood.
Impossible Malaysia celebrates National Day on
31 August.
Less likely The water level of the river increases during
More likely monsoon season.
A healthy diet can increase the body
mass.
Certain Getting an odd number from the roll of a dice.
Equally likely One day has 25 hours.
184 Teacher’s Notes 12.1 (ii)
Encourage pupils to write events about themselves, scientific facts, history or other
subjects. Emphasise the five likelihood.
SMART PROJECT
MIND MAP OF LIKELIHOOD OF EVENTS
Tools/Materials MS Word Impossible Equally likely
Participants 4 pupils in a group
Certain
LIKELIHOOD
Task More likely Less likely
Each group creates a mind map of the five likelihood of events. Present the
mind map. Then, print and compile them into a scrapbook.
FUN ADVENTURE THE EVEN AND ODD GAME
END
Tools/Materials dice, a piece of 5 × 5 squared
card, 4 different coloured counters
Participants 4 pupils in a group
Rules of game
1 Determine the turns. Place the counters on START
START.
The game shows the
2 Throw the dice. The rules to move the counter are: likelihood to get even or odd
a if it is an EVEN number, move the counter one numbers. For example, if
box above. the grid is changed to 3 × 3
b if it is an ODD number, move the counter one boxes, the likelihood to win
box to the left. is if you get an EVEN number
c if the counter is moved out of the game card, followed by EVEN, ODD, and
go back to START. ODD numbers.
3 Take turns. Repeat step 2.
4 The first player to reach END wins.
What is the likelihood of winning if the squared
card is changed to 8 × 8 or 10 × 10? Discuss.
12.1 (ii) Teacher’s Notes 185
Discuss ways to calculate the likelihood of winning with a 5 × 5 grid and
a 10 × 10 grid.
SMART PROJECT
Tools/Materials A3 paper, 4 different coloured pens, Template
a few pieces of 15 cm × 2 cm Likelihood of Events
manila cards, sticky tape
Certain
Participants 4 pupils in a group
More likely
Task
1 Prepare a likelihood of events template as shown. Equally likely
Write Certain at the top, followed by More likely, Less likely
Equally likely, Less likely, and Impossible.
2 Each player is given a different coloured pens and Impossible
3 pieces of manila cards.
3 Each player will write an event on each of the 3 pieces of manila cards
as shown.
Night will occur tomorrow. Earthquake in Malaysia. A triangle has 5 sides.
Malaysia is a developed country. It will rain. A month has 29 days.
4 Each player takes turns to paste the event Certain Example of work
cards on the template.
Likelihood of Events
5 Each group will present their work and teacher
provides feedback. Night will occur tomorrow.
6 Display the work at the mathematics corner. Malaysia is a developed country.
More likely
Equally likely It will rain.
Less likely Earthquake in Malaysia.
Impossible
A month has 29 days.
A triangle has 5 sides.
186 Teacher’s Notes 12.1 (ii)
Guide pupils to write events of the five likelihood and determine the level of
likelihood of each event.
SKILFUL MIND Likely to
1 Match. happen
Eclipse of the moon occurs each year. Unlikely to
One day the cats will have horns. happen
A year has 366 days.
Robots will become waiters in restaurants.
August has 32 days.
2 Colour the impossible events in yellow and events that are certain in pink.
What is the hidden pattern?
My family January has It will
is going on 31 days. not rain today.
a holiday.
There are There are
I passed 12 months in 7 days in
the test. a week.
a year.
I will fall
The Sun does when I cycle.
not set.
I failed All living I will
the test. things will die. grow taller.
I will I will go
grow old. sightseeing.
My teacher It will rain
will scold me. tomorrow.
My mother
cooks chicken.
The pattern is .
12.1 (i) Teacher’s Notes 187
12.1 (ii)
Make copies of activity 2 for practice.
Surf http://www.ixl.com/math/grade-6/probability-of-one-event for extra
practices and learn about likelihood.
SELF-ASSESSMENT
A Answer all the following questions.
1 Diagram 1 consists of several 4 Look at Diagram 2.
squares of the same size. Write the ratio of:
a. the number of shaded Diagram 2
parts to the number of
unshaded parts.
Diagram 1 b. the number of unshaded parts to
the number of shaded parts.
State the ratio of the number of blue
parts to the number of white parts. c. the number of shaded parts to
the total number of parts.
2 Look at the Cartesian plane below. 5 Draw and measure the angles in:
Find the horizontal distance and a. an octagon.
vertical distance between the b. a heptagon.
points:
y E G 6 Write two events for each likelihood
4 in the following table.
3 D Fx
Likelihood Event
2C 234 5 67
Certain
1 More likely
O1 Equally likely
Less likely
a. C and F. Impossible
b. D and E.
c. D and F . 7 The incomplete pictograph shows
d. D and G. the sales of ice creams for four days.
e. G and F.
Sale of Ice Creams
3 Match the following likelihood of
the events. Monday
The temperature in Unlikely Tuesday
Malaysia reaches 40°C. to
Wednesday
A baby can walk the happen Thursday
moment he is born.
Likely to Key: represents 10 ice cream cones.
North Pole becomes happen
South Pole. The number of ice creams sold on
Thursday is 25% of the total sales
in 3 days. Find the difference in
the number of ice creams sold on
Monday and Thursday.
188
8 Measure the angles of the polygons 12 Look at the Cartesian plane below.
below. Name the polygons and Which of the following is the
write their characteristics. possible coordinates of P?
a. b. A. (9, 11) B. (9, 9)
C. (11, 9) D. (11, 11)
y
c. d. 20
10 P
9 Draw a hexagon on a squared x
paper. Measure the angles. O 10 20
10 The height of two basketball 13 The incomplete bar chart shows
players, Malik and Janang, is 1.43 m 80 recyclable items disposed in a
and 150 cm respectively. State the week. The number of glass items
ratio of Janang’s height to Malik’s disposed is 2 times the number
height. of paper. Calculate the number of
plastic materials disposed.
11 The pie chart shows the
percentage of the sales of concert 32 Disposal of Recyclable
tickets based on its price. The 28
percentage of the RM180 ticket is Number of items Items in a Week
not shown. The total number of
tickets sold is 800 pieces. How 24
many RM180 tickets were sold? 20
Sale of Concert Tickets 16
12
RM120 RM60
25% 30% 8
RM180 RM80 4
30% Tin Paper Glass Plastic
Types
14 The coloured area 6 cm
on both sides of
#
the folded paper
is cut. What is the 5 cm 5 cm 4 cm
shape of the cut
piece? Calculate the 3 cm
perimeter, in cm,
and area, in cm2, Diagram 3
of the remaining part.
189
B Solve the following problems.
1 Madam Swee Hong wants to 5 Table 1 shows three types of
paint her room. She was told by vegetables picked by Mr Tan from
the salesman that 1 ℓ of paint is his farm to be sold in Pasar Tani.
needed to paint an area of 2.5 m2.
If the area of her room is 50 m2, Type of Mass (kg) Price
what is the volume, in ℓ, of paint vegetables 1 kg (RM)
needed to paint her room?
Cucumber 24 kg RM3.50
2 The area of the base of an aquarium Tomato 70% of the mass RM5.25
Long bean of cucumbers
is 640 cm2 and the height is 36 cm. 2 of the mass of RM7.50
2 3
3 of the aquarium is filled with tomatoes
water. Calculate the volume, in Table 1
cm3, of water in the aquarium. What is the total sales, in RM, if Mr Tan
sold all the vegetables?
3 Diagram 4 shows a composite
of 2 cuboids of the same size. 6 Mother cooks 120 pieces of meat
Calculate, in cm2, the shaded puffs. The number of sardine puffs
surface areas. is 31 more than the number of
meat puffs.
3 cm
6 cm a. Calculate the ratio of the number
of sardine puffs to the number of
7 cm 3 cm meat puffs.
Diagram 4 b. What is the ratio of the number of
meat puffs to the total number of
4 The measurement of a picture puffs?
to be pasted on a cardboard is
shown in Diagram 5. 7 Diagram 6 consists of a square
WXYZ and a right-angled triangle
30 cm YVZ. Calculate:
50 cm a. the perimeter, in W 10 cm X
Diagram 5 cm, of the shaded 8 cm V
region. 6 cm
a. State two possible ratios of
the length to the width of the b. the area, in cm2, of Z Diagram 6 Y
cardboard. the shaded region.
b. Calculate the area, in cm2, and
perimeter, in cm, of the cardboard
based on the answers in (a).
190
8 Diagram 7 shows a cube Y with a 11 The bar chart below shows the
length of 3 cm. How many similar mass of 20 pupils.
cubes are needed to form a cuboid
with a length of 18 cm, a width of Mass of Pupils in Year 6 Maju
6 cm, and a height of 12 cm?
34 kg
Mass 31 kg
3 cm Y 28 kg
Diagram 7 25 kg
9 The pictograph below shows the 0 24 68
number of tomatoes based on the Number of pupils
packets sold by Jasni.
Find the: a. mode. b. median.
c. range. d. mean.
Sale of Tomatoes 12 Diagram 9 shows two regular
3 pieces pentagons. The length of FG is
4 pieces 3 cm longer than the length of AB.
5 pieces
6 pieces a. State the ratio of the length of FG
to the length of AB.
Key: represents 1 packet of tomatoes.
b. What is the ratio of the perimeter
of pentagon ABCDE to the
Find the: a. mode. b. median. c. range. perimeter of pentagon FGHIJ?
H
10 Diagram 8 shows a net of a CG I
composite three-dimensional J
shape built by Dania. What is the: B D
18 cm M
a. perimeter, in cm, of the net?
AE F
b. surface area, in cm2, of the net?
Diagram 9
4 cm 4 cm 13 Diagram 10 shows the composite
of 2 cuboids, S and T. Calculate
2 cm 2 cm 4 cm 4 cm the surface area, in cm2, of the
4 cm composite three-dimensional
shape.
4 cm
4 cm
12 cm
2 cm 2 cm 4 cm
4 cm ST
8 cm 5 cm
Diagram 8 Diagram 10
191
14 The pie chart shows the sales of 17 Mark the possible position of Ikram (A)
20 bottles of pomegranate juice and Davin (B) on the Cartesian
on Saturday. plane below if the horizontal
distance and the vertical distance
Sale of Pomegranate Juice between them is 6 units and
on Saturday 4 units respectively.
30% 40% Key: y
20%
0.25 ℓ 7
10% 0.5 ℓ 6
0.75 ℓ 5
1.0 ℓ 4
3
a. What is the difference in 2 x
percentage between the highest 1
sales and the lowest sales?
O 1 23 4567
b. Find the:
i. mode. ii. range. iii. mean. 18 Diagram 11 shows an isosceles
15 The incomplete pie chart shows the triangle ABC and a rectangle DEFG.
heights of 10 pupils.
a. What is the percentage of pupils ADEB is a straight line. The length of
with the height of 142 cm?
b. Find the mode, range, and mean. AD is equal to the length of EB and
The Heights of Pupils the length of AB is equal to 30 cm.
The perimeter of DEFG is 50 cm. The
height of triangle ABC is equal to the
144 cm length of GF. A D E B
140 cm 30%
GF
50% 20 cm
142 cm 25 cm
16 The picture shows the mass of a C
packet of jelly powder. To prepare
a perfect jelly, a packet of jelly Diagram 11
powder needs 1.5 ℓ of water. Sara
wants to cook 75 g of jelly. What a. Calculate the perimeter, in cm, of
is the volume, in mℓ, of water the shaded region.
needed?
b. Calculate the area, in cm2, of the
25 g shaded region.
19 Neema has 16 pieces of foreign
stamps. Aninah has 15 pieces of
foreign stamps and 20 pieces of
local stamps. What is the ratio of
the number of foreign stamps to all
the stamps that they have?
192
Unit 1: Numbers and Operations Unit 6: Time
Brain Teasers (pg 101)
Brain Teasers (pg 7) Brain Teasers (pg 9) Brain Teasers (pg 13)
day month hour minute second
17 and 71, 37 and 73 1 001 557 =1 2 6 03 1 7 : 48 : 59
Skilful Mind (pg 20) Skilful Mind (pg 110)
1) 12-hour system 12:45 p.m. 9:15 p.m. 12:30 p.m. 6:05 p.m. 7:10 a.m. 12:10 a.m.
1) (a) 785 (b) 1 631 327 (c) 5 171.298 (d) 8 853.172
24-hour system 1245 hours 2115 hours 1230 hours 1805 hours 0710 hours 0010 hours
2) (a) The number pattern is adding 850. The method is press 1 204 500
+ 850. Then, press the = button 5 times. The answer is
1 208 750. 2) Start Time End Time Duration
2 days 8 hours 50 minutes
(b) (i) 8 (ii) 88 5:15 a.m., 7/1/2015 2:05 p.m., 9/1/2015
77 888 2130 hours, Tuesday 1910, hours Wednesday 21 hours 40 minutes
666 8 888 1:20 a.m., 6/9/2016 10:30 p.m., 20/12/2016 3 months 14 days 21 hours 10 minutes
5 555 88 888 2030 hours 0300 hours 6 hours 30 minutes
3) 83, 89, 97 4) (a) 2 370 000 (b) 4 625 000 (c) 8 019 000 0015 hours 0515 hours 5 hours
5) (a) 0.5 million / 1 million (b) 3.4 million / 32 million (c) 9.75 million / 9 3 million 27 August, 7:00 p.m. 30 August, 11:00 p.m. 3 days 4 hours
2 5 4
16 January, 2015 16 May 2017 2 years 4 months
6) (a) 330 000 km2 / 0.33 million km2 (b) 580 000 / 0.58 million 3) (a) Not appropriate because the time in Sweden is 0200 hours or 2:00 a.m.
(b) 1345 hours
Unit 2: Fractions (c) (i) V = 150 000 (ii) W = 50 000 (c) (i) 9 days (ii) 9/3/2016, 5:45 a.m.
Brain Teasers (pg 29) Self-assessment (Unit 1 – Unit 6)
Yes, each pupil gets 2 of the pizza. Section A (pg 111)
5
1) One million twenty thousand four hundred and nine 2) 370 000
Brain Teasers (pg 31)
3) 6 000 000 4) 7 5) 900 000 6) 4 404 004 7) 17 553.28
(a) 1 + 1 = 1 ÷ 1 or 1 + 1 = 1 ÷ 1 and others. When A = B (B – 1). 10
2 2 2 2 6 3 6 3
8) 49 9) 3 + 17, 7 + 13 10) 977 883 11) RM756 12) 1 1
(b) 1 – 1 = 1 × 1 or 1 – 14= 1 × 1 and others. When D – C = 1. 8
2 3 2 3 3 3 4
13) Press the calculator 3 × 6 =. Then, press = 4 times more.
Skilful Mind (pg 38)
P = 1 458.
7 2 5 17 19 2
1) (a) 9 (b) 7 (c) 12 (d) 1 35 (e) 5 20 (f) 6 3 14) 187.5 15) 11 16) 4 400 000 17) 106 009 18) 1100 hours/11:00 a.m.
45
1 1 4 1 1
2) (a) 1 27 (b) 32 (c) 27 (d) 5 (e) 12 (f) 12 19) 3 1 20) 23, 29, 31, 37, 41, 43, 47, 53, 59 21) 16 22) 5 1 23) 43.52
5 2
2 2 4 4 1 3 5 4 1
3) (a) 3 × 7 = 21 (b) 5 ÷ 5 = 4 (4) (a) 40 (b) 6 cups (c) 11 5 kg (d) 2 2 times 24) 1 × 2 = 2 25) 142.2 26) 12:10 a.m. 27) 2 months 25 days
3 3 9
Unit 3: Decimals 28) (a) Property, jewellery (b) Car instalment, income tax
Brain Teasers (pg 40) Brain Teasers (pg 41) 29) 1 day 11 hours 45 minutes, 8 days 9 hours 40 minutes
3 × 1.6 ÷ 4 = 1.2 0.1 kg 30) Telephone: Discount RM135 (Discount: 15%)
Brain Teasers (pg 49) Brain Teasers (pg 51) Camera: Original price RM1 500 (Discount: 20%)
12 m 6.9 or 0.69 Laptop: Selling price RM2 700 (Discount: RM900)
Skilful Mind (pg 52) 31) (a) RM280.20 (b) RM4 950.20 32) RM3 250
1) (a) 11.28 km (b) 6.68 kg (c) 0.534 ℓ (d) RM10.03 (e) 1.89 33) Car: RM46 290 34) 2 years 10 months 16 days
(f) 125.1 (g) 1.76 (h) 10.065 House: RM322 600
2) Perform division operation first, then the quotient is multiplied by 5. 35) The possible number of tickets 36) 4.081 million + 0.919 million
* Accept any reasonable answers. 2-D 3-D 5 million ÷ 2
140 10 3 × 2 500 000
3) (a) 64 × 9.037 ÷ 8 = 72.296 (a) (estimation method) 131 15
72
(multiply first, then divide) 3 122 20 7 500 000
(b) 38.34 × 9 ÷ 2.7 = 127.8 8 ) 576 113 25 * Accept any reasonable answers.
64
(multiply first, then divide) ×9 – 56 37) 3 200 000
(b ) 328..734 × 1 = 127.8 (cancellation method) 576 16 38) 0.1818, 0.2727, 0.3636, 0.4545 (pattern of adding 0.0909)
9 – 16
0.3 0 39) 10 hours 30 minutes 40) Numbers involved in the calculation
4) (a) 0.802 m (b) (i) 0.35 ℓ (ii) 15.4 ℓ 1 have been cancelled.
2
(c) RM0.15 million 41) 4 42) RM426 43) 7 hours 11 minutes
Unit 4: Percentage 44) (a) 5 × 5 = 3 1 (b) 5 × 6 = 10 45) 1 , 1 1 , 10 , 3
8 8 9 7 21 2 2 3 8
Brain Teasers (pg 59)
RM0.18 million 46) - 0820 hours to 1045 hours 2 hours 25 minutes
Skilful Mind (pg 62) 5:00 p.m. to 7:30 p.m. - 2 hours 30 minutes
1) (a) RM8 256 (b) 56 days (c) RM2 190.04 (d) RM20 000 (e) Li Suan
2) (a) RM725 (b) 80 durians 12:20 a.m. to 8:10 a.m. 0020 hours to 0810 hours -
Unit 5: Money
Brain Teasers (pg 65) Section B (pg 114)
The sales of bags in Kedai Sukan Ria because the profit of a bag is RM5.35 1) 42 000 2) RM2 400 3) Several answers: 3, 11, 23 or 5, 13, 19 or 7, 13, 17
whereas the profit of a bag in Kedai Sukan Kita is only RM3.45. 4) 25 5) RM7 800 6) 2 weeks or 14 days 7) 5:55 a.m., Wednesday
48
Skilful Mind (pg 88)
(a) RM45 (b) RM400 000 (c) RM4 680 (d) RM53 (e) RM1 160 000 8) Accept any reasonable answers. 9) 45% 10) RM120
(f) 20% (g) 1 packet of biscuits = RM9.60 (h) Encik Hamid From 6 000 to 6 333 (2nd day) 11) 150 000 and 350 000
1 packet of milk powder = RM19.20 RM565 382 From 12 000 to 12 666 (1st day) 12) (a) 75% (b) 50 minutes
193