MathematicsDLP Tahun 6

RUKUN NEGARA

Bahawasanya Negara Kita Malaysia
mendukung cita-cita hendak:

Mencapai perpaduan yang lebih erat dalam kalangan
seluruh masyarakatnya;

Memelihara satu cara hidup demokrasi;

Mencipta satu masyarakat yang adil di mana kemakmuran negara
akan dapat dinikmati bersama secara adil dan saksama;

Menjamin satu cara yang liberal terhadap
tradisi-tradisi kebudayaannya yang kaya dan pelbagai corak;

Membina satu masyarakat progresif yang akan menggunakan
sains dan teknologi moden.

MAKA KAMI, rakyat Malaysia,
berikrar akan menumpukan

seluruh tenaga dan usaha kami untuk mencapai cita-cita tersebut
berdasarkan prinsip-prinsip yang berikut:

KEPERCAYAAN KEPADA TUHAN
KESETIAAN KEPADA RAJA DAN NEGARA

KELUHURAN PERLEMBAGAAN
KEDAULATAN UNDANG-UNDANG
KESOPANAN DAN KESUSILAAN

(Sumber: Jabatan Penerangan, Kementerian Komunikasi dan Multimedia Malaysia)

RUKUN NEGARA.indd 1 4/10/16 8:55 PG

i

ii

iii

iv

v

3

SKILFUL MIND

SMART PROJECT

SMART CALCULATION

BOOST
YOUR KNOWLEDGE

vi

1

A Let’s Learn to Use a Calculator

1 Types of layered cakes Amount exported

Cheese 752 168

Strawberry 319 504

Sweet and sour 186 097

Chocolate 85 479

Calculate the total number of cheese and chocolate layered This is a
cakes exported. 12-digit

752 168 + 85 479 = calculator.

Steps: 'On' button

• Press the on button
c

• Press the buttons

752168 +

• Press the buttons Number Operation
85479 buttons buttons

• Press the = button. The screen displays the answer .

The sign ’ means leave a space between the digits.

752 168 + 85 479 = 837 647

The total number of cheese and
chocolate layered cakes exported
is 837 647.

1.1 (i) State the two types of cakes 1
with a total export of less
than half a million.

Teacher’s Notes
Provide various examples of daily situations in transactions involving addition,
subtraction, multiplication, and division.
Explain the function of the operation buttons on a calculator.
Surf http://www.ehow.com/way_5519128_basic-calculator-instructions.html and
http://www.bbc.co.uk/bitesize/ks2/maths/number/using_calculator/read/5/

2 235 620 − 1 740 ÷ 60 =

Remember! Division must be done first.

STEP 1 STEP 2

Press Calculator Press Calculator
on screen display on screen display
c c
0 0

1740 1 740 2 3 5 6 2 0 235 620

÷ 60 60 – 2 9 29

= 29 = 235 591

235 620 − 1 740 ÷ 60 = 235 591

3 98 130 + 675 × 14 =

Multiply first, Press the buttons 0
then add.
Add first, then 9 8 1 3
Press the buttons
67 5 × 14 multiply.

+675

Yi Wen +98 1 3 0 = × 14 =
and the answer on the
and the answer on the Ramesh screen is 1 383 270 .
screen is 107 580 .

Why are their answers different?
Explain.

Teacher’s Notes

2 Remind pupils of the concept that involves mixed operations and brackets. 1.1 (i)

Stress on the calculation in vertical form and other calculation methods

learned.

4 107 896 ÷ 19 =
Round off the answer to: a one decimal place.

This is an b two decimal places.
8-digit

calculator.

Press the on button. Press ÷ 1 9 . Press = and
c get the answer.

Press 1 0 7 8 9 6 .

a Round off 5 678.7368 to 1 decimal place which is 1 digit after the decimal point.

5 678.7368 =

1 decimal place Look at the second digit after the decimal
point which is 3. 3 is less than 5. Retain
digit 7. Ignore the digits 3, 6 and 8.

5 678.7368 ~ 5 678.7

107 896 ÷ 19 = 5 678.7

b Round off the answer to 2 decimal places.

2 decimal places 6 is more than 5. Add 1 to
digit 3. Ignore the digits
5 678.7368 = 6 and 8.

+1

5 678.74 BOOST
107 896 ÷ 19 = 5 678.74 YOUR KNOWLEDGE

Explain how you round off Round off to the nearest
5 678.7368 to the nearest thousandth means rounding off a
number to three decimal places.
thousandth.

1.1 (i) Teacher’s Notes 3

Introduce several types of calculators such as the 8-digit or 12-digit calculator.
Ask pupils to use different calculators and compare their answers. Then, discuss.

5 What is the answer on Adam’s card for the number pattern below? Use a

calculator to help you. Adam

20 468 40 936 61 404

The numbers above 11 1 11 The number sequence
are arranged in an above increases by the
ascending order. 20 468 40 936 same value. So, the
+ 20 468 + 20 468 number pattern is add
20 468.
40 936 6 1 404

Steps:
• Press the buttons 2 0 4 6 8 + 2 0 4 6 8 =

• The answer on the screen is 40 936 . Is this a suitable way to
• Press the = button again. get the 8th number for the
• The answer on the screen is 61 404 . number pattern above?

8 × 20 468 =

• Press the = button 5 times.
The number on Adam’s card is 163 744 .

6 Look at the number pattern below. What are the next three numbers?
Use a calculator to get the answers.

1, 2, 6, 24, , ,

1=1×1= 1 Think of the 5th to 7th numbers in this
1×2=1×2= 2 number pattern. Explain your answer.

1 × 2 × 3 = (1 × 2) × 3 = 2 × 3 = 6

1 × 2 × 3 × 4 = (1 × 2 × 3) × 4 = 6 × 4 =

Teacher’s Notes

Guide pupils to write the number in each empty box every time the = button is
4 pressed to avoid confusion or mistakes. 1.1 (i)

Use a strategy to identify the number pattern without using a calculator.

7 Explain the pattern.

Diagram 1 Diagram 2 Diagram 3 Diagram 4
Pattern 0 + 1 = 1 1 + 2 = 3 3 + 3 = 6 6 + 4 = 10

The 50th diagram in the sequence above has 1 275
balls. How many balls will there be in the 51st diagram?

8 999 999 , 888 888 , 777 777 , ....... , ....... , ....... , ....... , .......

63 × 15 873 56 × 15 873 49 × 15 873 × 15 873

What is the value in to get the 8th number above?
Explain and complete the number pattern.

SELF-PRACTICE

1 Use a calculator to calculate the answers.

a 503 627 + 819 543 + 37 508 = b 9 700 100 – 34 295 + 6 457 =

c 26 × 84 072 ÷ 16 = d 586 840 + 935 510 ÷ 34 =

e 867 549 + 3 168 × 4 = f 4 102 120 – 847 008 ÷ 12 =

2 Use a calculator to solve the following. Round off the answer to two decimal
places.

a 203 568 divided by 37. b Divide 4 hundred thousand by 29.

3 Use a calculator to complete the number patterns below.

a 1 × 8 + 1 = b 9×9=

12 × 8 + 2 = 98 × 9 =

123 × 8 + 3 = 987 × 9 =

1234 × 8 + = × 9 =

×+ = ×=

1.1 (i) Teacher’s Notes 5

Provide a variety of questions to enhance pupils’ skills to use calculators in their
calculation and to identify the number patterns.
Ask pupils to identify the number patterns in their daily lives. For example,
a home address in a housing estate.
Surf http://www.ask-math.com/worksheet-on-whole-numbers.html

B Recognise Prime Numbers

1 2÷1 2 31
25 2÷2 31 ÷ 31
The numbers on the
branches which have 31 ÷ 1 17
2 fruits are numbers
that are divisible by 1 25 ÷ 5 25 ÷ 25 17 ÷ 1 17 ÷ 17
and itself. 4 25 ÷ 1 1

4÷1 4÷4 4÷2 1÷1

They are prime numbers.

2 BOOST
YOUR KNOWLEDGE
2÷2 = 1 2÷1 = 2 Prime numbers are
numbers that must be:
divided by itself divided by 1 ✓ greater than number 1.
✓ only divisible by 1 and
2 is more than 1. itself.
2 can be divided by 2.
2 can be divided by 1. Why are the numbers 1
Therefore, 2 is a prime number. and 25 not prime numbers?
Discuss.
17 and 31 are more than 1. They
can be divided by 1 and itself only. AMAZING FACTS
In 200 BC, Eratosthenes
Therefore, 17 and 31 are created an algorithm to get
prime numbers too. prime numbers known as
The Sieve of Eratosthenes.
4 can be divided by 1, 2 and 4. The
divisor exceeds two numbers.

Therefore, 4 is not a prime number.

Teacher’s Notes

6 Use number cards to identify prime numbers. Emphasise the concept of 1.2 (i)
prime numbers.

SMART PROJECT

Tools/Materials MS Word, pencil, rubber, picture of an animal from used
reading materials, a pair of scissors, glue

Participants 2 pupils in a group

Task Work sample

1 Launch MS Word. Click Insert Table
and select 10 × 10 table.

2 Key in numbers from 1 to 100. Print.

3 Colour the number 2. Mark (/) on all the
multiples of 2 with a pencil.

4 Colour the number 3. Mark (/) on all the 1 2 3 4 5 6 7 8 9 10
multiples of 3 with a pencil. 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
5 Colour the number 5. Mark (/) on all the 31 32 33 34 35 36 37 38 39 40
multiples of 5 with a pencil. 41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
6 Colour the number 7. Mark (/) on all the 61 62 63 64 65 66 67 68 69 70
multiples of 7 with a pencil. 71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
7 Colour all the remaining numbers. 91 92 93 94 95 96 97 98 99 100

8 What are these numbers?

9 How many prime numbers are there
from 1 to 100?

10 Decorate your work.

2 State two prime numbers
with a sum of 28.
What are the prime
numbers from 40 to 70?

11 + 17 5 + 23

BRAIN TEASERS

Reverse the digits 1 and 3 in 13 to become 31. 13 and 31 are prime numbers.
List other prime numbers that have the same characteristic.

1.2 (i) Teacher’s Notes 7

Challenge pupils with questions involving prime numbers and the four basic
operations.
Surf http://www.teachingideas.co.uk/maths/prime.html

FUN ADVENTURE
Recite the following poem.

PRIME NUMBER POETRY

Looking for prime numbers?
1 to 100, are they?

But not all are prime numbers,
So let’s check and see.

Two rules to remember are,
Greater than number 1,

And only divisible by itself,
But remember, one is not a prime number.

Now state all the prime numbers,
There are twenty-five in all,
Now let’s think smart,

What is the 25th prime number?

SELF-PRACTICE 1
1 Say the prime numbers and rewrite them in ascending order.

5 36 23 9 83 2

94 73 41 11 64 17 58

2 List the prime numbers from 10 to 30.

3 Complete the number sentences below with prime numbers.



a 8 = + b 100 = + c 60 = +
+
d 30 = + e 24 = + f 52 =

Teacher’s Notes

Carry out group activities such as riddles or puzzles involving prime numbers.
8 Surf http://www.mathsisfun.com/prime-composite-number.html 1.2 (i)

C Seven-Digit Numbers
Exploration of seven-digit numbers

1 BH Saturday, 4 October 2014 The total number of

The pupils’ data on 31 August 2014 showed pupils is two million

that the total number of primary school seven hundred and
pupils was 2 708 211. They consisted of eight thousand two
1 391 785 boys and 1 316 426 girls. hundred and eleven.

Look at the chart below. Say the place value and digit value for 0, 2 and 1.

Digit 2 7 0 8 211

Place value Millions Hundred Ten Thousands Tens Ones
thousands thousands

Digit value 2 000 000 700 000 8 000 200 10

I partition 1 391 785 according to the place value.

1 391 785 = 1 millions + 3 hundred thousands + 9 ten thousands + 1 thousands +
7 hundreds + 8 tens + 5 ones

Partition 1 316 426 according to its digit value. Discuss.

1 316 426 when rounded off to the nearest hundred thousand becomes 1 300 000.
1 300 000 can be written and read as 1.3 million or 1130 million.

2 Round off 1 600 000 to the nearest million.
1 600 000

1 000 000 1 500 000 2 000 000

1 600 000 is closer to 2 000 000.
1 600 000 when rounded off to the nearest million becomes 2 000 000.

1 600 000 is equal to 1.6 million or 1 3 million.
5

BRAIN TEASERS

07 15

Write the smallest 7-digit number using the cards given. Each card can only
be used twice.

1.3 (i) Teacher’s Notes 9

Use flash cards and digital display to get the pupils to say aloud or write the
numbers in millions either in numerals or words. Partition the numbers and
round off the numbers to the nearest million.
Surf http://www.onlinemathlearning.com/place-values.html

Decimal millions and fraction millions NUMBER FACT

1 A total of 0.051 million people attended a dinner Each day the
in conjunction with the Opening of Dataran blood in the
Malaysia in Pantai Klebang, Melaka organised human body
by the Melaka State Government on 4 February flows 0.18
2010. million km.

Source: The Malaysia Book Of Record Source: http://halaqah.net/v10/
index.php?topic=1648.15;wap2
ones tenths hundredths thousandths
0 Zero point
05 1 eighteen million.
Am I right? Why?

zero point zero five one million

2 A queen bee lays about Almost 1 million
3 million eggs in a year. 2
4 candidates sat for the

http://phhp.com.my/leaflet/royal_jelly.pdf UPSR in 2013.

The diameter of the sun is http://www.mysumber.com/
jumlah-calon-dapat-5a-
approximately 1 2 million km. upsr-2013.html
5

http://imzaroncikgusains.blogspot.

com/2010/03/matahari-sun.html

Three-fourths of a One over two million.
million or three-quarters
of a million.

3 million 1 million
4 2

Can 1 million be read as half a million? State 1 2 million in words.
2 5

Teacher’s Notes 1.3 (ii)

Use numeral or word cards randomly to get pupils to read aloud decimal

10 millions and fraction millions.

Produce a scrapbook by collecting facts from the Internet involving decimal
millions up to 3 decimal places and fraction millions.

3 Two point zero eight One, five over eight million

five million 15 million
2.085 million 8

Which number is written
correctly? Discuss.

SELF-PRACTICE

1 Write the numbers in words or numerals.

a 7 000 000 b 6 032 400 c 9 501 101

d Four million and two thousand e Eight million six hundred thousand and thirteen

2 Correct the mistakes in the yellow boxes.

a 5.109 million Five point one hundred nine million
Seventy three over eight million
b 7 3 million
8 Four over three million
Six million point one two
c 3 million
4

d 6.12 million

3 List the fraction millions and decimal millions. Write them in words.

A3wc5i1tchomrtdihllieniognptopopesuotlpaaltteiiso.tTnichseofflraoarmgbeotshutetrap53coepmiunlialtlhitoieonnstpaceetenowspulaess. itnIhne20KS1aa0dr,aaSwzaaabnka,dhuthhsauedinr
population was 2.5 million people. The largest race in Sarawak was
the Ibans that was about 0.7 million people.

Source: http://pdomba2domba.blogspot.com/2012/11tahukah-anda-populasi-malaysia-mengikut.html

4 Partition the following numbers according to the place value and digit value.

a 4 021 006 b 5 904 107 c 6 130 750

5 Round off the following numbers to the:

Number Nearest ten Nearest hundred Nearest million
thousand thousand

a 7 514 039

b 3 962 750

1.3 (ii) Teacher’s Notes 11

Collect data or interesting facts from various sources involving decimal millions
and fraction millions. Carry out activities such as reading and writing the numbers
in numerals and words.
Surf http://malaysianreview.com/42585/jumlah-rakyat-malaysia-cecah-30-
juta-hari-ini-2014

D Relationship of Decimal Millions and Fraction Millions
with Whole Numbers

1 There are about 950 000 species of insects on Earth.

Source: http://www.factmonster.com/ipka/A0934288.html

Convert 950 000 to decimal millions.

950 000 = million

950 000 is less than 1 000 000. Therefore, the digit value for millions is 0.
Then, place a decimal point between the digit for millions and the digit for

hundred thousands. Ignore all the digit 0 after the digit 5. Then, write the

word ’million’. 0.95 million

Number 950 000

Place value Millions Hundred Ten Thousands Hundreds Tens Ones
thousands thousands

Digit value 0 9 5 0 0 00

950 000 = 0.95 million There are 30 000 species of fish on Earth.
What is this value in decimal millions?

2 State 6 500 000 in decimal millions.
6 500 000 = million

METHOD 1 METHOD 2
6 500 000 = (6 500 000 ÷ 1 000 000) million
6 500 000 = ( 6 500 000 ) million
1 000 000 = 6.5 million

= ( 65 ) million Why should the decimal point
10 be moved 6 places to the left?

= 6.5 million

6 500 000 = 6.5 million

3 Each year, an estimate of 0.1 million sea turtles and marine animals around
the world die due to ingestion of plastic bags which are mistaken for food.

Source: http://www.wdcs.org/wdcskids/en/story_details.php?select=879

Convert 0.1 million to a whole number. SMART CALCULATION
0.1 million = × 1 000 000

0.1 million = 0.100000 × 1 000 000 Decimal Whole

= 100 000 millions number

0.1 million = 100 000 ÷ 1 000 000

Teacher’s Notes

Surf http://news.discovery.com/earth/plants/874-million-species-on- 1.3 (iii)
12 earth-110823.htm

4 7.09 million = State 0.709 million

7.09 million = 7.090000 × 1 000 000 as a whole number.

= 7 090 000

7.09 million = 7 090 000

5 Convert 250 000 to fraction millions.

250 000 = million 1 000 000 = 1 million
250 000 = (1205000000000 ) million
500 000 = 1 million 500 000 = 1 million
2 2

= 25 ÷ 25 million 250 000 = 1 million 250 000 = 1 million 250 000 = 1 million 250 000 = 1 million
100 ÷ 25 4 4 4 4

1 million
8
1
= 4 million

1 Convert 1 million to a whole number.
4 8
250 000 = million

6 State 1 300 000 in fraction 7 Identify the cards of the same value.
millions. Discuss.

1 300 000 = million

1 300 000 = (11 300 000 ) million 1 million 200 000
000 000 5

= 13 million
10

150 000 0.2 million

= 1 3 million
10

1 300 000 = 1130 million

BRAIN TEASERS 1
8
4. 25 million = 4 million = 4 25 000

Find the value of .

1.3 (iii) Teacher’s Notes 13

Carry out a quiz by matching the decimal millions and fraction millions to whole
numbers.
Use fraction charts, fraction stripes or number lines to describe the relationship
between whole numbers, fraction millions, and decimal millions.

8 1 7 million = 9 2 5 million =
10 8

1170 million = 1 million + 7 million 2 5 million = 21 million
10 8 =
8 125 000
= 1 000 000 + 700 000 21
× 1 000 000
8
= 1 700 000 1

1170 million = 1 700 000 = 2 625 000

2 5 million = 2 625 000
8

Use both the methods above to state 6 1 million as a whole number.
2

10 P
0 0.15 million 0.3 million

Look at the number line above. State the value of P as a whole number.

P = 0.3 million + 0.15 million + 0.15 million

= 0.6 million 0.15 million − 0 = 0.15 million
0.6 million = 0.6 000 00 × 1 000 000 0.30 million − 0.15 million = 0.15 million

= 600 000

The value of P is 600 000 .

State 600 000 in fraction millions.

SELF-PRACTICE

1 Convert the decimal millions and fraction millions to whole numbers.

a 0.7 million = b 3.05 million = c 2.61 million =

d 1 3 million = e 4 million = f 8 9 million =
4 5 10

2 Complete the table given.

Number 400 000 1 750 000 9 875 000
Decimal millions

Fraction millions

3 Look at the number line. What

is the value of K? State the 0 K 1 million 3 million 1 million
value in decimal millions. 4 8 2

Teacher’s Notes

Vary the questions on the relationship between decimal millions and fraction 1.3 (iii)
14 millions with whole numbers and vice versa.

E Solve the Problems

1 At the end of 2014, several states in Malaysia
were hit by severe floods. The table below
shows an estimated number of flood victims
in three states.

State Estimated number of victims (people)

Kelantan 45 400

Terengganu 10 800 less than Kelantan

Pahang 33 000

Source: http://www.bernama.com/bernama/v3/bm/news_lite.php?id=1096406

Calculate the total number of flood victims in the three states. Give the
answer in decimal millions.

Solution

Given Number of flood victims as in the table.

Asked for Total number of flood victims

Operations Addition and subtraction

Solve 45 400 + 45 400 − 10 800 + 33 000 =
11
45 400 123 800

45 400 − 10 800

+ 33 000 11 3 000

123 800

113 000 ÷ 1 000 000 = 0.113 million

Check the answer using a calculator.

45 400 + 45 400 − 10 800 + 33 000 = 0.113 million

The total number of flood victims in the three states was 0.113 million.

The estimated number of flood victims in Sarawak was 0.107
million less than the total number of flood victims in the three
states. What was the number of flood victims in Sarawak?

1.3 (iv) Teacher’s Notes 15

Guide pupils to find the keywords in the question.
Give plenty of oral exercises in constructing number sentences based on the
problems given with story cards.

2 In 2015, a factory produced 1 million kg of cooking
5
oil for export. Production targets in each subsequent

year is 2 times the mass of cooking oil compared to

the previous year. What is the mass, in kg, of cooking

oil produced in 2019?

Solution Form a table based on the information given.

1 million = 200 000
5

Year Mass (kg) 3 200 000 million = 32 million
2015 200 000 1 000 000 10
2016 2 × 200 000 = 400 000
2017 = 3.2 million
2018 2 × 400 000 = 800 000
2019 2 × 800 000 = 1 600 000
2 × 1 600 000 = 3 200 000

Check Press the buttons 2 × 2 0 0 0 0 0
and then press the buttons × 8 = .

The mass of cooking oil produced in 2019 is 3.2 million kg.

The mass of cooking oil produced in 2017 is 800 000 kg. The mass of
production for each month is the same. What is the mass of cooking oil
produced in the first 3 months?

a. 800 000 ÷ 12 × 3 = Which is the correct number sentence
b. 800 000 × 3 ÷ 12 = to solve the problem given?
Discuss.

Teacher’s Notes

Guide pupils to determine and choose a simple problem-solving strategy such 1.3 (iv)
16 as drawing a diagram or attempting a simpler problem.

3 A company published reading materials consisting of reference books,
3
comics, and magazines. The number of comics is 10 million copies less than

the reference books and 21 800 copies more than the magazines. Calculate

the total number of reference books that have been published if the number

of magazines is 0.405 million copies.

Solution Important information

➙ ➙
➙
3
Comics are 10 million less Comics are 21 800 more 0.405 million
than the magazines magazines
than the reference books

Solve

3 million = 3 × 1 000 000 0.405 million = 0.405 × 1 000 000
1 0 10 = 405 000

= 300 000 Use the information to draw a diagram.

Reference + 300 000 + 21 800
books
Comics Magazines
? 405 000

Working − 300 000 − 21 800 = 405 000
backwards
405 000 + 21 800 + 300 000 =

4 05 000
21 800

+ 300 000
726 800

726 800 − 300 000 − 21 800 = 405 000

Check the answer by subtracting 300 000 and 21 800 from 726 800.

The total number of reference books published is 726 800 copies.

If the number of reference books for Bahasa Melayu,
English, Mathematics, and Science is the same, calculate
the number of Mathematics books.

1.3 (iv) Teacher’s Notes 17

The working backwards strategy is used in solving problems and to verify
the answers.
Discuss other appropriate strategies such as creating a systematic list.

4 A total of 1.392 million canned pineapples is exported to five

countries, P, Q, R, S and T. The number of canned pineapples
1
exported to countries P, Q and R is 4 million each. The

number of canned pineapples received by country S is more

than country T. Calculate the possible number of canned

pineapples received by country S and country T respectively.

Solution Do logical reasoning

Given The total number of canned pineapples received by 5 countries is
1
1.392 million. Each country P, Q and R received 4 million canned pineapples.

The number of canned pineapples received by country S is more than country T.

Solve P + Q + R + S + T = 1.392 million

1 million + 1 million + 1 million + S + T = 1.392 million
4 4 4

Remember! 1 million = 0.25 million. 3 million + S + T = 1.392 million
4 4

Therefore, 3 million = 0.75 million. 0.75 million + S + T = 1.392 million
4

0.75 million – 0.75 million + S + T = 1.392 million – 0.75 million
S + T = 1.392 million – 0.75 million
S + T = 0.642 million

Therefore, the possible number of canned pineapples received by
countries S and T are as follows:

Country Possible number of canned pineapples received (million)

S Option A Option B Option C
T 0.342 0.4 0.5
Total 0.242 0.142
0.3
0.642 0.642
0.642

Check Use subtraction to check the answers.

Possible number of canned pineapples received by country T (million)

A B C
0.642 0.642 0.642

− 0.342 − 0.400 − 0.500

0.300 0.242 0.142

How many cans of pineapples could countries S and T possibly
received other than options A, B and C? Discuss.

Teacher’s Notes

Pose more reasoning questions involving units of measurement to assist pupils 1.3 (iv)
18 achieve Performance Level 6.

SELF-PRACTICE

Solve the following problems. State Population
(people)
a The table shows the population in 1 732 000
two states in 2013. Calculate the total Wilayah
Persekutuan 3 428 000
population in both states. Give the Kuala Lumpur
answer in decimal millions.
Sabah

Source: http://www.rurallink.gov.my/c/document_library/get_file?uuid=edecc38f-c183-456d-b503-
a32ab6afad6e&groupId=977333

b Factory P produces 1.4 million bottled orange juice and apple juice. The
number of bottled orange juice is 36 740. Calculate the number of bottled
apple juice produced.

c The government distributes a total of RM3 000 000 to flood victims in a particular
district. A total of 3 000 families received an equal amount. Calculate the total
amount of money received by each family. Give the answer in decimal millions.

d The largest sports stadium in the world, the Rungrado May Day in North
Korea, can accommodate 0.15 million spectators. In a particular sporting
event, the number of adult spectators was the same as the number of
teenage spectators. Calculate the number of adult spectators.

Source: http://www.arsenal4u.com/2012/09/artikel-10-stadium-sukan-terbesar-di-dunia/

e A tree planting campaign was launched in 2010 with the theme “Green

The Earth: One Citizen, One Tree”. The number of trees planted in

2011 was 4 708 482, which was 1 046 727 trees less than in 2010 and

195 798 less than in 2012. Calculate the total number of trees planted

in 2010 and 2012.

Source: http://www.forestry.gov.my/index.php/my/kempen-menanam-26-juta-pokok-2010-2014-3

f A total of 6 1 million tourists visited a resort island from January to April. In
8 February, the number of tourists was
3 1
January and 1 4 million and 1 2 million

respectively. The number of tourists in March was less than in January, but

more than in February. How many tourists possibly visited the island in

March and April?

1.3 (iv) Teacher’s Notes 19
Surf http://www.onlinemathlearning.com/number-to-words.html

SKILFUL MIND

1 Use a calculator to calculate the answers. Round off the answers of c and d

to the nearest thousandth.

a 890 024 – 68 403 × 13 = b 1 621 320 + 240 168 ÷ 24 =

c 243 051 ÷ 47 = d 513 484 ÷ 58 =

2 a What is the number pattern for the number line below? Explain how to

find the value of P using a calculator.


1 204 500 1 205 350 1 206 200 P

b Complete the following number patterns. Use a calculator to find the answers.

i. 9 – 1 = ii. 9 × 9 + 7 =

98 – 21 = 98 × 9 + 6 =

987– 321 = 987 × 9 + 5 =

9 876 – 4 321 = 9 876 × 9 + 4 =

3 List all the prime numbers from 80 to 100.

4 Convert the decimal millions and fraction millions to whole numbers.
5
a 2.37 million = b 4 8 million = c 8.019 million =

5 Write the following numbers as decimal millions and fraction millions.

a 500 000 = b 3 400 000 = c 9 750 000 =

6 Solve the following problems.

a State States in Peninsular Malaysia Sabah and Sarawak

Area 0.132 million km2 66 000 km2 more than
Peninsular Malaysia

Source: http://www.basic-mathematics.com/whole-numbers-test.html

Based on the table above, calculate the total area of Malaysia.

b A factory produces 1.08 million biscuits of various flavours such as

coffee, chocolate, and lemon. The number of coffee and chocolate

flavoured biscuits is 250 000 pieces each. Calculate the number of

lemon flavoured biscuits.

c The number of voters in Polling Station V is 3 times the number
of voters in Polling Station W. The difference in the number of
voters is 0.1 million people. How many voters are in:

i. Polling Station V? ii. Polling Station W?

Teacher’s Notes

20 Provide more of such questions to enhance pupils' understanding of all skills 1.1-1.3
learned in this topic.

2

A Multiplication of Fractions Akim, one part is for you.
Please enjoy the cake.
Multiply proper fractions with proper fractions

1

Let’s cut half
of this cake
into three
equal parts.

Thank you, aunty.
Let’s eat, Eisha.

What fraction of the cake is given to Akim?

1 × 1 = The overlapping shaded
3 2 part is 1 .

1 × 1 = 1 1 1 6
3 2 6 2 3
1 of 1
1 1 1 1 32
3 2 6 6
× =

1 of the cake is given to Akim.
6

If initially, there was 1 1 1 of 1
3 6 23
of the cake, the fraction
1 11
for 2 of the cake would 32 1 × 1 = 1
2 3 6
be as shown in the

following diagram. Why is the product of 1 and 1 the same
3 2
1 1
as the product of 2 and 3 ? Discuss.

2.1 (i) Teacher’s Notes 21

Carry out simulation activities with transparencies and paper foldings to show
the concept of multiplying two proper fractions.

2 Ca lcula te the product of 2 and 3 .
3 4

2 × 3 = 2
3 4 3

Method 1

2 × 3 = 6
3 4 12
3
S implify. = 6 ÷6 4
12 ÷ 6
Shade 3 parts.
= 1 4 • Shade 2 parts.
2 3
This is cancellation
method. • The overlapping shaded

Method 2 parts is 6 .
12
11

2 × 3 = 2 × 3
3 4 3 4
12 BOOST
YOUR KNOWLEDGE
= 1
2 • Multiply the numerator
with the numerator.
2 × 3 = 1
3 4 2 • Multiply the denominator
with the denominator.
Use paper folding to show the
• Give the answer in the
simplest form.

product of 2 and 1 .
3 5

3 3 × 2 = 4
5 7
What is the number
3 × 2 = 3× 2 multiply the numerators sentence for the
5 7 5× 7 multiply the denominators diagram?

= 6
35

3 × 2 = 6
5 7 35

Teacher’s Notes 2.1 (i)
Guide pupils to solve multiplication of proper fractions using diagrams.

22

Multiply proper fractions with mixed numbers

1 Mr Leong has 3 1 packets of fertilizer. He wants to
2
use 2 of the fertilizer to fertilize his hibiscus plant. How

5

many packets of the fertilizer will he use to fertilize the

hibiscus plant?

2 × 3 1 =
5 2
1 Shade 321
2 × 3+ 1 = 2 × 7 Convert 3 2 to an parts. 3 21
5 ×2 5 2
improper fraction. 14
10
Multiply the fraction.
1140 = 152
= 14 Shade 2
10 5

= 1 1 40 ÷÷ 22 Sfriamcptiolifny. the of each

part.

= 1 2 Group the
5 overlapping
shaded
2 × 3 1 = 125 parts.
5 2

Mr Leong will use 1 2 packets of the BOOST
5 YOUR KNOWLEDGE
fertilizer to fertilize his hibiscus plant.
• Convert the mixed numbers
Try solving using the cancellation to improper fractions.
method. Compare the answers
and discuss. • Give the answer in the
simplest form.

Use squared paper as in the

example. Shade to get the

answer for 2 × 1 1 .
3 3

2.1 (i) Teacher’s Notes 23

Remind pupils to identify the numerators and denominators that can be
simplified.
Emphasise the multiplication concept through paper folding activities.

2 2 1 × 2 =
4 5
BOOST
2 1 × 2 = 9 × 2 Shade YOUR KNOWLEDGE
4 5 4 5 2 parts. The diagram for
2 5 the larger fraction
18 ÷ 2 5 is drawn first.
= 20 ÷ 2
18
= 9 2 1 20
1 0 4
 Group the 18
2 1 × 2 = 9 shaded parts
4 5 10
that overlap.
 Shade 2 1 parts.
4

Try solving using
cancellation method.

Multiply mixed numbers with mixed numbers

1 1 1 × 121 =
3

( ) ( ) 1 1 21 1 1 21
1 3 × = 1 × 121 + 3 × +

( ) 1 31 =1+ 1 = 1 1 + 1 × 3 1 × 1 1 1 × 1 1
3 2 3 2 2 3 2

= 1 1 + 3 ÷ 3 +
2 6 ÷ 3
Partition it.

= 1 1 + 1
2 2

=2

1 1 × 121 = 2
3

Teacher’s Notes 2.1 (i)
Surf www.k5learning.com to do multiplication exercises on fractions.

24

2 4 1 × 5 2 =
6 5

59 Simplify 25 and 5.
1 5 52 25 27 Simplify 27 and 6.
4 6 × = 6 × 5

21 Denominator

= 5 × 9 Multiply 5 and 9.
2 1
Multiply 2 and 1. 22 Whole
number
45 Convert the improper fraction 2 45
= 2
to a mixed number. –4

= 22 1 05
2
–4

4 1 × 5 2 = 22 1 1 Numerator
6 5 2

3 191 × = 3 2
3

1 1 × = 3 2
9 3
9
10 × = 11 Multiply with 10
9 3
on the right so that

Multiply 10 with 9 11 3 the equation is
9 10 9 10 11 9
so that only the 10 × 9 × = 3 × 10 balanced.

unknown is on 11 1

the left. = 11 × 3
1 10
BOOST
= 33 YOUR KNOWLEDGE
10
A fraction multiplied by the
= 3130 inverse of the fraction will give
the answer 1.

191 × 3130 = 323 2 × 9 = 18
9 2 18

=1

Calcula te the value o f Q. 3 1 = Q × 2 1
2 2

2.1 (i) Teacher’s Notes 25
Explain how the equation is balanced on the left and right.

SMART PROJECT

Tools/ 1 set of cards with proper fractions, 1 set of Participants 2 pupils in

Materials cards with mixed numbers, transparencies, a group
ruler, coloured pens, stapler

Task Example:

1 Each pupil takes 2 fraction cards and sufficient The first The second
transparencies. fraction fraction

2 Draw a diagram of the same size to represent the first 1 1 1
and second fractions. Colour or shade the parts that 3 2

represent the first and second fractions.

3 Overlap the diagram of the first fraction with the second The product

fraction.

4 Look at the shaded parts that overlap. Write the product

of the fractions. 4 = 2
5 Collect and display your work at the mathematics corner. 6 3

SELF-PRACTICE
1 Multiply.

a 7 × 9 = b 3 × 6 = c 1 1 × 1 =
9 10 4 7 6 2

d 2 1 × 3 = e 9 × 4 3 = f 1 × 8 1 =
7 5 10 4 3 10

g 1 1 × 2 5 = h 3 5 × 4 1 = i 8 5 × 3 1 =
2 8 7 5 6 9

2 Draw and shade the following diagrams to get the answers.

a 4 × 1 = b 1 5 × 2 =
5 2 6 3

3 Draw the diagrams and shade them to get the answers.

a 3 × 5 = b 3 7 × 3 = c 2 1 × 1 4 =
8 9 9 10 3 7

4 Solve the following.

a × 8 = 1 b 5 4 × = 1 1 c 3 3 × = 2 1
9 6 7 2 5 4
26
Teacher’s Notes 2.1 (i)

Guide pupils to carry out the Smart Project. Prepare enough transparencies.
Remind pupils to relate the idea of equivalent fractions to draw diagrams of the
same size for both the fractions.

B Division of Fractions

Divide proper fractions by whole numbers

1 Puan Anis distributes 9 of a Swiss roll equally into
10
3 containers. What fraction of the cake is in each

container?

9 ÷ 3 = Remember! 3 = 3 . Use multiplication
10 and inverse 1 .
3 1
9 9 3 1 to 3
10 10 1
÷ 3 = ÷

9 1
10 31 ÷ 3 1
= × 1 × 3 9
10
= 9×1 ÷ 1 1
10 3
=39 × 1 9 divided by 3
10

10 31 333
10 10 10
= 3
10

9 ÷ 3 = 3 There is 3 of the cake in each container.
10 10 10

2 4 ÷ 8 = 3 Divide 7 by 21.
5 9

4 ÷ 8 = 4 ÷ 8 Convert the divisor to a fraction 7 ÷ 21 =
5 5 1 with a denominator of 1. 9

1 Change the operation and 7 ÷ 21 = 7 ÷ 21
4 1 then inverse the divisor. 9 9 1
= 5 × 8 Do cancellation.

2 = 7 ×
9
= 1 × 1 Multiply the numerators.
5 2 Multiply the denominators.
=
1
= 10

4 ÷ 8 = 1 7 ÷ 21 =
5 10 9

2.2 (i) Teacher’s Notes 27

Explain the inverse relationship between division and multiplication.
Surf https://www.khanacademy.org/math/arithmetic/fractions/
dividing-fractions-tutorial/

Divide proper fractions by proper fractions

1 Cut the wood
into a few pieces,
This is only 3 m. each measuring
4 1 m.
8

How many pieces of wood measuring 1 m were cut by Paramjit’s father?
8

3 m ÷ 1 m =
4 8

Method 1 3 m 3 m = 6 m
4 4 8

= 6 × 1 m
8

1 m 1 m 1 m 1 m 1 m 1 m There are 6
8 8 8 8 8 8 pieces of wood
measuring 1 m.
06 5 4 3 2 1 3 m
4 8

Method 2 BOOST
YOUR KNOWLEDGE
2 Change division
• Change division to
3 ÷ 1 = 3 × 8 to multiplication. multiplication.
4 8 4 1
1 Inverse 1 to 8 . • Inverse the divisor.
8 1 • Simplify the numerator

=6 and the denominator by
cancellation (if any).
3 m ÷ 1 m = 6
4 8

The number of wood measuring 1 m cut by Paramjit’s father is 6 pieces.
8

Teacher’s Notes

Remind pupils about the repeated subtraction method when dividing. Stress
28 that inversion only involves the divisor and the operation. 2.2 (i)

Train pupils to identify the numbers that can be simplified.

2 5 ÷ 1 =
6 3

Method 1 Method 2

5 There are 15 1 1
6 5 3 5 3
parts of 1 . 6 ÷ = 6 × 1
6
2

15 = 5
6 2

grouping = 2 1
2

2 3
6

2 21 65 ÷ 1 = 221
3

Divide mixed numbers by whole numbers

1 179 ÷ 4 = 2 6 2 ÷ 22 =
7

1 7 ÷ 4 = 1 7 ÷ 4 6 2 ÷ 22 = 6 2 ÷ 22
9 9 1 7 7 1

4 2

= 16 × 1 = 44 × 1
9 4 7 22
1 1

= 4 = 2
9 7

1 7 ÷ 4 = 4 6 2 ÷ 22 = 2
9 9 7 7

BRAIN TEASERS

2 4 of pizzas are given equally to 7 children. Will each child get more than
5 the pizza? Prove it.
1 of
5

2.2 (i) Teacher’s Notes 29

Use paper folding and shading diagram methods to solve the division of two
fractions. Emphasise on how to obtain the correct answer.

Divide mixed numbers by proper fractions

1 How many 1 are there in 121 ?
5

121 ÷ 1 = 1 3
5 2 2
1
15
1 21 ÷ 1 = 3 × 5 2
5 2 1
There are a total
= 15 7 21 1 of 15 parts of 1 .
2 2
2
= 7 1 Rearrange the
2 shaded parts

121 ÷ 1 = 721
5

2 1 1 ÷ 3 = Use the common 3 ÷ 1 = 6 3
4 7 multiple of the 3 4
denominators to
1 1 ÷ 3 = 5 ÷ 3 solve this. ÷ 1 = 6 3
4 7 4 7 3 4

= 5 × 7 ÷ 3 × 4 ÷ 1 = 27
4 × 7 7 × 4 3 4

= 35 ÷ 12 × 3 = 27
28 28 1 4

19
1 3 1 27 1
× 1 × 3 = 4 × 3
= 35 × 28
28 1 2 11

1 9
4
=

= 1
4
= 2

= 2 1 ÷ 1 = 6 3
4 3 4

1 1 ÷ 3 = 2 1 ÷ = 3130 . Solve it.
4 7 5

Teacher’s Notes

Carry out enrichment activities such as cross number games.

30 Encourage pupils to check their answers by recalculating using the answers obtained. 2.2 (i)

Surf http://www.themathpage.com/arith/division.htm#ex16 and
http://www.dadsworksheets.com/worksheets/fraction-division.html

BRAIN TEASERS

Replace the same letters with the same digits to make this number sentence
true. Discuss.

a 1 + 1 = 1 ÷ 1 b 1 – 1 = 1 × 1
A B A B C D C D

SELF-PRACTICE

1 Calculate.

a 3 ÷ 1 = b 5 ÷ 2 = c 4 ÷ 3 =
4 2 6 3 9 7

d 5 ÷ 65 = e 7 ÷ 63 = f 1 1 ÷ 24 =
8 10 3

g 2 7 ÷ 1 = h 3 3 ÷ 9 = i 4 4 ÷ 8 =
8 4 7 10 5 9

2 Solve the following.

a How many 1 are there in 2 ? b How many 2 are there in 2 130?
3 9 5

c How many 1 are there in 6 ? d How many 1 are there in 8 ?
2 7 4 9

3 Calculate the quotients horizontally and vertically.

a 3 ÷ 7 b 4 1 ÷ 7 c 1 5 ÷ 4
4 8 2 9 8 5

÷÷ ÷÷ ÷÷

7 ÷ 2 5 ÷ 1 9 ÷ 30
10 7 6 3 10

4 Complete the given number sentences.

a 1 ÷ = 4 b 2 2 ÷ = 4 c ÷ 3 = 5 1
5 15 7 7 5 3

2.2 (i) Teacher’s Notes 31

Focus on questions that are similar to question 4.
Surf http://www.dadsworksheets.com/worksheets/fraction-division.html

C Solve the Problems

1 The recipe to make chocolate chip muffins A Recipe for Chocolate Chip Muffins
3 (12 pieces)
is as shown. 4 cup of chocolate chips is
2 cups of flour

needed to produce 12 muffins. Zara tried 1 cup of granulated sugar
2

the recipe and used only half of the flour. 3 teaspoons of baking powder
How much chocolate chips did Zara use?
1 teaspoon of milk
2

1 cup of cooking oil
3

Solution 1 egg

Give n 43 cup of chocolate chips for 3 cup of chocolate chips
12 muffins. 4

3 tablespoons of white sugar

2 tablespoons of brown sugar

Zara used 1 of 3 cup of chocolate chips.
2 4

Aske d for Ho w much chocolate chips did she use?

Operatio n Multiplication

S olv e 1 × 3 =
2 4

1 × 3 = 1 ×3
2 4 2×4

= 3
8

Check Draw and shade the diagram. 3 .
The overlapping shaded parts is 8

3 1 × 3
4 2 4

1 × 3 = 3
2 4 8

Zara used 3 cup of chocolate chips.
8

Teacher’s Notes 2.3 (i)

Guide pupils to understand the questions carefully and write the steps of the

32 solution according to the Polya method or other suitable methods.

Encourage pupils to draw diagrams to represent the problem.

Damia has 1 m of ribbon. She uses part
2 Dofamthiea hriabsbo1n21 mto odfercibobraotne. Shheer umseosthe31r’osf

tphreesreibnbt obnoxt.oWdheactoirsattheehleenr gmthotohferrib’sbognift
bdoidx.shWehuasteis? the length of the ribbon that
she uses?

Solution

Given Original length of the ribbon is 1 1 m. 1 of it is used.
2 3

Asked for The length of ribbon used

Operation Multiplication Draw and shade the diagram. The overlapping

Solv e 1 × 1 1 m = 1
3 2
shaded parts are 1 × 1 1 m = 3 m = 1 m.
32 6 2

2

1 1 1 × 121 1
2 3 2

Rearrange

1 1 1
CS ehme cak k 13 × 2 3 3
1 = × 2

1

= 1 × 1
1 2

= 1
2

1 × 1 1 m = 1 m
3 2 2

Damia uses 1 m of the ribbon.
2

2.3 (i) Teacher’s Notes 33
Carry out simulation activities with transparencies to multiply fractions.

3 Encik Yusof has a piece of land measuring

2 1 acres. He wants to plant a few types of fruit
4
trees. He decides that the area for each type

of tree is 3 acre. How many types of trees can
4
be planted?

Solution A piece of land of 2 1 acres.
Given 4
3
Each type of tree will be planted over an area of 4 acre.
Asked for
The number of trees that can be planted
Operation
Division

Solve

Me thod 1 Method 2 Trial and error

2 1 acres ÷ 43 acre = 3 acre × ? = 2 1 acres
4 4 4

2 1 ÷ 3 = 9 ÷ 3 Number of 1 2 3
4 4 4 4 types of
trees

31 3 3 1 3 3×3=9
9 4
= 4 × 3 Area 44 × 2 = 24 4
(acres)
11 2 1
4
=3 = 2

3 multiplied by 3 is equal to 2 1 .
44
Therefore, 3 types of trees can be planted.

Check Use a calculator. Press the buttons 9 ÷ 4 × 4 ÷ 3 = 3

2 1 acres ÷ 3 acre = 3
4 4

Encik Yusof can plant 3 types of fruit trees.

Teacher’s Notes

34 Diversify strategies such as drawing a diagram to represent and solve 2.3 (i)
a problem.

When using the calculator, remind pupils to change the operation and inverse

the divisor.

4 Siew Ting has a jug filled with 1 1 ℓ of water. How
2

many glasses with a capacity of 1 ℓ could be filled
8

with water from the jug?

Solution

Given BJwToAahifligajteuh81ng1n1gℓ2u1.o2a1mℓfnℓ1bago2ei1refrdlwℓaoitfsaougftbaelwaenrsrgaisskteeiasprnaiwskdieptuhod81auarℓcleaaydmpaianngcgteoitlydaaasopgbfalea81trsℓdissitiihwspaikaittahdcnuaa n 81 cba ℓep. afilcleitdy


Asked for


Operation Division

Solve Method 1 Method 2 Simulation: Prepare 1 1 ℓ of
2
1 81 ℓ = water in a jug and
1 2 ℓ ÷ a few glasses with a

1 1 ÷ 1 = 3 ÷ 1 1 1 ℓ capacity of 1 ℓ.
2 8 2 8 2 8

4 1 1 1 1 1 1
3 8 8 8 8 8 8 8
= 2 × 1 ℓ ℓ ℓ ℓ ℓ ℓ

1 1 ℓ 1 ℓ 1 ℓ 1 ℓ 1 ℓ 1 ℓ
8 8 8 8 8 8
= 12

3 1 3 12 glasses with a capacity
8 2
Check 12 × ℓ = ℓ of 1 ℓ can be filled.
8
2
= 1 1 ℓ
2

1 1 ℓ ÷ 1 ℓ = 12
2 8

12 glasses with a capacity of 1 ℓ can be filled with
8
water from the jug.

If only half of the volume of water from the jug is poured,

how many glasses with a capacity of 1 ℓ is needed?
8

2.3 (i) Teacher’s Notes 35
Carry out simulation activities to assist pupils understand the concept of division.

5 Only 3 of a durian cake baked by Nina left. The
4

remaining parts of the cake were shared equally

among her friends. Each of her friend got 1 part.
20

How many of her friends had a piece of cake?

Solution

G iv en B2ET1haa0 eckbhirakefhermiake agndidinuairngniaogkntep2k431a0.rbtpsaaohrtfaodgfuiatrhinae.nScceaatkiakeep. woraasng34 r.akan dapat
1
A ske d fo r The number of friends who got 20 part of the cake

Operation Division A is an unknown. This is the number
of friends who got 1 part of the cake.
S olve 43 ÷ A = 1
20 20

3 × 1 = 1
4 A 20

11 1
3 4 1 1 4
4 × 3 × A = 20 × 3

1 1 5

1 = 1 The numerators are the same, 1 = 1.
A 15 Therefore, the denominators are also equal, A = 15.

Therefore, A = 15.

Ch eck 3 ÷ 15 = 3 ÷ 15 Create a story from this
4 4 1 number sentence and
solve it.
1 1
3 15
= 4 ×

5 7 ÷ = 1
8 16

= 1
20

3 ÷ 15 = 1
4 20

15 of Nina’s friends had a piece of cake.

Teacher’s Notes 2.3 (i)
Use the algebraic method to find the unknown.

36

SELF-PRACTICE

Solve the following problems.

a Puan Marziah buys 4 kg of anchovies. She prepares a few dishes using
of the 5
1
4 of the mass anchovies. Calculate the mass of the anchovies used.

b Encik Azhar has 3 1 ℓ of cooking oil. He uses 3 of the volume of the oil to fry
5 8

fish crackers. Calculate the volume of the cooking oil used.

c 1 1 kg of flour is needed for a muruku batter. Puan Kavita needs to prepare
8
2 1 times of the muruku batter for a party. What is the mass of flour
2

needed by Puan Kavita?

d Siew Kheng only has 3 hour left before the end of
4

her Mathematics test. She has 15 more questions

to answer. How long, in hour, does she have to

answer each question?

e Lisa’s mother wants to make mooncakes to be donated to

several charities. She has 5 1 kg of flour. Each mooncake
4 can Lisa’s
1
needs 8 kg of flour. How many mooncakes

mother make?

f Bee Choo brings 3 1 pieces of pancakes to school. During
2

recess, she divides the pancakes equally for her 7 friends.

How much of a pancake will each get?

2.3 (i) Teacher’s Notes 37

Surf http://studentaccessible.com/worksheets/word-problems-for-multiplying-
and-dividing-fractions-worksheet for extra exercises.

SKILFUL MIND

1 Solve these.

a 7 × 8 = b 4 × 1 = c 1 1 × 1 =
8 9 7 2 4 3

d 4 × 1 6 = e 1 7 × 3 1 = f 5 2 × = 36
5 7 10 2 5

2 Calculate.

a 8 ÷ 6 = b 5 ÷ 20 = c 2 2 ÷ 15 =
9 7 8 9

d 4 1 ÷ 9 = e 1 1 ÷ = 15 f 2 ÷ =8
2 10 4 3

3 Write the number sentences for the following diagrams.

a b

×=

4 Solve the following. ÷=

a 3 of a number of pupils took part in hurdles. 1 of them also took part in
10 4

long jump. Calculate the fraction of pupils who participated in long jump.

b Adlina has 2 1 cups of fine sugar. She sprinkles 1 of the fine sugar on
2 3

chocolate biscuits and the remainder on strawberry biscuits. How many

cups of fine sugar were sprinkled on the chocolate biscuits?

c A total of 82 3 kg of rice will be repacked to be donated to 7 welfare
5
homes around Bandar Jaya. What is the mass of rice received by each

welfare home?

d Encik Khairi and Encik Hashim have a rope each. Encik Khairi has a rope

measuring 3 2 m. The total length of their rope is 9 1 m. How many times
3 6
more is the length of Encik Hashim’s rope compared to the length of Encik

Khairi’s rope?

Teacher’s Notes

Provide grid papers to pupils during the multiplication and division of fractions 2.1 - 2.3
38 activities.

3

A Multiply and Divide Decimals

1 I pour 2 bottles of milk
equally into 8 glasses of
1.5 ℓ similar size.

1.5 ℓ

1.5 ℓ

1.5 ℓ

1.5 ℓ

Calculate the volume of milk in a glass.
2 × 1.5 ℓ ÷ 8 =

STEP 1 First, calculate the total STEP 2 Then, divide the answer
volume of milk in the by 8 to find the volume of
1 2 bottles. 0.375 ℓ
8 3 .000 ℓ milk in a glass.
1.5 ℓ –0
× 2
30
3.0 ℓ –24

2 × 1.5 ℓ ÷ 8 = 0.375 ℓ 60
– 56

40
– 40
0

The volume of milk in a glass is 0.375 ℓ.

3.1 (i) Half of the volume of milk in one of the bottles 39
spilled. The remaining milk in the bottle is divided
equally among 3 siblings. How much milk, in ℓ, will
each sibling get? Discuss.

Teacher’s Notes
Conduct a simulation using a graduated container such as a measuring cylinder
or a beverage container to enhance pupils’ understanding.
Conduct reading aloud of the number sentences involving multiplication and
division of decimals with whole numbers.

2 Azim’s family jogs a distance of 17.5 km in a week. They jog the same
distance each day.

A healthy body leads to
a keen mind.

What is the distance jogged by Azim and his family in 9 days?
17.5 km ÷ 7 × 9 =

Divide, then STEP 1 STEP 2
multiply.
2.5 km 4
7 17.5 km 1 decimal place
2 .5 km
– 14 ×9
35
22.5 km
– 35
0 BOOST
YOUR KNOWLEDGE
17.5 km ÷ 7 × 9 = 22.5 km
When multiplying, the number of
decimal places must be the total
number of decimal places in the
numbers that are being multiplied.

The distance jogged by Azim and his family in 9 days is 22.5 km.

If they jogged a distance of 20.4 km in 8 days, can Azim’s family attain a
distance of more than 30 km on the 12th day? Discuss.

BRAIN TEASERS
Move a matchstick so that the number sentence below is correct.

× ÷=

Teacher’s Notes

40 Use other methods such as unitary or proportion to encourage pupils to solve 3.1 (i)

daily problems involving multiplication and division of decimals with whole

numbers.

3 I saved 80 sen a day in

January. Great! Now, you can buy

4 storybooks of equal price.

Calculate the price of a storybook based on the situation above. January
There are 31 days 2016
in January.

31 × RM0.80 ÷ 4 = S MTWT F S

METHOD 1 RM 6.2 0 31 1 2
4 RM2 4.8 0 34 5 6 78 9
2
– 24 10 1 1 12 13 14 15 16
RM0.8 0 08 17 18 19 20 21 22 23
× 31 24 25 26 27 28 29 30
–8
0 80 00 METHOD 2
+ 24 00
RM2 4.8 0 –0 RM0.20
0
31 × RM0.80 = 31 × RM0.20
4

1

= RM6.20

31 × RM0.80 ÷ 4 = RM6.20

The price of a storybook is RM6.20.
BRAIN TEASERS

The Bee Hummingbird is the smallest bird in the world with
a mass of 0.002 kg. The mass of 100 Bee Hummingbirds
is equal to the mass of 2 newborn panda. What is the
mass of a newborn panda, in kg, if each of them has the
same mass?

Source: http://www.factzoo.com/birds/bee-hummingbird-smallest-
bird-in-the-world.html

http://enchantedlearning.com/subjects/mammals/panda/

3.1 (i) Teacher’s Notes 41

Use daily situations such as buying or selling groceries and paying
instalments for furniture, house or car as additional questions to enhance
pupils’ understanding.

4 The table shows the mass of 6 cans Number Mass (kg)
of spaghetti sauce. Find the mass of of cans 2.73 kg
15 similar cans of the spaghetti sauce. 6 cans
?
METHOD 1 15 cans

2.73 kg ÷ 6 × 15 = METHOD 2 6 cans — 2.73 kg
3 cans — 2.73 kg ÷ 2
0.455 kg 1 can — 0.455 kg 1. 365 kg
6 2.730 kg 2 2.7 30 kg = 1.365 kg
–0 15 cans — 15 × 0.455 kg –2
11
27 2 22 07
–24 –6 2.730 kg 6 cans
0.455 kg 2.730 kg 6 cans
33 × 15 13 + 1.365 kg 3 cans
– 30 – 12 6.82 5 kg 15 cans
1
30 10
– 30 2 275 – 10
0 + 04 550 0

6.8 2 5 kg

2.73 kg ÷ 6 × 15 = 6.825 kg

The mass of 15 similar cans of the spaghetti sauce is 6.825 kg.

5 30 × 4.03 ÷ 26 = 6 16.9 hours × (5 ÷ 10) =

4.0 3 4.65 5 ÷ 5 = 1 Do the operation in
× 30 26 120.90 10 ÷ 5 2 the brackets first.

0 00 – 104 16.9 hours × 1 = 16.9 hours
+ 120 9 0 169 2 2

120. 9 0 – 156 8. 4 5 hours
130 2 16.90 hours
– 16
–130
0 09
–8
30 × 4.03 ÷ 26 = 4.65
10

– 10

Divide, then multiply. 0
What is the answer? 16.9 hours × (5 ÷ 10) = 8.45 hours

Teacher’s Notes

Provide a variety of questions and show a variety of calculations such as
42 cancellation or multiplication to get the answer. 3.1 (i)

7 A piece of batik fabric as shown in the
 
diagram is cut into four equal parts to be
framed. Calculate the area, in m2, for each 1.5 m
part of the batik fabric.
3.8 m
METHOD 1 1.425 m2
 

3.8 m × 1.5 m ÷ 4 = 4 5.700 m2 BOOST
–4 YOUR KNOWLEDGE
4
17 The process of multiplication
3.8 m 1 decimal place and division of decimals is
the same as multiplication
× 1.5 m 1 decimal place –16 and division of whole
2 decimal places 10 numbers.
1
–8
190 20

+ 380 – 20
5.70 m2

3.8 m × 1.5 m ÷ 4 = 1.425 m2 0

METHOD 2

Divide the length by 2 and the width by 2.
Then, multiply the quotients.

3.8 m = 1.9 m 64
2
0.75 m
1.5 m = 0.75 m
2 × 1.9 m

11

675

+ 0 750
1.42 5 m2

The area for each part of the batik fabric is 1.425 m2.

8 36.27 × 5.9 ÷ 13 = 16.4 6 1
13 213.9 93

31 3 2 decimal places −13 BOOST
52 6 1 decimal place 83 YOUR KNOWLEDGE

36.2 7 3 decimal places − 78 The decimal places for
× 5.9 59 the product is the total
number of decimal
1 326 4 3 −52 places of the two
79 numbers multiplied.
+181 3 5 0
−7 8
21 3.9 9 3

36.27 × 5.9 ÷ 13 = 16.461 13
−1 3

Teacher’s Notes 0

3.1 (i) Emphasise the position of the decimal places for the product is based on the 43
total number of decimal places of the numbers multiplied.
Surf http://www.k5learning.com/free-math-worksheets/sixth-grade-6
decimals-division