Core activity 12.1: The decimal system
Resources: Place value grid photocopy master (chapter 1, p7). (Optional: 0–9 digit
cards photocopy master (CD-ROM).)
Display the Place value grid (used in chapter 1). Point to the digits that make up the
number 0.46.
“What is this number? Write it down. What is the value of the 4? What is the value o
Establish that there are many ways of saying/writing the number (examples are in the ta
Repeat for other numbers.
Write the number 0.46 on a place value grid. H T U• t
0 •4
Demonstrate to learners how you multiply 0.46 by 10. (The value of each digit is 10×
Show on grid:
H T U• t h 0.46
0 •4 6 0.46 × 10 = 4.6
4 •6
Multiply by 10 again:
HT U• t h 0.46
4 0 •4 6 0.46 × 10 = 4.6
4 •6 4.6 × 10 = 46
6•
Establish that multiplying by 10, then multiplying by 10 again, is equivalent to multip
100
0.46 × 100 = 46
110 Unit 2A 12 Decimals
LB: p46
t cards photocopy master; Multiplication and division loop
Ways of saying and writing 0.46
of the 6?” Saying Writing
able to the right).
zero point four six 0.46
h zero point four and zero point zero 0.4 + 0.06
6 six
× greater)
four tenths and six hundredths 46
+
10 100
forty six hundredths 46
100
Look out for!
Learners who may need reminding of column
headings. Encourage them to draw tables
whenever they need to.
plying by
Repeat for division by 10 and 100.
38 ÷ 10 = 3.8
38 ÷ 100 = 0.38
Emphasise the need for 0 as a place holder in the units column.
H T U• t h
3 8•
3 •8 8 38
0 •3 38 ÷ 10 = 3.8
38 ÷ 100 = 0.38
Repeat for other numbers.
Ask learners to look at these numbers:
348 36 3.65 34.8 3.48
“Which numbers is the smallest? How do you know?” (Answer: 3.48)
“Which number is the largest? How do you know?” (Answer: 348)
“Which number is 3.7 rounded to the nearest tenth?” (Answer: 3.65)
“Which number is 3 rounded to the nearest whole number?” (Answer: 3.48)
“What is 3.48 rounded to the nearest tenth?” (Answer: 3.5)
Emphasise that the number of digits in a decimal number does not determine its
example, 3.65 is NOT greater than 36.
36 = 30 + 6
3.65 = 3 + 0.6 + 0.05
s value, for
Core activity 12.1: The decimal system 111
Summary
Learners consolidate work on the number system to include working with decimals u
to two decimal places.
Notes on the Learner’s Book
The decimal system (p46): gives learners an opportunity to practise working with dec
The questions are not presented in any particular order and learners are expected to u
prior knowledge, for example, in question 2, reference is made to inequality signs. It
important that learners are able to make connections like this but, if they find it chall
teachers may suggest that they work with a partner.
More activities
Rounded or not (individuals or pairs)
You will need newspapers with articles involving numbers.
Collect examples of numbers used in news items or reports. Say whether each numbe
Smallest number (pairs)
You will need 0–9 digit cards photocopy master (CD-ROM).
Shuffle the digit cards and lay them face down on the table.
Player one takes three digit cards at random and uses them, together with the decimal
a zero before the decimal point so 0.49, for example, is not allowed.
Player two does the same.
The player with the smallest number wins the round.
Multiplication and division loop (whole class)
You will need Multiplication and division loop cards photocopy master (CD-ROM)
There are 16 cards in the set and all must be used. The game can be played with a cla
learners; or with a smaller group where a learner holds more than one card. To start,
expression reads the whole of their card. Play continues until the loop is formed.
Games Book (ISBN 9781107667815)
Decimal in-between (p28) is a game for two players. It focuses on ordering numbers a
112 Unit 2A 12 Decimals
up Check up!
cimals. “What is the value of each digit in the numbers 1.04 and
use 13.79?”
is
lenging, “Write the number six tenths and four hundredths as a
decimal.”
“Which is larger, 4.2 or 0.428? How do you know?”
er is exact or has been rounded.
l point card, to make the smallest possible number. The number must not have
).
ass where two learners hold one card between them; or with a group of 16
a learner says the second part of their card and the learner with the equivalent
and recognising a decimal number in-between two given numbers.
Blank page 113
Core activity 12.2: Operations with decimals
Resources: (Optional: 0–9 spinner (CD-ROM).).
Present the problem:
“The school cook is preparing school dinners. She weighs 7.8 kg of rice. She needs 1
how much more rice she needs.”
Allow thinking time, then discuss learners’ methods which may include:
7.8 + 0.2 gives 8, then 2 more makes 10
+0.2 +2
7.8 8 10
7.8 + = 10 so 10 − 7.8 =
10. 10.0
− 7.8 − 7.8
Put a 0 after the decimal point so that both numbers show the same number of decim
Remind learners that it is useful to be able to recall decimal facts to 10:
3.2 + = 10 (Answer: 6.8)
10 − = 6.3 (Answer: 3.7)
Repeat the activity but this time with two numbers that have two decimal places, addi
cook has 0.23 kg of sugar but needs 1 kg. How much more does she need?”
Again, collect the learners’ methods, which may be similar to those above.
Remind learners that it is also useful to be able to recall two place decimal facts to 1:
0.27 + = 1 (Answer: 0.63)
1 − = 0.71 (Answer: 0.29)
114 Unit 2A 12 Decimals
10 kg. Work out LB: p48
Teaching point!
Work with decimal calculations to 1 and 10 mentally
for a short time as a ‘lesson starter’.
Look out for!
Learners who may forget to carry the 1 in situations
where the units add to 9 and the tenths add to 1.
Show them some examples,
such as:
6.3 + 3.7 = 10 (6 + 3 = 9 and 0.3 + 0.7 =1)
or
4.8 + 5.2 = 10 (4 + 5 = 9 and 0.8 + 0.2 = 1)
mal places.
ing up to 1: “The
:
Now ask learners to work out an addition involving numbers with different numb
places:
“How much rice and sugar together does she have to begin with?” (i.e. What is 7
Collect learner’s methods.
Now present a new problem:
“The cook has 7.8 kg tomatoes but she needs twice that amount. How much does
To help with this question we could think about what double 78 is. Ask learners
out. Discuss the learners’ different methods:
double 80 = 160 so double 78 is 4 less
double 70 is 140, double 8 = 16 so double 78 = 140 + 16
jottings: 78
140 16
156 (Answer: double 78 ÷ 10 = 156
“How, then, could I work out double 7.8?”
“How could I work out double 0.78?” (Answer: double 78 ÷ 100 = 1.56)
Summary
Learners add and subtract decimals using an appropriate method that may incl
working mentally, with jottings or using more formal methods.
They find doubles of decimal numbers by initially doubling a whole number, t
dividing by 10 or 100.
Notes on the Learner’s Book
Operations with decimals (p48): the learner book provides a variety of questions
related to addition and subtraction of decimals. They include straight-forward ca
missing numbers and examples using mathematical language including total and
difference. Question 9 requires learners to reiterate the fact that the units add to 9
tenths add to 1 and question 10 is presented as a puzzle.
bers of decimal
7.8 kg + 0.23 kg?)
s she need?”
to work this
6 ÷ 10 = 15.6) Look out for!
Encourage estimation when deriving facts. Double
7.8 is approximately double 8 which is 16.
lude Check up!
then “Find the missing number in these number sentences:
s mainly 2.7 + = 10 10 − 8.3 = ”
alculations,
d “If I know that double 79 is 158, how can I find double 7.9?
9 and the
What is double 0.79? Explain your method.”
“A bdul sets out a calculation like this:
7.4
- 3.68
What advice would you give him?”
Core activity 12.2: Operations with decimals 115
More activities
Spinner subtraction
(pairs)
You will need 0–9 spinner (CD-ROM).
Player one spins the spinner three times and makes a number with two decimal places
larger number. Player two does the same.
The winner of the round is the player with the smaller answer.
The overall winner is the player who wins most rounds.
116 Unit 2A 12 Decimals
es. Repeat to make a second number. Subtract the smaller number from the
Blank page 117
Core activity 12.3: Decimals in context
Resources: Money word problems photocopy master (p120).
Challenge learners to write as many metric units as they can in two minutes.
Collect results, tabulating them by length, capacity and mass to include at least:
Length Capacity Mass
millimetre
centimetre millilitre gram
metre litre kilogram
kilometre
Underline ‘kilo’, ‘centi’ and ‘milli’.
Explain that: ‘kilo’ means thousand (from Greek)
‘centi’ means hundredth (from Latin)
‘milli’ means thousandth’ (from Latin).
Use these to build up conversions, for example 1 litre = 1000 millilitres
Write a measurement on the board and discuss the value of the digits, for example, 4.
“H ow many whole litres in this?” (Answer: 4)
“W hat is the 7 worth?” (Answer: 7 or 0.7 of a litre)
10
“H ow many millilitres are the same as 4.75 litres?” (Answer: 4750 millilitres)
Demonstrate on place value chart.
Th H T U • t h
4 •7 5
4 7 5 0• 4.75
4.75 × 1000
118 Unit 2A 12 Decimals
LB: p50
.75 litres:
Learners work in pairs to solve the money problems. Each problem is presented on
Learners must find the question and the information required to solve it. Some infor
redundant and must be discarded.
(Answers: A = $4.38, B = $8.50, C = $3.00, D = $15.15)
Review answers.
Suggest that learners follow guidelines if they are unsure of what to do:
read the question carefully
pick out important words and phrases
calculate the answer
check that the answer makes sense in the context of the problem.
Summary
Learners apply knowledge to ‘real life’ situations involving money and measu
They develop problem solving skills.
Notes on the Learner’s Book
Decimals in Context (p50): the investigation invites learners to use trial and imp
solve a problem. Encourage them to work systematically, showing their working
Questions 1 to 10 involve measures in everyday life and questions 11 to
18 involve working with money. The questions are not in order of difficulty. If le
them challenging suggest that they work in pairs.
More activities
Measures diary (individual)
Keep a diary of the occasions you use money or measures over a period of a we
write two sentences explaining your results.
n seven cards.
rmation is
ures. Check up!
provement to “How many centimetres are the same as 1.25 metres?
g clearly. How many millimetres?”
earners find
“A book shop is offering $2.25 off a book priced at
$15.50. What is the new selling price?”
eek. Write down what you do each time. At the end of the week look back and
Core activity 12.3: Decimals in context 119
Money word problems
A A box of four balls costs $2.96 B Motor boats can be hired for
$1.50 for 15 minutes.
A Dinesh and Gopal buy three boxes B Rowing boats can be hired for
of balls between them. $2.50 for an hour.
A Dinesh pays $4.50 B Mr Wood takes David and Andrew
to the boating lake.
A Dinesh and Gopal are both 11 B David goes on a motor boat for
years old. one hour.
Dinesh and Gopal share the cost B Andrew goes on a rowing boat for
one hour.
A of the balls but Dinesh pays more
than Gopal.
A Dinesh and Gopal enjoy playing Mr Wood pays for David and
tennis, cricket and football. B
Andrew to go on the boating lake.
A How much does Gopal pay for the How much does Mr Wood pay
balls? B to hire the boats for David and
Andrew?
C A shop sells three types of candle: D The price of an adult’s theatre
plain, star and stripe. ticket is $17.95
C Plain candles cost $0.35 each. D The price of a child’s theatre ticket
is $8.45
C Star candles cost $0.60 each. D Zina takes two children to the
theatre.
C Stripe candles cost $0.85 each. D Zina pays for the tickets with a
$50 note.
There is a special offer at the shop,
D Zina and the two children go to
C ‘Buy 10 candles and get $0.50 off’. the afternoon performance.
Sapna chooses four star candles D The theatre seats 240 people.
C and two stripe candles. Josh D How much change does Zina get
from her $50 note?
chooses ten plain candles.
How much does Josh pay for ten
C plain candles at the special offer
price?
Instructions on page 118 Original Material © Cambridge University Press, 2014
2A 13 Positive and negative
numbers
Quick reference
Core activity 13.1: Positive and negative numbers (Learner’s Book p)
Learners revise work from previous stages, extending it to find differences betwe
negative numbers and between two negative numbers.
Prior learning Objectives* – please note that listed objectives
taken as a whole
This chapter builds on work
in Stage 5 where learners 2A: Numbers and the number sy
ordered and compared 6Nn11 – Order and compare positive numb
positive and negative
numbers on a number line 2A: Calculation (Addition and subtr
or temperature scale. They 6Nc13 – Find the difference between a posi
calculated rises and falls in
temperatures. as temperature or on a number line
2A: Problem solving (Using unde
6Ps3 – Use logical reasoning to explore an
6Ps6 – Make sense of and solve word pro
diagrams or on a number line; use
*for NRICH activities mapped to the Cambridge Primary
Vocabulary
positive negative zero
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
Positive and negative numbers Vocab ulary
een positive and Let’s investigate positive: a po sitive
T he differe nce between tw o nu mber s is 3 . number is greater
One n umber is 2. tha n zero.
Wha t c ou ld the o ther n umb er b e ? negative: a negative
number is les s than
Draw a number linee.. zero.
We us e a − sign to
show a negative
1 Wh ich numbers are marked w ith crosses o n num be r.
the number lines ?
zero: is ano ther na m e
for ‘n oth in g’ or
(a) ‘n oug ht’.
5
0 5
On a number line
(b) it is the p oin t where
50 0 50 numbers change from
positive to
2 The table shows the min imum temperature on negative.
four day s. negative po sitive
Day Te m pe r at ure ° C num ber s nu mber s
M on day 2 10 0 10
T u es d ay 1
W e d ne s d ay 3
T h ur s day 4
Write the temperatures in order, starting with the co ldes t
te m pe ra ture .
3 Write each se t of temperatures in order, starting w ith the co ldes t
te m pe ra ture .
(a) 4 °C 1 °C 8 °C 2 °C 3 ° C (b) 2
°C 4 °C 7 °C 1 3 °C 13 ° C (c) 6 °C 6
°C 0 °C 7 °C 4 ° C
52
s might only be partially covered within any given chapter but are covered fully across the book when
ystem
bers to one million, negative numbers to an approximate level.
raction)
itive and negative integer, and between two negative integers in a context such
e.
erstanding and strategies in solving problems)
nd solve number problems and puzzles.
oblems, single and multi-step (all four operations), and represent them e.g. with
e brackets to show the series of calculations necessary.
objectives, please visit www.cie.org.uk/cambridgeprimarymaths
Unit 2A 121
Core activity 13.1: Positive and negative numbers
Resources: My time line photocopy master (p124). Blank number lines photoco
NOTE: Before working on the core activity, ask learners to find out dates of signific
starting ten years before they were born and ending at the present time. These may in
personal dates such as births of family members or wider events such as scientific d
signif icant events locally or globally.
Display the My time line photocopy master or similar for the whole class to see.
Mum and dad get My brother I go to
married is born secondary school
10 0 10 20
I start
schoo
l
“This is part of my time line. 0 represents the year I was born and every division repr
one year.
How old was I when my brother was born? (Answer: two)
How long had my mum and dad been married when I was born? (Answer: five ye
How long had my mum and dad been married when my brother was born?” (Answ
Establish how to label the points on this line. Mark −5, 5 and 15.
Ask learners to use their own information to make a personal time line. If they know any
interesting facts these can also be added. You may choose to provide blank number line
Learners work in pairs. They sit ‘back to back’. One learner describes their time line, def
positive and negative numbers so the other can replicate it. Compare, then swop roles.
122 Unit 2A 13 Positive and negative numbers
LB: p52
opy master (p125). Vocabulary
cant events
nclude positive: a positive number is greater than zero.
discoveries,
negative: a negative number is less than zero.
resents We use a − sign to show a negative number.
ears)
wer: seven years) zero: is another name for nothing or nought. On a
number line it is the point where numbers change
y other from positive to negative.
es for support.
fining the negative numbers positive 10
numbers
10 0
Look out for!
Learners who have difficulty drawing a number
line.
They need to consider the following:
how to draw the number line (length, number
of divisions)
how to place numbers on the line
how to give clear instructions using the words
positive, negative and zero.
Look out for!
Learners who say minus when they should say
negative. Always use the correct vocabulary for
numbers, for example −7 is ‘negative 7’.
Display one of the learner’s time lines and ask questions, for example:
“H ow old was Ahmed when he started school?
Which two events were six years apart?
What is the difference in age between Ahmed’s brother and sister?”
Ensure that you work across zero to include positive and negative numbers. Inv
questions that can be answered from the time line. You could extend this wor
lines from history.
Show a number line marked in ones from −10 to +10. This time ask questions th
context free, for example:
“What is the difference between -5 and +3? (Answer: 8)
What is the difference between −5 and −2?” (Answer: 3)
Summary
Learners use positive and negative numbers in and out of context.
They use number lines con dently including using them to find differences be
negative numbers and between two negative numbers.
Notes on the Learner’s Book
Positive and negative numbers (p52): the investigation requires learners to consid
‘difference’ and possibly draw a number line to help solve the puzzle. Note that a
be positive.
This is a topic where many learners find a context useful. The examples are a mi
and out of context.
More activities
Making loop cards (individual)
Here are two cards from a set of ‘follow me’ cards.
I am −4 Which number is 3 more than −1? I am +2 Which numbe
Make eight more similar cards that, together with these cards, would form a loop
must be −2.
(Try your game out with a group of friends.)
Games Book (ISBN 9781107667815)
Order five (p31) is a game for two players. It focuses on ordering and comparing
Opportunities for display!
Make a poster of the learners’ time lines.
vite learners to ask
rk to include time
hat are
etween positive and Check up!
der the word “G ive me two temperatures that lie between
a difference will always 0 °C and −10 °C. Which one is warmer?”
ixture of questions in
“If the temperature was −8 °C and it increases
by ten degrees what is the new temperature?”
er is 2 less than 0?
p. The answer to the last question must be −4 and the answer on the rst card
g positive and negative numbers.
Core activity 13.1: Positive and negative numbers 123
My time
Mum and dad get My brother
married is born
10 0
I star
schoo
Instructions on page 122
e line
I go to 20
secondary school
10
rt
ol
Original Material © Cambridge University Press, 2014
Blank nu
10 0
10
Instructions on page 122
umber lines 20
10
0 10
Original Material © Cambridge University Press, 2014
126 Blank page
2A 14 Multiples, factors and men
Quick reference
Core activity 14.1: Common multiples (Learner’s Book p54)
Learners extend their understanding of multiples and factors to include common
multiples.
Core activity 14.2: Mental strategies for addition and subtraction (Learner’s
Learners develop and re ne mental strategies for addition and subtraction and lea
the most appropriate and efficient strategy, depending on the numbers involved.
Core activity 14.3: Mental strategies for multiplication (Learner’s Book p58)
Learners develop and re ne mental strategies for multiplication and learn to use
most appropriate and efficient strategy.
Prior learning Objectives* – please note that listed objectives might only
fully across the book when taken as a whole
This chapter builds
on, and refines work 2A: Numbers and the number system
done previously. 6Nn6 – Find factors of two-digit numbers.
Learners should 6Nn7 – Find some common multiples (e.g. for 4 an
know what multiples
and factors are. 2A: Calculation (Mental strategies)
Can enter data into a 6Nc4 – Use place value and number facts to add o
Carroll diagram.
subtract three-digit multiples of 10 and pair
6Nc6 – Add/subtract a near multiple of 10, 100 or 1
3127 + 4998, 5678 – 1996.
2A: Calculation (Multiplication and divisio
6Nc8/6Nc14 – Multiply pairs of multiples of 10, e.g. 30 ×
6Nc15 – Multiply near multiples of ten by multiplying
6Nc16 – Multiply by halving one number and doubli
6Nc17 – Use number facts to generate new multipli
tables.
6Nc22 – Know and apply the arithmetic laws as the
terms commutative, associative or distribut
2A: Problem solving (Using techniques a
6Pt1 – Choose appropriate and efficient mental or
addition, subtraction, multiplication or divis
2A: Problem solving (Using understandin
6Ps1 – Explain why they choose a particular meth
*for NRICH activities mapped to the Cambridge Primary objectives,
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
ntal strategies using them
Mult iples and factors Vocab ulary Mental strategies for add it io n and Mental strateg ies fo r mu ltip licat ion
subtraction (1)
Let’s investigate common multiple: a There are ma ny way s in wh ic h y o u ca n us e ta b le
A lig h t f la sh es every f o ur min u te s a n d a be ll r ing s multiple tha tis shared 1 Here are some examples of calc ulatio n strategies. Some are facts to he lp y o u mu ltip ly lar ger num ber s. Vocab ulary
every five min ute s. by two or more
n If the lig h t f la sh es a s the be ll r ing s a t the sam e time, numbers. For example, correct but s ome are no t. Reme m ber : a lway sas k 1 2 3 45 6 7 8 9 near multiple of 10 :
ho w lon g w ill it be u ntil th is ha p pe ns a ga in ? k y o urse lf, ‘Ca n I do 10 a number either s ide
s Book p56) • 1 2 is a commo n Say whether each example is correct or no t. If it is 1 1 2 3 45 6 7 8 9 of a multiple of 10.
earn to use T hin k a bo u t th e multiple of 2 an d incorrect, give a correct strategy . th e ca lc ulatio n 10
multiple s of 4 a n d.5 3 because 12 is a menta lly ?’ before 2 2 4 6 8 10 1 2 1 4 16 1 8 For example, 20 is
) 5 multiple of both 2 2 0 a multiple of 1 0 s o
the 1 Fin d two factors of 24 that total 11. and 3. wr itin g a ny th in g
19 an d 21 are near
• 12 is a common
multiple of 6 an d 4 2 Copy and complete the s pider diagrams to s how other facts tha t docwann .be derived 3 3 6 9 12 15 18 21 24 27 30 mu ltiple s of 1 0.
because 12 is a
multiple of both 6 from the fact in the centre of the d iagram. 4 4 8 12 16 20 24 28 32 36 40
and 4.
Calc ulation St r ate g y 5 5 10 15 20 25 30 35 40 45 50
• 12 is a common 3456 2000 3
multiple of 2, 3, 4 (a) 3456 1997 427 200 1 6 6 12 18 24 30 36 42 48 54 60
and 6. ( b) 4 2 71 9 9 4865 300 1
(c) 4865 299 4824 3000 3 7 7 14 21 28 35 42 49 56 63 70
9843 8000 3
( d) 4 8 2 4 2 9 9 7 8 8 16 24 32 40 48 56 64 72 80
(e) 9843 7997
730 – 280 = 450 9 9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100
2 Choo se three of the se numbers to make a correct number sentence. 21 2 2 2 3
24 2 5 2 6 27 28 29
? ? ? a multiple of 1 0 63 + 28 = 91 73 – 28 = 45 1 Here is a method for calcu lating 15 7 : 10 7
3 Vijay is th in kin g o f a num ber. 6.3 + 2.8 = 9.1 70
It is unde r 20. 5 7 35
It is a mu ltip le of 15 7 105
3. It is a Use this method to wor k out:
multiple of 5 .
(a) 18 7 ( b) 19 5 (c) 13 7 (d) 17 9
( g) 16 8 (h) 15 5
(e) 17 8 (f) 18 9
What number is Vijay thin kin g of?
56
54 58
y be partially covered within any given chapter but are covered
e
nd 5). Vocabulary
or subtract two-digit whole numbers and to add or common multiple
rs of decimals, e.g. 560 + 270, 2.6 + 2.7, 0.78 + 0.23. near multiple
1000, or a near whole unit of money, and adjust, e.g.
on)
40, or multiples of 10 and 100, e.g. 600 × 40.
g by the multiple of ten and adjusting.
ing the other, e.g. calculate 35 × 16 with 70 × 8.
ication facts, e.g. the 17× table from the 10× + 7×
ey apply to multiplication (without necessarily using the
utive).
and skills in solving mathematical problems)
r written strategies to carry out a calculation involving
sion.
ng and strategies in solving problems)
hod to perform a calculation and show working.
, please visit www.cie.org.uk/cambridgeprimarymaths
Unit 2A 127
Core activity 14.1: Common multiples
Resources: Carroll diagram photocopy master (p134). Robot steps photocopy ma
Display a Venn diagram and ask, “Can you give multiples of 4 mu
me a number that would fit in the box?”
Establish that the number, for example, 20 is a 4 15
multiple of 4 and a multiple of 5. We say 20 is a
common multiple of 4 and 5. 4 and 5 are factors
of 20.
Work with learners to place other numbers on the
diagram. “What type of number would be placed in the
box, but not in either of the circles?”
(Answer: a number that is not a multiple of 4 or 5.)
Give learners a copy of the Blank four cell Carroll diagram photocopy master or get
draw a Carroll diagram themselves. Ask learners to label the diagram as shown below
work in pairs to place the numbers 1 to 20 into the correct cells.
multiples of 2 not multiples of 2
multiples
of 5
not multiples
of 5
As a class, discuss the characteristics of the numbers in each part of the diagram. “W
numbers are multiples of 2 and multiples of 5?” (Answer: 10 and 20.)
Reinforce that 10 and 20 are common multiples of 2 and 5.
Learners to work in pairs on the investigation Robot steps.
128 Unit 2A 14 Multiples, factors and mental strategies using them
aster (p135). LB:
ultiples of 5 p54
5
Vocabulary
common multiple: a multiple that is shared by two or
more numbers.
12 is a common multiple of 2 and 3 because 12 is a
multiple of both 2 and 3.
12 is a common multiple of 6 and 4 because 12 is a
multiple of both 6 and 4.
12 is a common multiple of 2, 3, 4 and 6.
t them to Look out for!
w and then
Learners that need further practice of sorting
Which these numbers in to Carroll diagrams or
identifying common multiples. Remind them
that multiples of two are even numbers and that
multiples of f i ve always end in a five or a zero.
Get them to point to some multiples of two and/or
five on a 100 grid.
Learners who finish the activity quickly. Change
the headings of the Carroll diagram to multiples
3/5, not multiples 3/5 and so on.
Review activity emphasising that the three heels are in line again at 30 when:
Little has taken 15 steps
Robo has taken 10 steps
Large has taken six steps.
30 is a common multiple of 2, 3 and 5.
2, 3 and 5 are factors of 30.
Summary
Learners extend their understanding of multiples and factors to include common
Notes on the Learner’s Book
Multiples and factors (p54): the learner’s book provides a miscellany of items re
and factors. Questions 3, 4 and 8 relate specifically to common multiples. Each
is presented in a different format including using Carroll diagrams, so linking w
handling curriculum.
n multiples. Check up!
“Give me some common multiples of three and seven.”
elated to multiples
of these questions
with the data
Core activity 14.1: Common multiples 129
Core activity 14.2: Mental strategies for addition and
Resources: (Optional: 0–9 spinner (CD-ROM.)
Ask learners to calculate the answers to these questions and discuss their strategies wi
63 + 29 = (Answer: 92)
290 + 630 = (Answer: 920)
2.9 + 6.3 = (Answer: 9.2)
0.63 + 0.29 = (Answer: 0.92)
6300 + 2900 = (Answer: 9200)
Ask learners how they worked out 63 + 29 and collect various methods which may in
(60 + 20) + (3 + 9)
use or imagine a number line
count on from 63, the larger number
63 + 30 then subtract 1.
Say that 29 is a near multiple of 10 (30 − 1). When we add or subtract a near multip
100 or 1000, it is often easier to add/subtract the multiple then adjust.
Example:
5678 − 1996 (mentally or with jottings).
4 2000
3678 3682 5678
Practise with different numbers.
Now look at the other calculations from the beginning of the lesson. Review work do
the connection between the calculations, “Once you know the answer to the first one
all the others are 92 multiplied or divided by 10, 100 or 1000.”
Display an addition fact, for example, 17 + 4 = 21
Ask learners for the three associated facts they can get from rearranging the addition:
4 + 17 = 21 21 − 4 = 17 21 − 17 = 4
130 Unit 2A 14 Multiples, factors and mental strategies using them
d subtraction LB: p56
ith a partner: Vocabulary
nclude: near multiple: a number either side of a multiple
of 10. For example, 20 is a multiple of 10 so 19 and 21
are near multiples of 20.
ple of 10,
one, stressing A reminder.
as 92, then inverse: opposite in effect.
The inverse of adding 9 is subtracting 9.
: The inverse of multiplying by 5 is dividing by 5.
Ask learners for any addition/subtraction facts that they can deduce from the fac
4 = 21, like they did in the first task. E.g. we can multiply 21 − 4 = 17 by 10 to
40 = 170
Allow thinking time then collect and display facts, for example:
Addition Subtraction
17 + 4 = 21 21 − 4 = 17
170 + 40 = 210 210 − 40 = 170
1700 + 400 = 2100 2100 − 400 = 1700
1.7 + 0.4 = 2.1 2.1 − 0.4 = 1.7
0.17 + 0.04 = 0.21 0.21 − 0.04 = 0.17
and so on and so on
Practise with different numbers.
Summary
Learners develop and re ne mental strategies for addition and subtraction.
They use mental, jottings or formal methods depending on the numbers involv
Notes on the Learner’s Book
Mental strategies for addition and subtraction (1) (p56): the learner book contai
examples only as the core activity is about using mental strategies.
More activities
Sums and differences
(pairs) need 0–9 spinner (CD-ROM).
You will
Players spin a 0–9 spinner four times, then create the largest possible number. Th
produce the score for the round.
The first player to pass a score of 100 wins the game. For example:
The cards: Are rearranged to:
4607 7640
7640 − 1997 = 5643
Score 5 + 6 + 4 + 3 = 18
ct that 17 + Look out for!
o get 210 −
Learners who think that subtraction can be done in any
ved. order. Show them, with examples, that addition can be
ins a few done in any order but subtraction cannot.
17 + 4 = 4 + 17
17 − 4 ≠ 4 − 17
This work relates to mental strategies, it is best to
revisit mental strategies frequently in small amounts
of time.
Check up!
“If you know 13 + 8 = 21 how can you nd the
answer to:
1300 + 1.3 + 0.8?”
“H ow8d0o0e?s knowing 17 + 6 = 21 help you to
calculate 0.17 + 0.06?”
hey subtract 1997 from their number and then add the digits of the answer to
Core activity 14.2: Mental strategies for addition and subtraction 131
Core activity 14.3: Mental strategies for multiplicatio
Resources: Table square photocopy master (p136). (Optional: 0–9 spinner (CD
Ask the class to work out 13 × 50. After some time for working, ask, “How did you w
out?” Take some suggestions from the class, for example:
13 × 100 = 1300 and 1300 ÷ 2 = 650
13 × 5 = 65, then 65 × 10 = 650
“Using 13 × 50 = 650 what else can you work out?”
13 × 51 = 650 + 13 = 663
13 × 49 = 650 − 13 = 637
51 and 49 are near multiples of 50.
Practise working out other related facts.
For example, start with 27 × 40 then work out 27 × 41 and 27 × 39.
Display the table square. Talk to the learners about what it shows.
Work with learners to identify relationships, for example:
4× table is double the 2× table,
8× table is double the 4× table and so on.
Ask, “How can I use the 10× table and the 7× table to give the 17× table?”
“I want to know 8 × 17 I know that 8 × 10 = 80
8 × 7 = 56
so 8 × 17 = 80 + 56 = 136”
Work with other similar relationships, for example 8 × 19 using the 9× and 10× tables
Ask learners to work out 7 × 8? (Answer: 56)
“What is 70 × 8?” (Answer: 560)
“How about 80 × 7?” (Answer: 560)
(Remind learners that multiplication can be done in any order so 7 × 8 = 8 × 7)
132 Unit 2A 14 Multiples, factors and mental strategies using them
on LB: p58
D-ROM).) As this work is related to mental strategies it is best
work it done frequently, but in small amounts. Once the
strategies are learned they should be practised
regularly.
s.
“How can I use the result, 70 × 8 = 560, to help work out 35 × 16?”
(Answer: double one number and halve the other number to give the same
answer.)
70 × 8 = 560
so 35 × 16 = 560
Practise with other similar examples.
Summary
Learners extend their store of mental strategies for multiplication concentratin
known facts to work out other facts quickly.
Learners use a range of strategies including using multiples of 10 then adjusti
near multiples of 10, doubling one number and halving the other and combini
table facts.
Notes on the Learner’s Book
Mental strategies for multiplication (p58): the learner book exemplifies mental
for multiplication and challenges learners to practise using the strategies.
More activities
Mental multiplication
(pairs)
You will need 0–9 spinner (CD-ROM).
Players spin a 0–9 spinner three times and use the digits to make a 2-digit numbe
write down the answer.
Check the result with your partner and discuss the best strategy to work out the a
Games Book (ISBN 9781107667815)
Mathematical bingo (p31) is a game for the whole class or a group. It focuses on
ng on using Check up!
ing to multiply by
ing “If I know that 27 × 60 = 1620 how can I work out
27 × 59? …. and 27 × 61?”
strategies
“If I know that 13 × 10 = 130 and 13 × 7 = 91 what
is 13 × 17? How did you work out your answer?”
er and a 1-digit number. Multiply the numbers together using any method and
answer.
n mental facts and strategies, not just for multiplication.
Core activity 14.3: Mental strategies for multiplication 133
Blank four cell C
Instructions on page 128
Carroll diagram
Original Material © Cambridge University Press, 2014
Robot steps
A number investigation
Here are three robots, Little, Robo and Large.
They make footprints on the floor.
Little has size 2 boots.
Robo has size 3 boots. They are 1.5 times larger than Little’s boots.
Large has size 5 boots. A size 2 boot and a size 3 boot together is the
same length as a size 5 boot.
They start with the heels of their boots in a line.
They each walk heel to toe.
When will all three robots be in line again?
Explain your answer.
Instructions on page 128 Original Material © Cambridge University Press, 2014