M) and a set of Decimal cards (CD-ROM).
practice in finding pairs of decimals, with two decimal places, that
Blank page 221
Core activity 24.2: Multiplication and division
Resources: Related facts photocopy master (p225). (Optional: 0–9 spinner (CD-R
Mental strategies are best developed when learners are able to discuss strategies and p
others. The activities that follow should be done as a class, and discussion and collabo
of ideas should be encouraged. Count in threes starting at 0, then count in steps of 0.
Complete a chart to show the results as you go.
×12345678
3 3 6 9 12 15 18 21 24
0.3 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4
“Use the table to help you work out:”
4 × 0.3 = × 6 = 1.8 2.4 ÷ 3 =
(Answers: 1.2, 0.3, 0.8)
“If 3 × 7 = 21 what will 0.3 × 7 be?” (Answer: 2.1)
“If 15 ÷ 3 = 5 what will 1.5 ÷ 3 be?” (Answer: 0.5)
Remind learners that if they know one multiplication or division fact they can find ot
0.3 × 6 = 1.8
3 × 6 = 18
3 × 0.6 = 1.8
18 ÷ 3 = 6 1.8 ÷ 3 = 0.6
Work in pairs on part one of the activity sheet Related facts.
Review the work done.
Say that, “We can extend these ideas to doubling and halving.”
double 24 = 48 double 2.4 = 4.8
222 Unit 3A 24 Mental Strategies
LB: p91
ROM).)
problem-solve with
oration
.3 starting at 0.
8 10
27 30
2.7 3.0
ther related facts.
half 24 = 12 half 2.4 = 1.2
Work in pairs on the second part of the activity sheet Related facts.
Review the work done.
Ask learners to look at the first column of the last table:
“How can I tell that all the numbers are divisible by 2?” (Answer: the numbers
“Which numbers are also divisible by 4? How can you tell?” (Answer: 28, 48 an
Summary
Learners ref i ne mental methods for multiplication and division with an emphas
decimals.
Notes on the Learner’s Book
Mental strategies for multiplication and division (p91): learners solve multiplica
calculations, including ‘empty box’ problems. These should be used as examples
or whole class discussion about mental methods. The Let’s investigate puzzle inv
relationship between multiplication and division to work out the value of each sy
as a class activity learners should discuss the possible ways of finding
the solution and as such develop strategies that can be used mentally.
More activities
Decimal multiplication
(pairs)
You will need a 0–9 spinner (CD-ROM).
Each player draws the game board:
.× =
Each player spins the spinner three times, placing a number in one of the boxes e
The winner of the round is the player with the largest answer.
Play ten rounds.
s are all even.)
nd 64. When halved the number is still even.)
sis on Check up!
ation and division “If I know that 40 ÷ 8 = 5 what related decimal
s to encourage group multiplication and division facts can you work out?”
volves using the
ymbol, when done “What is the missing number × 7 = 2.1?”
“What is double 3.6? What is half of 3.6?”
each time. They then complete the calculation.
Core activity 24.2: Multiplication and division 223
Addition and subtraction 1
1. 6 + 8 + = 21
2. 30 + 50 + = 140
3. 57 + 43 + = 126
4. 30 + 80 + 50 = 210
5. 25 + 13 + 52 + = 169
Addition and subtraction 2
1. 4.6 + 3.7 + 2.4 =
2. 4.8 - 3.7 =
3. 0.7 + 0.35 =
4. 5.8 + 4.9 =
5. 1.6 + 1.7 =
Instructions on page 218 Original Material © Cambridge University Press, 2014
Related facts
Part 1: Multiplication and division
Complete each column in the table. The first one has been started for you.
Fact Related facts 0.6 × 3 =
0.3 × 6 =
1. 3 × 6 = 18
2. 7 × 5 =
3. 6 × 8 =
4. 5 × 9 =
5. 7 × 8 =
6. 6 × 7 =
Complete the table. The first one has been started for you.
Fact Related fact
1. 36 ÷ 4 = 9 3.6 ÷ 4 =
2. 72 ÷ 8 =
3. 32 ÷ 8 =
4. 49 ÷ 7 =
5. 24 ÷ 6 =
6. 56 ÷ 8 =
Instructions on page 222 Original Material © Cambridge University Press, 2014
Part 2: Doubling and halving
Complete these tables. The first one in each table has been started for you.
Fact Related fact
double 1.4 = 2.8
1. double 14 = 28
2. double 24 =
3. double 32 =
4. double 49 =
5. double 35 =
6. double 27 =
Fact Related fact
half of 3.2 =
1. half of 32 = 16
2. half of 28 =
3. half of 16 =
4. half of 30 =
5. half of 84 =
6. half of 96 =
Instructions on page 222 Original Material © Cambridge University Press, 2014
3A 25 Addition and subtraction
Quick reference
Core activity 25.1: Addition and subtraction (2) (Learner’s book p92)
Learners have an opportunity to revisit objectives covered earlier in the stage, esp
subtraction of decimals. Teachers should use their knowledge of learners when d
unit needs to be covered in its entirety.
Prior learning Objectives* – please note that listed objectives mi
fully across the book when taken as
This chapter revises
objectives covered earlier in 3A: Calculation (Addition and subtr
the stage. 6Nc12 – Add or subtract numbers with the sa
amounts of money.
6Nc13 – Find the difference between a positiv
integers in a context such as temper
3A: Problem solving (Using techniq
6Pt1 – Choose appropriate and efficient me
involving addition, subtraction, multip
*for NRICH activities mapped to the Cambridge Primary ob
Vocabulary
No new vocabulary.
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
pecially the addition and Add it ion and subtraction
deciding whether or not the
Let’s investigate
Arran ge the dig its 0 , 1, 2, 3 , 4, 5 , 6 a n d 7 to ma ke tw o n umber s w ith tw o
dec im al p lac es s o tha t:
(a) the s um of the n um bers is a s c lo se a s po s s ib le to 4 0
( b) th e d iffere nce be tween the nu mber s is a s clo se as p o s sib le to 10 .
T he zer o m u s t no t be p lac ed in the te n s or hu ndre dth s po s itio n.
? ? ● ? ? a nd ? ? ?● ? U se d igit .
cardss.
1 For each pair of numbers nd :
(i) the larger number
(ii) the d ifference between the numbers.
(a) 5 and 1 (b) 4 an d 6 (c) 9 a nd 2
(d) 5 an d 4 (e) 2 and 12 (f) 0 an d 6
2 Complete these calculations.
(a) 14.8 5. 6 (b) 13. 26 17. 64 (c) 4 5.83 31. 04
52. 75 (f) 70. 34 49.7 8
(d) 56 .1 2 6.6 (e) 68 .63
3 There are three bags for sa le in a sh op.
Bag A – $16 .50 Bag B – $13. 35 Bag C – $1 1.8 0
Mira buy s bag B and bag C. How
much did s he spen d ?
92
ight only be partially covered within any given chapter but are covered
s a whole
raction)
ame and different numbers of decimal places, including
ve and negative integer, and between two negative
rature or on a number line.
ques and skills in solving mathematical problems)
ental or written strategies to carry out a calculation
plication or division.
bjectives, please visit www.cie.org.uk/cambridgeprimarymaths
Unit 3A 227
Core activity 25.1: Addition and subtraction
(2)
Resources: Temperature chart photocopy master (p230). (Optional: 0–9 spinners (
Display the Temperature chart, showing temperatures at different times during three d
“Draw a number line from -10 °C to +10 °C for each day. Mark on the four temperatures.
one with the time of day.”
Ask questions related to the chart/number lines, for example:
“What was the lowest temperature?
What was the rise in temperature between 6:00 am and midday on Tuesday?
What was the difference in temperature between 6:00 pm and midnight on Monday
(Answers: -6 °C, 4 °C, 3 °C)
Model finding the difference between two temperatures on a number line:
difference between a positive and negative integer, for example +4 and 3
7
-5 0 +5
-3 -4
difference between two negative numbers, for example -1 and 4.
3
-5 0 +5
4 -1
Remind learners that we can use a number line to help us add and subtract decimal nu
including amounts of money.
228 Unit 3A 25 Addition and subtraction
LB: p92
(CD-ROM). Decimal subtraction grids (CD-ROM).)
days. Look out for!
s. Label each Learners who do not use the correct vocabulary for
negative numbers, for example:
y?”
–7 (a number) is ‘negative seven’
–7° C (a temperature) is ‘minus seven’.
Encourage them to repeat with the correct word.
umbers,
Example: Add 31.8 to 3.47 (remember we can write 31.8 as 31.80 to avoid confu
with decimal places).
Subtract: 31.8 − 3.47
Remind learners that there are other ways of doing these calculations and they sh
always choose the most efficient method they can cope with.
Summary
Learners consolidate earlier learning about addition and subtraction of decima
and negative numbers.
They apply the rules to temperatures.
Notes on the Learner’s Book
Addition and subtraction (p92): the learner book provides a variety of examples
forward, some in context and some with a strong problem solving element.
More activities
Decimal subtract (pairs)
You will need 0–9 spinners (CD-ROM). Decimal subtraction grids (CD-ROM)
Players take turns to spin the spinner and each player places the digit on their gri
bottom tens number is larger than the top tens number, players should spin again
When the grids are complete each player finds the difference between their two n
can vary the game so the player with the smallest answer wins the point.
Games book (ISBN 9781107667815)
Find the total (p44) is a game for two players. It provides an opportunity for lear
usion +3 +0.2 +0.27
hould 31.80 34.80 35 35.27
3
0.07 -0.4
28.33 28.40 28.80 31.80
als Check up!
s, some straight “Give me two temperatures between 0 °C and -10 C.
Which is the colder? What is the difference between
them?”
).
id (check that the top number is larger than the bottom number – generally, if the
n).
numbers. The player with the largest answer wins the point for the round. You
rners to practise adding decimals with different numbers of decimal places.
Core activity 25.1: Addition and subtraction (2) 229
Temperatu
6:00 am midday
Monday 3 °C 8 °C
Tuesday 0 °C 4 °C
Wednesday 1 °C 2 °C
Instructions on page 228
ure chart midnight
6:00 pm
1 °C 4 °C
2 °C 6 °C
0 °C 5 °C
Original Material © Cambridge University Press, 2014
3A 26 Multiplication and division
Quick reference
Core activity 26.1: The laws of arithmetic (Learner’s book p94)
Learners understand that multiplication can be done in any order (commutative law
be partitioned and reordered (associative law). Though they do not use the name th
the distributive law and how it helps with mental calculations so, for example, if t
tables they can work out the 17 times table.
Core activity 26.2: Fractions and division (Learner’s book p96)
Learners understand the relationship between fractions and division and recognis
which a division problem may be presented. They divide by single digit and 2-di
Prior learning Objectives* – please note that listed objectives might only be
are covered fully across the book when taken a
This chapter
builds on work 3A: Calculation (Multiplication and division)
in Stage 5 6Nc17 – Use number facts to generate new multiplicati
where learners
were introduced 10× + 7× tables.
to brackets as a 6Nc19 – Divide three-digit numbers by single-digit num
way of ordering
operations. remainder and to divide three-digit numbers by
Learners
understand and including sums of money.
have used the 6Nc20 – Give an answer to a division as a mixed numb
relationships
between the 4, 5, 10 or 100).
operations 6Nc21 – Relate f i nding fractions to division and use th
of addition,
subtraction, including several tenths and hundredths of qua
multiplication 6Nc22 – Know and apply the arithmetic laws as they ap
and division.
necessarily using the terms commutative, asso
3A: Problem solving (Using techniques and s
6Pt1 – Choose appropriate and eff i cient mental or w
calculation involving addition, subtraction, mul
6Pt5 – Estimate and approximate when calculating e.
3A: Problem solving (Using understanding a
6Ps1 – Explain why they choose a particular method
working.
*for NRICH activities mapped to the Cambridge Primary objectives, ple
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
n (3)
The laws o f arith met ic Fractions and d iv is ion
Let’s investigate 3 4 12 is a lloewd, d , Let’s investigate 2of 15 50 o1f 20 3 of 20
U se a ny of the nu mber s 1, 2, 3 a n d 4 3 5 2 4
to ge ther w ith brac ke ts a n d the opera tion b ut 1 2 3 is no t Fin d a ll the ca lc ulation s 30 3 60 4
sig n s to ma ke a s ma ny nu mber s fr om 1 1 th at ha ve a n a ns wer 20 60 45
to 2 0 as y o u c an . of 1 0. 2 45 5
1 1 14 3
Ho w ma ny n um bers ca n y o u ma ke u s in g allowe d. T his is an e xam ple o f of 45 of 3 0 of 6 0 20 2
1, 2 , 3 a n d 4 in the ca lc ulation ? u sin g a ll f our nu mber s: 3 3 1
w) and that multiplications can 50 5 4
hey realise the importance of of 50
they know the 7 and 10 times (4 2) ( 3 1) 1 6 5
ise different ways in The order of operations is important in daily life. 1 Wor k o ut the calculations. Give y our answer as a mixed number.
igit numbers. WWhhiicchh ddoo yyoouu ddoo r st, crasttc,hcathtcehstchhe schobuols or wa ke up ?
Re me m ber: (a) 227 4 ( b) 42 9 7 (c) 5 25 9
br ac kets
rst ! (d) 38 9 5 (e) 3 15 6 (f) 45 9 8
2 Wor k ou t the calculations. Give y our answer as a decimal.
(a) 491 4 (b) 37 5 2 (c) 4 68 5
1 Wor k ou t the answers to the follow ing: 3 Answer these ques tio ns mentally .
(a) How many forties are there in four hu ndred?
(a) (3 4) 6 ? (b) 7 (1 1 6) ? ? (b) What is two third s of ninety n ine ?
(c) ( 14 7) 1 3 ? (d) ( 2 3) ( 4 − 1) (c) Four times a number is four hundred. What is the number?
(d) What is the remainder when 2 8 is divide d by ve?
(e) ( 14 6) ( 3 1) ? (f) (2 7 9) 4 ? (e) Six times a number is three tho usan d. What is the n umber?
(f) What is o ne fth of a tho usan d?
2 Are the following statements tr ue or false? (g) When a number is div ide d by seven, the answer is tw o, remainder two.
Exp lain y our decisio ns to y our partner. (a) (3 What is the number?
6) 2 3 (6 2) (h) What is three quarters of forty four?
(b) 6 (8 2) 6 8 2 (i) Halve twenty ve.
(c) (7 2) 3 7 (2 3) (j) Div ide ve hundred by twenty ve.
(d) (3 4) 5 3 (4 5)
(e) 5 3 5 (5 3) 5 4 Calculate. ( b) 78 2 1 7 (c) 7 77 2 1
(a) 630 14 (e) 6 96 1 2 (f) 858 22
(d) 8 55 1 9
94 96
e partially covered within any given chapter but
as a whole
ion facts, e.g. the 17× table from the
mbers, including those leaving a Vocabulary
y two-digit numbers (no remainder)
No new vocabulary.
ber, and a decimal (with divisors of 2,
hem as operators to find fractions
antities.
pply to multiplication (without
sociative or distributive).
skills in solving mathematical problems)
written strategies to carry out a
ltiplication or division.
.g. use rounding and check working.
and strategies in solving problems)
to perform a calculation and show
ease visit www.cie.org.uk/cambridgeprimarymaths
Unit 3A 231
Core activity 26.1: The laws of arithmetic
Resources: (Optional: Target number cards (CD-ROM); Order matters game card
Ask learners to work out these calculations:
3+4×5 (3 + 4) × 5
Allow thinking time, then take responses:
3 + 4 × 5 = 23 (convention says that we do multiplication before addition)
(3 + 4) × 5 = 35 (convention says that we deal with brackets first)
“We can use brackets to show stages in thought processes, for example, I want to mul
12 × 40
= 12 × (4 × 10) think of 40 as 4 lots of 10
= (12 × 4) × 10
We know that we can multiply in any order. So we can move the brackets here and mu
× 4 first, making the multiplication easier to calculate. So multiply 12 by 4 then mult
result by 10.”
= 48 × 10
= 480
“Work through some similar examples:
1. 16 × 50 2. 21 × 30 3. 73 × 20
6. 40 × 38”
4. 54 × 70 5. 38 × 40
(Answers: 800, 630, 1460, 3780, 1520, 1520)
Discuss methods with learners.
“What is interesting about the answers for questions 5 and 6? Why are these an
((Answer: The order of multiplication does not affect the answer) – this is the co
“We can extend the idea of multiplying in any order to help us multiply by numbers
know the 10× table and the 7× table. We can separate 17 into 10 and 7 and multipl
12 separately, then add the products together.”
12 × 17
= 12 × (10 + 7)
= (12 × 10) + (12 × 7)
= 120 + 84
= 204
232 Unit 3A 26 Multiplication and division (3)
d (CD-ROM).) LB: p94
ltiply 12 by 40: Look out for!
ultiply the 12 Learners who may need reminding of a way of
tiply the remembering the conventions for order of
operations. An acronym that may help learners
remember is:
BIDMAS
brackets
indices
division
multiplication
addition
subtraction
nswers the same?” NOTE: Learners are not required to deal with indices
ommutative law.) at this stage.
For reference only: Index (indices)
s like 17 where we
ly each number by A small number placed to the upper right of a number
which shows the number of times the start number is
multiplied by itself, for example 43 = 4 × 4 × 4
For teacher reference – the laws of arithmetic as
they apply to multiplication. Learners are not
expected to know the names of these laws.
Example of commutative law
65 × 78 = 78 × 65
Example of the associative law
8.4 × 60 = 8.4 × (10 × 6) or (8.4 × 10) × 6
Continues on next page ...
“Work through some similar examples:
1. 16 × 17 2. 19 × 17 3. 14 × 17
4. 14 × 19 5. 18 × 19”
(Answers: 272, 323, 238, 266, 342)
Summary
Learners understand and use mathematical conventions for calculation.
Notes on the Learner’s Book
The laws of arithmetic (p94): the Learner’s Book stresses the importance of foll
mathematical conventions emphasised in questions 2, 3 and 4. Question 7 focuse
distributive law to help mental calculation.
More activities
Target number (whole
class)
You will need a set of Target number cards (CD-ROM).
Shuffle the 100s and the units cards and randomly choose six cards: two multiple
100 500 3 6 8 9
Display the cards or write them larger on the board. Ask a learner to give you an
Allow five minutes for learners to use the cards with any mathematical operation
necessary to use all the cards but no card can be used more than once.
Example: (100 × 3) + (8 × 9) - 6 = 366 (a difference of two)
Order matters (whole class)
You will need Order matters game card (CD-ROM) a 0–9 spinner (CD-ROM).
Teacher uses a spinner to generate a set of random numbers. As each number is c
boxes are filled each player works out the four answers and finds the total. The w
lowing Example of distributive law
es on the 36 × 97 = 36 × (100 − 3)
= (36 × 100) − (36 × 3)
= 3600 − 108
= 3492
Check up!
“Why do these calculations give different answers?
(3 × 4) + 6 3 × (4 + 6)
What are the answers to the calculations?”
“H ow could you multiply 16 × 9 using your
knowledge of the 10× and 6× tables?”
es of 100 and four single digits, for example:
ny 3-digit number. This is the target number, for example, 364.
ns and brackets to make a number as close to 364 as possible. It is not
.
called out players place it in one of the boxes on their ‘game card’. When all the
winner is the player with the highest total.
Core activity 26.1: The laws of arithmetic 233
Core activity 26.2: Fractions and
division
Resources: (Optional: two 1–6 dice.)
Remind learners that earlier in the stage we divided by single digit numbers. We are g
extend this to dividing by two-digit numbers.
Involve learners in working through the example 560 ÷ 24 using the guidance below:
Move on to talking about different ways of writing divisions and the answers to divis
Stages for division
Start by multiplying 24 by multiples of 10 to get an estimate. As 24 × 20 = 480 and 24
answer lies between 20 and 30. Then use the repeated subtraction method,
24 560
– 480 24 ×
80
–72 24 ×
8
Answer: 23 remainder 8
Then show learners how to calculate the answer using the long division algorithm wh
digits of the answer are recorded above the line,
23
24 560
–480
80
–72
8
Answer: 23 remainder 8
“What can you say about these expressions?”
5 28 28 ÷ 5 28 1 of 28.
5
5
234 Unit 3A 26 Multiplication and division (3)
LB: p96
going to
:
sions.
× 30 = 720, the
here the
Look out for!
Learners who think that a large denominator makes a
large number. Take two groups of four counters.
Ask the learner to split the first group into two equal
groups (halves) and the second into four equal groups
(quarters). Let them see that the group for a half (2) is
bigger than the group for a quarter (1). Show also that
three-quarters (3) is bigger than a half and that the size
of a fraction depends on a combination of the
numerator and denominator.
(Answer: they are equivalent. Finding 1 of a quantity is the same
5
as dividing by 5.)
“What is the result of dividing 28 by 5?” Learners discuss in pairs. Review the an
below as a class.
5 remainder 3
53 (Ensure that learners recognise this as a mixed number.)
5
5.6
Ask, “Which is larger, 3 of 24 or 2 of 30?” Learners discuss in pairs. Review the
43
given below as a class.
1 of 24 = 24 ÷ 4 = 6 so 3 of 24 = 6 × 3 = 18
4 4
1 of 30 = 30 ÷ 3 = 10 so 2 of 30 = 10 × 2 = 20
3 3
so 2 of 30 > 3 of 24
34
“We can use this method to find a fraction of any number.”
Summary
Learners understand the relationship between fractions and division and recog
different ways in which a division problem can be presented.
They divide by single digit and 2-digit numbers.
Notes on the Learner’s Book
Fractions and division (p96): the Learner’s Book covers all the forms of division
wealth of practise questions in and out of context.
nswers given
e answers
gnise Check
n and provides a
up“!What is the result of dividing 48 by 5? Give your
answer in different ways.”
“Divide 636 by 12 and explain your method.”
Core activity 26.2: Fractions and division 235
More activities
Fraction dice (pairs)
You will need two 1–6 dice.
Learners take turns to roll the dice. When a learner rolls two different numbers they m
that fraction of 60. The answer is their score for the round. If they roll a double they
Play five rounds. The player with the largest total is the winner.
Find the largest and smallest answers (individuals or pairs)
Choose any five digits.
Arrange them to make a division calculation HTU ÷ TU
Work out the answer.
Try other arrangements using the same five digits.
Which arrangement gives the largest answer?
Which arrangement gives the smallest answer?
Games Book (ISBN 9781107667815)
Finding fractions of numbers (p44) is a game for two players. It focuses on improving
a number is the same as dividing the number by 10 and 3 of a number can be fou
10
236 Unit 3A 26 Multiplication and division (3)
make a proper fraction (numerator is smaller than the denominator) and find
score 60 points for that round.
g understanding of how fractions relate to division. For example, 1 of
und by dividing by 10 and then multiplying by 3.
10
3A 27 Fractions
Quick reference
Core activity 27.1: Fractions (Learner’s book p98)
Learners revise and consolidate previous learning. They recognise equivalence a
Core activity 27.2: Mixed numbers and improper fractions (Learner’s book p
Learners convert between mixed numbers and improper fractions.
Prior learning Objectives* – please note that listed objective
book when taken as a whole
This chapter builds on work
in Stage 5 where learners 3A: Numbers and the number s
ordered fractions and 6Nn21 – Compare fractions with the same
worked on the equivalence. 6Nn22 – Recognise equivalence between
Learners have changed 6Nn24 – Order mixed numbers and place
improper fractions to mixed
numbers and become 6Nn25 Change an improper fraction to a
familiar with the vocabulary,
including numerator and 6Nn26 – Reduce fractions to their simples
denominator.
3A: Problem solving (Using und
6Ps6 – Make sense of and solve word pr
e.g. with diagrams or on a numbe
*for NRICH activities mapped to the Cambridge Primar
Vocabulary
equivalent fraction simplify cancel improper fraction mixed number
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
Fractions Vo c abula r y Mixed nu mb ers and improper fractions
Let’s investigate 1 equivalent fractions: Let’s investigate Voca bula ry
“A ll fractio ns tha t are e q uivale nt to ha ve a are equal in value.
For example, Ro ll tw o 1 – 6 dice a n d us e th em to ma ke mixed number:a
d en o m in ato r th at is an ev en n u m b er .” 2 an im pro per fractio n. If y o u r oll a ‘d ou b le ’ wh ole number a nd a
2 y ou will have a w ho le n umb er. proper fraction
Inve stiga te w he th er th is stateme n t is c orrec t? 3 Cha n ge the im pro per fractio n s to m ix ed ceoxmambipnleed, 1. 4Fo.r3
6 num be rs.
and simplify fractions.
p100) Wr ite a b ou t y o ur f in d ing s . 5 10 Inve stiga te h o w m any differ en t improper fraction: an
1 Complete the equivalent fractions in the se 2 mixe d numbers can be made . improper fraction is a
fraction where the
spider diagrams. cancel or s implify : Be sy setma tici.c. numerator is greater than
16 30
means to reduce the or equa l to the
6 2 4 10 numerator and denominator. For number.
43 23
denominator of a 1 Cha nge each improper fraction to a mixed
fraction to the e xa m ple , 5 3( ve
6 s ma lle s t numbers 5
13 7 11
poss ible. 41 (a) (b) (c) 5 thirds) and (three
30
For example, 1 2 2 4 5 3
11 3 thirds) are imopprer
9 (f)
(d) (e) 8 fractio ns.
24 40 24 12 7
15 20 2 Change each mixed number to an improper fraction.
(a) 1 1 (b) 2 1 2 2 2 1
(c) 3 (d) 4 (e) 5 (f) 4 6
2 Simplify the se fractions by cancelling. 4 3
3 5 3
14 2 6 ? 24 2
(a) ( b) (c) ?
3 Four p izzas are each cut into quarters.
21 ? 30 5 36 ? (a) How many pieces are cut?
4 (b) Five pieces are eaten. How much p izza is lef t?
15 ? 15 3 12
(d) (e) (f)
20 4 25 ? 16
3 Write these fractions in the s imple st form. Write y our answer as an improper fraction and a mixed
num be r.
14 9 15 9
(a) (b) (c) (d) 4 What are the two mis sing mixed
numbers on th is number line? 1 2 3
28 36 45 15
4
10 0
98
es might only be partially covered within any given chapter but are covered fully across the
system
e denominator and related denominators, e.g. 3 with 7.
48
fractions, e.g. between 1 s, 1 s and 1s.
100 10 2
between whole numbers on a number line.
a mixed number, e.g. 17 to 21.
88
st form, where this is 1, 1, 3 or a number of fifths or tenths.
42 4
derstanding and strategies in solving problems)
roblems, single and multi-step (all four operations), and represent them
er line; use brackets to show the series of calculations necessary.
ry objectives, please visit www.cie.org.uk/cambridgeprimarymaths
Unit 3A 237
Core activity 27.1: Fractions
Resources: Strips of paper. (Optional: Ordering fractions cards (CD-ROM); 1–
Show a strip of paper 12 squares long. Say that the strip represents a whole one and
establish that each square represents 1.
12
Fold in half lengthwise then open up. Establish that 1= 6
2 12
Repeat to show 1 = 3
4 12
Explore other equivalent fractions with strips of different lengths.
Look at a fraction family (when all fractions are equivalent) such as, 4 , 3, 2, 1
12 9 6 3
Establish that 1 shows the fraction that is reduced to its simplest form and that this
3
achieved by cancelling the common factors.
Show that it is correct to simplify fractions by cancelling to the lowest terms in succe
stages but it is more efficient to spot the highest common factor and cancel to lowest
single stage.
Single step: 4 = 1 (cancelling by a factor of 4 from numerator and denominator).
12 3
Multiple steps: 4 = 2 (cancelling by a factor of 2) then 2 = 1 (cancelling by a fa
12 6 63
Present the following problem and allow time for learners to discuss.
How much pizza do you want?
12 1 3
4 3 12 6
238 Unit 3A 27 Fractions
LB: p98
–9 digit cards photocopy master (CD-ROM).)
may be Vocabulary
essive equivalent fractions: are equal in value.
t terms in a Example:
×2
3 6
5 10
×2
cancel or simplify: means to reduce the numerator
and denominator of a fraction to the smallest numbers
possible.
Example: 6 → 1
12 2
Focus on identifying common factors when cancelling
(reducing fractions to their lowest terms).
actor of 2).
Review answers, presuming that they like pizza and want the most possible (you
change to something different).
Make sure that the following points are covered:
it is the relationship between the numerator and denominator that fixes the siz
fraction
2 is the largest fraction
3
3 is equivalent to 1
62
it is easier to compare fractions when they have the same denominator.
To make it easier to see which is the highest fraction you could change all the de
12, so we find the equivalent fraction for each that has 12 as its denominator:
1= 3 2= 8 1 3= 6
12
4 12 3 12 6 12
Summary
Learners understand the relationships between fractions.
Th ey recognise when fractions are equivalent
Th ey use a common denominator to compare and order fractions and simplify
when it is appropriate to do so.
Notes on the Learner’s Book
Fractions (p98): the Learner’s Book provides practice dealing with equivalent fr
questions 5 and 6 learners are required to use their knowledge of equivalence to
problems.
may need to Look out for!
ze of the
Learners who think that the fraction with the largest
enominators to denominator has the largest value. Ensure that
learners focus on the relationship between
numerator and denominator rather than the size of
the numerator and denominator. This could lead
them to mistakenly say, for example, that 10 is greater
12
than 5 because 10 is greater than 5 and 12 is greater
than 66.
Check up!
y fractions “Here are three equivalent fractions:
ractions. In 12 4
o solve
5 10 20
Could 3 be in this set? How do you know?”
15
“Give me a fraction equivalent to 34with a
denominator of 12. How did you work it out?”
Core activity 27.1: Fractions 239
More activities
Ordering fractions (whole
class)
You will need Ordering fractions cards (CD-ROM).
A number of learners are given a card with a fraction on it. Ask them to arrange them
then display the cards on a number line.
Using fraction notation (whole
class)
You will need 0–9 digit cards photocopy master (CD-ROM); do not use the ‘0’ card
Each learner has a set of digit cards. Explain that you are going to ask questions that
one card above the other. Ask questions such as:
“What is the largest fraction you can make?”
“Show me a fraction equivalent to one-half. Can you show me a different answer?”
Games Book (ISBN 9781107667815)
Equivalent fractions dominoes (p48) is a game for two or four players. Gives practice
240 Unit 3A 27 Fractions
mselves in order. Equivalent fractions stand behind each other. Discuss
d.
may involve fractions and show how fractions can be made by holding
”
e in identifying equivalent fractions.
Blank page 241
Core activity 27.2: Mixed numbers and improper frac
Resources: (Optional: two 1–6 dice or spinners (CD-ROM).)
Draw a number line on the board.
012 3
Ask learners, “Where would you place the following fractions?”
3 15 2 1 21
8 8
88
Ask, “How did you work out your answers?”
“Which is larger 118or 118? How do you know?”
Revise converting improper fractions to mixed numbers:
11 11 ÷ 8 = 1 remainder 3 or 13
88
Show how to convert a mixed number to an improper fraction:
181 = 1+ 1 = 88+ 1 8= 9
8
8
Practise other examples.
“Discuss in pairs how to order these mixed numbers:”
134 132 1172
242 Unit 3A 27 Fractions
ctions LB: p100
Vocabulary
mixed number: a whole number and a proper fraction
combined.
Example: 13
4
improper fraction: an improper fraction is a fraction
where the numerator is greater than or equal to the
denominator.
Examples: 5 (five thirds) and 3 (three thirds)
33
aimreproper fractions.
Look out for!
Learners who do not use the correct terminology.
Reinforce: mixed numbers and improper fractions
(not ‘top heavy’ fractions).
Review different ways of solving the problem:
mark fractions on a number line divided into twelfths
1 2
1172 1 2 134
3 19
1182 12
change to improper fractions, then find equivalent forms
13 = 7 = 21 12 = 5 = 20 1 7 = 19
4 4 12 3 3 12 12 12
Summary
Learners change mixed numbers to improper fractions and vice versa.
They understand when it is useful to work in different forms.
Notes on the Learner’s Book
Mixed numbers and improper fractions (p100): gives an opportunity to practise
mixed numbers and improper fractions and to order mixed numbers on a number
More activities
Make the larger number
(pairs)
You will need two 1–6 dice or spinners.
Learners each roll both dice and make a whole number (roll a ‘double’) or an im
wins the round.
NOTE: 1, 2, 3, 4, 5 and 6 are all factors of 60 but depending on the two fraction
2
converting Check up!
r line. “Place these fractions in order of size, explaining your
method as you work:”
13 12 15
438
mproper fraction. They compare the fractions. The learner with the larger number
ns it may be possible to use a smaller common denominator.
Core activity 27.2: Mixed numbers and improper fractions 243
244 Blank page
3A 28 Fractions, decimals and pe
Quick reference
Core activity 28.1: Fractions and decimals (Learner’s book p101)
Learners extend their knowledge of fractions and decimal. They start to convert
fractions to decimals.
Core activity 28.2: Percentages (Learner’s book p102)
Learners extend their knowledge of fractions and percentages. They start to conv
fractions to percentages and vice versa.
Prior learning Objectives* – please note that listed o
covered fully across the
This chapter builds on work done
in Stage 5 where learners began to 3A: Numbers and the nu
work with equivalences of fractions, 6Nn23 – Recognise and use the eq
decimals and percentages. 6Nn27 – Begin to convert a vulgar f
Learners have changed improper 6Nn28 – Understand percentage as
fractions to mixed numbers and 6Nn29 – Find simple percentages o
worked with simple percentages.
3A: Problem solving (Usi
6Ps8 – Solve simple word problem
*for NRICH activities mapped to the Cambridge
Vocabulary
No new vocabulary.
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
ercentages Fractions and deci mals Pe rcentages D iv ide the threesec tio n s
s into s quare s.
vert Let’s investigate Let’s investigate
7 A wealthy f arme r ha s f o ur so n s a n d a 3- sec tio n T he or ien tatio n of the
farm as s ho wn in the dia gram. s on s ’ farm s d oe s n ot
K iki say s 1 0. 7 is eq uivale nt to . have to be the sa me.
10 T he f arme r wan ts to give his f arm to h is s o n s so
th at eac h rece ive s 2 5 %. Bu t he say s tha t e ac h so n
Davy say s it s h o uld be 0 .7 a nd M o ha mme d say s it s h o uld be 1 . 07 . mu s t ha ve a far m th e sam e s ha p e a s the or ig ina l.
D o y o u a gree w ith a ny o f the se stu de nts ? Ho w is the far m div ide d ?
Draw a pic ture to he lp e x pla in why . 1 Copy and complete the table sho wing equivalent
fractions and percentages.
1 Convert the se fractions to decimals.
1 1 2 3
(a) (b) (c) (d)
10 2 5 5
5
3 3 4 (h) Fra ctio n Per ce n ta g e
(e) (f) (g) 8
4 8 5 •
2 Ahme d ha s the d igit cards 0, 1, 2 an d 5 . 63 40%
Arrange the d igit cards to ma ke this correct (a)
1 100
3 Which n umber is the greater or 0. 4 ?
(b)
4
Exp lain how y ou kno w. 1
(c)
4 Loo k a t this gr id.
4
7 1 0.8 (d) 10%
0.3 2
1 9
10 10 (e)
2 100
0.6 0.2
2 Find each percentage. Start by nding 10% of each amou nt.
5
Which number sa tis e s a ll the criteria: (a) 1 0% of 5 0 ? so 20 % of 5 0 ?
1 (b) 10 % o f $ 70 ? so 30 % o f $ 70 ?
2
The fraction is les s than The fraction is no t equ 3 (c) 10 % of 75 0 mm ? so 70 % of 75 0 mm ?
al to
(d) 10 % of 4 0 cm ? so 5% of 4 0 cm ?
10
The fraction is greater than 0.1 The fraction is n ot equ
1
al to
5
10 1 10 2
objectives might only be partially covered within any given chapter but are
e book when taken as a whole
umber system
quivalence between decimal and fraction forms.
fraction to a decimal fraction using division.
s parts in every 100; express 1, 1, 1, 1 , 1 as percentages.
of shapes and whole numbers2. 4 3 10 100
ing understanding and strategies in solving problems)
ms involving percentages, e.g.find discounted prices.
e Primary objectives, please visit www.cie.org.uk/cambridgeprimarymaths
Unit 3A 245
Core activity 28.1: Fractions and decimals
Resources: Decimal number square photocopy master (p251). (Optional: Using
Work with learners to demonstrate fraction and decimal equivalence. Using fraction w
number lines.
Use a fraction wall to make links first to tenths and then to the decimal equivalent, fo
example:
2 = 4 = 0.4
5 10
11 11 11 11 11
10 10 10 10 10 10 10 10 10 10
1 1 1 1 1
5 5 5 5 5
1 1
2 2
Use number lines to reinforce the decimal equivalent of fractions, first in tenths, then
fifths, for example:
1 2 3 4 5 6 7 8 9 10
10 10 10 10 10 10 10 10 10 10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Establish that a fraction can be converted to a decimal by dividing the numerator by t
denominator so, for example:
2 = 2 ÷ 5 = 0.4
5
3 = 3 ÷ 8 = 0.375
8
246 Unit 3A 28 Fractions, decimals and percentages
LB: p101
g digit cards to make equivalent fractions and decimals (CD-ROM).)
walls and
or
n halves and Look out for!
0 Learners who confuse fraction and decimal
0
1
the
equivalents. For example, remind that 0.4 is 4
and not 1. 10
4