1A 1 The number system (1)
NOTE: the objectives that form the basis of this chapter revise, consolidate and e
what they know already and use this information to target work appropriately. Yo
them to produce a poster showing what they know. Alternatively, you could ask t
means of formative assessment.
Quick reference
Core activity 1.1: Place value (Learner’s Book p2)
Learners revise and consolidate work on place value up to 1 000 000 and down to
decimal places.
Core activity 1.2: Ordering, comparing and rounding numbers (Learner’s B
Learners revise and consolidate work on ordering and rounding. Work in this cha
concentrates on whole numbers as the objectives are repeated later in the year wh
decimals are the main focus.
Prior learning Objectives* – please note that listed objectives might onl
covered fully across the book when taken a
This chapter builds on
the work done in Stage 1A: Numbers and the number system
5 on five- and six-digit 6Nn2 – Know what each digit represents in whole
numbers. 6Nn3 – Know what each digit represents in one- a
6Nn8 – Round whole numbers to the nearest 10,
6Nn10 – Make and justify estimates and approxima
6Nn12 – Use correctly the symbols for >, < and =.
6Nn13 – Estimate where four-digit numbers lie on a
– 1A: Problem solving (Using techniques a
6Pt5 – Estimate and approximate when calculatin
1A: Problem solving (Using understandin
6Ps9 – Make, test and refine hypotheses, explain
results or conclusions orally.
Vocabulary *for NRICH a
million please visit w
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
extend number work covered in Stage 5. We suggest that you ask your learners
ou could give the learners a list of the objectives covered in this chapter and ask
them to complete the The Number System photocopy master (CD-ROM) as
Number Ordering,comparing and rounding numbers
Place value Vo c abula r y Let’s investigate D o not a tetmp tpt
There are 11 8 7 stu de n ts in a lar ge c ity s c ho o l. to w or k o ut ann
Let’s investigate million : equal to one There are 42 clas se s in th e sc ho o l.
Ra phae l ha s eig ht d ig it cards. tho usan d th ousa nds accurate a n swer.
and written a s Ap pro ximate ly , h o w ma ny s tu de n ts are in
o two 4 7 1 00 0 0 00. eac h c la ss ?
2
1 millio n 1 0 10 E xp la in to a fr ie n d ho w y o u ma de y o ur dec isni.o n.
10 1 0 10 10
He u se s th e car ds to ma ke two f o ur- dig it n um bers. 1 Draw a line 1 0 centimetres long. Mar k 0 an d 1 0 000 at the e nd points. 0 10 0 00
He u se s e ac h card o n ly on ce.
Th in k a bo u t the larges t Estimate the p ositions of the fo llowin g numbers. Mar k
He f in ds the differe nce betwee n his tw o n um bers . st an d smalles tunmber s each one with an arrow an d its letter: 60 00 marked A
s y ou can ma ke. 3500 marked B
W ha t is the large st d ifference h e ca n ma ke ? 9050 marked C
1 Write the numbers shown on the se charts in words an d gures. 2 Ro un d these numbers to the nearest hundred.
(a ) 100 000 200 000 300 000 400 000 500 000 600 000 70 0 (a) 45 6 78 ( b) 2 4 05 5 (c) 50 5 05
10 000 20 000 60 000 70
00 0 800 000 900 000
Book p4) 30 000 40 000 50 000 3 Roun d these numbers to the nearest th ousa nd.
apter
hen 0 00 80 000 90 000 (a) 147 950 ( b) 65 507 (c) 15 7 84 6
1 00 0 2000 3 0 00 4000 5 0 00 6000 4 Order the following sets of numbers from smallest to largest.
(a) 54 7 54 55 475 5 5 54 7 54 7 75 55 447
7 00 0 8000 9 0 00 (b) 4 5 05 4 45 5 40 45 504 45 04 5 4 5 50 0
(c) 456 0 65 45 0 56 6 455 65 6 456 565 4 50 6 66
1 00 200 3 00 400 50 0 600 Use any of the numbers in par t (c) to complete these inequalities.
7 00 800 9 00
10 20 30 40 50 60
70 80 90
12 34 5 6 7
89
?? ??
0.1 0.2 0.3 0.4 0.5 0 .6
0.7 0.8 0.9 4
0 .01 0.02 0 .0 3 0.04 0.0 5 0.06
0 .07 0.08 0 .0 9
( b) 100 000 200 000 300 000 400 000 500 000 600 000 70 0
00 0 800 000 900 000
10 000 20 000 30 000 40 000 50 000 60 000 70
0 00 80 000 90 000
1 00 0 2000 3 0 00 4000 5 0 00 6000
7 00 0 8000 9 0 00
1 00 200 3 00 400 50 0 600
7 00 800 9 00
10 20 30 40 50 60
ly be partially covered within any given chapter but are 70 80 90 6
as a whole 12 34 5 7
89
0.1 0.2 0.3 0.4 0.5 0 .6
0.7 0.8 0.9
0 .01 0.02 0 .0 3 0.04 0.0 5 0.06
0 .07 0.08 0 .0 9
2
e numbers up to a million.
and two-place decimal numbers.
100 or 1000.
ations of large numbers.
an empty 0–10 000 line.
and skills in solving mathematical problems)
ng e.g. use rounding and check working.
ng and strategies in solving problems)
n and justify methods, reasoning, strategies,
activities mapped to the Cambridge Primary objectives,
www.cie.org.uk/cambridgeprimarymaths
Unit 1A 1
Core activity 1.1: Place value
Resources: Place value grid photocopy master (p7); large version for class disp
The Number System photocopy master (CD-ROM).)
Display the Place value grid photocopy master for the whole class to see,
and write the number 2002.2 so that everyone can see it.
Mark it on the grid and say, “Two thousand and two point two.”
Ask questions about each digit in turn:
“What is the value of this digit?”
“How many times larger or smaller is the value of this two than this two?”
When discussing place value in terms of how many times larger or smaller
one digit in the number is relative to another digit, the learners may find
it easier to visualise it as follows:
× 1000 U• t h
. 2
Th H T
÷ 10
Repeat with other numbers up to two decimal places, for example 3003.33 and also w
numbers that don’t have the same digit repeated e.g. 2450.12.
“What is the largest number I can show on this chart?” (Answer: 999 999.99)
“How do I read this number?” (Answer: nine hundred and ninety nine thousand,
hundred and ninety nine point nine nine.)
“What happens if I add 0.01 to this number?” (Answer: I get one million (accept ‘1
zeros’); learners might not be familiar with the term ‘million’ even if they can id
the answer will be a 1 followed by six digits, if this is the case, introduce them to
terminology.)
“How do I write this number?” (Answer: 1 000 000)
2 Unit 1A 1 The number system (1)
LB: p2
play. (Optional: Match the numbers photocopy master (CD-ROM);
200 000 Vocabulary
20 000
2000 million: equal to one thousand thousands;
200
20 1 000000 = 10 × 10 × 10 × 10 × 10 × 10
2
0.2 Look out for!
0.02 Learners who do not read decimal numbers
with correctly. Explain why the number, for example,
1.23, is read as ‘one point two three’ and NOT as
nine ‘one point twenty three’; it is because the digits 2
and 3 represent two tenths and 3 hundredths NOT
1 and six 2 tens and 3 units. Similarly, 1.02 is read as ‘one
dentify that point zero two’.
o this Learners who may use different notation for
writing large numbers. For example, 1 million is
written as 1 000 000 or 1,000,000 in different parts
of the world.
Opportunities for display!
Learners collect examples of numbers from
newspapers, magazines and other sources for display.
Each number could be written in words and figures
and displayed on a place value chart.
Summary
Learners use a place value chart for numbers up to a million and down to two
decimal places.
They read and write numbers and recognise a million written in figures.
They understand that the position of a digit affects its value.
They read decimal numbers correctly.
Notes on the Learner’s Book
Place value (p2): contains examples that provide practice in reading and writing
1 million. Further work related specifically to decimals can be found on pages 1
with The decimal system, (chapter 12).
More activities
Match the numbers
(pairs)
You will need the Match the numbers photocopy master (CD-ROM).
Cut out the cards from the activity sheet and lay them face up on the table. Learn
number. Their partner checks the answer. Repeat until all the cards have been us
numbers.
Self assessment (individual)
You will need The Number System photocopy master (CD-ROM).
This is a self assessment sheet where learners practise the skills from the core ac
identify what they want to get better at.
Games Book (ISBN 9781107667815)
Place value challenge (p1) is a game for two players. It gives practice writing (an
Check up!
Read this number, 1234.05. “What is the value of
the digit 1; and the digit 5?”
“W rite the number six hundred and fifty six point
six in figures.”
numbers up to
109 to 119 (starting
ners work in pairs. They take turns to pick up two matching cards and say the
sed. This will act as a check to see that learners have understood how to say large
ctivity. They identify skills that they can do and which they need help on. They
nd reading) large numbers.
Core activity 1.1: Place value 3
Core activity 1.2: Ordering, comparing and rounding
Resources: Blank number lines photocopy master (p8); large version for class
Display the Blank number lines photocopy master, so that the whole class can see the
number line marked from 0 to 10000.
0 10000
Ask learners what number goes in the middle of the line? (Estimate the halfway ma
mark 5000 on the line.) “How did you work it out?” (Learners should know, or be
reason from previous work, that 5000 is half of 10 000.)
Mark and label other divisions on the line: start with by marking and labelling every
Then ask learners to show the position of some four-digit numbers, for example 4500
and 4800.
4500
0 4000 5000 10 000
Ask learners to order the following numbers starting with the smallest: 4300, 4500, 4
4100, 4200; they can use the number line for help if they need to. (Answer: 4100, 42
4500, 4800)
Ask learners to use two of the numbers from the list above to complete the number se
below, then read them aloud:
> <
Repeat with other sets of numbers.
4 Unit 1A 1 The number system (1)
g numbers LB: p4
s display. (Optional: 0–9 spinner (CD-ROM).)
e empty Always ask, “How did you decide?” giving learners
the chance to explain their methods using correct
mathematical vocabulary.
ark, and Look out for!
e able to
1000. Learners who do not know the conventions for
0, 4200 rounding. To round to the nearest thousand look
4800, at the hundreds digit:
200, 4300,
entences if it is less than 5 round down
if it is 5 or more round up. 213 000
to the nearest thousand
213 241
To round to the nearest hundred look at the tens
digit:
if it is less than 5 round down
if it is 5 or more round up.
to the nearest hundred
213 241 213 200
To round to the nearest ten look at the units digit:
if it is less than 5 round down
if it is 5 or more round up.
to the nearest ten
213 241 213 240
Using the number line ask learners to round each number (4300, 4500, 4800, 41
4200) to the nearest thousand. (Answer: 4100, 4200, 4300 round to 4000 and 4
and 4800 round to 5000.)
Emphasise how a number line can help learners to visualise, for example, 4200 i
nearer to 4000 than to 5000. Repeat with other sets of numbers
Ask where they would place the number 4155 on the number line. (Answer: just
half way between 4100 and 4200.)
Support learner to:
round 4155 to the nearest thousand (Answer: 4000)
round 4155 to the nearest hundred (Answer: 4200)
round 4155 to the nearest ten. (Answer: 4160)
Repeat with other numbers.
Draw a new number line from 0 to 1000000. Tell learners that 1000000 is 1 mill
which is 1 thousand thousands. Ask learners to discuss what number they think
be positioned in the middle of the number line and to justify their answer (Answ
500000, 5 hundred thousand). Mark 5000000 on the number line. Ask learners
suggest some other six-digit numbers and to estimate where those numbers wou
placed on the number line. Suggest the number 843791. Ask some of the learne
mark with a dot where they think the number would be positioned, and ask the o
learners which dot they think most accurately places the number and justify thei
answer. Encourage learners to use rounding and approximation to help their estim
and reasoning.
100,
4500
is
t over
lion,
would
wer:
s to
uld be
ers to
other
ir
mate
Core activity 1.2: Ordering, comparing and rounding numbers 5
Summary
Learners confidently round numbers to the nearest 10, 100 or 1000 using
mathematical conventions.
They use a number line when appropriate to position numbers and understand that
a number line may be useful when ordering or rounding numbers.
They use the signs < and > to compare numbers.
Notes on the Learner’s Book
Ordering, comparing and rounding numbers (p4): the investigation challenges learne
use their knowledge to answer a different type of question and explain their decision
partner. They should round each number to work out an approximation, for example:
1200 ÷ 40 = 30
Questions 1 to 7 provide practise related to the Core activity. Part (c) of question 8
requires learners to think about all the numbers that could round according to
two criteria.
More activities
Rounding up
(pairs) need 0–9 spinner (CD-ROM).
You will
Player one ‘spins the spinner’ five times to create a 5-digit number. The player choos
Player two rounds the number to the nearest thousand.
If the number ‘rounds up’ player two scores a point.
If the number ‘rounds down’ player one scores a point.
The first player to score five points is the winner.
Games Book (ISBN 9781107667815)
More or less (p1) is a game for two players. It provides an opportunity to use the sym
6 Unit 1A 1 The number system (1)
t Check up!
ers to “Round 512 345 to the nearest thousand. How did you work
n to a out your answer?”
:
“Order this set of numbers: 41 325 43 521 45 123
43 324 43 512. What did you look for when making your
decisions?”
“Use < or > to complete this inequality: 45 123 45 213”
ses which of the five digits to put in each box.
mbols < and > using one place decimals.
Place v
100000 200000 300000 400000 5000
10 000 20 000 30 000 40 000 50 0
1 000 2 000 3 000 4 000 50
100 200 300 400
10 20 30 40
1234
0.1 0.2 0.3 0.4
0.01 0.02 0.03 0.04
Instructions on page 2
value grid
000 600000 700000 800000 900000
000 60000 70000 80000 90000
000 6000 7000 8000 9000
500 600 700 800 900
50 60 70 80 90
56789
0.5 0.6 0.7 0.8 0.9
0.05 0.06 0.07 0.08 0.09
Original Material © Cambridge University Press, 2014
Blank num
0
0
0
Instructions on page 4
mber lines
10 000
10 000
10 000
Original Material © Cambridge University Press, 2014
1A 2 Multiples, factors and prime
Quick reference
Core activity 2.1: Factors and multiples (Learner’s Book
Lp6e)arners consolidate previous learning related to multiples and factors.
Core activity 2.2: Odd and even numbers (Learner’s Book p8)
Learners extend their work on odd and even numbers as they explore rules relate
addition, subtraction and multiplication.
Core activity 2.3: Prime numbers (Learner’s Book p10)
Learners are introduced to prime numbers and the definition of a prime number,
can recite the prime numbers less than 20.
Prior learning Objectives* – please note that listed obje
when taken as a whole
This chapter builds on previous
work on odd and even numbers, 1A: Numbers and the numb
multiples and factors. 6Nn6 – Find factors of two-digit num
Prime numbers have not been 6Nn7 – Find some common multiples
formally introduced in previous 6Nn17 – Recognise odd and even num
Stages, so this might be the first 6Nn18 – Make general statements ab
time learners have encountered 6Nn19 – Recognise prime numbers up
the definition.
1A: Problem solving (Using
6Ps3 – Use logical reasoning to expl
6Ps9 – Make, test and refine hypoth
orally.
*for NRICH activities mapped to the Cambridge P
Vocabulary
factor multiple odd even prime number
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
es
Mult iples and factors Vocab ulary Odd and even numbers Vocab ulary Prime numbers Vo c abula r y
Let’s investigate factor: a whole Let’s investigate odd : od d numbers Let’s investigate prime number: a
You ne e d 1 3 co un ters an d a 5 by 5 gr id. aren ot div is ible by
T he se qu en ce below u se s the n umber s 1 to 4 s o tha t num ber tha t d ivide s eac h 2.They en d in 1, 3, 5, 7 L oo k at th is s ta te me nt. E very eve n n um ber prime n umber ha s
or 9. For example, 768 9 greater tha n 2 is the s um of e xactly tw o d ifferent
n um ber is e ithe r a fa ctor or a m u ltiple of th e exactly in to a n othe r is an o dd n umber.
n um ber . prev io us Here are tw o e xam ple s: two prime factors; itse lf an d 1 .
n u m be rs .
number. For example, NOTE: 1 is nota prime
number. It has o nly
1, 2, 3 an d 6 are the o nefactor ( 1).
factors of 6. even: even numbers are 6 3 3 (3 is a p rime n um ber ) Examples of prime
numbers: 2 , 3, 5, 7 ,
1 6 62 3 6 div isible by 2. 1 2 5 7 ( 5 an d 7 are p rime nu mber s) 11…
They end in 2, 4, 6,
2
8 or 0. For example,
factor factor factor factor ● P lace 1 3 co un ters o n the grid s o that there is an od d 6 5 78 is a n e ven ● Chec k if the s ta te me nt is tr ue f or all the e ve n
num ber of c ou nter s in each r o w, co lum n an d on bo th num ber s to 3 0.
num be r.
multiple: a number
Each n umb er is us ed o nce on ly . U se car ds thatt dia go nals. O n ly o ne co u nter ca n be place d in e ach cell. ● Ca n y o u f ind a n eve n n um ber tha t d oe s n ot s atisfy
Fin d a s im ilar se q ue nce tha t can be e as ily tha tca n be divide d ● P lac e 1 0 co un ters o n the gr id s o th at there is an th e r u le ? Try so me n umber s greater tha n 3 0.
use s the num bers 1 to 6. move d ar o un d. exactly by another
number is a multiple of even n um b er of c ou nter s in eac h r o w,
tha t number. Start
co lu m n a n d o n b o th diag o na ls. O n ly o ne 1 L is t all the prime numbers between 1 0 an d 20.
at 0 a nd c oun t u p
1 Which of these numbers are multiples of 8 ? in s teps of the same can b e p la ce d in eac h cell. There is more 2 I dentify these prime numbers from the clues.
(a) It is less tha n 3 0.
18 24 4 8 5 6 68 7 2 size an d y ou will n d co u nter th an o ne The sum of its dig its is 8.
numbers that are 1 Which of these numbers are even? ans wer. (b) It is between 3 0 an d 6 0. The
multiples of the step sum of its digits is 10.
ed to 2 Which of the se numbers are factors of 30 ? 9 1 1 26 3 3 57 187 2 00 2
, and size. For example,
4 5 6 1 0 20 60 Exp lain to a partner how y ou kno w.
3333
3 U se each of the d igits 5, 6 , 7 and 8 once to 0 3 6 9 12 2 An dre makes a three-digit number. A ll the 3 Copy an d complete these n umber sentence by placing a prime numbe
make a total tha t is a multiple of 5 . dig its are od d. r in each box.
The sum of the d ig its is 7.
? ?+ ? ? 3, 6, 9, 12 . . . are What cou ld A ndre’s number be ? ? ? ? 30
multiples of 3 .
4 Fin d all the factors of: ? ? ? 50
(a) 24 (b) 3 2 (c) 2 5.
3 O llie makes a three-digit number using the d igits 2, 3 an d 6 . H is number ? ? ? 70
is odd .
5 My age th is y ear is a multiple of 8. My age 4 I den tify the prime numbers represented by ? an d ? .
next y ear is a multip le of 7. How old am The hundreds digit is greater tha n 2 . What
I? could Ollie’s number be ? (a) ? 2 (c) ? 2 52
49
8 (b) ? 1 2 9 (d) ? ? 2 0
10
6
ectives might only be partially covered within any given chapter but are covered fully across the book
ber system
bers.
s (e.g. for 4 and 5).
mbers and multiples of 5, 10, 25, 50 and 100 up to 1000.
bout sums, differences and multiples of odd ad even numbers.
p to 20 and find all prime numbers less than 100.
g understanding and strategies in solving problems)
lore and solve number problems and puzzles.
heses, explain ad justify methods, reasoning, strategies, results or conclusions
Primary objectives, please visit www.cie.org.uk/cambridgeprimarymaths
Unit 1A 9
Core activity 2.1: Factors and
multiples
Resources: There are no specific resources required for this activity.
Revise learners’ knowledge of factors and multiples by creating a whole class ‘Mathe
orchestra’. Sit five learners on chairs facing the class. Number these learners 1 to 5, a
them that they are members of the mathematical orchestra. Each learner has their ow
orchestra. Explain that you are the conductor and your job is to slowly count the beat
up/down movements of your hand.
Instruct learner ‘number 1’ to stand and immediately sit down on each beat, then inst
‘number 2’ to stand and immediately sit down on beats 2, 4, 6 … (the multiples of 2)
learner ‘number 3’ to stand on beats 3, 6, 9 … (the multiples of 3), learner ‘number
on beats that are multiples of 4, and learner ‘number 5’ to do so on beats that are mu
Practise together first so that you are con dent the learners understand. Practise the c
beats, or beyond. If you were counting 8 beats, the learners would stand/sit as per the
below on beat 1, 2, 3, 4, etc, to beat 8.
stand stay seated
1: 2: 3: 4:
5: 6: 7: 8:
Direct questions to the rest of the class such as:
“Can you predict how many learners will stand on count 10?” (Answer: 3)
“Which numbers do they represent?” (Answer: 1, 2, and 5)
“What is the relationship between these numbers?” (Answer: they are factors of 1
Establish the definitions of multiple and factor, then issue a challenge, “When is the
four learners will stand together?” (Answer: 1, 2, 3 and 4 will stand on 12.) Act ou
mathematical orchestra to demonstrate this. You could try similar challenges that are
to act out with the mathematical orchestra.
10 Unit 1A 2 Multiples, factors and primes
LB: p6
ematical Vocabulary
and inform
wn part in the multiple: a number that can be divided exactly by
t with another number; start at 0 and count up in steps of the
same size and you will find numbers that are multiples
truct learner of the step size. For example,
). Then tell
4’ to do so +3 +3 +3 +3
ultiples of 5.
count to 8 0 3 6 9 12
e diagram
3, 6, 9, 12 . . . are multiples of 3.
10.)
first time factor: a whole number that divides exactly into
ut with the another number. For example, 1, 2, 3 and 6 are the
also possible factors of 6.
1×66 2×36
factor factor factor factor
general statement: a statement that does not use
particular examples, e.g. ‘Two odd numbers added
together give an even number’.
counter-example: an example that shows a general
statement is wrong.
Ask learners from the orchestra to sit back down and ask the class to discuss in grou
first time all five learners will stand together? How do you know?” (Answer: 60; it is
2, 3, 4 and 5.) Discuss answers and reasons as a class.
Tell learners that it can be useful to know all the factors of a number. Ask learne
strategies they would use to find all the factors of 24. If the learners cannot com
ideas of their own, then suggest the following:
“Start with the lowest number, 1 is always a factor of a whole number.”
“What is the ‘partner’ of 1?” (Answer: 24) “The number itself will also alwa
“What about 2?” (Answer: It must be a factor because 24 is an even numb
“What is 2’s ‘partner’?” (Answer: halve 24 to get 12.)
“What about 3?” (Answer: 24 ÷ 3 = 8 so 3 and 8 is another factor pair.)
“Continue in this way to nd all possible pairs.” (Answer: 1 & 24, 2 &12, 3 &
“Is there any point in continuing any further? Why not?” (Answer: 5 is not a
already been found.)
“Record the results systematically, for example, {1 24} then {1, 2 12
until you reach the ‘middle’.”
If necessary, remind them of factor bugs (Stage 5, chapter 4).
Ask groups of learners to discuss the general statement, ‘The larger the number, the
has. Ask them to consider some test pairs, for example 16 and 27, 12 and 20. Lear
p’ redict which number will have the most factors, and then they can calculate all the
number to test the statement.
Discuss the results as a class. Is the general statement correct? Or did someone f
counter-example?
Summary
Learners revise and extend previous work on multiples and factors, using them to
puzzles and problems.
Notes on the Learner’s Book
Multiples and factors (p6): learners are presented with a selection of straight-for
and puzzles. The puzzles involve both multiples and factors together, so learners
about the definitions of the words. Useful links are made with data handling obj
both Carroll diagrams and Venn diagrams are used in questions 6, 7 and 9.
ups, “When is the The ‘Mathematical orchestra’ activity can be adapted
s a multiple of 1, for use with the whole class by:
ha ving more learners in the row.
ers what ha ving groups of learners seated around tables with
me up with
each table representing a number. An alternative
ays be a factor.” to standing up and sitting down is to wave arms up
ber.) and down.
& 8, 4 & 6.) Look out for!
factor, 6 has Learners who confuse factors and multiples
(a common error). Remind these learners of the
2, 24} and so on def i nitions and give them some examples of factors
and multiples, particularly where a factor is not also
a multiple. For example, 2 is a factor of 8 but is not
a multiple of 8.
e more factors it Opportunities for display!
rners should first Ensure that the words ‘multiple’ and ‘factor’ are
factors for each clearly displayed with their definitions. Add the
posters made in the More activities section to the
find a display
o solve Check up!
“Here are four numbers: 3, 4, 7 and 12. Which of
rward questions
s have to think these numbers are factors of 12?” (Answer: 3, 4
jectives, as and 12.)
“I am thinking of a number between 20 and 40. It is a
multiple of 5 and a multiple of 7. What number am 11
I thinking of?” (Answer: 35)
“How can you be sure you have found all the factors of
a number?”
Core activity 2.1: Factors and multiples
More activities
Make a poster (individual)
Learners design a poster about factors and multiples, remembering to include the def
Puzzles (pairs)
Solve these puzzles, then write similar puzzles for your partner to solve:
1. What is my number? It is even, a multiple of 4, a factor of 24, and between 10 an
2. What is my number? It is a factor of 24, a factor of 40, and a factor of 52 but is n
Games Book (ISBN 9781107667815)
Factors in a row (p5) is a game for two players. The game provides practice in ndin
12 Unit 1A 2 Multiples, factors and primes
finitions of each word.
nd 20 (Answer: 12)
not the number ‘2’ (Answer: 4)
ng all the factors of 2-digit numbers.
Blank page 13
Core activity 2.2: Odd and even
numbers
Resources: Blank 3 by 3 grid photocopy master (p18).
By Stage 6 learners should be con dent of what is meant by odd and even numbers b
necessary, reinforce that odd numbers cannot be divided by 2 without leaving a rem
This point is important because some learners might be confused by previous work o
division where odd numbers were divided by two to leave a remainder.
Show an example of how you can arrange the numbers 1, 2, 3 and 4 in a 2 by 2 grid s
sum of the numbers horizontally and vertically is odd and the sum of the numbers d
is even. One solution is:
21 odd
34 odd
even even
odd odd
Ask learners to find a different solution.
As a class, discuss the different solutions learners found. Ask, “How did you decide whe
numbers? Can you give us any rules about adding odd and even numbers?” (Answe
evens or two odds need to go along the diagonal (based on the rules: even + even
odd+ odd = even); then the other two numbers can be placed to ll the gaps so tha
and columns there will be one odd and one even number (based on the rule, e
odd.)
Learners work in pairs on a different investigation. Ask them to place the digits 1, 2, 3, 4
8 and 9 in the 3 by 3 blank grid photocopy master so that each line horizontally, verticall
diagonally add up to an odd number.
Review work and arrive at the rules for adding three numbers that gives the following
pattern for solutions to the problem:
EOE
OOO
EOE
(Answer: either three odds have to add together or two evens and an odd.)
14 Unit 1A 2 Multiples, factors and primes
LB: p8
but if Vocabulary
mainder.
on odd numbers: are not divisible by 2 without a
remainder; they end in 1, 3, 5, 7 or 9. For example,
so that the 4689 is an odd number.
diagonally
even numbers: are divisible by 2, without a remainder;
they end in 2, 4, 6, 8 or 0. For example, 7578 is an even
number.
ere to put the Opportunities for display!
er: either two
n = even and Display the rules for:
hat in all rows • adding odd and even numbers:
even + odd =
even + even = even
4, 5, 6, 7, odd + odd = even
ly and even + odd = odd
odd + even = odd
g • subtracting odd and even numbers:
even - even = even
odd - odd = even
even - odd = odd odd
- even = odd
• multiplying odd and even numbers:
even × even = even
even × odd = even
odd × even = even
odd × odd = odd.
Move on to discussing multiplying odd and even numbers, asking, “What are the
multiplying odd and even numbers?”
Allow the learners thinking time then collect their ideas. (Answer: odd × odd =
= even, even × even = even.)
Summary
• Learners work confidently with odd and even numbers.
• They find and use general statements and solve increasingly challenging puzz
Notes on the Learner’s Book
Odd and even numbers (p8): learners are familiar with odd and even numbers so
designed to encourage them to think about the properties of odd and even n
exception of question 1). Links are made with place value, multiples and calcula
Learners who need support could work in pairs.
More activities
Squirrels nut store (pairs)
Squirrels hide nuts to eat in the winter. Three squirrels hide 25 nuts altogether. E
many nuts did each squirrel hide? Find as many different ways as you can.
(Answer: There are 10 different solutions. [1, 3, [1, 5, 19] [1, 7, 17] [1, 9,
21]
e rules for
= odd, even × odd
zles. Check up!
o the questions are Mira says, ‘I can add three odd numbers to get a total of
numbers (with the 30.’
ation. “Is she right? Explain your answer.”
Each of them hides a different odd number of nuts. How
15] [1, 11, 13] [3, 5, 17] [3, 7, 15] [3, 9, 13] [5, 7, 13] [5, 9, 11])
Core activity 2.2: Odd and even numbers 15
Core activity 2.3: Prime numbers
Resources: Sieve of Eratosthenes photocopy master (p19).
Remind learners of the definition of a multiple and a factor.
Give each learner a copy of the Sieve of Eratosthenes photocopy master. Learners will u
explore patterns for the 2, 3, 4, 5, 6 and 7 times tables. First they should cross out the nu
then they should cross out the multiples of 2 except 2, the multiples of 3 except 3, the m
4, the multiples of 5 except 5, the multiples of 6 and the multiples of 7 except
7. Before they cross out the multiples of each number, they should try to predict what pa
might emerge and try to explain the patterns they’ve found after doing the crossing out.
You could use the following questions as a guide:
(2× table) “What do you notice? Can you explain what you see?” (Every other nu
crossed out, because consecutive numbers have a pattern of odd-even-odd-eve
and all multiples of 2 are even numbers.)
(3×table) “Can you predict what will happen? Will you shade any numbers that ar
shaded? If so, which ones?” (Even multiples of 3 will already be crossed out as
also multiples of 2.)
(4× table) “Do you need to shade these multiples? Why not?” (No, because all mu
are also multiples of 2; they are all even numbers.)
(5× table) “Can you explain the pattern?”
(6× table) “Do you need to shade the multiples of 6? Why not?” (No, because all m
6 are even and therefore are also multiples of 2.)
(7× table) “Which numbers were not already shaded in?” (49, 77, 91)
At the end they should look at the grid and say what is special about the numbers tha
haven’t crossed. Establish that they are the prime numbers: 2, 3, 5, 7, 11, 13, 17, 19,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Explain what is meant by a p
number. Explain that the method they just carried out is called the ‘Sieve of Eratosth
was devised by a historical mathematician, called Eratosthenes, to identify all the prim
less than 100.
16 Unit 1A 2 Multiples, factors and primes
LB: p10
use this to Vocabulary
umber 1,
multiples of prime number: a prime number has exactly two
atterns different factors, 1 and the number itself. For example,
2, 3, 5, 7, 11 are all prime numbers.
umber is NOTE: 1 is not a prime number; it has only one factor (1).
en-odd etc.
re already Look out for!
they are Learners who think that 1 is a prime number.
ultiples of 4 Emphasise that prime numbers always have two
different factors and 1 only has one factor.
multiples of
at they
, 23, 29,
prime
henes’ and
me numbers
Repeat the mathematical orchestra activity from Core activity 2.1. You might wis
multiples: 6, 7 . . . Ask learners which ‘beats’ had two people standing together (
and 5) and what is special about these numbers? (Answer: prime numbers.)
Summary
Learners know the definition of a prime number and can recite the prime numbe
Notes on the Learner’s Book
Prime numbers (p10): the investigation links work on odd and even numbers wit
numbers. This is followed by examples focusing on identification and use of pri
More activities
Eratosthenes (individuals or pairs)
The ‘Sieve of Eratosthenes’ is the 10 by 10 grid that learners used to find all the
Eratosthenes. Learners find out as much as they can about his life and work.
sh to add extra
(Answer: beats 2, 3,
ers less than 20. Check up!
“Which of these are prime numbers?”
th prime 11, 21, 31, 41, 51, 61
ime numbers.
e prime numbers less than 100. The mathematician who devised the sieve was
Core activity 2.3: Prime numbers 17
Blank 3 by 3 grid
Instructions on page 14 Original Material © Cambridge University Press, 2014
Sieve of Eratosthenes
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
1. Cross out the number 1.
2. On the grid, cross out all the multiples of 2 except 2.
What do you notice? Can you explain what you see?
3. On the same grid, cross out all the multiples of 3 except 3.
Can you predict what will happen? Will you cross out any numbers
that are already crossed out? If so, which ones?
4. Explore what happens for multiples of 4, 5, 6 and 7.
(4× table) Do you need to cross out these multiples? Why not?
(5× table) Cross out all the multiples of 5 except 5. Can you explain
the pattern?
(6× table) Do you need to cross out the multiples of 6? Why not?
(7× table) Which numbers were not already crossed out?
Now look at your grid. What is special about the numbers that
you haven’t crossed out?
Instructions on page 16 Original Material © Cambridge University Press, 2014
20 Blank page
1A 3 Multiplication and division
(1)
Quick reference
Core activity 3.1: Multiply and divide by 10, 100 and 1000 (Learner’s Book
Lp1e1ar)ners multiply and divide whole numbers by 10 and 100, extending to multip
1000.
Core activity 3.2: Mental strategies for multiplication (Learner’s Book p13)
Learners develop and re ne mental strategies for multiplication, including worki
near multiples of 10, halving and doubling.
Prior learning Objectives* – please note that listed objectives might only be p
when taken as a whole
This chapter revises
work done in Stage 1A: Numbers and the number sys
5 when learners 6Nn4 – Multiply and divide any whole numb
multiplied and divided
whole numbers by 10 1A: Calculation (Multiplication and
and 100, and worked 6Nc8/6Nc14 – Multiply pairs of multiples of 10, e.g.
on mental strategies
for multiplication. 6Nc15 – Multiply near multiples of ten by mul
6Nc16 – Multiply by halving one number and
1A: Problem solving (Using techniq
6Pt1 – Choose appropriate and efficient me
subtraction, multiplication or division
1A: Problem solving (Using unders
6Ps1 – Explain why they choose a particula
– Use ordered lists or tables to help solve nu
6Ps6 – Make sense of and solve word prob
*for NRICH activities mapped to the Cambridge Primary objectives
Vocabulary
near multiple of 10
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
Mult ip lying and d ivid ing by 10, 100 and 1000 Vocab ulary Mental strategies for multiplication
Let’s investigate near multiple of 10 : Let’s investigate
Che n g is th in kin g of a n u mb er. a n umber either s id e Fin d differe nt way s of c om p le tin g th is ca lc u la tio n.
W ha t n um ber is Che n g thin kin g of ?
of a multiple of 10. ? ? ? 24 a
I m ultiply my n um b er by 1 0 0 , th e n For example, 20 is
plying and dividing by d iv id e by 1 0 , the n m ultiply by 1 0 0 0 . multiple of 1 0, so
ing with multiples and My a nswer is one 19 an d 21 are near
multiples of 10.
h u n dr e d a n d s e ve n ty th o u s a n d .
1 U se the given fact to derive a new fact and then explain y our method . Copy and
complete the table, the rst o ne ha s been d one for y ou.
F a ct De rive d f act M e tho d
1 Copy and complete th is set of m is sing n umbers. 2 5 10 0 a 7 9 63 7 18 126 18 is double 9 so double the
? ? 10 0 25 0 an s w er
b 7 3 21 70 3
? 1 0 25 00 25 0 1 0 ? c 5 7 35 50 70
2 What is the mis sin g n umber? 10 0 d 6 8 48 6 16
e 8 13 104 4 13
10 1 0 00 0 ? f 6 7 42 6 70
3 A decago n ha s 1 0 sides. g 5 9 45 5 91
h 6 9 54 6 89
i 4 7 28 39 7
j 39 30
2 79 1
2 U se table facts to help y ou wor k ou t the following:
(a) 30 × 70 (b) 5 0 × 9 (c) 20 × 6
(d) 50 × 80 (e) 8 × 90 (f) 70 × 6 0
What is the perimeter of a regular decagon w ith s ides 1 7 3 Wor k ou t the following u sin g a mental strategy : (a) 29 ×
centimetres long?
6 (b) 4 1 × 5 (c) 19 × 7 ( d) 2 1 × 8
4 Milly say s,“Every multiple of 10 00 is d ivisible by 100.” Is
she righ t? (e) 49 × 6 (f) 51 × 4
Exp lain y our answer.
Exp lain to y our partner how y ou wor ked o ut the answers.
For more questio ns, turn the pa ge .. .
13
11
partially covered within any given chapter but are covered fully across the book
stem
ber from 1 to 10 000 by 10, 100 or 1000; explain the effect.
division)
. 30 × 40, or multiples of 10 and 100, e.g. 600 × 40.
ltiplying the multiple of ten and adjusting.
doubling the other, e.g. calculate 35 × 16 with 70 × 8.
ques and skills in solving mathematical problems)
ental or written strategies to carry out a calculation involving addition,
n.
standing and strategies in solving problems)
ar method to perform a calculation and show working. 6Ps4
umber problems systematically.
blems and represent them.
s, please visit www.cie.org.uk/cambridgeprimarymaths
Unit 1A 21
Core activity 3.1: Multiply and divide by 10, 100 and
1000
Resources: True or false multiplication and division cards photocopy master (
Start the session by revising work on multiplying and dividing by 10 and 100.
Learners work in pairs using the True or false multiplication and division cards cut from t
master. They take a card in turn and say whether the statement is true or false.
If the statement is true they explain how they know.
If the statement is false they give the correct answer.
Review work done, reminding learners of the following rules:
To multiply by 10, move each digit one place value Tth Th H
to the left. Zero may be needed as a place holder.
To multiple by 100, move each digit two place 3
values to the left; 0 might be needed as Tth Th H
a place holder.
“Following this pattern, how can I multiply by 1000?” 3 40
(Answer: move digits three places to the left e.g. 34 × 1000 = 34
000)
Emphasise that multiplying by 1000 is equivalent to multiplying by 10, then 10 again
again.
Repeat for division by 10, 100 and 1000. Ask, “Following this pattern, how can I div
1000?” (Answer: move digits three places to the right, e.g. 58 000 ÷ 1000 = 58) E
that dividing by 1000 is equivalent to dividing by 10, then 10 again, then 10 again.
22 Unit 1A 3 Multiplication and division (1)
LB: p11
(p26).
the photocopy
HTU Example:
34
To divide by 10, move each digit one place value
3 40 to the right.
HTU
To divide by 100, move each digit two place
34 values to the right.
000
n, then 10
vide by
Emphasise
Tth Th H T U 58000 10
5 8000 5800 10 (58 000 100)
580 10 (58 000 1000)
5800
580
58
Work with learners to complete these number sentences, asking each time for an
34 × 1000 =
× 78 = 78 000
63 000 ÷ 1000 =
36 000 ÷ = 36
Summary
Learners confidently multiply and divide any whole number by 10, 100 or 1000
Notes on the Learner’s Book
Multiplying and dividing by 10, 100 and 1000 (p11): learners apply their knowle
division by 10, 100 and 1000 to problems set in different contexts. Ensure that le
multiply by 1000 they can multiply by 10, then 10 again, then 10 again or they c
either order. The investigation and questions 4 and 7 can be used to illustrate this
More activities
Make a poster (individual)
Design a poster that shows how to multiply and divide by 10, 100 and 1000.
Illustrate it with examples, including drawings, pictures or photographs.
1 metre is 100 times as long as 1 centimetre.
1 cent is 100 times smaller than 1 dollar.
n explanation:
0 and explain the effect. Check
edge of multiplication and up“!What is 48 000 ÷ 1000? How did you
earners understand that to work it out?”
can multiply by 10 and 100 in
s relationship. “Complete these number sentences:”
× 42 = 42 000 ÷ 1000 = 6
Core activity 3.1: Multiply and divide by 10, 100 and 1000 23
Core activity 3.2: Mental multiplication strategies
Resources: Large sheet of paper; one per pair of learners.
Ensure learners are con dent with mental strategies for multiplication by revising me
in Stage 5.
“I’m going to start with a multiplication fact and use different strategies
to find out other related facts.” Write 7 × 8 = 56 for the whole class
to see.
Ask what other facts can be found.
Take one response, for example, 7 × 4 = 28 (4 is half of 8 so
halve the answer).
Learners work in pairs with a large sheet of paper to work out as many facts as they can
8 = 56’ in a few minutes.
Review work done, recording facts on a ‘master’ diagram. Ensure that the following s
covered:
doubling
Examples: 14 × 8 = 112 (double 7); 7 × 16 = 112 (double 8)
halving
7 × 4 = 28 (halve 8); 7 × 2 = 14 (halve 4)
“What happens if I double one number and halve the other?”
(Answer: the answer stays the same, e.g 7 × 8 = 56 so 14 × 4 = 56)
using multiples of 10
7 × 80 = 560 (multiply 7 × 8 by 10)
70 × 8 =560 (multiply 7 × 8 by 10)
70 × 80 = 5600 (multiply 7 × 8 by 10 and 10 again, or by 100)
24 Unit 1A 3 Multiplication and division (1)
LB: p13
ethods they used Spend a few minutes every day revising mental facts
and developing mental strategies.
7 × 8 56
n for ‘7 × Vocabulary
strategies are
near multiple of 10: 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.
Opportunities for display!
Use ‘spider diagrams’ to show related facts.
using multiples of 10 when the calculation involves near multiples of 10
“How could I use 7 × 80 = 560 to find 7 × 81?”
(Answer: add another 7 so 7 × 81 = 560 + 7 = 567)
“How could I use 7 × 80 = 560 to find 7 × 79?”
(Answer: subtract 7 so 7 × 79 = 560 − 7 = 553)
Say that, “We refer to numbers like 79 and 81 as near multiples of ten – in th
Remind learners that building up a store of table facts and using mental strategie
Summary Check u
Learners revise their store of multiplication strategies to ‘“I kn
include doubling, halving and using multiples of 10.
(Answ
They understand how to adapt answers of multiplying by 10 “I kno
to multiplying by a near multiple of 10.
… 80
Notes on the Learner’s Book …8×
Mental strategies for multiplication (p13): learners develop their
mental strategies through oral work, best done frequently a little larger
at a time. The learner book therefore contains a limited number of … 80
examples.
I know
… 41
… 39
More activities
Mental mathematics (whole class)
Ensure that you do frequent oral activities to revise and consolidate the various s
Create spider diagrams (individuals or pairs)
As per the start of the core activity, start with a multiplication fact and derive ot
Games Book (ISBN 9781107667815)
Domino multiplication (p5) is a game for two or four players. Learners practise m
his case it means a ‘near to 80’, which is 10 × 8.”
es can often help work out multiplication calculations.
up!
now that 7 × 13 = 91. How can I work out 14 × 13?”
wer: double the answer as 14 is double 7.)
ow that 8 × 11 = 88. What is … ? How do you know?”
× 11? (Answer: 880; multiply answer by 10 as 80 is ten times larger than 8.)
× 110? (Answer: 880; multiply original answer by 10 because 110 is ten times
r than 11.)
× 110? (Answer: 8800; multiply either of the previous answers by 10.)
w that 40 × 7 = 280. What is …? How do you know?”’
× 7? (Answer: 287; add 7 to the original answer, so 40 × 7 = 280 + 7 = 287)
× 7? (Answer: 273; subtract 7 to the original answer, so 40 × 7 = 280 – 7 = 273)
strategies of mental multiplication.
ther facts.
multiplying multiples of 10 and 100.
Core activity 3.2: Mental multiplication strategies 25
True or false multiplicati
✂ cards
789 × 10 = 7890 610 ÷ 10
4350 ÷ 10 = 435 675 × 10 =
4010 × 10 = 410 866 × 10
1940 ÷ 10 = 194 4500 ÷ 10
5200 ÷ 100 = 520 302 × 100 =
Instructions on page 22
ion and division
= 61 407 × 100 = 40 070
= 6705 21 × 100 = 2100
= 860 150 × 10 = 1050
0 = 45 6000 ÷ 100 = 6
= 30 200 45 600 ÷ 100 = 456
Original Material © Cambridge University Press, 2014