Show learners the Rolling cars sheet. Calculate the mean of Ramp A results together
written, mental or calculator methods (Answer: 6.1). They should record the number
sentence needed to calculate the mean time for each ramp. Learners should comple
resource sheet and then discuss the different solutions learners have found for question
Small groups of learners should carry out their own experiment (you may wish to do this
separate session), either the same experiment as on the Rolling cars sheet or choose a
for example:
How far do different designs of paper aeroplane fly?
How long does it take for an object to drop to the ground with different designs of para
from an upstairs window?
The experiment must be repeatable and the data must be numerical for this activity. Gro
devise their own table to record the results and then use average and range to describe
conclusions.
Ask learners to revisit the sticky notes. They should refl ect on and, as necessary, imp
their defi nition of average.
Summary
Learners have found the mode, median, mean and range of sets of data.
They have used brackets to show the series of calculations necessary to work out t
set of data.
Notes on the Learner’s Book
Average and range (p80): learners calculate the mode, median, mean and range of se
data. They make sets of numbers that have a particular, mode, median and mean.
192 Unit 2C 21 Statistics
r, using Opportunities for display!
r
Groups can make a display of their experiment, carried
ete the including photographs of the experiment being
3. out, the table of results and written
s in a
another one, conclusions using average and range.
achute
oups should
e their
mprove
the mean of a Check up!
ets of
G enerate a set of data by throwing a dice five times.
Ask learners to find the mode, median, mean and
range for the set of data.
Tell learners that earlier you threw the die five times
and the mode was 2, the median was 3, the mean was
3.2 and the range was 3. Ask them to work out what
five numbers you threw.
More activities
About my class (groups)
Small groups of learners can investigate, and find the averages and range, for inf
nu mber of pets
nu mber of siblings
ag e of all siblings of learners in the class
shoe size
distance from home to school.
Games Book (ISBN 9781107667815)
The mean game (p106) is a game for two to four players. Players select numbers
formation about their classmates, e.g.
from a grid to try to make a set of data with a mean closest to 15.
Core activity 21.1: The three averages 193
Core activity 21.2: Using statistics to persuade
Resources: The Daily Gossip photocopy master (p200). 1 to 20 cards photocopy
(Optional: newspapers (or access to online news sites); access to online hotel review
Tell learners that they are going to investigate what is meant in statements that use av
provide information.
Give the class the following statements to discuss, one at a time:
Cars in our car park hold 4.3 people on average.
Ask learners to discuss with their partners, ‘How many people can travel in an average c
car park?’
On average, learners in Class 6 watch 1.8 films at the cinema per year.
Ask learners to discuss with their partners, ‘Does this mean that most learners in Class 6
cinema before the end the second time that they go in a year?’
Average number of legs for human beings is 1.99.
Ask learners to discuss with their partners, ‘How many legs does the average person ha
For each of these examples, discuss what the ‘average’ means. Tell learners that these
averages are all mean averages. The mean average is a very useful statistic for giving
understanding of a set of data, but it does not necessarily relate to a ‘real-life’ typical
example of a value in a set of data.
Display The Daily Gossip sheet with the headline ‘Half of all students get a low
average mark in National Tests!’ Tell learners that the intention of the newspaper s
that we should be shocked by how poor the students’ marks were in the test. A
learners, “Is this shocking?”
In small groups, learners investigate different marks, by generating a number each using
numbered 1 to 20 (card replaced each time), they might have got in a test with 20 marks
should work out the mean, mode and median averages and decide whether, with those
of them would have got lower than the average score. Ask groups to write a statement e
whether or not they should be shocked by the headline.
194 Unit 2C 21 Statistics
LB: p82
master (p201). Hotel reviews photocopy master (p202).
w sites; Changing the World with charts (CD-ROM).)
verages to Vocabulary
car in our statistics: the collection, organisation, presentation,
interpretation and analysis of data.
6 leave the
Look out for!
ave?’ Learners who need support organising their
e investigation. Suggest they follow this sequence:
an shuffle the 1 to 20 cards
l take one card, write down the number, then
wer than replace the card
story is continue taking and replacing cards until six
Ask the
numbers are written
g cards work out the mode, median and mode of the
s. They
marks, half numbers
explaining 3 is half of 6, so check if three of the numbers
written down are lower than each of the averages
repeat with further sets of six cards
discuss the outcomes of the investigation and try
to generalise the results.
Assign one of the hotels on the Hotel reviews sheet to each pair of learners. They s
case for visiting that hotel, rather than the others, by choosing data to represent in a
and presenting an argument using the statistics, including at least one measure of a
learners to think carefully about the scale on their graph, so that it will be most persu
not be misleading). Pairs can make a poster showing why their hotel is the best.
Summary
Learners have explored how statistics are used, and used statistics themselves to
persuasive argument.
Notes on the Learner’s Book
Using statistics (p82): learners read bar charts of rainfall and then calculate the
and mean averages, choosing the average that best supports an argument for each
work together in a group to produce a project. They use data collection and repre
written persuasive statements to argue for an improvement to the school or local
should argue a Look out for!
a graph or chart Learners who have difficulty starting on this
average. Remind
uasive (but it must challenging activity. Give them one piece of
information from the resource sheet about their
hotel e.g.
The Seaview Hotel has the highest average score
for location.
The Riverside Lodge has the highest average
score for service.
The Plaza Hotel has the highest average score
for rooms.
Opportunities for display!
Display the persuasive posters. Make arrow labels that
point to where and how learners have used graphs,
charts or statistics in their arguments.
form a Check up!
mode, median Ask learners to talk about the statistics they used in
h graph. Learners their poster and how they chose to represent the
resentation and data. Ask them to ref l e ct aloud about how they could
l area. improve how they communicate the data on the poster.
Core activity 21.2: Using statistics to persuade 195
More activities
Statistics in the news (individuals or pairs)
You will need newspapers (or access to online news sites).
Ask learners to look through print or online newspapers for stories with statistics that
them in the classroom to discuss.
More hotels (individuals or pairs)
You will need access to online hotel review sites.
Look at a hotel review website and use real data (reviews) to compare hotels/holiday/
Change the world (pairs or small groups)
You will need Changing the World with charts (CD-ROM).
Tell learners that they are going to explore how to use statistics to make a persuasive
Changing the World with charts, which has information about the 19th century statist
together to write some statements about the data presented in the chart and explain w
196 Unit 2C 21 Statistics
t are used to argue a point of view. Make a collection of these and display
/tourist attractions.
argument. In pairs, or small groups, ask learners to look closely at the chart on
tician and founder of modern nursing, Florence Nightingale. They should work
why this chart might persuade people to improve hospital conditions.
Finding mode and median averages
Here are nine sets of number cards:
56
20 14
9
48
1 12
1. Find the sets of data that matches these mode and median averages:
(a) mode 6, median 6
(b) mode 12, median 12
(c) mode 12, median 13
(d) mode 6, median 10
(e) mode 7, median 7
(f) mode 18, median 13
(g) mode 7, median 4
(h) mode, 18, median 18
Instructions on page 191 Original Material © Cambridge University Press, 2014
2. One of the sets of cards does not match the mode and median in any of
the questions.
What is the mode and the median of that set of cards?
3 . Choose two numbers between 1 and 20. The first number is the mode,
the second the median.
Challenge a partner to write a set of numbers that would have that
mode and median.
4. Investigate with a partner whether it is possible to choose a mode and
median where a set of numbers cannot be written.
Instructions on page 191 Original Material © Cambridge University Press, 2014
Rolling cars
You could use a calculator to complete this sheet.
Class 6 investigated a toy car
rolling down a ramp covered
with different materials. They
were testing which materials
had the most and least friction.
Ramp Trial 1 Trial 2 Trial 3 Trial 4 Mean
average
A 6.4 seconds 5.8 seconds 6.6 seconds 5.6 seconds
B 7.7 seconds 8.2 seconds 8.2 seconds 7.9 seconds
C 5.7 seconds 5.3 seconds 5.4 seconds 5.6 seconds
D 5.7 seconds 6.4 seconds 5.9 seconds 6.0 seconds
1. Calculate the mean average for each car ramp.
2. Write a sentence explaining which ramp was the quickest, using the
data to support your argument.
The class tried a fifth ramp. The mean average was calculated as 7.0
seconds. Trial 1 for the ramp was measured at 6.0 seconds.
3. Find three times that would make a mean average of 7.0 seconds.
Ramp Trial 1 Trial 2 Trial 3 Trial 4 Mean average
E 6.0 seconds 7.0 seconds
Instructions on page 192 Original Material © Cambridge University Press, 2014
The Daily
Half of all students get a lower than average mark in N
Everyone was shocked today as these frightening statistics
Instructions on page 194
Gossip
National tests!
s were revealed about our failing children.
Original Material © Cambridge University Press, 2014
✂ 1 to 2
1 2
6 7
11 12 1
16 17 1
Instructions on page 194
20 cards 4 5
10
3 15
20
89
13 14
18 19
Original Material © Cambridge University Press, 2014
Hotel reviews
Five families have reviewed each of these hotels. They have given a score
out of 5 for these different parts of their stay, with 1 as the lowest score
and 5 as the highest.
The Seaview Hotel
Value Location Entertainment Rooms Cleanliness Service
Family A 4 4 3 34 4
Family B 5 5 3 33 5
Family C 5 5 5 55 5
Family D 5 5 3 53 5
Family E 4 5 5 55 4
The Riverside Lodge
Value Location Entertainment Rooms Cleanliness Service
Family F 5 3 5 45 5
Family G 4 3 5 34 5
Family H 5 5 5 55 5
Family I 5 5 4 45 5
Family J 5 5 4 55 5
The Plaza Hotel
Value Location Entertainment Rooms Cleanliness Service
Family K 5 5 5 555
Family L 4 5 5 555
Family M 4 5 3 544
Family N 3 1 4 441
Family O 5 5 5 555
Instructions on page 195 Original Material © Cambridge University Press, 2014
2C 22 Probability
Quick reference
Core activity 22.1: Language of probability (Learner’s Book p84)
Learners use the language of probability to discuss the likelihood of events and
Learners create equally likely outcomes and compare the likelihood of an outcom
circumstances.
Prior learning Objectives* – please note that listed
covered fully across th
Describe the occurrence of familiar
events using the language of chance or 2C: Handling data (Proba
likelihood. 6Db1 – Use the language associa
and risk, including those w
*for NRICH activities mapped to the Cambrid
Vocabulary
probability • likelihood • chance • likely • unlikely • impossible • evens • equally likely
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
outcomes. Pro bab ility 6
me in different
Let’s investigate
What is the chance of spinning a ‘3’ on the
hexagonal spinner?
What is the chance of spinning a ‘6’ on the
pentagonal spinner?
Whatisthechanceof spinninga‘2’onthehexagonalspinner?
Vocabulary
chance:thelikelihood that aparticular outcomewill occur.
probability:the chancethataparticularoutcome willoccur,considering thetotal possible
outcomes.
likelihood:thechance thata particular outcome will occur.
impossible:anevent has no chanceof occurring,ithasa probability of0. certain:an event
that will de nitely occur,it has a probability of1. 1
unlikely:anevent
likely:anevent that will probablyhappen,it has aprobability between 2 and1.
what number will
that willprobably not 1
happen,ithas a probability between 0and 2 .
evens:aneventwherethere isthe samechance 1
of it occurringas not occurring,ithasa probabilityof 2 .
equallylikely: when thechance of differentoutcomesis the same,forexample,
show whenyouroll adice.
Asteroid could Road accident risk
hit Earth in 2036 doubles w hen children
A 100-metre wideasteroid will brush are 11 and 12 y ears old
past the Earthata distance of 30 000
km. There is littlechance of it hitting us Research suggest that the risk of being injured or killed
now, but a‘tiny but real’ likelihood
othnat it the roads rises at the point whenchildren who had
previously
may hit us in 2036. been accompanied to and from school begin tomake
their own way there and back.
84
d objectives might only be partially covered within any given chapter but are
he book when taken as a whole
ability)
ated with probability to discuss events, to assess likelihood
with equally likely outcomes.
dge Primary objectives, please visit www.cie.org.uk/cambridgeprimarymaths
• certain
Unit 2C 203
Core activity 22.1: Language of probability
Resources: Coloured cubes (red and yellow or any two different colours) and bags
scissors.)
Display the word ‘impossible’. Ask learners in pairs to think of sentences that includ
‘impossible’. Display one or two of these sentences that use ‘impossible’ meaning a
something happening. Repeat with the word ‘certain’.
Draw a probability line from ‘impossible’ to ‘certain’.
impossible
Name some events that are likely but not absolutely certain (e.g. ‘We will do PE this
unlikely but not completely impossible (e.g. ‘Our class will be on television this even
learners to debate where these events should be placed on the probability line by cons
they are to impossible or certain.
Ask learners in small groups to discuss, jot down and then share with the class, any oth
think of that describe how likely something is to happen.
Make a class list of the words and tell learners that they will add to this list as they w
activities. Ensure that ‘likely’ and ‘unlikely’ are on the list, and add these terms to the
(‘likely’ to the right of the centre and ‘unlikely’ to the left). Ask learners how the mid
half way between ‘impossible’ and ‘certain’, should be labelled. Label the line ‘evens
this means that, just like tossing a coin and calling ‘heads’, there is an even chance th
on heads as there is that it will not land on heads.
Give pairs of learners some red and yellow cubes and a bag. Ask them
to put three red cubes and one yellow cube in the bag. Ask two or three
learners to say whether it would be more likely for them to randomly
select a red cube or a yellow cube from the bag and why. In pairs learners
should take out, and replace, a cube from the bag ten times and record
how many times it is a red cube and how many times it is a yellow cube.
204 Unit 2C 22 Probability
LB: p84
s to put them in. (Optional: Dice chance (CD-ROM); a 1–6 dice;
de the word Vocabulary
zero chance of
chance: the likelihood that a particular outcome
certain will occur.
afternoon’) and probability: the chance that a particular outcome
ning’). Ask will occur, considering the total possible
sidering how close outcomes.
her words they can likelihood: the chance that a particular outcome
will occur.
work through the
e probability line impossible: an event has no chance of occurring,
ddle line, which is it has a probability of 0.
s’ and explain that
hat the coin will land certain: an event that will definitely occur, it has a
probability of 1.
likely: an event that will probably happen, it has a
1
probability between 2 and 1.
unlikely: an event that will probably not happen, it
has a probability between 0 and 1 .
2
evens: an event where there is the same
chance of it occurring as not occurring, it has a
probability of 1 .
2
Opportunities for display!
Display the probability line with events linked to
different parts of the line. Encourage learners to
add to the display during the session.
Ask each pair to say how many red cubes and yellow cubes they pulled from th
on the board. Discuss whether the results of their experiment match how likely
be to pull a red or yellow cube from the bag. Ask three or four learners to
probability line they would put the probability of taking a red cube and the p
yellow cube.
Ask learners to change the cubes in their bag so that there is an even chance of r
a yellow cube. Pairs should share their solution with the class. Make a list of the
by the class to make the even chance, all the bags should have the same number
cubes, but this could be any number.
Ask learners to change the cubes in their bag so that it is more likely that they w
cube. Collect each pair’s solution. Ask learners to indicate on the probability line
is that they will take a yellow cube from their bag. (This should be somewhere b
‘evens’.)
Give two learners these bags of cubes:
Bag 1 has 3 red cubes and 6 yellow cubes.
Bag 2 has 2 red cubes and 2 yellow cubes.
The two learners should show the class the contents of their bag. Tell the learner
am more likely to take a red cube than you because I have 3 red cubes in my bag
in your bag.”
Tell the learner with Bag 2 to say, “I am more likely to take a red cube than you b
cubes in my bag that are not red, I have 2 and you have 6.”
Ask learners to discuss in groups and give reasons as to which learner is correct ab
likely to take a red cube from their bag.
Show learners that in Bag 2 there is an even chance of taking a red, or not red cu
would be placed at the ‘evens’ point on the probability line. In Bag 1 there are tw
cubes as red cubes, so taking a red cube is ‘unlikely’ and the probability is lower
taking a red cube from Bag 2.
Divide the class into two groups. The learner with Bag 1with one group, the learne
other group. The groups should experiment with the probability of pulling a red cub
taking, recording and replace a cube from the bag 30 times.
Discuss the outcomes as a class.
he bag and make a list Look out for!
they thought it would
suggest where on the Learners who try to use numbers to describe the
probability of taking a
probability. It is not necessary for learners, at this
removing a red cube or
e different options used stage, to assign a fraction to a probability but, if it
of red cubes as yellow
is appropriate for the learners, describe the chance
will remove a yellow
e where the probability of taking a yellow cube as 1 in 4, and the chance of
between ‘certain’ and
taking a red cube as 3 in 4. Comment that 1 4 of
r with Bag 1 to say, “I the cubes in the bag are yellow, and 3 of the cubes
g, and you only have 2 4
in the bag are red.
because there are less
Look out for!
Learners who may be confused between things
that are highly unlikely and impossible, and also
highly likely and certain. Explain that it is
possible (not impossible) that one group might
have pulled 10 yellow cubes from the bag and no
red cubes, but highly unlikely. Every time they
pulled a cube from the bag there was a chance of
pulling out a yellow cube.
bout who is more Opportunities for display!
ube, so this probability Display groups of cubes in transparent bags.
wice as many yellow
r on the line than Learners should write probability statements to go
with the groups of cubes e.g. ‘It is impossible to
er with Bag 2 with the pull a blue cube from this bag.’ Or, ‘It is likely that I
ube from each bag by will pull a green cube from this bag.’
Core activity 22.1: Language of probability 205
Summary
Learners have used the language of probability to discuss the likelihood of events
and outcomes.
Learners have created outcomes that have an even probability and compared the
likelihood of an outcome in different circumstances.
Notes on the Learner’s Book
Probability (p84): learners find and record probability language in newspaper reports
decide which bag has the best and worst chances of pulling out a particular colour m
and they investigate whether a game is fair.
More activities
Display (groups or whole class)
Make a display showing events that are impossible, unlikely, equally likely, likely and
Is it fair? (groups)
In small groups, learners can design and make games that are either fair or unfair. Ot
unlikely, equally likely, likely or certain for each player to win.
Roll the dice (pairs)
You will need Dice chance (CD-ROM). A1-6 dice. Scissors.
Give learners a dice and a copy of the Dice chance sheet. Learners should discuss th
sheet together. Encourage them to look at the real dice to help them decide the proba
statements difficult encourage them to use the statements in question 1 as templates.
Games Book (ISBN 9781107667815)
The equally likely game (p110) is a game for three to eight players. Players each have
describes three of the six numbers, so when the dice is rolled they have an equal chan
matches their card, learners move their counter forward. The first learner to get their
206 Unit 2C 22 Probability
Check up!
Ask learners to match these statements to the
probabilities:
s. They Statement Probability
marble impossible
Picking a red card from a pack of
playing cards.
The sun will set this evening. unlikely
Cherries growing on an apple tree. even chance
Winning a lottery. likely
Rolling a number higher than 2 on a dice. certain
d certain.
ther groups should play the games and decide if it is impossible,
he questions and vocabulary on the resource sheet in pairs and complete the
ability of throwing a number, or set of numbers. If learners find writing the
e a card describing some of the numbers on a 6-sided dice. Every card
nce of getting a number that matches their card. When the number on the dice
counter to the end of the board is the winner.
3A 23 The number system (3)
Quick reference
Core activity 23.1: The number system (2) (Learner’s book
p88)
Counting on and back in decimal steps.
Core activity 23.2: History of number (2)
Learners explore the Hindu/Arabic number system.
Prior learning Objectives* – please note that listed objectives m
across the book when taken as a w
This chapter builds on work
on the number system 3A: Numbers and the number sys
done earlier in the stage.
It includes work on whole 6Nn1 – Count on and back in (e.g.) 1s, 0.1
numbers and decimals.
3
6Nn5 – Multiply and divide decimals by 10 o
6Nn9 – Round a number with two decimal p
6Nn11 – Order and compare positive number
6Nn14 – Order numbers with up to two decim
6Nn16 – Recognise and use decimals with u
6Nn20 – Recognise historical origins of our n
3A: Problem solving (Using unders
6Ps3 – Use logical reasoning to explore and
*for NRICH activities mapped to the Cambridge Primary ob
Vocabulary
No new vocabulary.
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
Number
The nu mber system (2) 3
Let’s investigate Be sy stema tic .
U se the se f o ur car ds.
Ma ke a s ma ny n um bers as p o s sib le
between 0 a nd 4 0 .
You m us t u se a ll f o ur car ds eac h time .
1 Place these numbers in order of s ize s tarting with the smalle st.
(a) 7.3 3. 7 0. 37 7. 03 3.0 3
(b) 4 50 45 0 54 0 45 54 4 05 450 00 5 4 50 4 05
2 Use < a nd > to sho w the relationsh ip between the se pa irs of numbers
(a) 7.3 4 and 7. 43 (b) 1.23 and 1.2
(c) 0.34 a nd 0.0 5 (d) 1. 78 an d 1. 9
3 The tallest s tructure in the world is the Burj Khalifa in Duba i. It is
829. 84 metres hig h.
What is this he igh t to the nearest who le n umber?
4 Haibo p ut a number in her calculator. She multiplies the n umber by 10 a nd
the calculator sh ows:
What number did Haib o pu t in the calculator?
5 Write the number tha t is three ones, four tenths an d ve hundredths.
6 In the number 65 .43 which digit is in the tenths p lace?
88
U nit 3 A : C o r e a ct iv it y 2 3 . 1 The n u m be r s y st e m ( 2)
might only be partially covered within any given chapter but are covered fully
whole
stem
1s and repeated steps of whole numbers (and through zero).
or 100 (answers to two decimal places for division).
places to the nearest tenth or to whole number.
rs to one million, negative numbers to an approximate level.
mal places (including different numbers of places).
up to three places in the context of measurement.
number system; understanding something of its development.
standing and strategies in solving problems)
d solve number problems and puzzles.
bjectives, please visit www.cie.org.uk/cambridgeprimarymaths
Unit 3A 207
Core activity 23.1: The number system
(2)
Resources: Decimal arrow cards photocopy master (p213); one set for each learner.
master (p214); one for each learner. (Optional: 0–9 spinner (CD-ROM).)
Ask learners to count on and back in repeated steps, for example:
“Start at 1.5 and count on in steps of 0.1
Start at 2.4 and count on in steps of 0.2
Start at 4.5 and count back in steps of 0.5”
Choose other starting numbers and count on and back in decimals and whole number
Draw a zero to ten number line.
0123456789
Draw a circle round the section from 0 to 1 to represent a magnifying glass and anoth
0 to 1 under it (line should be long enough to be able to fit in answers to the followin
“Which numbers lie between 0 and 1?” (Fill in the tenths: 0.1, 0.2, 0.3 . . .)
“Which numbers lie between 0 and 0.1?” (Fill in hundredths: 0.01, 0.02, 0.03. . .
Give learners a set of decimal arrow cards each or between pairs. Show how you can
using the cards. For example 0.47 can be made from 0 + 0.4 + 0.07:
0
.4 0 .4 7
.0 7
Assemble with the card showing the smallest value at the bottom and arrows overlapp
Ask learners to use the cards to show:
208 Unit 3A 23 The number system (3)
LB: p88
. The Number System record sheet (self assessment sheet) photocopy
rs.
10
her, longer number line from
ng questions).
.)
n make a decimal number
pping.
“a number between 0 and 1”
“a number between 7 and 8”
“the largest number using your cards that is between 5 and 6” (Answer: 5.99)
“a number between 4.2 and 4.3” (Answer: any of, 4.21, 4.22 ….. 4.29)
“a number that is 10 × 0.06” (Answer: 0.6)
“a number that is the answer to 401 ÷ 100.” (Answer: 4.01)
Display a list of numbers.
1.13 3.91 10.1 1.3 3.03 3.0
Ask learners:
“W hich number is the smallest? How did you work it out?” (Answer: 1.13)
“W hich number is the largest? How did you work it out?” (Answer: 10.1)
“W rite the numbers in order starting with the smallest. What could you do to h
(Answer: 1.13, 1.3, 3.03, 3.09, 3.91, 10.1)
Remind learners that it helps to use zeros so that all decimals have the same num
decimal places. For example re-write 10.1 as 10.10. Remind them that these num
exactly the same.
Ask learners to work individually on the Number System record sheet self-assessm
You can use this to assess the learners’ understanding of the number work in this s
earlier number sections in Stage 6. This includes knowledge of counting on and ba
rounding, estimating and ordering.
Review (either with the whole class, group or individually) the responses to the a
sheet. Recap any problem areas that may be identified from the assessment.
) Look out for!
09 Learners who struggle with the difference between,
for example, 3.09 and 3.9. Tell them to write both
help you?” decimals with two decimal places so it is easier to
mber of see which is larger.
mbers are
3.9 = 3.90
ment sheet. 3.9 > 3.09
section and
ack, place value,
assessment
Core activity 23.1: The number system (2) 209
Summary
Learners count on and back in decimal steps.
Th ey can identify a decimal number in-between two given decimals.
Th ey assess their own progress and confidence with many of the whole numb
topics taught (the results of which can be used by the teacher to plan further t
decimal arrow cards to consolidate work on place value, ordering and rounding).
Notes on the Learner’s Book
The number system (2) (p88): learners haves an opportunity to practise working with
whole numbers and decimals. The questions are not presented in any particular order,
sometimes give additional support. Teachers may wish to use the results of the self a
sheet to prioritise certain questions for individual learners.
More activities
Rounding decimals
(pairs)
You will need 0–9 spinner (CD-ROM).
Each player draws a grid.
12345
6 7 8 9 10
Take turns to spin the spinner three times, then use the digits to make a number with
write it on your grid.
Example: digits 5, 6 and 4 could make 4.56 which rounds to 5. Write 4.56 on the grid
The first person to complete their grid wins.
Games Book (ISBN 9781107667815)
Place your numbers now (p41) are two games for two players. Game 1 focuses on ord
rounding decimals.
210 Unit 3A 23 The number system (3)
er and decimal Check up!
tasks using the
“Say a number between zero and 0.1; between zero
h and 0.01.”
, as this can
assessment “G ive me a number which lies between 5.15 and 5.18.”
“P ut these numbers in order, starting with the
smallest:
5.1 5.05 0.55 0.5 0.15”
“Give me a number with two decimal places that
rounds to five.”
two decimal places. Round the number to the nearest whole number and
d next to 5
dering numbers to create a number in a given range. Game 2 focuses on
Blank page 211
Core activity 23.2: History of number (2)
Resources: Bengali numbers photocopy master (p216)
Explore with learners the Hindu/Arabic number system. Ask them to do some resear
provide them with the following information:
The number system we use was developed in India 2000 years ago and was introdu
nations by Arab traders. The marks we use to represent numbers are called numeral
They are made up using symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0 which are known as d
Example: The digits 3 and 8 form the numeral 38 for the number ‘thirty-eight’ and
for the number ‘eighty-three’.
This system is more useful than the ancient Egyptian system because:
it uses only 10 digits to construct all numbers
it uses the digit 0 or zero to show an empty place
it has a place value system where digits represent different numbers when placed
place value columns.
Look at patterns on the Bengali a hundred square (part of the Bengali numbers photo
master 216 on page 19) focusing on relationships in the rows and columns, for examp
. . all have a zero in the units place and 11, 12, 13, 14 . . . all have a 1 in the tens place
Activity
The Bengali hundred requires learners to use their knowledge of place value to put it
212 Unit 3A 23 The number system (3)
ch and/or
uced to European
ls.
digits.
d the numeral 83
d in different
ocopy
ple, 10, 20, 30, 40 .
e.
t together.
Decimal Arrow Cards
✂
.0 0 .0 0
.0 1 .1 1
.0 2 .2 2
.0 3 .3 3
.0 4 .4 4
.0 5 .5 5
.0 6 .6 6
.0 7 .7 7
.0 8 .8 8
.0 9 .9 9
Instructions on page 208 Original Material © Cambridge University Press, 2014
Name: ………………………………………………….
The Number System record sheet
Look at each of the statements below.
Draw a smiley face by those statements that you feel confident about.
The questions will help you to decide.
I can do this on my own
Whole Numbers
6Nn1 I can count forwards and backwards in repeated steps.
Count on in 5s from 23.
6Nn2 I know what each digit represents in whole numbers up to
a million.
Which number is fifty thousand and five?
5005 50005 5000005 5000005
6Nn8 I can round whole numbers to the nearest 10, 100 or
1000.
Round 678765 to the nearest 10, to the nearest 100 and to
the nearest 1000.
6Nn11 I can order and compare positive numbers to 1 million.
Put these numbers in order, smallest first.
400400 440404 40404 44444 414040
6Nn12 I use correctly the symbols <, > and =
Use <, > or = to make this number sentence true:
50000 half a million
6Nn13 I can estimate where 4-digit numbers lie on an empty 0 –
10000 number line.
Here is a number line from 0 to 10000:
0 10000
Draw an arrow () to show the position of 3500.
Instructions on page 209 Original Material © Cambridge University Press, 2014
Decimals
6Nn1 I can count on and back in steps of 0.1.
6Nn3 I know what each digit represents in one- and two-place
decimal numbers.
What is the value of the digit 5 in these numbers?
5.04 1.05 3.45 4.54
6Nn5 I can multiply and divide decimals by 10 and 100.
Here are four number cards.
A B C D
3.330 33.03 33.3 333
6Nn9 Which card shows the number ten times as big as 3.33?
I can round a number with two decimal places to the
nearest tenth or whole number.
Draw lines to show each number rounded to the nearest
tenth.
The first one has been done for you.
to the nearest tenth
3.3
3.52 3.4
3.5
3.77
3.6
3.35 3.7
3.8
6Nn14 I can order numbers with up to two decimal places.
Put these numbers in order, smallest first:
0.45 0.55 4.55 4.5 5.45 4.05 5.4
Instructions on page 209 Original Material © Cambridge University Press, 2014
Bengali numbers
Here are the Bengali numerals for 1, 10 and 100.
Number 1 10 100
Bengali numeral
Cut along lines to make a jigsaw from the Bengali hundred square. Make
the pieces interesting shapes, for example:
Then reassemble the jigsaw using your knowledge of place value to help you.
Bengali hundred square
Instructions on page 212 Original Material © Cambridge University Press, 2014
3A 24 Mental Strategies
Quick reference
Core activity 24.1: Addition and subtraction (1) (Learner’s book p90)
Learners refine their work on mental strategies for addition and subtraction with
working with decimals.
Core activity 24.2: Multiplication and division (Learner’s book p91)
Learners refine their work on mental strategies for multiplication and division w
working with decimals.
Prior learning Objectives* – please note that listed objectives m
taken as a whole
This chapter builds on work
done earlier in the stage. 3A: Calculation (Mental strategies)
6Nc1 – Recall addition/subtraction facts for nu
6Nc2 – Derive quickly pairs of one-place deci
+ 0.22.
6Nc3 – Know and apply tests of divisibility by
6Nc4 – Use place value and number facts to
of 10 and pairs of decimals, e.g. 560 +
6Nc5 – Add/subtract near multiples of one wh
6Nc6 – Add/subtract a near multiple of 10, 10
e.g. 3127 + 4998, 5678 – 1996.
6Nc7 – Use place value and multiplication fac
6Nc9 – Double quickly any two-digit number e
3A: Problem solving (Using techniqu
6Pt1 – Choose appropriate and efficient men
multiplication or division.
*for NRICH activities mapped to the Cambridge Primary ob
Vocabulary
No new vocabulary.
Cambridge Primary Mathematics 6 © Cambridge University Press 2014
Mental strategies for add it io n and Mental strategies for multiplication and
subtraction (2) div ision
Let’s investigate 15 Let’s investigate 1 T here are n o ‘c arry ’ num ber s.
U se the num ber s 1 to 8 s o tha t Rep la ce e ac h sy m bo l w ith a dig it
eac h s id e o f the sq uare a dd s u p to ma ke the d iv is ion correc t. 2 A ll the sq uares are the sa me nu mber
to 1 5 . an d a ll the tr ia ng le s are th e s ame
U se dig it card sth aat t num ber.
can be m ov ed aro u nd.
h an emphasis on 1 Write the numbers 1, 2, 3, 4, 5, 6 an d 7 in the circles 3 Us e the re latio ns hip b etween mu ltip lica tion a n d
with an emphasis on so that each line add s up to 12. U se each number ??
only once. div isio n to h elp y o u.
? ?? Answer these q uestio ns mentally as qu ickly as y ou ca n.
1 Calculate 1.3 multiplied by 4.
2 D ivide 4 .2 by 6.
3 D ivide 3 .5 by 10.
4 Do ub le 1 5.5.
??
5 What is 10 .5 div ide d by 5 ?
2 Use the numbers 1, 2, 3 , 4 a nd 5 to 12 6 Multiply 3. 5 by 4.
complete this star pattern. All lines of
7 What is half of 1 .6 ?
numbers must add u p to 2 4.
8 What is the product of 7 and 0. 6?
9 D ivide 7 .5 by 3.
10 ? ? 9 10 W hat is d ou ble 1.8 ?
?? 11 Find the m iss in g numbers.
?
(a) 0. 7 9 ? (b) 0 .7 8 ? (c) 0. 7 5 ?
86 (f) ? 6 4 .2
90 (d) 0.2 ? 1. 8 (e) ? 9 5.4
12 Find the m iss in g numbers.
(a) 7. 2 6 ? (b) 4 .8 8 ? (c) 8. 1 ? 0. 9
(f) ? 6 0 .8
(d) 3.6 ? 0. 4 (e) ? 7 0.2
91
might only be partially covered within any given chapter but are covered fully across the book when
umbers to 20 and pairs of one-place decimals with a total of 1, e.g. 0.4 + 0.6.
imals totalling 10, e.g. 7.8 and 2.2, and two-place decimals totalling 1, e.g. 0.78
2, 4, 5, 10, 25 and 100.
add or subtract two-digit whole numbers and to add or subtract three-digit multiples
+ 270, 2.6 + 2.7, 0.78 + 0.23.
hen adding numbers with one decimal place, e.g. 5.6 + 2.9, 13.5 – 2.1.
00 or 1000, or a near whole unit of money, and adjust,
cts to multiply/divide mentally, e.g. 0.8 × 7, 4.8 ÷ 6.
e.g. 78, 7.8, 0.78; derive the corresponding halves.
ues and skills in solving mathematical problems)
ntal or written strategies to carry out a calculation involving addition, subtraction,
bjectives, please visit www.cie.org.uk/cambridgeprimarymaths
Unit 3A 217
Core activity 24.1: Addition and subtraction (1)
Resources: Addition and Subtraction 1 and 2 photocopy master (p224). (Optional
cards (without the 0) and Decimal cards (CD-ROM).)
Learners work through the activity Addition and subtraction 1 individually, attempting to
Work through each question in turn asking how they found the missing number using
“How did you work it out?”
“Which operations have you used?”
“Is there another way to do this?”
“Are there any particular numbers that made the calculation easy/hard? For exam
57 and 43 are complements of 100 in question 3
30 + 50 = 80, then double 80 = 160 in question 4
5+ 3 + 2 = 10 in the units parts of question 5.”
(Answers: 7, 60, 26, 50,
79)
Work with learners to list, then display useful mental strategies for addition and subtr
reordering: 54 + 29 + 46 = 54 + 46 + 29
partitioning using multiples of 10 and 100: 284 − 153 = 284 − 100 − 50 − 3
partitioning bridging through multiples of 10: 297 + 148 = 297 + 3 + 145
using near doubles: 321 + 487 = double 400 − 79 + 87
Learners work through the activity Addition and subtraction 2 attempting to answer each
Review as for work with whole numbers, emphasising that the same strategies can als
“Q uestion 1: re-order so 4.6 and 2.4 are together
Qu estion 2: partition 4.8 − 3 − 0.7
Qu estion 3: bridge through multiple of 10 to give 0.7 + 0.3 + 0.05
Qu estion 4: compensating 5.8 + 5 − 0.1
Qu estion 5: near doubles to give double 3.2 + 0.1.”
Emphasise that there is no single correct method for any calculation.
(Answers: 10.7, 1.1, 1.05, 10.7, 3.3)
218 Unit 3A 24 Mental Strategies
LB: p90
l: 0–9 spinners and Dice difference scoring grids (CD-ROM); 0–9 digit
answer each question mentally.
g these questions as prompts:
mple:
raction, for example: Opportunities for display!
Display examples of different
h question mentally. mental strategies and then extend
the display in future lessons.
so be used for decimals:
Summary
Learners revise and consolidate a range of mental strategies for addition and s
They understand that there are different ways of working out an answer.
Notes on the Learner’s Book
Mental strategies for addition and subtraction (2) (p90): mental strategies are be
using activities with the whole class, not by using a printed page. The activities p
book are restricted to problem solving ones.
More activities
Decimal addition (whole class)
Write a set of decimal numbers on the board, for example:
4.6 2.7 3.4 7.5 3.3 2.1 2.6 1.8 2.2 3.9
Ask learners to find pairs that make a whole number.
Extend the activity to finding pairs of numbers that make a whole number of ten
example 0.07 and 0.03 make 0.1)
0.07 0.06 0.03 0.05 0.04 0.09 0.01 0.05
Dice difference (pairs)
You will need two 0–9 spinners (CD-ROM). Dice difference scoring grids (CD
Player one spins twice and makes a decimal number. For example, if the spinner
first number. Repeat to form the second number. Player
two does the same.
Both players find the difference between the two numbers and record in the diffe
The winner of the round is the player with the smallest answer. That player is aw
point.
Continue playing further rounds. The overall winner is the player with most poin
The game can be varied so that players add their two numbers.
The game can be altered so the winner is the player with the largest amount.
subtraction. Check up!
est developed by “W hat is 0.78 + 0.23? How did you work out your
provided in the answer?”
“W hat is 5678 − 1996? How did you work out your
answer?”
Player 1 Player 2
nths (for First number
Second number
Difference
D-ROM).
r shows 1 and then 6 then the number could be 1.6 or 6.1. Record as the
erence box.
warded one
nts.
Core activity 24.1: Addition and subtraction (1) 219
Compensating cards
(pairs)
You will need a set of 0–9 digit cards (without the 0) photocopy master (CD-ROM
Place the 1 to 9 number cards and the decimal cards face down (in different piles).
Players take it in turns to choose a whole number and then a decimal card.
Add the two numbers and explain method used. Partner checks the answer.
Alternative versions: Subtract numbers or choose two decimal number cards.
Games Book (ISBN 9781107667815)
Sum to one snap (p41) is a game for small groups of two to four players. It provides p
add to one.
220 Unit 3A 24 Mental Strategies