Download Document - Eldwick Primary School

6
Year
Helping your child with Maths
Book 1 2015
Please look after this booklet
Book1
1
Theme
Pictograms and Grouped Data
Your child will already have been introduced to bar graphs when they were younger but in Years 5 and 6 they will work
with more complex graphs including line graphs and pie charts, demanding a more sophisticated understanding of data
and how it can be represented.
A Pictogram is simply a way of representing a number of items using pictures. For example:
In the picture on the left we have a pictorial
representation of the number of pets counted in a
survey. The person has decided that each
individual picture of an animal = 2 pets.
Therefore the results of the survey are as follows:
Dogs = 7
Rabbits = 4
Cats = 9
Other = 4
Can you work out the results if each picture stood
for 10 animals?
Grouped data, quite simply, is data that has been sorted into ranges of values rather than
recording each single value.
Example:
Let’s say we decided to carry out a survey of the ages of people at a football match. As the people
left we would ask their age and record their answers in a tally chart like this:
AGE
FREQUENCY
Below 10
llll llll llll
14
10-19
llll llll llll llll llll llll lll
33
20-29
llll llll llll llll llll llll llll llll llll llll llll
54
30+
llll llll llll llll
19
1
This data would then be plotted onto a bar graph or pie chart. As we have grouped the data there
would only be four bars in the bar graph or four slices in the pie. Why not get a piece of paper and
draw the graph? How could you accurately draw a pie chart to represent this data? Clue 1:
Protractor! Clue 2: 120 people watched the football match and a circular pie is equivalent to an
angle of 360°. Clue 3: 3 x 12 =36. Still stuck? Ask your teacher!
That’s all there is to it really. Of course the main skill expected of Year 6 pupils is to be able to
interpret information from graphs including line graphs and pie charts. In Year 4 and 5 children
have become confident at carrying out simple surveys and representing the information in a graph.
In Year 6 children need to be a little more ‘flexible’ in the way they tackle graph work. Certainly in
recent years KS2 SATs questions have featured a wide variety of graphs, demanding a good
understanding of different scales and the ability to interpret the data presented. For a ‘different’
type of data handling question, access the electronic version of this booklet on the school website
and click here.
NOTES:
2
2
Theme
Understanding Metric and Imperial Units of
Length and Weight
This section looks at different units of length and weight. The emphasis is to gain a good understanding of how to convert
between different metric units of measurement.
LENGTH
Facts to LEARN:
1m = 100cm (and so 100 is the ‘key’ number when converting between the two)
1m = 1000mm (and so 1000 is the ‘key’ number when converting between the two)
1cm = 10mm (and so 10 is the ‘key’ number when converting between the two)
1km = 1000m (and so 1000 is the ‘key’ number when converting between the two)
Once these facts have been ‘internalised’ then converting from one unit to another is
straightforward – as long as you remember the ‘key’ number!
Converting between Metres and Centimetres (key number = 100)
To change m to cm simply MULTIPLY by 100. For example:
1.23m = 1.23 x 100 = 123cm
To change cm to m simply DIVIDE by 100. For example:
1056cm = 1056 ÷ 100 = 10.56m
Converting between Metres and Millimetres (key number = 1000)
To change m to mm simply MULTIPLY by 1000. For example:
9.75m = 9.75 x 1000 = 9750mm
To change mm to m simply DIVIDE by 1000. For example:
3
234mm = 234 ÷ 1000 = 0.234m
Converting between Centimetres and Millimetres (key number = 10)
To change cm to mm simply MULTIPLY by 10. For example:
1.5cm = 1.5 x 10 = 15mm
To change mm to cm simply DIVIDE by 10. For example:
246mm = 246 ÷ 10 = 24.6cm
Converting between Kilometres and Metres (key number = 1000)
To change km to m simply MULTIPLY by 1000. For example:
3.06km = 3.06 x 1000 = 3060m
To change m to km simply DIVIDE by 1000. For example:
1029m = 1029 ÷ 1000 = 1.029km
Imperial Units of Length
Some familiarity with imperial measures is expected. Children should be able to understand and
use approximate equivalences between metric units and imperial units such as inches, pounds and
pints. They should also be able to convert between miles and kilometres.
If there is a question in SATs it is usually something like:
“If 4km is equivalent to 2.5 miles, how many km are there in 17.5 miles?”
or
0_______________________________________________________________50 miles
0_______________________________________________________________80km
“Approximately how many kilometres are equivalent to 35 miles?”
WEIGHT
Facts to LEARN:
1kg = 1000g (and so 1000 is the ‘key’ number when converting between the two)
1 Tonne = 1000kg (and so 1000 is the ‘key’ number when converting between the two)
4
We are helped here by the fact that the ‘key’ number is always 1000, and therefore there is less
chance for confusion!
Converting between Kilograms and grams (key number = 1000)
To change kg into g simply MULTIPLY by 1000. For example:
1.15kg = 1.15 x 1000 = 1150g
To change g into kg simply DIVIDE by 1000. For example:
874g = 874 ÷ 1000 = 0.874kg
Converting between Tonnes and Kilograms (key number = 1000)
To change tonnes into kg simply MULTIPLY by 1000. For example:
1.03 Tonnes = 1.03 x 1000 = 1030kg
To change kg into tones simply DIVIDE by 1000. For example:
1607kg = 1607 ÷ 1000 = 1.607tonnes
Imperial Units of Weight
If there is a question in SATs it is usually something like:
“If 2.2lb(pounds) = 1kg, how many pounds are equivalent to 12kg?”
or
“If 1oz(ounce) = 30g and there are 16oz in 1lb, what weight in lbs is approximately equivalent to
5kg?”
Clearly if children are to convert comfortably between different METRIC units of measure then
the ability to multiply/divide by 10/100/1000 is crucial. When your child was in Year 5 they learnt
how to do this by moving the digits to the LEFT (multiplying) or to the RIGHT (dividing). It works
like this:
Multiplying by 10 – move the digits ONE place left like this:
TH
H
T
1
U
.
t
1
.
2
2
.
0
So -
1.2 x 10 = 12
5
Multiplying by 100 – move the digits TWO places left like this:
TH
1
H
3
T
U
.
t
h
1
3
.
2
5
2
5
.
So -
13.25 x 100 = 1325
Multiplying by 1000 – move the digits THREE places left like this:
TTH TH
1
6
H
0
T
U
.
t
h
1
6
.
0
5
5
0
.
So -
16.05 x 1000 = 16050
Notice here we need to put a zero in the units column to make the answer correct
Dividing by 10 – move the digits ONE place right like this:
TH
H
T
U
.
2
5
6
.
2
5
.
t
So -
256 ÷ 10 = 25.6
6
6
Dividing by 100 – move the digits TWO places right like this:
TH
H
T
U
.
1
6
.
0
.
t
h
So -
16 ÷ 100 = 0.16
1
6
Dividing by 1000 – move the digits THREE places right like this:
TH
2
H
T
0
5
U
.
0
.
2
.
t
h
th
So -
2050 ÷ 1000 = 2.05
0
5
0
Notice that we do not need to put the final zero in the answer.
Need more practice? Use the grid below to do just that!
TTH
TH
H
T
U
.
.
.
.
.
.
.
.
.
.
7
t
h
th
3
Theme
Angles
Measuring and Estimating Angles
Angles are measured in degrees. Children have already used protractors in Year 5, but firstly we
need to make sure they can remember how to use them! It is also important that children are able
to estimate the size of any given angle. In order to help them ‘estimate’ intelligently, they need to
first of all recognize angles as ‘acute’, ‘obtuse’ or ‘reflex’.
Acute: angles that are LESS than 90˚ are called acute angles.
8
An angle of exactly 90˚ is called a ‘right-angle’ and looks like this:
Obtuse: angles that are GREATER than 90˚ but LESS than 180˚ are called obtuse angles.
An angle of exactly 180˚ is a straight line! I don’t think we need to draw one of those for you!
Reflex: angles that are GREATER than 180˚ but less than 360˚ are called reflex.
9
An angle of 360˚ is a full turn or a circle – I don’t think we need to draw you one of those either!
Using a Protractor
Protractors can be 180˚ or 360˚. Your child will be taught how to use both. Some children prefer
one to the other. In their tests they can choose to use the one they are most comfortable with.
Protractors can be used to measure angles and draw angles. Your child will be shown how to do this.
For a good ‘on-line’ tutorial on how to use a protractor, find this page on the electronic version of
this booklet on the school website and click here . When the tutorial loads choose option 3 –
‘Measuring Angles’.
10
Calculating Angles
Children in Year 6 also need to be able to calculate angles using the rules they know about the
number of degrees in a triangle, on a straight line etc. Again it would be really useful if your child
could learn by heart these few important rules:
(i)
Angles that meet at a point total 360˚
The simplest way to think about
this is to look at a cross. Here we
have 4 right angles meeting at a
point, and 4 x 90˚ = 360˚
Here is an example where none of the angles are right angles:
140˚ + 87˚ + 80˚ + 53˚ = 360˚
So in the following example we can calculate the missing angle by applying the rule:
110˚ + 75˚ + 63˚ + 50˚ = 298˚
298˚ + Angle C = 360˚
Angle C = 360˚ - 298˚ = 62˚
11
Looking at angles in relation to a clock face is quite a common way of testing a Year 6 pupil’s
understanding of this topic, so it’s probably worth having a quick look at this now. Here is a clock
face showing 4 o’clock. Can you calculate the obtuse angle that is made by the two hands?
The key to this is knowing that each section of the clock face must equal 30˚. (12 x 30 = 360˚)
So the answer to the question is 4 x 30˚ = 120˚. Now try this one: Imagine the time has advanced
to 4.45 – How many degrees will the minute hand have moved through?
Answer: The minute hand will turn through 9 ‘sections’ of the clock face to reach 4.45, and
therefore the angle = 9 x 30˚ = 270˚
(ii)
The three angles of a triangle total 180˚
All triangles contain three angles that total 180˚. So if we know two of the angles we can easily
calculate the third:
In this example we can see that Angle A =
40˚ and Angle B = 60˚. So Angle C must =
80˚ because 40˚ + 60˚ + 80˚ = 180˚
Equilateral Triangles
Equilateral triangles are triangles where all the sides (and therefore the angles) are all the same
size. So each angle of an equilateral triangle will equal 60˚. (60 + 60 + 60 = 180)
Isosceles Triangles
Isosceles triangles are triangles where two of the sides (and therefore two of the angles) are
the same size. Look at this example of an isosceles triangle:
12
Notice that the two equal angles are at the end
of the equal sides. 72˚ + 72˚ + 36˚ = 180˚.
Notice the = sign on the two equal sides. If you
know that a triangle is ‘isosceles’ then you only
need to know one of the angles to be able to work
out the other two. Look at the next example.
Right angled IsoscelesTriangle
C
42˚
A
B
In this isosceles triangle sides A and B
are of equal length and one of the
angles is 42˚. Can you work out the
other two angles?
Solution:
If A and B are equal in size then the angle between sides C and B must be equal to the angle
between A and C (ie 42˚). So if we add these two angles together we get 42 + 42 = 84.
Therefore the third angle must be 180 – 84 = 96˚
Some space for your own jottings/notes:
13
Scalene Triangles
A scalene triangle is one where all the sides (and therefore all the angles) are different in
size. For example:
(iii)
Angles on a straight line total 180˚
150˚
x˚
In this example we can see that angle x must equal 30˚
Because 150 + 30 = 180˚.
Now try this one. Can you calculate the value of angle K?
A
41˚ K
B
20˚
We can see that AB is a straight line and that it has three angles sat on it. We are told the
size of two of these angles (41˚ + 20˚ = 61˚). Therefore angle K must be 180˚ - 61˚ = 119˚.
If you log on to the school website and access the electronic version of this booklet you can
click here and you will be able to view a powerpoint presentation entitled ‘Missing Angles’. This
should clarify things if you are still a bit unsure!
14
4
Theme
The Four Rules (Again!)
Yes folks – it’s here again! For the next couple of weeks your child will be reminded about various written and mental
strategies that they could use in adding, subtracting, multiplying and dividing. Your child will probably already have a
favourite method for each of these operations, but will their method be the most appropriate for the job? Furthermore, can
they apply their calculation strategies effectively to solve written problems?
Adding and Subtracting
During the course of KS2 your child has learnt various methods for adding and subtracting
numbers, including formal written methods (column addition/subtraction). However we cannot
stress enough the importance of mental strategies, and indeed the vast majority of KS2 test
questions can be answered through a combination of mental calculations and written jottings.
Example: Mr Smith is a poor teacher and so he has been trying to earn some extra money in the
school holidays by washing his friends’ cars. He earned £45 in week 1, £38 in week 2, £27 in week
3 and £35 in week 4. He now has enough money to enjoy himself during the last two weeks! How
much did Mr Smith earn altogether?
Now you may be sat at home reading this thinking the best way to add these up is to write them
out in a column addition. Certainly if you were taught in a primary school in the 1960’s or 70’s, then
this ‘formal’ way of adding numbers would have been the obvious method. Like this:
45
38
27
+35
145
2
This method is fine if you are able to write the sum down neatly and remember to add on the
number you have ‘carried’. Some children are not good at this. Therefore they may prefer a
more ‘informal’ way of adding:
15
45 + 38 + 27 + 35
We begin by adding the ‘tens’, looking for number bonds as we go:
40 + 30 + 30 = 100 + 20 = 120
This leaves:
5+8+7+5
5 + 5 = 10
8 + 7 = 15
10 + 15 = 25
Then 120 + 25 = 145
So Mr Smith earns £145 by washing cars.
Compensatory addition and subtraction
This is an important mental strategy, and one that many children fail to use.
Example:
5246 + 2997
Again many ‘older’ people would simply write one number on top of the other and perform a
column addition without giving it a moment’s thought. Others might ‘partition’ the numbers into
thousands, hundreds, tens etc. and add up separately before combining all the totals at the
end. The clever mathematicians, however, can see that 2997 is just three less than 3000. So:
5246 + 3000 = 8246
Now by adding 3000 we have actually added 3 too many, and therefore we now need to
subtract 3 to make the answer correct:
8246 – 3 = 8243
and therefore: 5246 + 2997 = 8243
Here is another example, this time in a ‘take-away’ sum:
8506 – 398
In this example the smart way is to initially subtract 400. So
16
8506 – 400 = 8106
But by taking away 400, we have actually subtracted 2 extra, so we must now adjust our
calculation accordingly by adding the 2 back on:
8106 + 2 = 8108
Therefore 8506 – 398 = 8108. Simple isn’t it!
See if you can use this ‘compensatory’ method to solve this next problem:
Mrs Smith went to Blackpool for the day. She had £100 in her purse. She bought a lilo for
£3.99 and a beach ball for £2.99. For her lunch she enjoyed fish and chips (£2.50) and a ‘99’
ice cream which actually cost £0.99! In the afternoon she bought a pass for the Pleasure
Beach costing £15.95. Having been sick on the ‘Pepsi Max’ she decided to call it a day and go
home – but not before having 3 rides on Denis the Donkey (99p per ride – ‘buy two get one
free!)
How much money did Mrs Smith have left out of her original £100?
Use this box for your jottings:
£
17
Adding and Subtracting Decimals
It’s only very small but that decimal point can create total chaos in the minds of our young
mathematicians! And yet there is no need to panic – we can use our knowledge of whole number
calculations to help us. Here is an example:
What is 11.38 - 7.69?
Step 1: make a rough estimate = 11 – 7 = 4
Step 2: ignore the decimal point so that the sum reads 1138 – 769
Step 3: subtract 769 from 1138 using any method you like. Answer = 369
Step 4: put the decimal point into the answer so that it confirms your estimate = 3.69
Another way of looking at it is to say that because there are 2 digits after the decimal point
in the numbers in the question, then there will be 2 digits after the decimal point in the
answer.
BANANA SKIN!!
What if we are faced with a question like this?
What is 17.45 – 13.2?
Here you can see that there are 2 digits after the dp in the first number but only 1 digit
after the dp in the second. This is what you have to do:
Step 1: still make an estimate = 17 – 13 = 4
Step 2: make the 13.2 into 13.20 (this doesn’t change the value of the number but it does
make both numbers in the question have 2 digits after the dp)
Step 3: ignore the decimal point so that the sum now reads 1745 – 1320
Step 4: subtract 1320 from 1745 using any method you like. Answer = 425
Step 5: put the decimal point into the answer so that it confirms your estimate = 4.25
Counting On
Your child has been taught this method of subtraction in Year 4 and 5. We think it is really
quite a safe method. Many Year 6 pupils make errors in subtraction calculations because they
have forgotten how useful this method is! Here is a reminder of how it works:
Example: Mrs Asady has just bought a new pair of pom-poms for her cheerleading club. They
cost £7.65. She paid for them with a £20 note. How much change did she receive?
+5p
+30p
+£12
£7.65_____£7.70____________£8.00________________________________£20.00
So you can see that we have counted on from £7.65 to £20 in three stages:
18
£7.65 up to £7.70
£7.70 up to £8
£8 up to £20
= a jump of 5p
= a jump of 30p
= a jump of £12
So altogether we have jumped £12.35. So this must be Mrs Asady’s change.
Now let’s try another one:
Calculate 7.01 – 1.82
Estimate first: 7 – 2 = 5
Ignore the decimal point to give us 701 – 182
Now count on from 182 to 701
+8
+10
+500
+1
182______190_________200__________________________700____701
So altogether we have jumped on 519.
Now put the decimal point back in so that it confirms your estimate = 5.19
Multiplying and Dividing
This section of work begins again with a ‘refresher’ of ideas and methods covered in Year 5.
Again we must reiterate the importance of ‘Times Tables’ knowledge. If your child is able to
learn these multiplication/division facts well, then they will have an important tool in their
mathematical kitbags!
DOUBLING AND HALVING
This is an important mathematical skill and one which will help your child in many areas of
multiplication and division. Here are some top tips:

X2 means you double the number
X4 means you double and double again
X8 means you double, double and double again.

÷2 means halve the number
÷4 means halve and halve again
÷8 means halve, halve and halve again.

The x16 table is the x8 table doubled
The x 18 table is the x9 table doubled and so on…

The starting point for multiplying by 25 or 50 is x 100. For example:
19
What is 13 x 25?
13 x 100
13 x 50
13 x 25
= 1300
= 650
= 325
We can also use this method to calculate multiples close to 50 or 25. For example:
There are 24 rows of seats in a theatre and each row contains 17 seats. How many seats are
there altogether in the theatre?
If there were 25 rows of seats, we could work it out like this:
100 x 17
50 x 17
25 x 17
= 1700
= 850
= 425
But there were 24 rows not 25, so that is one row less (ie 17 seats less)
So 24 x 17 = 408

Another useful tip to know is that in any multiplication sum, if you double one side and
halve the other the answer will be the same. For example:
14 x 35 is the same as 7(half of 14) x 70(double 35)
7 x 70 = 7 x 7 x 10 = 490
Here is another example:
18 x 40 =
9 x 80 =
9 x 8 x 10 = 720
It is worth remembering this little ‘trick’ as it can turn quite difficult and scary looking sums
into little pussycats. Miaow!
20
5
Theme
Mixed Numbers and Improper Fractions
This mini theme is a reprise of a topic covered in Year 5, so your child should know something already – famous last
words! Last year we did lots of visual representations of Mixed Numbers and Improper Fractions; this year we’ll try a
quicker way of moving between the two!
First of all let’s make sure we all understand what the two terms mean.
A MIXED NUMBER is where you have a whole number and a fraction together. For example:
3½ is a mixed number
An IMPROPER FRACTION is where the numerator (top number) is greater than the denominator
(bottom number). For example:
7
2
is an improper fraction
Year 6 children will be taught how to convert from mixed numbers to improper fractions and viceversa. Here are the two rules in a nutshell:
(i) “To turn a mixed number into an improper fraction multiply the whole number by the
denominator, add the numerator and then put the total over the denominator”
For example
3½
Start by taking the whole number (3) and multiply it by the denominator (2)
So 3 x 2 = 6
Next add the numerator (1), so that makes 7
Finally put this total on top of the denominator (2), so this gives us 7
21
2
(ii) “To change an improper fraction into a mixed number simply divide the numerator by the
denominator. Any remainder then becomes the numerator over the original denominator”
For example: 7
2
Start by dividing the numerator (7) by the denominator (2)
7 ÷ 2 = 3 remainder 1
So our answer of 3 becomes the whole number and the remainder becomes
So
7
2
=3
½
Why not try a few examples of your own in this box!
22
½
6
Theme
Equivalent Fractions
Here we go with another aspect of fractions work. It is really important that Year 6 children are able to recognise the
common fraction equivalents (including decimals and percentages) and that they have some strategies to use if they are not
sure.
All fractions where the numerator is
numerator) are equivalent to
1
of the denominator (the denominator is double the
2
1
:
2
1 2 3 4
= = = etc.
2 4 6 8
All fractions where the numerator is
1
of the denominator (the denominator is four times the
4
1
numerator) are equivalent to :
4
1 2 3 4
= =
=
etc.
4 8 12 16
All fractions where the numerator is
numerator) are equivalent to
1
of the denominator (the denominator is three times the
3
1
:
3
1 2 3 4
= = =
etc
3 6 9 12
You can work out equivalents of
1 1
,
etc in exactly the same way.
5 6
23
This fraction wall will help you spot common equivalents:
Here is another way of working out equivalent fractions. If you take any fraction and multiply or
divide the numerator and denominator by the same number you will create an equivalent fraction.
For example:
28
40
If we multiply top and bottom by 2 we get
56
28
. This fraction is therefore equivalent to
.
80
40
As long as we multiply top and bottom by same number we will create an equivalent fraction. If we
84
28
multiply by 3 we get
, and so this fraction is equivalent to
.
120
40
We can also create equivalent fractions by dividing top and bottom by the same number. Let us
28
14
stick with
as our example. Let us divide by 2: 28 ÷ 2 = 14, and 40 ÷ 2 = 20. Therefore
is
40
20
7
equivalent. We could divide by 2 again: 14 ÷ 2 = 7, and 20 ÷ 2 = 10. This gives us
. Some of you
10
bright sparks may have realised that we could have divided by 4 at the start instead: 28 ÷ 4 = 7,
and 40 ÷ 4 = 10.
24
7
we can see that we are unable to divide any further. When this
10
happens we say that the fraction is in its simplest form.
If we now look at this fraction
Look at the table below. Can you change the fractions into their simplest forms? Remember you
will have to divide numerator and denominator by the same number. Can you spot the fraction that
is already in its simplest form?
2
4
4
16
3
9
200
400
150
600
6
12
8
32
3
12
4
5
3
24
5
10
6
18
4
20
5
20
3
15
4
12
2
8
5
20
10
40
11
22
7
21
60
180
7
28
10
30
6
24
9
18
12
36
50
100
25
75
80
240
Now have some fun using this interactive resource on equivalent fractions: (You will need to access
the electronic version of this booklet via the Learning Platform)
http://www.learningplanet.com/sam/ff/index.asp
25
7
Theme
Ordering Fractions
The next strand in our ‘Fractions’ work is all to do with ordering fractions from smallest to largest or vice versa. The skills
we have gained through our work on equivalence will be put to good use here!
Ordering fractions that have the same denominator is simple. It is when fractions have different
denominators that problems might be encountered. Rule number 1: Don’t panic! You already have
the skills and the knowledge to tackle any question to do with ordering fractions.
Let’s look at a typical SATs question:
Put these fractions in order starting with the smallest:
7
10
2
3
7
15
1
2
4
5
Now, as you can see, each of these fractions has a different denominator. So what we need to do
is to change them so that they all have the same denominator. But how do we do that? Simple.
Start by looking at the denominators. We need to find a number that is a common multiple of 10, 3,
15, 2 and 5. In other words we need a number that all these denominators will divide into exactly.
Have you worked it out yet? That’s right – it’s 30.
The next step is to take each of the fractions in turn and change them into thirtieths – fractions
out of 30. Here’s how to do it:
7
?
=
So the denominator of 10 has become a denominator of 30. In other words we have
10 30
multiplied the denominator by 3. Therefore we must multiply the numerator by 3 as well to give an
equivalent fraction out of 30. Well 7 x 3 = 21, so
7
21
=
10 30
Now let’s move onto the next one:
26
2
?
=
This time the denominator of 3 has been multiplied by 10 to make 30, so we must multiply
3 30
the numerator by 10 also:
2 20
=
3 30
7
?
=
multiply by 2 to give:
15 30
7
14
=
15 30
1 15
=
I hope you spotted this one without too much trouble! And finally –
2 30
4
?
=
This time we need to multiply by 6 to give:
5
30
4 24
=
5 30
So our results are:
Original
Fraction
7
10
2
3
7
15
1
2
4
5
Fraction out
of 30
21
30
20
30
14
30
15
30
24
30
So the original fractions arranged from smallest to largest are:
7
15
1
2
2
3
27
7
10
4
5