Download Maths Booklet for Parents - St John of Beverley RC Primary School

St. John of Beverley R.C. Primary School
Maths Calculation
Booklet for parents
Contents
Introduction
4
Overview of expectations
5
Addition
Initial mental methods
Addition on a number line
Expanded columns method
Standard columns method
7
8
9
9
Subtraction
Partitioning
Subtraction on a number line
Expanded columns method
Standard columns method
10
10
11
12
Multiplication
Partitioning
Multiplication on a number line
Grid multiplication - TU x U and TU x TU
Expanded columns method
Standard columns method
13
13
14
15
16
Division
Division by grouping
Division on a number line
Division by ‘chunking’
Short division
Long division
18
18
20
21
23
Calculations in context
26
How can you help at home?
27
3
Introduction
o
This booklet is intended to explain the ways in which your children
are taught to solve calculations.
o
These methods have been revised to fit in with the 2014 National
Curriculum.
o
The aim is that your children are more confident in their Maths and
that they enjoy what they do because they have a genuine
understanding.
o
If you have any questions or you wish to have some guidance with
any of the methods, please see your child’s class teacher.
4
Overview of expectations
Maths in Foundation Stage and KS1:
o
Mental maths emphasis (counting, ordering, number bonds, finding
halves and quarters)
o
Informal jottings are encouraged
o
Children use some formal written methods
KS2:
o
Throughout your child’s journey in KS2, there is a strong emphasis
on developing mental methods in conjunction with more formal
written methods
o
As children progress, they will be introduced to a range of written
methods that can be selected from to solve a variety of problems
When faced with a calculation, no matter how large or difficult the
numbers may appear to be, all children should ask themselves:
Can I do this in my
head?
If I can’t do it
wholly in my head,
what do I need to
write down to
calculate the
answer?
Do I know the
approximate size
of the answer?
Will the written
method I know be
useful?
5
Children in KS2 will be asked to attempt the following calculations using
the standard columns methods:
NB: d = digits, dp = decimal place
Addition and Subtraction
Multiplication
Division
6
Addition
Initial mental methods
Partitioning
Partitioning is another word for separating the digits. This is a method
for carrying out mental calculations by separating out the tens and
units, it can also be a simple written method.
Example:
Partition both numbers and recombine. Gradually develop to
partitioning only the second number.
36 + 53 =
refined to:
36 + 53 =
6+3=9
53 + 6 = 59
50 + 30 = 80
59 + 30 = 89
80 + 9 = 89
NB: When extending to hundreds, it is vital that the children understand the importance
of zeroes as place holders E.g. 102 = 1 hundred, 0 tens and 2 units.
Compensation method
Children who are confident with rounding may use the compensation
method. This means they add on more than is needed, then adjust to fit
the calculation.
Example:
358 + 19 = ?
Round the 19 up to the nearest 10 first:
358 + 20 = 378
Then adjust by taking away the extra 1:
378 – 1 = 377
7
Addition on a number line
Children are first taught to count on in ones from any number, for
example: 7, 8, 9, 10, 11. They then progress on to counting on in tens
from any number, for example: 10, 20, 30, 40, extending to: 12, 22, 32,
42, 52.
The empty number line is a method that children are taught to help
them with their mental calculations.
Method:
Write the first number at the left hand end of the line.
Partition the number you are adding into tens and units.
Add the units to the bigger number, drawing the jumps as you do
so.
Add the tens, again drawing the jumps.
Progression - Once children become confident with this method, they
may be able to add 3 tens in one jump, i.e. + 30 rather than + 10 + 10 +
10.
8
Expanded columns method
The expanded method is sometimes used when children are first taught
standard written methods. This is taught to develop their understanding
about the value of digits.
Example:
NB: Often children will move straight onto the contracted method of column addition if
they have a secure understanding of place value.
Standard columns method
Once children are secure with place value, the expanded method can be
contracted to:
If the 2 numbers added together are larger than ten
e.g. 7 + 9 = 16 the unit is placed in the relevant column
and the ten is carried over to the bottom of the next
column.
Once children are secure with this method, they will learn to apply it to
larger numbers, including numbers with various decimal places.
9
Subtraction
Partitioning
Partitioning is another word for separating the digits.
In this method, you need to partition the second number from the
calculation and take this away a step at a time from the first number.
Example:
42 - 15 =
Partition 15 into 10 and 5
42 - 10 = 32
32 - 5 = 27
Therefore: 42 - 15 = 27
Subtraction on a number line
In the early stages of KS2, children will use a number line to support
them with subtraction. They will start at the end of the number line and
jump backwards. They are finding the difference between two values by
counting on.
Method:
The larger number is written at the end of the number line and the
smaller number is taken away by counting back.
Again, this calculation method requires children to be able to
partition 2 digit numbers.
78 - 53 =
78 - 3 (units) = 75
75 - 50 (5 tens) = 25
10
Progression - Once children become confident with this method, they
may be able to subtract 5 tens in one jump, i.e. - 50 rather than - 10 - 10
- 10 - 10 - 10.
Expanded columns method
Children who are very secure with using a number line, will move onto
more formal written methods for subtraction.
To begin with, children may be taught to expand their columns method
to ensure they understand the calculations that are taking place.
NB: At this stage, children will be using numbers that do not require ‘borrowing’, i.e. the
digits in the top number are larger than those in the bottom number.
Example:
An important thing to bear in mind is that we are very careful in the
language we use. We talk about subtracting 50 from 80 and not 5 from
8. This is to make sure that children recognise the value of the digits in
preparation for the next step.
11
Standard columns method
As children develop their confidence with subtraction, they will be
introduced to more complicated examples.
These may not be written in the ‘expanded’ layout and may require
some ‘borrowing’.
Example:
Step 1 – Look at the units. You cannot subtract 6 from 3, so you go to the
next place value column (‘tens’) and borrow a ten.
Step 2 – The 5 in the tens column is crossed out and reduced to a 4 and 1
ten is carried over into the units column.
Step 3 – There are now 13 units in the units column. Therefore we can do
the calculation 13 - 6 = 7
Step 4 – Next, go to the tens column. Nine tens cannot be subtracted from
4 tens so you go to the next place value column (‘hundreds’) and borrow a
hundred.
Step 5 – The 5 hundred is crossed out and reduced to a 4 and 1 hundred is
carried over into the tens column.
Step 6 – There are now 14 tens in the tens column and therefore 14 tens 9 tens = 5 tens.
Step 7 – Lastly, go to the hundreds column. 4 hundreds - 3 hundreds = 1
hundred.
As with addition, once children are secure with this method, they will
learn to apply it to larger numbers, including numbers with various
decimal places.
12
Multiplication
It is essential that all children learn their times tables facts. They are
used in many areas of maths and need to be learnt by heart.
Children need to understand that multiplication is based on repeated
addition.
Once this is understood, various methods of helping them to multiply 2
numbers and derive an accurate answer are taught.
The children will need to:



Solve number sentences e.g. 14 x 8 = ?
Use their multiplication knowledge across many areas of maths
e.g. calculating area of shapes and for division.
Apply their knowledge of multiplication to solve word problems.
Partitioning
Using partitioning with multiplication helps children solve their
calculation in easier steps. Children will be taught this as a jotting for
mental calculations.
Example:
Multiplication on a number line
In the early stages of KS2 this is the most appropriate way of teaching
multiplication.
13
Think of the number sentence 5 x 4 as 5 lots of 4.
Start at zero, on the left hand end of the number line. Make 5 jumps in
‘lots of’ 4.
Grid multiplication
Grid Multiplication enables children to see each stage of their
calculation. This method also uses partitioning but is set out in a grid.
It requires children to have an understanding of multiplying by 10 and
100.
Example 1:
23 x 7 = ?
x
20
3
1
+
7
140
21
1
4
2
6
0
1
1
Step 1 – Partition the 2-digit number into ‘tens’ and ‘units’.
Step 2 – Multiply 20 by 7 and fill in the answer (140).
Step 3 – Multiply 3 by 7 and fill in the answer (21).
Step 4 – Add the answers together using column addition.
The same method can be extended to larger numbers and then on to
decimals
14
Example 2:
56 x 27 = ?
x
50
6
20 1000
120
7
42
350
1
0
1
3
1
5
+
1
0
2
5
4
1
0
0
0
2
2
When the grid method is understood, children can move onto the
columns method of multiplication.
Expanded columns method
When children are first introduced to the columns method for
multiplication, they may well be asked to express each aspect of their
calculation in order to aid their understanding of the steps involved.
Example 1:
Here, the 7 is multiplied by the 5 to give 35 first, then the 30 is multiplied
by 5 to give 150 and finally these products are added together to give the
answer.
Example 2:
7x7
30 x 7
7 x 30
30 x 30
15
Standard columns method
This method requires children to have a secure understanding of place
value and times table facts. Children must make sure digits are in the
correct columns.
Example 1:
Step 1 – Start with the units. 5 x 5 = 25 (carry the 2 tens over to the tens
column).
Step 2 – 3 x 5 = 15. Add the 2 (carried over) to give 17.
Example 2:
Step 1 – Position the digits in their place value columns.
Step 2 – Multiply the top units by the lower units: 5 x 6 = 30. The ‘0’ goes
in the units column and the ‘3 tens’ are carried over to the top ten.
Step 3 – Multiply the top tens by the lower units: 8 x 6 = 48. Add the 3
tens carried over which makes 51. The ‘1’ is placed in the tens column and
the ‘5’ is carried over to the hundreds column.
Step 4 – Multiply the top hundreds by the lower units: 4 x 6 = 24. Add the
5 (carried over) = 29.
Step 5 – Write a zero in the units column below the first answer to show
that all the answer is multiplied by 10.
16
Step 6 – Multiply the top units by the lower tens: 5 x 1 = 5. Write 5 in the
tens column.
Step 7 – Multiply the top tens by the lower tens: 8 x 1 = 8. Write the
answer in the hundreds column.
Step 8 – Multiply the top hundreds by the lower tens: 4 x 1 = 4. Write 4 in
the thousands column.
Step 9 – Lastly, add the two products together using column addition:
2910 + 4850 = 7760.
Step 10 – Check your workings.
This method can sometimes cause confusion with children being unsure
which digits to carry and which to multiply. Ensure that each step is
considered carefully.
17
Division
Children have many ways of interpreting what division means. They
often talk about ‘sharing’ or ‘grouping’.
Children will be taught a range of written methods to support their work
on division. Having a secure knowledge of all times table facts is crucial
to support their work on division.
Division by grouping
One of the first methods used when discussing division will be the
physical grouping of things. With this method, you are asking the child
to find out how many lots/sets/groups of that number there are.
Example:
Division on a number line
In the earlier stage of KS2, the children will be taught to use an empty
number line to help with their division.
Example 1:
30 ÷ 5 can be modelled as:
Grouping - How many 5’s make 30?
18
Draw jumps of 5 along a number line. After each jump write the multiple
of 5 you land on. Complete this until you reach 30. Children to count the
number of jumps they have made. This shows you need 6 jumps of 5 to
reach 30.
As the children become more confident with using the number line, they can
begin to make more efficient jumps.
Example 2:
10 lots of 6 = 60
86 ÷ 6 = 14 remainder 2
It would take a long time to jump in 6’s to 86 so children can jump in
bigger jumps using their tables facts to help them. A jump of 10 lots of 6
takes you to 60. A jump of 4 lots of 6 takes you to 84 with a remainder of
2. Altogether that is 14 (10 + 4) jumps of 6 with a remainder of 2.
19
Division by ‘chunking’
This method is known as ‘chunking’. Using this method, you are
effectively taking away chunks of the number at a time to see how many
‘lots’ of the divisor the starting number contains.
Due to the process illustrated below, children can only make use of this
method once they are confident at using standard column subtraction.
The children are taught to look for bigger chunks (i.e. chunks of 10 or
more) to start with to make the method simpler and easier.
Example 1:
Step 1 – 10 ‘lots’ of 3 = 30. This is less than 55 so this can be subtracted
using standard column subtraction, leaving an answer of 25.
Step 2 – We can then take 8 ‘lots’ of 3 away from 25 as 8 x 3 = 24. This is
taken away from the 25 to leave an answer of 1.
Step 3 – You cannot take any more ‘lots’ of 3 away from 1 so this is left as
a remainder.
Step 4 – Count up how many ‘lots’ of 3 were taken away altogether. 10 + 8
= 18 therefore the answer to 55 ÷ 3 is 18 remainder 1.
20
As children become more confident with this method, they will be given
more challenging calculations to undertake.
Example 2:
Step 1 – Here the children should spot that they can take 100 ‘lots’ away
initially as 100 x 3 is less than their starting number. Subtracting this
leaves an answer of 155.
Step 2 – The children could then work out that 50 ‘lots’ of 3 could be taken
away to leave an answer of just 5.
Step 3 – One more ‘lot’ of 3 can be taken away from 5 to leave a
remainder 2.
Step 4 – Again, count up how many ‘lots’ of 3 have been subtracted
altogether.
Short division
A step by step guide to short division can be found at:
http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths
/school_booster/busstopdivision.html
This method is sometimes referred to as the ‘bus stop’ method.
21
The aim with this method is to see how many times the divisor will
divide into each digit.
Example:
Step 1 – In this method, you always start with the biggest place value so
our first job is to see how many times 4 divides into 3. 4 ÷ 3 = 1 remainder
1. We write the 1 directly above the 4 and carry the remainder 1 onto the
next digit.
Step 2 – With the addition of the carried over 1, the digit 5 is changed into
15. 15 ÷ 3 = 5. The 5 is written above this column.
Step 3 – The last step is to divide 5 by 3. 5 ÷ 3 = 1 remainder 2. The 1 is
written above the column and the remainder 2 is written next to the
completed answer.
As children progress, they may be asked to express remainders as a
decimal or a fraction.
Expressing as a decimal
This method begins in an identical way to that detailed above. The
children start with the highest place value digit and calculate how many
times the divisor can be divided into that digit. They then move through
the digits, making sure to carry over any remainders.
When they have calculated the last digit, if there is a remainder, the
process alters.
Example:
22
Step 1 – Write a decimal place behind the last digit both above and inside
the ‘bus stop’.
Step 2 – Place the remainder in front of this digit.
Step 3 – Calculate how many times the divisor can be divided into this
newly-formed number. 10 ÷ 2 = 5. Therefore 5 is written above the ‘bus
stop’ behind the decimal place to give a final answer of 455 ÷ 2 = 227.5.
This process can be repeated to fill in as many decimal places as there
are remainders.
Expressing as a fraction
Example:
2
3
Again this method proceeds in exactly the same way as standard short
division, until you need to deal with the remainder.
Here, the children are taught to use the remainder as a numerator (2)
and the divisor as the denominator (3) making
2
3
.
Long division
This method is used when you have to divide larger numbers (beyond
their times table knowledge).
23
Example:
432 ÷ 15 = 28.8
Step 1 – 4 ÷ 15 is not possible, so we look at the first two digits together.
43 ÷ 15 = 2 (because 2 x 15 = 30, 3 x 15 = 45 which is too big).
Step 2 – The 2 goes in the tens column above
the bus stop.
Step 3 – To calculate the remainder from the
first calculation, write the answer to 2 x 15
underneath like this:
Step 4 – Now we subtract the bottom number
from the top number: 43 – 30 = 13 (this is the
remainder from the first division).
Step 5 – Next, we bring down the next digit
of the dividend (2) to give 132.
Step 6 – Divide this number by the divisor: 132 ÷ 15 = 8 (because 8 x 15 =
120, the nearest multiple of 15 below 132).
Step 7 – The whole number answer (8) is
placed at the top of the bus stop. To find the
remainder, write the 120 below the 132.
24
Step 8 – Now, we subtract the bottom number
from the top number: 132 – 120 = 12.
Step 9 – Bring down the next digit of the
dividend (0). Here a zero has been added
behind the decimal point to enable us to obtain
a final answer as a decimal.
Step 10 - Divide this new number by the divisor:
120 ÷ 15 = 8 (no remainder). Place the whole
number (8) at the top of the bus stop.
Step 11 – Again, the number used in the
division is written below and subtracted
from the top number. There is no
remainder this time so we have our answer
– 28.8 (remember to include the decimal
place in the answer).
25
Calculations in context
All the methods in this booklet support children in using their mental
and written skills to solve calculations. Children need to be encouraged
to use the method that they understand and can use confidently.
It is important that children are able to choose the most appropriate
method for the calculation. For example:
4003 – 3998
These numbers are very close together and so counting up on a
number line (actual or imagined) would be the most efficient
method.
200 ÷ 4
Dividing by 4 is the same as halving and halving again. As it is easy to
halve 200 and easy to halve 100, this would be the most efficient
method.
Using and applying appropriate skills is very important, when
calculations are needed to solve a problem.
4 CDs at £2.99 each – How much altogether?
£2.99 is almost £3.00 and so round up, multiply, then adjust:
4 x £3.00 = £12.00
£12.00 – 4p = £11.96
26
How can you help at home?






Practise recalling times tables
Count using money
Tell the time
Weigh and measure out cooking ingredients
Guess the shape
Play puzzles and other games that develop numerical and logical
reasoning
Websites
There are several good websites for practising Maths at home and
developing skills with number and reasoning. You may like to look at:
http://resources.woodlands-junior.kent.sch.uk/maths/
http://www.mathsisfun.com/ - Covers all areas of Maths. Lots of good
logic puzzles!
http://www.coolmath4kids.com/ - Covers all areas of maths
http://www.bbc.co.uk/bitesize/ks2/maths/ - Covers all areas of maths
http://www.transum.org/Software/SW/Starter_of_the_day/index.htm
- Good for years 5 and 6.
http://www.maths-games.org/times-tables-games.html - Good website
for grouping games for all areas of maths from various websites.
http://www.mad4maths.com/ - Fun games for KS2 children.
http://www.crickweb.co.uk/ks2numeracy.html - Good variety of maths
games.
http://www.ictgames.com/resources.html - Wide variety of games for
KS1 and Lower KS2.
27
http://www.topmarks.co.uk/Flash.aspx?f=SpeedChallenge - Speed
challenge activities for practising times tables, rounding, number bonds.
http://mathszone.webspace.virginmedia.com/mw/add_sub/3d_3d_add
.swf - Column addition.
http://www.amblesideprimary.com/ambleweb/mentalmaths/pyramid.h
tml - Pyramid addition.
http://mathsframe.co.uk/en/resources/resource/48/column_subtractio
n - Various maths practise.
Practising number facts
 Find out which number facts your child is learning at school
(addition facts to 10, times tables, doubles etc). Try to practise for a
few minutes each day using a range of vocabulary.
 Have a ‘fact of the day’. Pin this fact up around the house. Practise
reading it in a quiet, loud, squeaky voice. Ask your child over the
day if they can recall the fact.
 Play ‘ping pong’ to practise complements with your child. You say a
number. They reply with how much more is needed to make 10.
You can also play this game with numbers totalling 20, 100 or 1000.
Encourage your child to answer quickly, without counting or using
fingers.
 Throw 2 dice. Ask your child to find the total of the numbers (+),
the difference between them (-) or the product (x). Can they do this
without counting?
 Use a set of playing cards (no pictures). Turn over two cards and
ask your child to add or multiply the numbers. If they answer
correctly, they keep the cards. How many cards can they collect in 2
minutes?
 Play Bingo. Each player chooses five answers (e.g. numbers to 10 to
practise simple addition, multiples of 5 to practise the five times
28
tables). Ask a question and if a player has the answer, they can
cross it off. The winner is the first player to cross off all their
answers.
 Give your child an answer. Ask them to write as many addition
sentences as they can with this answer (e.g.
multiplication or subtraction.
 Give your child a number fact (e.g. 5 + 3 = 8). Ask them what else
they can find out from this fact (e.g. 3 + 5 = 8, 8 – 5 = 3, 8 – 3 = 5, 50
+ 30 = 80, 500 + 300 = 800, 5 + 4 = 9, 15 + 3 = 18). Add to the list
over the next few days. Try starting with a ‘x’ fact as well.
Real life problems
 Go shopping with your child to buy two or three items. Ask them to
work out the total amount spent and how much change you will
get.
 Buy some items with a percentage extra free. Help your child to
calculate how much of the product is free.
 Plan an outing during the holidays. Ask your child to think about
what time you will need to set off and how much money you will
need to take.
 Use a TV guide. Ask your child to work out the length of their
favourite programmes. Can they calculate how long they spend
watching TV each day / each week?
 Use a bus or train timetable. Ask your child to work out how long a
journey between two places should take? Go on the journey. Do
you arrive earlier or later than expected? How much earlier/later?
 Help your child to scale a recipe up or down to feed the right
amount of people.
 Work together to plan a party or meal on a budget.
29
Counting ideas
 Practise chanting the number names. Encourage your child to join
in with you. When they are confident, try starting from different
numbers - 4, 5, 6 . . .
 Sing number rhymes together - there are lots of commercial tapes
and CDs available.
 Give your child the opportunity to count a range of interesting
objects (coins, pasta shapes, buttons etc.). Encourage them to
touch and move each object as they count.
 Count things you cannot touch or see (this is more difficult!). Try
lights on the ceiling, window panes, jumps, claps or oranges in a
bag.
 Play games that involve counting (e.g. snakes and ladders, dice
games, games that involve collecting objects).
 Look for numerals in the environment. You can spot numerals at
home, in the street or when out shopping.
 Cut out numerals from newspapers, magazines or birthday cards.
Then help your child to put the numbers in orders.
 Make mistakes when chanting, counting or ordering numbers. Can
your child spot what you have done wrong?
 Choose a number of the week e.g. 5. Practise counting to 5 and on
from 5. Count out groups of 5 objects (5 dolls, 5 bricks, 5 pens). See
how many places you can spot the numeral 5.
30