1. What and how do we teach it?
Key Stage 2 Maths
What and how do we teach it?

2. Aims of the meeting tonight
To help you to understand more of what we do in maths at Key Stage 2
To enable you to support your child more confidently in maths
To give you more information about what we teach and how we teach maths at Key Stage 2

3. By the end of Key stage 2 we hope that children will….
Develop a reliable, accurate method to calculate in all 4 operations
Be able to record their working in a compact way
Apply basic maths skills to calculations
Apply the maths they know to new situations
Make links between different areas of maths
Develop their own strategies for problem solving, choosing the maths they need to use and how they will record their working.
Key stage 2 maths moves from using concrete apparatus to more abstract working and formal recording
To do this successfully, children need a firm grasp of basic mental maths skills and the concept of the number system.
The key to a good understanding of written methods of calculations is based on mental strategies.

4. Basic skills in maths Use of the number line and 100 square
Counting – forwards, backwards, different steps, decimals
Complements
Crossing over ‘boundaries’
Multiplication facts and linked division facts
Use of money, finding change
Time
Doubling halving
Multiplying and dividing by 10, 100, 1000
Use of calculator
Measures – conversion and practical reading of scales

5. Using number lines
“The number line is a powerful and sophisticated linear model of the number system. It embodies all learning styles; visual, auditory and kinaesthetic. It evolves into an internalised mental representation which can be used when children are able to dispense with the actual line.”
Numeracy Team 2002

6. The four kinds of track or line
Number tracks
Numbered lines
Partly numbered line
The empty number line 10 9 8 7 6 5 4 3 2 1 5 6 10 9 8 7 12 13 14 15 16 11
Number lines 3
There are four types of number line/track. They will be explored in this presentation.
First we will look at progression from tracks to lines. 5 10 15

7. Number lines used in Tests at KS2
More examples.
QCA have noted the increasing and successful use of number lines for subtraction problems in national tests.

8. Time lines
I started walking at 08:45 and finished at 12:25. How long did my walk take me?
Number lines 22
The use of a timeline can be very powerful for thinking through time calculations. This is an extension of work on number lines, therefore providing a familiar model for the child. It involves finding the difference by counting on from the earlier to the later time.

9. Written methods for the 4 operations
The key to a good understanding of written methods of calculations is based on mental strategies. Each of the four operations builds on mental skills which provide the foundation for jottings and informal written methods of recording. These skills lead on to more formal written methods.
The transition between stages depends on a child’s understanding of the method. Not all children will be ready to move on to the next stage at the same time and should not be hurried until they have a secure understanding of the method.

10. Subtraction
43 – 26
This calculation can be modelled two ways.

11. How can we model subtraction?
Both understandings of subtraction (take away and difference) can
be modelled on number lines.
43 – 26 = 17 23 33 43 17 20 -3
- 3
-10
-10
Subtraction 27:
(animated slide)
This shows how the same calculation can be modelled on number lines either using the idea of ‘take-away’ or counting on to find the difference.
The first example uses partitioning – but only the second number is partitioned. It involves counting back (in tens and ones) so the jumps are shown under the line.
The second example demonstrates how you can count on to find the difference. This is called complementary addition.
+ 10
+ 4
+ 3 26 30 40 43

12. Number lines with more complex calculations
381
624
400
+200
600
+19
+24
624 – 381 =
3.81
6.24
4.00
+2.00
6.00
+0.19
+0.24
6.24 – 3.81 =
Subtraction 28:
(animated slide)
Empty number lines are suitable for all ages (they are used in the KS3 Framework). These are examples of number lines used for 3 digit numbers and decimals.
They use knowledge of complements (these are taught systematically throughout Key Stage 1 and 2). The jumps taken can then usually be added mentally.
The second example is similar to the first but with numbers 100 times smaller and therefore two decimal places.
Understanding of the first example will support children when calculating the second.

13. Complementary addition vertically
Complementary addition can also be represented vertically.
624 – 381
19 (400) (600) (624)
4 (30) (40) (43) 17
Subtraction 29:
(animated slide)
This is the same calculation as on the previous slide (and a simpler version) but now the jumps are written in vertical format and added
The numbers in the bracket shows point reached by any jump.
This is the method of recording pupils in Year 8 and 9 will use as opposed to drawing a number line (this is shown in Leicestershire guidance for Pencil and Paper Procedures).

14. Multiplication Arrays
Arrays are important because they provide a good visual image of the calculation that links closely to the concept of repeated addition.
4 x 2 = 8
Multiplication 10
Arrays are an important image to use in Year 2 and beyond.
2 x 4 = 8

15. Other good images – number lines
Multiplication 12
Number lines are another good image. They help reinforce counting in equal steps/the link to repeated addition.
Note how this slide shows 2 multiplied by 5 and the next slide…
This image can be expressed as 2 multiplied by 5, two five times, 5 groups of 2, 5 lots of 2 and 5 hops of 2 on a number line.

16.  
This image can be expressed as 5 multiplied by 2, five two times, 2 groups of 5, 2 lots of 5 and 2 hops of 5 on a number line. It is also double 5 (5 x 2).
Multiplication 13
… shows 5 multiplied by 2. Multiplying by 2 should also be linked to doubling.

17. Partitioning 7 x 14 = (7 x 10) + (7 x 4) = 70 + 28 = 98 70 +70 +2870
+70
+28
7x10
7x4 98
Multiplication 20
The distributive law uses the concept of partitioning and is used in grid multiplication. It can also be modelled on an empty number line as shown in the slide.

18. Partitioning - continued
18 x 8 10 8
                
                  80 64 8
Multiplication 21
(animated slide)
This animation goes back to the idea of arrays. It shows how the array can be partitioned (using the distributive law) and how the grid method of multiplication is based on this.

19. 18 x 8 =144 10 8 80 64 8
Multiplication 22
Once the array is removed you are left with grid multiplication.

20. Grid method of multiplication
38 x 27 30 8 20
600
160
760
+ 266
1026
210 56 7 =
Multiplication 23
(animated slide)
For anyone unfamiliar with grid multiplication work through this two by two-digit example.

21. Skills needed
Skills needed to be able to carry out the grid method of multiplication when multiplying 2 two-digit numbers.
partition numbers into tens and units/ones
recall multiplication facts
multiply by ten
multiply by multiples of ten
add together two and three-digit numbers
decide whether the answer is sensible
Multiplication 25
In order to carry out grid multiplication (such as the one previously demonstrated) these are the key skills required.
Some of these are exemplified on the next slides.

22. Multiplying by 10
It is important that multiplying by 10 is not thought of as a case of ‘adding zeros’. It in an inappropriate expression because adding zero actually leaves a number unchanged and the ‘add a zero rule’ fails when, for example, 0.2 is multiplied by 10 (‘adding a zero’ results in 0.20).
Children need to understand that when you multiply by 10 the digits move one place to the left, leaving an empty space which is filled by zero (a place holder).
Multiplication 26
This outlines the importance of teaching x10, x100 etc correctly.
Children often refer to adding zeros when multiplying by 10 etc. It is worth checking that they understand what is really happening and ensure that when they describe calculations to others they use the correct explanation.

23. Multiplication facts
Children will struggle with multiplication if they can’t recall multiplication facts.
Knowing a multiplication table is much more than being able to recite it in order. It also means children should be able to respond quickly to oral or written questions phrased in a variety of ways, e.g.
What are six fives?
What is 3 times five?
5 multiplied by 3 is…
How many fives in 35?
What would I multiply by five to get 30?
Multiplication 27
There will be some multiplication facts/tables children struggle to recall. It is important to teach strategies for these, e.g. using a known facts such as 6 x 6 to establish that 7 x 6 = 42.
It is also important to make links between the tables (e.g. the four times-table is double the two times-table, the five times-table is half of the 10 times-table) and ensure that children realise that knowing one multiplication facts means they know three other facts (e.g. if you know 7 x 6 = 42 you also know 6 x 7 = 46, 46  7 = 6 and 46 6 = 7).

24. Division In Year 3 and 4 children need to know that:
16  2 does not equal 2  16
division reverses multiplication (the inverse) – this allows them to solve division calculations by using multiplication strategies (18  3 by counting the hops of 3 to 18)
there will be remainders for some division calculations (to be expressed as whole-number remainders).
relate division and fractions
use a written method for division (chunking).
Division 19
While children need to consider grouping and sharing in Year 3 and 4 they also need to understand the effect of dividing by 1; the fact that unlike multiplication division is not commutative, that multiplication facts can help with division equations and that some calculations will result in a remainder.

25. Teaching chunking - number line
72 ÷ 5 =
Grouping - How many 5’s are there in 72?
Adding groups of 5
5 x 10 or
10 groups of 5
5 x 4 or
4 groups of 5
Division 25
(animated slide)
Grouping is the concept used in chunking.
As illustrated in the slide the question 72  5 can be thought of as ‘how many fives are there in 72?’
One way of working this out would be to start at 72 and add 5 as many times as possible. Division is associated with repeated addition here but you could also use subtracting five as many times as possible.
This, however, is time-consuming and inefficient. Instead of adding individual fives it is possible to add ‘chunks’ of five. In the above example ten chunks of five have been added then another 4 chunks of 5 leaving a remainder of 2. This means there are 14 chunks of 5 in 72 with 2 left over, so the answer is 14 remainder 2.

26. Teaching chunking - vertical
5 x 1 = 5
5 x 2 = 10
5 x 5 = 25
5 x 10 = 50
72  5 =
(5 x 10) 22
(5 x 4) 2
Answer: 14 remainder 2
Division 26
This shows the same example but written vertically. This is the written method referred to as ‘chunking’.
Children often find it helpful to record some key multiplication facts by the side of the calculation as a useful reference point. The most useful facts are generally 1 times, 2 times, 5 times and 10 times the divisor.
Initially it is helpful to pick a dividend (the number to be divided) that is more than 10x (but less than 20x) the divisor.

27. Teaching chunking - larger numbers
256  7 = (7 x 30) (7 x 6)
7 x 1 = 7
7 x 2 = 14
7 x 5 = 35
7 x 10 = 70
Division 28
Another example of chunking but with larger numbers Both the partitioning method and vertical methods are shown.
Again the multiplication facts can help provide a useful starting point.
Answer: 36 remainder 4

28. Real Life Problems
To make a box pieces of wood 135mm long have to be cut from a 2.5m length. How many lengths of wood can be cut?
Train fares cost £ I have £52. How many people can I take on the journey?
Division 32
Finally two more questions to solve using the calculator.
The first question involves quite complex mixed units and the need to interpret the display and make sense of the decimals.
The second example again requires the ability to interpret the answer and decide whether to round up or down (it is also possible to use the constant function to answer this).

29. Teach do we teach maths in school?
Abacus Maths scheme
Real life problems
Maths in context – links made with topics
Open ended investigations
Maths during CIL

30. How can you help at home?
Use information on our school website for ideas for support.
Refer to the calculation policy on the website
Support children with their homework – ask them to explain what they are doing and how
Reinforce the basic skills mentioned earlier – these will often be reflected in the targets set by teachers

31. Any Questions?
Please spend some time looking at the resources around the hall
– and have a go at some of the problems!