Working Together with Mathematics KS2 Workshop Thursday 8 th December – 08.00

Aims of the Workshop To consider briefly how mathematics is taught in school today. To understand different strategies for addition, subtraction, multiplication and division To discuss how you can support your children at home.

Which of these words would you use to describe Mathematics?

What Mathematics have you used this morning? Discuss…

How has Mathematics teaching and learning changed? “It’s not like it was when I was at school…” Interactive teaching It’s more active and collaborative ‘Having a go’ and ‘Talking maths’ are encouraged Emphasis on mental calculation Different approach to written calculations Maths through problem solving Maths is FUN!!!

Working towards Calculations Mental calculations are important Children need to have a secure understanding of number Need to be able to understand what they are doing to be able to enjoy maths.

Working towards Calculations At every stage, teachers first use examples that children can easily do mentally Children then see how the steps in a written procedure link to what they do in their heads They then move to using numbers that cannot easily be dealt with mentally, including money and decimal numbers

Working towards calculations The process Mental Recall Mental Calculations with jottings Informal Methods Expanded Written Methods Formal Written Methods Calculator

Progression in Addition Addition of numbers using objects or fingers = Using a number line to count on = Adding two numbers by keeping one in their head and adding on another =

Progression in Addition The empty number line The empty number line helps to record the steps on the way to calculating the total. The steps often bridge through a multiple of = = 84 or:

Partitioning The next stage is to record mental methods using partitioning. Partitioning both numbers into tens and ones mirrors the column method where ones are placed under ones and tens under tens. This also links to mental methods. Eg: = = = 123 or = = = 123 Partitioned numbers are then written under one another: Progression in Addition

Expanded method in columns Children can now move on to a layout showing the addition of the tens to the tens and the ones to the ones separately. Children should start by adding the ones digits first. The addition of the tens in the calculation is described as = 110 as opposed to = 11. Progression in Addition

Column method In this method, recording is reduced further. Carry digits are recorded below the line, using the words 'carry 10' or 'carry 100', not 'carry 1'.

Progression in Subtraction Subtraction of numbers using objects or fingers 10 – 5 = Using a number line to count backwards. 14 – 3 = Subtracting a number by counting back from the larger number in my head 27 – 3 =

The empty number line The empty number line helps to record the steps in mental subtraction. There are several ways to do this: Counting Back - a calculation like can be recorded by counting back 27 from 74 to reach 47. or Counting Up - the steps can also be recorded by counting up from the smaller number to find the difference or Progression in Subtraction

Partitioning Subtraction can be recorded using partitioning to write equivalent calculations that are easier to carry out mentally. For this involves partitioning the 27 into 20 and 7, then subtracting 20 and 7 in turn. 74 – 27 is the same as 74 – 20 – 7 74 – 20 = – 7 = 47 Progression in Subtraction

Expanded column method The partitioning stage is followed by the expanded column method, where tens and units are placed under each other. This is where the concept of ‘decomposition’ is introduced Example:

Progression in Subtraction Column method The expanded method is eventually reduced to:

Progression in Multiplication Initially multiplication is introduced as ‘counting’ in ‘groups’ or ‘sets’ of numbers e.g. 10, 20, 30, 40, 50, 60, etc 5, 10, 15, 20, 25, 30, 35, 40, etc 2, 4, 6, 8, 10, 12, 14, 16, etc

Progression in Multiplication Multiplication is then introduced as ‘repeated addition’ using vocabulary such as ‘lots of’ or ‘groups of’ real objects or pictures 3 lots of 3 = 9 leading to 3 x 3 = 9

Progression in multiplication This develops into understanding Multiplication as describing an ‘Array’   3 rows of 5  3 x 5 or 5 x 3

Progression in Multiplication Mental Multiplication using partitioning This allows the tens and ones to be multiplied separately to form partial products. These are then added to find the total product. Either the tens or the ones can be multiplied first but it is more common to start with the tens. This can look like......

Progression in Multiplication The Grid Method This links directly to the mental method. It is an alternative way of recording the same steps. For TU x TU, the partial products in each row are added, and then the two sums at the end of each row are added to find the total product.

Over to you! Have a go at solving these multiplications using the grid method. 65 x 8 74 x x 53

Progression in Multiplication The next step is to move the number being multiplied (38 in the example shown) to an extra row at the top. Presenting the grid this way helps children to set out the addition of the partial products in preparation for the standard method.

Progression in Multiplication Expanded short multiplication The next step is to represent the method in a column format, but showing the working. Attention should be drawn to the links with the grid method. Children should describe what they do by referring to the actual values of the digits in the columns. For example, the first step in 38 × 7 is ‘thirty multiplied by seven’, not ‘three times seven’, although the relationship 3 × 7 should be stressed.

Progression in Multiplication Short multiplication The expanded method is eventually reduced to the standard method for short multiplication. The recording is reduced further, with carry digits recorded below the line. The step here involves adding 210 and 50 mentally with only the 5 in the 50 recorded. This highlights the need for children to be able to add a multiple of 10 to a two-digit or three-digit number mentally before they reach this stage.

Progression in Multiplication Long multiplication This is applied to TU x TU as follows.

Progression in Division Initially division is introduced as ‘sharing’ using real objects or pictures. Share 10 apples equally between 2 children which eventually becomes 10 ÷ 2 = 5

Progression in Division Using a number line to count back using ‘repeat subtraction’ e.g. 12 ÷ 3 = – 3 – 3 – 3 – 3 = lots of 3 12 ÷ 3 = 4

Progression in Division Mental division using partitioning One way to work out TU ÷ U mentally is to partition TU into smaller multiples of the divisor, then divide each part separately. Informal recording for 84 ÷ 7 might be: In this example, using knowledge of multiples, the 84 is partitioned into 70 (most children will be secure with a multiple of 10) plus 14

Progression in Division Short division of TU ÷ U For 81 ÷ 3, the dividend of 81 is split into 60, the highest multiple of 3 that is also a multiple 10 and less than 81, to give Each number is then divided by 3. leading to

Over to you! Have a go at solving these divisions using the short method for TU ÷ U. 76 ÷ 8 91 ÷ 7 92 ÷ 4

Progression in Division Expanded method for TU ÷ U and HTU ÷ U This method, often referred to as 'chunking', is based on subtracting multiples of the divisor, or 'chunks'. It is useful for reminding children of the link between division and repeated subtraction.

Progression in Division Refining the 'Expanded' method for HTU ÷ U Initially children subtract several chunks, but with practice they should look for the biggest multiples that they can find to subtract, to reduce the number of steps. Once they understand and can apply the expanded method, children should try the standard method for short division.

Over to you! Have a go at solving these divisions using ‘chunking’. 86 ÷ ÷ ÷ 8

Progression in Division Long division for HTU ÷ TU The next step is to tackle HTU ÷ TU. The layout on the left, is in essence the 'long division' method. Conventionally the 23, or 2 tens, and the 3 ones forming the answer are recorded above the line, as in the second recording.

Supporting your child at home Use the same methods at home as in school Play board games Get your children to help with the shopping Television timetables Measuring ingredients, cooking Play cards and other number games Support your children in telling the time and using it often Use some of the ideas in the leaflets Ask questions!

Things to consider Do we create at home an atmosphere where exploration and ‘having a go’ is seen as more important than getting it right? Do we use opportunities to talk about maths?

Remember! Be Positive – even if you don’t feel it Ask your child to explain to you how they are doing their maths (It may be different to the way you were taught) ‘TALK’ to them about and involve them in everyday maths ‘ASK’ the teacher if you have any questions about the maths your child is doing Use the leaflets provided to support what you’re doing at home HAVE FUN!!!