Mathematics Kings Worthy Primary School Friday 20th September

Aims of the session To develop an understanding of how calculation strategies are developed from Key Stage 1 to Key Stage 2 To experience examples of Key Stage 2 maths investigations To gain an understanding of age-related expectations in maths at Key Stage 2 To give examples of home-learning activities 2

When you were at school….. How did you feel about maths? What were your experiences of maths like? How do you feel about maths now?

Maths in school today…. Maths is fun! Children are encouraged to “have a go”! They are not afraid to be wrong – mistakes are part of the learning Maths is made purposeful and interesting

Why has maths changed? Children used to learn “standard methods” – we were shown what to do before understanding was consolidated If you couldn’t remember, or hadn’t understood the methods…you went wrong Today, we teach methods that help children to understand the underlying maths and the basic concepts involved Children need to develop “number sense” – more insight into mathematics

Today’s techniques… …are not just about getting the right answer – but about knowing and understanding how you got there …are not new – many pre-date the techniques you learned …eventually join up with the ones you did in school – but the children understand them thoroughly …reduce the chance of mistakes being made …build a firm foundation for understanding more complicated mathematics later on

A balanced mathematical diet… Using and applying mathematics Counting and understanding number Knowing and using number facts Calculating Understanding shape Measuring Handling data

Place Value Our number system consists of ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 The place of each digit determines its value. For example, the “6” digit can represent for 6, sixty, six hundred. It depends where we place it… 6….65….653 Game

Counting We expect children at the start of Yr3 to be able to: count, read, write and order whole numbers to at least 100; know what each digit represents (including 0 as a place holder) partition 2 digit and some 3 digit numbers in different ways and use this to help solve problems know by heart all addition and subtraction facts for each number to at least 10, all pairs with totals to 20, all pairs of multiples of 10 with totals up to 100 add or subtract mentally a one digit number or a multiple of 10 to or from any two digit number begin to bridge through tens numbers using known number facts when adding and subtracting understand multiplication as repeated addition or as an array, solve simple x and ÷ problems by pictorial or abstract methods derive and recall facts for the 2, 5 and 10 multiplication tables use and interpret the symbols +/-/x/÷ and = in number sentences  

Counting We expect children at the start of Yr5 to be able to: read, write and order whole numbers to 10 and 100s of thousands, know what each digit represents and can partition 4 and 5 digit numbers into multiples of 1000, 100, 10 and 1 in different ways use diagrams to identify equivalent fractions; interpret mixed numbers and position them on a number line round fractions or decimals to the nearest and/or next whole number use mental recall of addition and subtraction facts up to 20 in solving problems involving larger numbers use informal methods to add two numbers together that have one decimal place use efficient written methods are used to add and subtract three digit and four digit whole numbers and £.p. know by heart 2,3,4,5,and 10X tables, use them to solve multiplication and division problems and begin to know 6,7,8 and 9X find pairs of factors of any number to 30 recognise two digit multiples of 2,5, 10 and three digit multiples of 2, 5, 10, 50 and 100 find remainders after division and round up or down after depending upon the context

Number Lines They help to develop an ability to order numbers, and give children a sense of where numbers sit in our number system They allow children to draw a picture – or model – in their heads when calculating Can be used in a range of aspects of mathematics – numbers and the number system, fractions, decimals, percentages, addition, subtraction, multiplication, division, measures, handling data Illustrate – use a labelled bead string? Clock? 0 – 60. Blank number lines

The four types of number tracks or number lines Numbered lines 10 9 8 7 6 5 4 3 2 1 5 6 10 9 8 7 12 13 14 15 16 11

The four types of number tracks or number lines… Partly numbered line The empty number line 10 5 15

Partitioning This means breaking up numbers into smaller numbers All numbers can be partitioned in many different ways If children understand about partitioning numbers in different ways, they will calculate more efficiently – and with understanding.

Typical at end of Yr 1/start of Yr 2 Jump 10’s first – Typical at end of Yr 1/start of Yr 2 26 + 23 +10 +10 +1 +1 +1 26 36 48 49 46 47 -10 -10 44 - 25 19 20 21 23 34 44 22 24

44 - 9 Jump and adjust 26 + 9 “Compensating” +10 26 35 36 -1 -10 44 34 Number lines 10 (Animated slide) Teaching idea: -10 to +10 number line – in pairs each child circles a number (not adjacent) and then they try to think of ways to get from one number to the next in two steps (could use decimal line, fractions, multiples of 7, etc.) 44 34 +1 35

Mental skills and strategies We constantly draw upon mental strategies that we know and have internalized when we carry out calculations doubling adding multiples of ten Partitioning compensating applying known facts Children need to learn and understand these strategies – and know how and when to apply them.

Partitioning 47 + 76 = 40 + 70 = 110 7 + 6 = 13 110 + 13 = 123 40 + 7 70 + 6 -------

Column 47 76 Adding 1s first ---- 110 (70+40) then… 47 13 ( 7+ 6) 76 13 ( 7+ 6) 76 ---- ----- 123 13 110 123

Column and exchange h t u 4 7 7 + 6 = 13 which is: 1 ten and 71 6 3 units -------- 1 2 3 We exchange 1 group of ten for ten units

Subtraction Expanded by partitioning: 563 – 241 then… 563- 500 + 60 + 3 241 200 + 40 + 1 ------ ------------- 322 300 + 20 + 2 = 322

Subtraction with exchange The previous method does not always work… 74 – 27 2 313411 - 2 6 8 670 + 14 ------ 20 + 7 0 7 3 ------- 40 + 7 = 47

Multiplication in KS1 Arrays are important because they provide a good visual image of the multiplication that links closely to the concept of repeated addition. 2 + 2 + 2 + 2 4 x 2 = 8 Multiplication 10 Arrays are an important image to use in Year 2 and beyond. 2 x 4 = 8 They are the visual image which leads directly into grid multiplication a key strategy in KS2

Arrays They show the link between multiplication and division – these are not concepts taught in isolation 2 x 4 = 8 4 x 2 = 8 8 ÷ 2 = 4 8 ÷ 4 = 2

Number lines for multiplication 2 + 2 + 2 + 2 + 2 0 2 4 6 8 10 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.

Multiplication on a number line There are 5 cakes in one box. How many cakes in 4 boxes?

Grid (2digit x 1digit) Encourage estimation first… 38 x 7 (approx 40 x 7 = 280) x 7 30 210 210 + 56 = 266 8 56

Expanded short Column but showing working… 38 x 7 then… h t u 30 + 8 3 8 7 7 --------- ------- 56 (7 x 8) 5 6 210 (7 x 30) 2 1 0 2 6 6

Short 38 x 7 grid for 2dp x 2dp 38 x then… 7 x 20 7 266 50 1000 350= 1350 5 6 120 42 = 162 1 5 1 2

Compact long (end of KS2) Children find this tricky!

Division in Key Stage 1 Division can be GROUPING or SHARING Sharing: We have 20 marbles in the marble run to share between 4 children. How many marbles do they get each? =5 marbles Grouping: We have 25 marbles and they come in packets of 5. How many packets were there? Putting 20 into groups of 5. Answer is how many in each group = 4 I have 20 cakes, I can fit 5 cakes in a box. How many boxes will I need?

KS1 strategies for division I have 12 lollipops and I share them among 4 people. How many lollipops will they have each? I have 12 lollipops and I want to put 4 lollipops in each bag. How many bags will I need?

Linking to tables Chunking 345 ÷ 3 300 (100 x 3) 45 (15 x 3) = 115 45

Short division See progression in calculation document

Long division http://www.mathsisfun.com/numbers/long- division-animation.html 425 ÷ 25 200 - (8 x 25) = 17 225- (8 x 25) 200 25 (1 x 25) 25- 00

Why are images important? Resources, models and images help children to visualise and understand mathematical concepts They build up – and remember - the mental picture in their minds They should be available throughout the primary years. Children will rely on them less and less TABLES!

When tackling a mathematical problem, children will ask themselves… What do I know about these numbers? Can I do this in my head? Do I know the approximate size of the answer? If I can’t do it all in my head, what do I need to write down to help me?

“Above all – have fun! Children often say that maths is their favourite subject and they get quicker and better at it when they understand what they are doing. There’s no need for you to “teach” your child: you help by helping them to explain their thinking and their understanding” From “Count on me – 200 ways to help with mathematics”

“It’s the school’s job to provide the structured learning… “It’s the school’s job to provide the structured learning….Your role is to nurture and support your child’s mathematical knowledge away from school, to bring it into their real lives and, most important of all, to turn it into an exciting adventure” From “Maths for Mums and Dads” – Rob Eastaway and Mike Askew

Have your questions been answered?