Starting points for considering arithmetic in Years 3 & 4 Dr Jenni Back Host Schools Project Lead

Key thresholds in mathematical development in arithmetic KS1 entry: conservation and counting KS2 entry: addition/subtraction, number bonds to 20, place value KS3 entry: multiplication/division, multiplication tables KS4 entry: proportional reasoning

The issues in years 3 and 4 Securing conservation and counting Additive reasoning, fluent recall and application of number bonds to 20, understanding place value Developing ideas of multiplication and division Intuitive understanding of proportional reasoning

The Big Ideas in arithmetic in Y3 & 4 Doing and undoing The number system and place value Securing and deepening understanding of addition and subtraction Building repertoire of known facts Developing understanding of multiplication and division

The foundations Counting Conservation Place value How do we count? What is involved? http://nrich.maths.org/8123 Conservation Counting all and counting on or back? Place value Exploring exchange and representing numbers Concrete, iconic and symbolic representations

Numicon and ‘subitising’ How does Numicon support children to understand and use number? An alternative: Hungarian number pictures Moving on to multiplication and division? Concrete, iconic and symbolic representations of number

Stretching higher attainers in Y3 Using problem solving approaches to teaching addition and subtraction Two examples: Strike it out Noah

Strike it out

Rich tasks: low threshold, high ceiling Accessible to all at the start of the lesson Allows the most confident/ competent to go as far as they can and offers the opportunity to ask their own questions Plenty of supporting activity of those who will benefit from it Allows different types of thinking and different approaches Article on NRICH website at http://nrich.maths.org/7701 Rich tasks offer a meaningful context that provokes the students to learn in order to solve it

Progression from EYFS to KS2 and beyond Recognising number symbols to 10 – restrict the number line and use concrete apparatus KS1 Developing skills in addition and subtraction of numbers to 20 Lower KS2 Recording results systematically Conjecturing about possible choices Relating addition and subtraction facts Upper KS2 Building fluency with number bonds to 20 Predicting results, developing strategies Beyond KS2 Developing mathematical thinking: exemplify, visualise, generalise, do/undo, conjecture, prove

Noah

Noah Can be found at http://nrich.maths.org/136 and the poster here 10/04/2019 Noah Can be found at http://nrich.maths.org/136 and the poster here Can be introduced at Key Stage 1 How might it be extended to stretch the highest attainers in KS2?

Progression from EYFS to KS2 and beyond 10/04/2019 Progression from EYFS to KS2 and beyond EYFS Using number names and counting concrete objects KS1 Counting accurately to 12 Applying notions of counting to a real context Making models and using them to solve mathematical problems Lower KS2 Recording results and reasoning about numbers Multiplication as successive addition and considering grouping Upper KS2 Extending task to higher targets, predicting results, posing questions Developing mathematical thinking in relation to numbers: exemplify, visualise, generalise, do/undo, conjecture, reason

Notions of multiplication 3 packs of oranges, each with 4 oranges (equal groups) 3 children each have 120ml of orange juice (equal measures) Amir walks at 5km per hour. In 2 hours he walks 10km (rate) Jack has 3 times as many sweets as his sister, Jodie. If Jodie has 5 sweets, how many does Jack have? (multiplicative comparison) The apple tree in our garden is 3 times as tall as it was 5 years ago. If it was 0.9m tall then, how tall is it now? (multiplicative change – scale) Crisps come in 3 flavours and 2 different sized bags. How many different types of bags do the crisp company make? (Cartesian product)

Models and images for multiplication Groups of objects Arrays The number line Area model Grid model Cuisenaire rods Dienes blocks Counters and …

Array without spaces

TU x TU Grid method 10 8 10 6

10 5 3 10 TU x TU Grid method but splitting the units further 5 1

Plates of strawberries

Array of tangerines

+5 +5 +5 +5 5 10 15 20 Jumps on the number line

Area model of multiplication

Plates of strawberries

+6 +6 +6 +6 +6 +6 +6 6 12 18 24 30 36 42 Jumps on the number line

Array with spaces

Beads on a rail

Another problem: factors and multiples: x and ÷ Give me a factor of 132, and another, and another Give me a factor of 108, and another, and another Give me a multiple of 4 bigger than 100, and another, and another Give me a multiple of 3 bigger than 100, and another, and another

Variations: adapting the task for different children Support Extension Prior learning assumed

Progression in this task EYFS Counting up and down KS1 Reciting sequences of odd or even numbers, counting in 10s and 5s Lower KS2 Derive and recall multiplication facts for the 2, 3, 4, 5, 6, and 10 times table Explore the properties of the numbers in these tables Upper KS2 Express general rules about the properties of multiples and factors of a give number verbally Beyond KS2 Generalise these rules and express them using algebra

The key place of talk Emphasis on prompts and questions that promote mathematical reasoning IRF or IRQRF? Children as problem posers

Division: different conceptions Consider the scenarios Each one illustrates a different way of thinking about division Consider how your children might approach answering the question What practical resources would help them? How might they demonstrate how they had worked out their answer?

Party scenarios How many apricots will we each have if 1kg is shared between 12 of us? There are roughly 30 apricots in one kilogram. Sharing by counting out and cutting There are 40 sandwiches and 12 people at the party. How many will each guest be able to have? What shall we do with the spares? Sharing by counting out and cutting Here is a cake, we need to share it between the 12 people at the party. How do we do it? Sharing by cutting into congruent shapes    

Party Scenarios continued There are 5 pizzas. How can we share them between the 12 people? Sharing by cutting successively into congruent shapes We have 3 litres of juice for a party of 12 people. How many 120ml glasses will it fill? How many glasses each will that give everyone? Sharing by pouring liquids I am making invitations for a party and want them to be 10cm by 12cm. How many will I be able to cut out of each A4 sheet of card that I have? How many sheets of card will I need to make 12 invitations? Sharing by fitting into a shape

Party Scenarios continued I am making bunting to hang across the room. How many flags will I need to make to reach across the diagonal of the hall measuring 6m by 7m? You will need to think about the size of the flags, their shape, the tape or string they are fixed to and the gaps between them. To put them across both diagonals and along all four of the sides, how many more will I need? Finding out how many lengths x there are in y I have a bag of 30 balloons to decorate the hall. How many groups of 4 can I make? Grouping in 4s

Party scenarios continued If 12 people come to the party, how can I divide them into teams? What would be the best way to do this? Grouping in 2s, 3s, 4s etc or using multiplication facts If 12 people come to the party and they all go bowling by car from the hall, how many cars will be needed if each car can take 5 people? How many cars will be needed if each car has to have an adult to drive it in addition to the party guests? Successive subtraction or grouping and rounding up

Division: what does it mean? Children need to make sense of it Crucial to focus on communicating ideas How would you explain it? Sharing Grouping Inverse of multiplication Successive subtraction What other concepts is it dependent on? Addition, subtraction and multiplication

How can we support children’s understanding of it? Easy access to a range of appropriate and useful representations Support them in applying their ideas to new problems and eventually new concepts

Next steps Identify a task based on some of the examples you have seen today to try with your class Consider the resources both practical and visual that might support the children’s thinking Elaborate a progression of possible responses or the development of concepts Consider how the children’s responses might reveal their journey through this progression Plan a lesson and teach it Reflect on this experience with your colleagues

Thank you for listening Dr Jenni Back jenni.back@ncetm.org.uk Telephone 07850100074