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

2. 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

3. 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

4. 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

5. The foundations Counting Conservation Place value
How do we count? What is involved?
Conservation
Counting all and counting on or back?
Place value
Exploring exchange and representing numbers
Concrete, iconic and symbolic representations

6. 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

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

8. Strike it out

9. 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
Rich tasks offer a meaningful context that provokes the students to learn in order to solve it

10. 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

11. Noah

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

13. 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

14. 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)

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

16. Array without spaces

17. TU x TU Grid method 10 8 10 6

18. 10 5 3 10
TU x TU Grid method but splitting the units further 5 1

19. Plates of strawberries

20. Array of tangerines

21. +5 +5 +5 +5 5 10 15 20
Jumps on the number line

22. Area model of multiplication

23. Plates of strawberries

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

25. Array with spaces

26. Beads on a rail

27. 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

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

29. 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

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

31. 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?

32. 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    

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. Thank you for listening
Dr Jenni Back
Telephone