1. Using rich tasks to support fluency and understanding in arithmetic
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: exchange
Equivalence and comparison
Building and using a repertoire of known facts

5. Strike it out

6. 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

7. 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

8. Rich tasks and high attainers
Rich tasks offer high attainers opportunities to think deeply about mathematics
High attainers are offered opportunities to tackle problems which can really challenge their conceptions and help them to develop higher order thinking skills such as working systematically or devising a winning strategy without moving beyond KS2 concepts
Collaborating with others and explaining their thinking helps them to cement their understanding through taking a teaching role
Posing their own questions develops their deep and broad understanding of mathematics

9. Understanding place value: a task
The task
Show your partner 3, 5, 12, 7 : take it in turns (everyone has 9 counters)
Now show them 16, 23, 18, 20
What can you do when you haven’t got enough counters?
Ask the children to choose their own numbers to make and explain their strategies

10. Progression from EYFS to KS2 and beyond
Accurate counting to ten
KS1
Partitioning numbers and using addition and subtraction facts
Developing an understanding of place value
Lower KS2
Secure understanding of place value
Using a range of know addition and subtraction facts
Solving problems involving number
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. Understanding place value: notes for teachers
Give every pair of children in the class a set of 9 double sided counters which are a different colour on the reverse.
Assigning places for tens, hundreds etc is analogous to assigning different values to the counters.
Key questions:
Have you run out of counters? What can you do now?
How about letting one blue counter be worth 2?
Explain to your partner how those counters show that number.

12. Noah

13. Noah Can be found at http://nrich.maths.org/136 and the poster here
17/09/2018
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?

14. Progression from EYFS to KS2 and beyond
17/09/2018
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

15. Another task: factors and multiples
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

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

17. 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

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

19. Getting stuck and ‘unstuck’
It is good for children to get stuck
The role of the teacher is to model being stuck and help children to develop strategies for getting ‘unstuck’
What might they be?

20. Getting ‘unstuck’ Talk about the problem with your friends
Use practical resources to model the problem
Try a simpler case
Try smaller numbers
Look at what you know and what you are trying to find out
Write down or talk about everything you know so far
Consider the kind of solution you are looking for – one, many, a proof
Convince yourself, a friend, a sceptic

21. Thank you for listening
Dr Jenni Back
Telephone