2. Reasoning, Justification and Proof
…a few of my favourite things…

3. Rich Mathematical Activity
Ahmed 1987:
It must be accessible to everyone at the start
It needs to allow further challenges and be extendible
It should invite children to make decisions
It should involve children in speculating, hypothesis making and testing, proving or explaining, reflecting, interpreting

4. Rich Mathematical Activity
It should not restrict pupils from searching in other directions
It should promote discussion and communication
It should encourage originality/ invention
It should encourage ‘what if’ and ‘what if not’ questions
It should have an element of surprise
It should be enjoyable

5. Nine Point Circles
A circle has points spaced equally around its circumference.
Join 3 points on the circumference to make a triangle
On a new circle, draw another.
What questions might we pose?

6. Nine Point Circles Some possible questions to explore:
How many different triangles can be drawn?
How can you be sure that you have them all?
What special triangles cannot be drawn in this way? Why?
What are the angles in each triangle?
If the radius of the circle is 10cm, what’s the side length of the equilateral triangle? And it’s area?
What’s the area of the smallest possible triangle?

7. ‘Types’ of problem or activity
Ones requiring systematic counting or a logical approach
Ones that require an understanding of mathematical structure
Puzzle or ‘magic’ type activities or problems
Some are more obviously linked to specific curriculum content

8. Several activities to have a go at
There is a question card with the basic problem on
Have a go at the basic problem, also keeping in the back of your mind what students might do
Jot down ideas of questions you might pose and prompts you might give
What reasoning skills will students practice?

9. All problems are rich, but some are more rich than others…
Which of the activities was particularly rich?
What made it rich for you?

10. Why might we use activities such as these?
The only way that students will improve their problem solving skills is through encountering a wide range of problems at regular intervals.
They need to be able to persevere with a problem…
… but also have enough experience to recognise when it is getting them nowhere.

11. About MEI
Registered charity committed to improving mathematics education
Independent UK curriculum development body
We offer continuing professional development courses, provide specialist tuition for students and work with industry to enhance mathematical skills in the workplace
We also pioneer the development of innovative teaching and learning resources