A collection of memorable mathematics lessons that conference and colloquia delegates had observed or taught which they felt were successful. Each account refers to one or more of the values and principles in the report.

Lesson Account 41 - Finding the Factors of Factorials

Written by

Pete Griffin

Organisation

NCETM

Age/Ability Range

Y8 Middle Ability

How was the session/task introduced?
Discussion of initial teacher’s question: “what are factorials?”.

2 examples offered 4! And 5!

Teacher says “so can we generalise this ?”

The initial response was multiply by the 3 numbers below it – teacher needed to push this.

Teacher waited very skilfully, reflected back the students’ comments to them, to prompt a considered response.

Pupils were asked to find all factors of 2!, 3!, 4!, 5!

They were seated in groups of 3 or 4 but asked to do this task in silence for 2 or 3 minutes before being asked to come together as a group.

This seemed a critical decision – group discussion was very lively and rich after this moment of silent, individual work.

2! = 2 1,2 No of factors = 2
3! = 6 1,2,3,6 No of factors = 4
4! = 24 1,2,3,4,6,8,12,24 No of factors = 8
5! = 120 1,2,3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30,40,60,120 No of factors = 16
6! = 720

All pupils asked to make a prediction for 720. Some excitement here that there seemed to be something going on, some generalisation, some overall structure governing these results.

Teacher – “How could we find out? How could we prove this?”

Some time working through factors of 720 – realisation from the pupils that it was a big job.

The teacher reminded them about writing numbers as product of prime factors (they had done this in a previous lesson).

Individual work doing this.

How was the session/task sustained?
Then pupils worked in groups to relate the number of factors to the product of primes format (with some input from teacher). The teacher chose 24 as an example.

1

2

2²

2³

1

1

2

4

8

3

3

6

12

24

24 = 2³ x 3¹
No of factors = 8.

Pupils worked on this in a mixture of small group work and whole class interactive teaching and came to the realisation that the number of factors was equal to the number of cells in the above grid.

The teacher then invited the pupils to re-consider 6! = 720 and whether this method of prime factorisation enabled them to arrive at an answer for the number of factors.

How was the session/task concluded?
Summary discussion of 6! = 720 = 24 x 32 x 51 and the grid which would represent this.

Pupils noticing that, as there are 3 prime numbers involved, the grid is 3-dimensional (i.e. a 5 by 3 by 2 grid) and that this shows that there are 30 factors and not the 32 earlier predicted.

Final discussion about whether the grid had to be drawn in order to determine the number of factors, i.e. whether just examining the powers of the primes in the prime factorisation would be enough.

What were the critical moments?
Working in silence prior to group talk and negotiation.

Excitement at being able to make a prediction for the number of factors of 6!

Realisation that two different ideas (no. of factors and prime factorisation) were related and converging together.

Interest in realisation that initial prediction of 32 for number of factors of 6! based on pattern spotting was incorrect.

What mathematics was learnt? (on plan and off plan) and what is the evidence of learning?
Very little content. Ability to reason. Questioning results, Searching for structure. Seeing beyond pattern to structure.

How was that mathematics learnt?
Through group discussion of results / findings arising from a situation they understood. The mathematical situation gave them the examples to create and work on. The mathematical structure provided the means of checking.

Other memorable outcomes
At end of lesson, sense of seeing a whole, complex problem and seeing all connections, reasoning. Appreciating a complete story.

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Values & Principles

Conceptual understanding and interpretations for representations

Appreciation of the power of mathematics in society