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Factors and multiples inquiry

There are four pairs of integers which have a highest common factor of 6 and a lowest common multiple of 180.
The prompt was devised by Mark Richards, a maths teacher at Lancaster Girls' Grammar School in Lancaster (UK). He reports here on using it with a new year 10 class:

I took the plunge and tried an inquiry with my year 10 high-attaining class. We were supposed to be revising lowest common multiple (LCM) and highest common factor (HCF) so I came up with the prompt. This was my first lesson with the class. In retrospect, perhaps, my first attempt with a completely new class was perhaps asking a bit much of all concerned.
Anyway, after seeing the prompt there were some comments and some questions, along the lines of 'Is it significant that 6 is a factor of 180?'  When I asked if they could see a question they might work on, nobody would volunteer anything, so I suggested that they work on finding the four pairs. At that point some students asked for a recap of HCF and LCM. Eventually, they all managed to find the four pairs. Some used a Venn diagram approach with the prime factors of 6 and 180; others wrote out all the factors of 180, then eliminated all of those which didn't have a factor of 6 and finally paired the remaining numbers starting with the smallest and largest (6 and 180), then the next smallest and largest (12 and 90), etc. 
I got two groups to run through their solutions at which point one student announced that both methods were essentially the same. I thought about that statement a fair bit and concluded that she is right. She also looked at the more general problem of how many solutions similar problems (with different LCMs and HCFs) would have and solved it. It's quite a neat little problem I thought: it turns out that the number of possible pairs is always a power of 2, which relates to combinations. I shall try another inquiry (and use the regulatory cards again) in the near future and see whether the students can take a bit more responsibility.
The prompt combines the potential for inductive exploration and deductive reasoning about the number of pairs that satisfy the conditions for the highest common factor and lowest common multiple. The four pairs of integers that satisfy the conditions in the prompt are 6 and 180, 12 and 90, 18 and 60, and 30 and 36. The inquiry could develop in different directions. Why, for example, are there no pairs for a HCF of 8 and a LCM of 140? When are there two pairs? Why is there never an odd number of pairs? Can there ever be more than four pairs?
Initiating inquiry
Students' questions and observations that have initiated classroom inquiry include:
  • What does 'integer' mean?
  • What does 'highest common factor' and 'lowest common multiple' mean?
  • How do you know there are four pairs?
  • Is it significant that 6 is a factor of 180?
  • The factors of 6 are 1, 2, 3 and 6.
  • How many factors of 180 are there?
  • Could we use prime factors to find the factors of 180?
  • Are there always four pairs?
  • If we changed the numbers for the HCF and LCM, would there still be pairs of integers?
  • What if we started with the pairs of numbers and found the HCF and LCM? Would that be easier?

Prompt sheet
Promethean flipchart    download
Smartboard notebook

An alternative prompt Multiples
 Multiples of 19
57     2 x 7 + 5 = 19
95     2 x 5 + 9 = 19
The prompt is the suggestion of Dave Fielding (a maths teacher at the Sir Robert Woodard Academy, Lancing, UK). The pattern can be shown to work for the first ten multiples, although, as Ed Bentley (a newly qualified maths teacher from Coventry) said in a twitter discussion, the next ten multiples give 38. When students inquire into multiples of other numbers, they begin to realise there are other related patterns:
54 is a multiple of 18, and 2 x 4 + 2 x 5 = 18
51 is a multiple of 17, and 2 x 1 + 3 x 5 = 17
However, the next two cases are different:
48 is a multiple of 16, and 2 x 8 + 4 x 4 = 32
45 is a multiple of 15, and 2 x 5 + 5 x 4 = 30

Dave Fielding can be followed on twitter @DaveFielding1 and Ed Bentley @edbentleymaths
Researching an inquiry approach
Mark Richards undertook a research project on introducing an inquiry-based approach into his maths department. The focus of
the research was on four year 7 classes. Mark's findings showed that students became more enthusiastic about maths from experiencing inquiry lessons, and were less likely to be discouraged by being stuck. For example, they were less likely to declare "I can’t do maths." This summary of his project was commissioned by the National Teacher Research Panel for the 2008 Teacher Research Conference. The quotes from his fellow teachers demonstrate the rich potential for inquiry in the classroom:
  • "What was surprising was the richness of the content of the work and how one piece of work could open the doors to many areas of mathematics."
  • "An open-ended approach to a topic can yield results which involve pupils doing higher level Maths than I thought possible."
  • "I have been consistently surprised by the insights pupils have shared when asked to find their own method to solve a problem, or when asked to justify a particular solution."

Mark Richards
is a teacher of mathematics at Lancaster Girls' Grammar School, Lancaster (UK).