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The prompt
The prompt
Mathematical inquiry processes: Find pairs of numbers that satisfy the condition; extend to other cases; generalise. Conceptual field of inquiry: Highest common factor and lowest common multiple.
The prompt was devised by Mark Richards, a teacher of mathematics at Lancaster Girls' Grammar School in Lancaster (UK). Mark undertook research into inquiry-based learning and introduced his department to inquiry (see the sections below).
If the class has learned about factors and multiples, the prompt can be used to introduce the concepts of highest common factor (HCF) and lowest common multiple (LCM). Students have the opportunity to derive new knowledge from their current understanding.
The prompt combines the potential for inductive exploration and deductive reasoning about the number of pairs that satisfy the conditions for the HCF and the LCM. 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 (see the illustration below from the slides).
Question, notice and wonder
Question, notice and wonder
During classroom inquiry students have made the following contributions during the question, notice, and wonder phase of the inquiry:
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 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?
Structured and open lines of inquiry
Structured and open lines of inquiry
The inquiry can be structured or open depending on the students' experience of inquiry. Structured lines of inquiry involve solving similar problems to the one in the prompt and applying a given method to generate more examples; open lines of inquiry involve devising the method and generalsing for all cases.
Lines of inquiry
Lines of inquiry
See the slides for more details of the lines of inquiry, including examples and methods to solve problems.
- More problems to solve
Students find pairs of integers to solve the same type of problem as the one in the prompt.
2. Generate examples
2. Generate examples
Students use the structured steps or devise their own method to create more examples.
3. Generalise
3. Generalise
Students consider how many solutions there are when given a highest common factor and a lowest common multiple.
Why are there no pairs of integers that satisfy the conditions of 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?
4. Sets of three integers
4. Sets of three integers
An example question: How many sets of three integers have a HCF of 4 and a LCM of 120?
One approach is to use the pairs of integers that have a HCF of 4 and LCM of 120. The pairs can form the basis of solutions to the new problem by considering the other factors of the larger number one-by-one.
Classroom inquiry
Classroom inquiry
Mark Richards reports on using the prompt:
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.
Collaborative research
Collaborative research
Mark Richards undertook a research project on introducing an inquiry-based approach into his mathematics department. The focus of the research was on four year 7 classes.
Mark's findings showed that students became more enthusiastic about maths after experiencing inquiry lessons. They were also less likely to be discouraged when they got stuck or to declare, "I can’t do maths."
The National Teacher Research Panel commissioned a summary of his project 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."
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