Ask a question the right way and it can launch a lesson. I'm fascinated by problems that inspire thinking, curiosity and a range of strategies. They're what make maths come alive and show that there are many legitimate and wonderful ways of doing maths.
A recent problem we shared for Maths Teacher Circles is no exception. Check it out. Which fraction would you say is largest?
Before you read on, take a moment to think about how you might tackle the problem.
Strategies used to tackle the problem
Strategy 1: Reduce to a common denominator
Fred Harwood noticed that both 555 and 777 are multiples of 111. So, reducing the fractions to a common denominator of 111 allows them to be meaningfully compared.
Another approach, similarly used a common denominator of 111 and then broke the two fractions into a sum of 100/111 + 'something else'. As the conversation below shows, it was a wonderful example of a strategy that came quickly to one person, but took some thinking from my end to work out:
Strategy 2: Compare with 1 whole
A few people, including Harjit Hundal, used a method of comparing how far each fraction was from 1 whole:
Strategy 3: Create equivalent fractions using common multiples
The last strategy that cropped up is the one we most frequently see in maths textbooks. As Mel Yeo's response shows, however, you're dealing with quite large numbers. So, in this context, it's not necessarily the most straightforward approach and perhaps for this reason, was rarely mentioned in conversation.
Why did this problem generate so much conversation?
This problem fantastically hooked people in and I was surprised by the breadth of responses given. I like to think about this question of what made this problem so juicy, so I can learn from it and take some of those features across to future problems.
When I reflected on the problem, here's what I noticed:
The question is simple and asks us to do something that we are familiar with - to compare the size of two items. You can give a gut response straightaway and, by doing so, your curiosity is piqued.
It's not straightforward to work out. So the problem invites different ways of thinking to the typical approach used in working with fractions. It challenges you to dig deep into your understanding of fractions and what they mean.
I'm curious about whether there are any other features of the problem that helped it to be a conversation-starter and which I haven't yet noticed. What do you think?
Ideas for the classroom
There are a few strategies I would suggest in running a task like this with students:
Start simple: First and foremost you want the problem to be accessible, so that there is buy in rather than disinterest or, worse, fear. Once students understand what a problem of this type is asking, then by all means, repeat it and start to build up the complexity. Depending on what your students are ready for, starting examples might be 4/11 vs 4/10, 3/4 vs 6/7 or 10/12 vs 6/9.
Annotate and name responses: Sharing students' responses on a whiteboard is a powerful way to highlight the depth and breadth of thinking that's going on. Quite often, people will have an approach to the problem that seems like the most obvious. Sharing different strategies will help to open up new possibilities and to also draw connections across mathematical ideas.
Stay neutral: Rather than passing judgement on different strategies, take students' ideas without judgement. What is an efficient strategy to one student, may not even make sense to another. Students will draw on strategies that make sense and connect to what they already know.
Encourage trying out others' strategies: Naming students' responses means that a particular strategy is easier to refer to, e.g., "For this next problem, see if you can use Harjit's strategy". When students have heard a new strategy explained by a peer and also seen it written down, it builds confidence in trying it out for themselves and exploring new territory.