Completeness - Fraction Multiplication

 

I recently came across a fun question (I think from White Rose Maths) but I thought it had missed a trick, so I amended it to allow for more explanation:

This first question proved much more difficult than the second one, given you have less numbers to use. I think I would have preferred the other way around first.

It was fascinating to see the different approaches to solving it. Most dived in with brute force, trying as many numbers as they could. One student was adamant it was impossible, so I asked her to prove it was impossible.

"How do I prove that it is impossible"

"Well, start with why you think it is impossible?"

"Because it is hard"

"What exactly is hard?"

"There are no fractions that work, I've tried them all"

"Show me you have tried them all then"

At this point she realised she hadn't considered improper fractions and so was focused in on a more rigid approach.

One student made an excellent discovery (we have only taught up to multiplication of fractions). 

"We can just flip the numbers and it still works"

"Which numbers did you flip? Do you have to flip all of them?"

"Just the fractions, but they both have to be flipped"

"Does that help when finding non-solutions"

"Maybe. Maybe all the answers are in pairs"

Turning a question into one of completeness and asking them to PROVE and EXPLAIN really gets the students thinking about how to strategise and solve complex problems. It also sowed the seeds of reciprocals and division, but I did not need to explain this here, it was of no significance to their process at this time and would cognitively overstretch them and take them away from their own strategies.