Possible pieces
Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?
Problem
We are going to look at possible jigsaw pieces.
Some of the most common jigsaw shapes are a bit like these three pieces below:
We will only use pieces that have at least one peg and one hole.
Challenge 1
Using pieces that have at least one peg and one hole, find all the possible ways of making a rectangular jigsaw three pieces wide and two pieces deep, with straight edges all the way around. All six pieces should be different.
Challenge 2
Find all the possible pieces that have at least one peg and one hole.
Challenge 3
Read all of this one before starting!
Find as many possible ways of making a two by four rectangular jigsaw, starting with this piece in the top left-hand corner.
Before you start making/drawing/constructing pieces, think carefully and then estimate how many possible arrangements there will be and be able to explain your reasoning to others.
Getting Started
How are you making sure you do not make any the same?
Here is a picture of the kind of jigsaw the problem is based on, in case you are not familiar with jigsaws. (This is just an example, this particular picture will not necessarily help you answer the challenges!)
Student Solutions
Yvonne from Zurich Switzerland wrote to say they found eight solutions to challenge 1 and 32 solutions to challenge 3. It would have been good to see pictures of them.
Millie from Nene Valley Primary sent in her work with a picture:
First I tried to make one puzzle that worked, and then I realised that there was a pattern of pegs and holes. The pegs went round in a circle and so did the holes, so only the middle peg or hole could change, or the circle could go round the other way.
A larger more legible version is available here.pdf
Well done Millie, and thanks to Yvonne for her two answers. If you have a go at this challenge, let us know what you find out.
Teachers' Resources
Possible Pieces
We are going to look at possible jigsaw pieces.
Some of the most common jigsaw shapes are a bit like these three pieces below:
We will only use pieces that have at least one peg and one hole.
Challenge 1
Using pieces that have at least one peg and one hole, find all the possible ways of making a rectangular jigsaw three pieces wide and two pieces deep, with straight edges all the way around. All six pieces should be different.
Challenge 2
Find all the possible pieces that have at least one peg and one hole.
Challenge 3
Read all of this one before starting!
Find as many possible ways of making a two by four rectangular jigsaw, starting with this piece in the top left-hand corner.
Before you start making/drawing/constructing pieces, think carefully and then estimate how many possible arrangements there will be and be able to explain your reasoning to others.
Why do this problem?
This activity requires little mathematics 'curriculum' knowledge (in terms of number/geometry etc), so learners can be 'freed up' to focus on their problem-solving and mathematical thinking skills. A certain degree of resilience and perseverance will be needed. It is a great context in which to give learners the experience of working on a challenge that takes a longer time than many and you may wish to offer them the opportunity to return to the task at a later date. It is likely that pupils will take up the chance to develop a systematic way of getting a solution.
Possible approach
You could leave this as a 'simmering activity' for children to contribute to during the week and then devote time at a later date to drawing their ideas together.
Key questions
Tell me how you're finding a solution.
How are you making sure you do not make any the same?
Tell me how you're trying to get solutions to challenge 2 ( or 3).
Possible extension
As mentioned above, because this activity requires little formal mathematics in terms of number/geometry etc, pupils will have the opportunity to really focus on their approach. You can encourage learners to reflect on the way/s they tackled the challenges and how they overcame any moments of being stuck.
Possible support
It may be appropriate only to share the first challenge initially so that learners have chance to focus on that without distractions.