Jigsaw pieces
How will you go about finding all the jigsaw pieces that have one peg and one hole?
Age
7 to 11
Challenge level
Problem
We have created a more recent version of this task here.
We are going to look at possible jigsaw pieces.
The kind of jigsaw we're looking at is rectangular, with straight edges.
Some of the most common jigsaw shapes are a bit like these four pieces below:
Some of the most common jigsaw shapes are a bit like these four pieces below:
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I've added the flag picture so you know which way up they go - the red at the top and the blue at the bottom. The two pairs of pieces above could look the same if you turned one around - but because the red needs to be at the top and the blue at the bottom, each piece in the pair are counted as different.
So, for each jigsaw piece we start with a square template, then some sides have a peg. The edge pieces of the jigsaw have one or two straight sides.
We will only use pieces that have at least one peg and one hole.
We will only use pieces that have at least one peg and one hole.
Challenge 1
Find all the possible pieces that have at least one peg and one hole.Here are three to start you off:
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Challenge 2
Again, only use pieces that have at least one peg and one hole and none the same.Find all the possible ways of making a rectangular jigsaw three pieces wide and two pieces deep, with flat edges all the way round.
Challenge 3
Read all of this one before starting!
Again, only use pieces that have at least one peg and one hole and none the same.
Find as many possible ways of making a two by four rectangular jigsaw, starting with this piece in the top left hand corner.Image
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!)
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!)
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Student Solutions
The solutions to the more recent version of this task are published on the Possible Pieces solution page.
Teachers' Resources
Jigsaw Pieces
We are going to look at possible jigsaw pieces.
The kind of jigsaw we're looking at is rectangular, with straight edges.
Some of the most common jigsaw shapes are a bit like these four pieces below:
Some of the most common jigsaw shapes are a bit like these four pieces below:
Image
Image
Image
Image
I've added the flag picture so you know which way up they go - the red at the top and the blue at the bottom. The two pairs of pieces above could look the same if you turned one around - but because the red needs to be at the top and the blue at the bottom, each piece in the pair are counted as different.
So, for each jigsaw piece we start with a square template, then some sides have a peg. The edge pieces of the jigsaw have one or two straight sides.
We will only use pieces that have at least one peg and one hole.
We will only use pieces that have at least one peg and one hole.
Why do this problem?
This activity challenges pupils to use their problem-solving skills and requires a certain degree of resilience and perseverance. 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
The pupils could start off by doing some quick jigsaws, or you could have a box containing jigsaw pieces out for them to see. Ask them what they notice about the pieces and encourage them to share their thoughts with a partner. Gather the whole group together to exchange ideas, listening out for those children who identify similarities and differences, such as "some pieces have bits cut out"
or "this one has some straight edges" etc. You could use this opportunity to introduce the language of 'peg' and 'hole' (as used in the task) to the group.
You can then present the challenge to the children. You may like to say very little else at that stage and let learners work in pairs or small groups on the task. (You may like to give out printed copies of the task.) After some time, bring them together again to share what they have done so far and to clarify the task. You could invite
some pairs/groups to talk about the way they are working so that a variety of approaches is highlighted. Some children may be drawing shapes on whiteboards or paper, others may be making them from card, some may have found more abstract ways to record what they are doing. You could discuss the advantages and disadvantages of each method. (It may be useful for learners to have some squared paper
but try to provide any resources which they request, even if you haven't anticipated them!)
Invite learners to explain how they are making sure their jigsaw pieces are all different from one another. If the children haven't had much experience of working in a systematic way, you could ask each pair/group to make the pieces out of card, then after a longer period of time, display the pieces somewhere easy to see. With the help of the children, group the pieces together, for example
all those with exactly one straight side; all those with just one 'hole'. In this way, the class will be able to identify pieces that are missing from the set.
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.
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).