Copyright © University of Cambridge. All rights reserved.

'Fitted' printed from http://nrich.maths.org/

Show menu

Fitted

If you are a teacher click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ...

Picture of some squares

Nine squares with side lengths $1, 4, 7, 8, 9, 10, 14, 15$ and $18$ cm can be fitted together with no gaps and no overlaps, to form a rectangle.

What are the dimensions of the rectangle?


 

Once you've had a chance to think about it, click below to see how three different pupils began working on the task.

 

This is how Anna started:

 

Here is what Brendan tried:

 

Here is Chandra's initial approach to the problem:

 

Can you take each of these starting ideas and develop them into a solution?

 


Why do this problem?

This problem is a challenging context in which to explore area. It can be solved in many different ways and reflection on different methods is encouraged through inclusion of the three sample approaches.  You may well need more than one lesson for this activity.

Possible approach

Pose the problem orally, or project the text onto a screen, without mentioning the examples of how some children started.  Give the class a few minutes to consider, individually, how they might go about tackling the problem, then pair them up and suggest that they talk to their partner about their ideas so far.  Try to stand back and observe, and resist the temptation to make helpful suggestions!

Allow pairs to work on the task so that you feel they have made some progress, but do not worry if they have not completed it or if they report being stuck.  The aim at this stage is for everyone to 'get into' the problem and work hard on trying to solve it, but not necessarily to achieve a final solution.   Make sure that the children have easy access to any resources that they are likely to need but don't put anything out on tables already otherwise this may lead them down particular routes.

At a suitable time, hand out this sheet to pairs.  Suggest to the class that when they've finished or can't make any further progress, they should look at the sheet showing three approaches used by children working on this task.  Pose the question, "What might each do next? Can you take each of their starting ideas and develop them into a solution?".  You may like pairs to record their work on large sheets of paper, which might be more easily shared with the rest of the class in the plenary. 

Allow at least fifteen minutes for a final discussion.  Invite some pairs to explain how the three different methods might be continued.  You may find that some members of the class used completely different approaches when they worked on the task to begin with, so ask them to share their methods too.  You can then facilitate a discussion about the advantages and disadvantages of each.  Which way would they choose to use if they were presented with a similar task in the future? Why?

 

Key questions

Tell me about this approach.  What do you think he/she was doing?
How do you think this will help to solve the problem?
What do you think he/she would have done next?
 

Possible extension

Tiles on a Patio is an investigation based on similar ideas which would make a good follow-up activity.

Possible support

It might help some pupils to have sets of squares cut out ready for use.  However, only give them out if children explicitly decide to approach the task in this way at the start, or if they are continuing Brendan's solution, and you think that they would spend too long accurately drawing and cutting the squares themselves.