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