Count the Trapeziums
How many trapeziums, of various sizes, are hidden in this picture?
Age
7 to 11
Challenge level
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How many trapeziums, of various sizes, are hidden in this picture?
(By the way, some of us here at NRICH think they should be called trapezia . What do you think?)
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Mark thinks it could be 33 trapeziums. Try counting the trapeziums and challenge Mark's answer if you disagree.
Why do this problem?
It's quite easy to hazard a guess about how many trapezia there are in this problem but to be absolutely certain (and convince someone else that you have all the possible solutions) requires some really systematic work.
Possible approach
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You could begin with a whole class challenge of a similar but simpler kind - for example how many trapezia in this shape?
Check that all the children know what a trapezium is and ask for a system for finding all possibilities in this diagram. Emphasise working systematically and what this means in practice - for example starting at the top and working clockwise.
Offer the recording sheet for those who want it - and scissors for them to cut out the variations and re-order them to check for missing diagrams. Working systematically does not come naturally to young children so being able to impose a structure onto randomly generated pictures can be a valuable step in learning how to be
systematic.
Key questions
Where will we start?
What story can we tell which will convince a friend that we have them all? How do we know?
Is there another way of arranging them in a pattern?
Possible extension
Adding another layer of triangles to the bottom of the diagram increases the complexity, but for children who are already working systematically this will be only slightly more challenging.
There is a collection of similar style problems here.
Possible support
Reducing the picture by one layer can be helpful for children who find visualising difficult. Provide a recording sheet so that they can model the same ordering strategy as in the main activity.