Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
How many rectangles can you find in this shape? Which ones are
differently sized and which are 'similar'?
Draw three straight lines to separate these shapes into four groups
- each group must contain one of each shape.
This problem can have more than one solution, depending on what you decide 'different' means. Thomas, Lydia and Bethan from Old Earth Primary School described a quadrilateral by the number of dots you move around the circle as you join its four corners. For example, a 1115 quadrilateral would look like this:
They thought there were eight different possible quadrilaterals although they didn't explain exactly how they found them. This way of recording the quadrilaterals is very useful - a good idea! Here is the list of eight that Thomas, Lydia and Bethan found:
Some children at Breckland Middle School also worked on this problem. They said:
Here is a picture of the solutions they found:
It is interesting now to compare these two sets of solutions. Can you match Thomas, Lydia and Bethan's solutions to the pictures above? So, if we decide that 'different' means that one shape cannot fit on another, even if it is rotated, then there would be ten as children at Breckland have drawn.
But, if we decide that reflections are the same too, then I think there are eight, as Thomas, Lydia and Bethan suggest. Here is a picture of the eight that Thomas, Lydia and Bethan found but I have drawn them in a particular order to make sure I didn't miss any out. Can you see what my system was?
If we define 'different' in one of the ways above, the problem is quite manageable. Some of you made life very challenging by opening up the possibilities. Kerim, working at home, and Maiya, from Marist College, approached the problem differently. Kerim wrote:
So, Kerim found thirty-five quadrilaterals which all have two corners next to each other. She then began to think about quadrilaterals where none of the corners was immediately next to another on the circle. She says:
Kerim has been very systematic in the way she has listed the quadrilaterals, which is very helpful. If you've had a go at this problem, how did you think about 'different' quadrilaterals? Do you agree with what Kerim has found? It is interesting that Kerim said she thought about the square because she'd played around with the problem to start with. It might be that using a
combination of drawings and other ways of describing the quadrilaterals (eg using numbers or letters) works well for this problem.