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:trapezium -1115
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:I figured this out by first labelling each point with a letter; A, B, C, D, E, F, G and H.
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:I knew that I could also draw a square not using points next to each other. I had discovered this when I was first experimenting with the problem. I could make two different squares this way. I played around with the circle and could not come up with any more quadrilaterals that didn't have two points next each other. I added these 2 squares to my 35 and got 37. I sure hope I'm right. I offered my dad a dollar if he could come up with more, but he has not met my challenge! Thanks for the problem. It was fun to solve.
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.In 2015 6GW at Godolphin Junior School, Slough decided there are many more solutions to the Quadrilaterals problem than were shown previously. That's great they have sensibly allowed themselves to have "anchor" points that were not at the vertices of the quadrilaterals. Well done all of you who worked on this in Godolphin School I like your thinking. In the light of this I wonder who can find the quadrilateral with the smallest area.