Why do this problem?
This problem introduces students to a surprising result which holds true for all quadrilaterals, although in the problem we invite them to explore convex quadrilaterals first. The GeoGebra interactivity provides a hook to engage students, and we hope the result will be intriguing to students, encouraging their curiosity and giving them a desire to persevere until they have proved the
You may wish to begin by inviting students to draw their own quadrilaterals on dotty paper, which you can print off from our Printable Resources
page. Once they have generated a few examples, invite them to explain anything they have noticed. This is a really good opportunity for discussion about the examples they chose - did their original quadrilaterals have any particular
symmetry properties that might have caused the results they are finding? Can they find an example that doesn't work?
At this point, you might like to encourage students to explore using dynamic geometry, either by using the interactivity in the problem, or constructing their own using software such as GeoGebra
, which is free to use.
Diagrams such as this one might be useful for helping students to construct a proof:
As well as considering the shape formed, students could also calculate the area of the original quadrilateral and the new one formed by the midpoints, and look for a relationship.
What is special about the new quadrilateral formed by joining the midpoints?
Does it always happen?
Does it help to draw in the diagonals?
Are there any congruent shapes in your picture?
Does that help you to work out the relationship between the areas?
What happens if you start with a concave quadrilateral?
Students could start by exploring what happens when you join the midpoints of a triangle.