Kite in a Square

Why do this problem?

Students often find geometric proofs quite intractable. In this problem, three different ways of proving the same result are presented, jumbled up, so that students can engage with the proofs without having to start from scratch.
 

Possible approach

This printable worksheet may be useful: Kite in a Square
These printable cards for sorting may be useful: Coordinates, Similar Triangles, Pythagoras

Show the image from the problem.
"ABCD is a square. M is the midpoint of AB. What fraction of the total area is shaded?"
Give students some time to have a go at the problem. While they are working, circulate and see the methods they are trying.

After a while, bring the class together again and acknowledge that the answer may not be immediately obvious.

"I've been given the methods used by three different people. Unfortunately each method has got jumbled up. Can you put the statements in the right order to build a logical argument?"

Hand out envelopes with each method (Coordinates, Similar Triangles, Pythagoras) to pairs or threes. Coordinates is the most accessible method, and Pythagoras the most challenging. It is a good idea to print each method on different coloured card to avoid them getting muddled up.
"With your partner, make sense of each step and put the cards in the right order.
Once you've completed the task, can you recreate each method for your partner without looking at the cards?"

Once students have spent enough time engaging with the three methods, making sense of them and recreating them for themselves, bring the class together. Invite students to present each method to the class, and finally discuss the merits and disadvantages of each.

The interactive proof sorters which are available can be used as an alternative to the printable cards, or for students to check their suggested arrangements of the cards.
 

Key questions

For Coordinates method:

• what are the equations of the lines?
• where do they intersect?

For Similar Figures method:

• which angles are the same?
• what lengths do we know?

For Pythagoras method:

• where are the right angles?
• what lengths do we know?

Possible support

Start by drawing a square on dotty paper (2 by 2 to start with) and explain that vertices and mid-points can be joined with straight lines.

Challenge students to find the different fractions of the square that can be shaded.

Possible extension

Enclosing Squares offers a follow-up activity linked to Coordinates.
Take a Square offers a follow-up activity linked to Similar Figures.
Pythagoras Proofs offers a follow-up activity linked to Pythagoras.