Overlap

Why do this problem?

This problem encourages students to appreciate that there can be more than one way to tackle a problem.  The methods suggested here use symmetry, congruence, and areas.  Each method has a picture which students are encouraged to use in order to complete a proof.  

Possible approach

This problem featured in an NRICH Secondary webinar in March 2022.

Project the geogebra app onto a white board.  Use the slider to rotate the red square around and ask students what fraction of the blue square they think is covered.  In particular, look at when the red square is horizontal and when it makes an angle of $45^{\circ}$ with the horizontal.  Do they think that the red square always covers the same fraction of the blue square?

There are 4 different proof methods to consider.  You might find it helpful to provide students with a printout of the 4 pictures behind the "hide and reveal" buttons.

These printable worksheets could also be useful: Method 2, Method 3, Method 4. Each worksheet consists of a series of diagrams which form a proof, and space for students to add the explanations necessary to complete the proof.