Sliding Rectangles

This problem has perhaps a slightly surprising result.  
 

Possible approach:

At the start, students could be shown the interactivity and asked to consider the areas of the rectangles $A$ and $B$ when the point $P$ is in the middle of the diagonal line.

What happens as the point $P$ slides up and down the diagonal?

Students could use dotty paper in order to investigate a particular rectangle, like the $16 \times 24$ rectangle shown in the getting started section. 

There are two possible proof methods shown behind "hide and reveal" buttons at the bottom of the problem.  They could be used as starting points for students to create their own proofs. 

• The first method uses areas of triangles and rectangles to prove the result, and is accessible to 11-14 students.
   
• The second method uses coordinates and straight line graphs, or proportional reasoning, along with some algebraic manipulation, to find the areas of the two rectangles. 

For the second method, students can be asked to explain why the diagram represents the same problem. Why do they think the diagonal has been drawn in the opposite direction?

Possible support:

Students could be asked to consider a point $P$, along the diagonal of a square, before moving on to the rectangle.