Why do this problem?

This problem was orginally published in December, hence the candle and the holly. Perhaps the light of understanding may for some 'come on' when they see the link between algebra and areas.

This question is about the difference of two squares but more generally it can be part of work on multiplying binomial expressions algebraically. The link between the algebra, and the areas of squares and other rectangles, and cutting up and rearranging the shapes, provides a good aid to visualising, understanding and remembering the algebra.

Possible approach

You might start with a drawing of a rectangle with sides $(a + b)$ and $(c + d)$, splitting it into four smaller rectangles and asking the class to give the areas of the large rectangle and the four smaller rectangles. Then the learners will be able to see how the formula for the expansion of $(a + b)(c + d)$ is derived. After practising the use of this formula the learners should have a sense of the link between areas and the algebra of multiplying binomial expressions and so they will be looking at the animation with some expectation that areas might be involved.

Key questions

What are the areas of the blue and yellow squares?
What happens when the yellow square is removed?
What is the area of the piece left behind?

Possible extension
See the problem Odd Squares.

Possible support
Try some numerical examples.