Doesn't add up

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

Doesn't Add Up printable sheet



Here is an 8 by 8 square that has been divided into four pieces:

 

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Square split into trapeziums and triangles

 
Here is a rectangle made from the same four pieces:

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Doesn't add up

You might like to try cutting out the shapes from a piece of paper.

It seems that the same pieces can make two shapes with different areas!

Can you explain where the extra area comes from?

Here are some questions you might like to consider:

Can other squares be split up and rearranged to make rectangles with a different area?

Are there other square/rectangle pairs where the areas differ by 1 square unit?

Is there a pattern in the sizes of squares that can be arranged in this way?