Task# Different Areas?

Illustrative MathematicsVaries

The goal of this task is to motivate a discussion of similarity and slope via a counterintuitive geometric construction where it appears as if area is not conserved by cutting and reassembling a simple shape. There are many viable approaches to this task including: Putting together the two congruent right 'triangles' to make a 'rectangle' and observing that they do not fit properly, leaving a long thin empty space which explains the discrepancy in areas. Calculating the slopes of the two line segments on the quadrilateral which appears to be a triangle: they are not the same and this means that the shape is not actually a triangle. Applying a dilation to one of the 3 unit by 8 unit triangles and observing that it does not make a 5 by 13 triangle.