The seventh graders noticed a problem when working with similar figures (figures that are the same shape but not necessarily the same size, as shown below).
The students could find a scale factor and use that to find missing side lengths but when they tried to use the scale factor to find an unknown area it didn’t work. They set out to study the problem in more detail.
First, the 7th graders made triangles with a scale factor of 2, 3, and 4 of the original. The students then repeated this with squares, rhombuses, and trapezoids and displayed their data in a large table. This confirmed what they knew- that the scale factor worked as a multiplier for the side length and the perimeter. It also confirmed the problem – when they multiplied the area of one triangle times 2 (the scale factor) the area they got was 2. But, when they counted the triangles they had 4. When they did times 3, the area was 9 triangles. Times 4 was 16 triangles. It was true for the squares, rhombuses, and trapezoids too. They realized there was something they could multiply by.
By squaring the scale factor, they found a multiplier that would help them solve unknown areas!
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