To make the most of their construction (and a good problem), I then asked them to find the ratios among all the pieces that they saw.
They also realized that even though they may not have created congruent parallelograms among themselves, they all came up with the answer of 0.17 or 1/6. But it is a parallelogram, so make sure you have the same side lengths." Without much trouble, they found the answer. Soon one student asked, "What should the angles be?" I replied, "It doesn't say in the problem, so I don't know. I then asked them to construct this parallelogram using Geometer's Sketchpad (GSP) and find the answer to the question posed. They were pretty sure the diagonals of a parallelogram cut it into 4 triangles of equal area. They weren't seeing the key pieces (at least to me they were key), so I told them to look at the "relationships" among the pieces. Without a clear way to find the height, frustration mounted.īut major props to them for persevering as you can tell from their boards that they tried to dissect the parallelogram into even more pieces in hoping they'd find what the shaded piece would be equal to. So I showed them the parallelogram below just to be less helpful and to remind them about assuming something is isosceles because it looks it. (I can't get AB and DC to be exactly 6.00 cm.) Then they could use the Pythagorean theorem to find AY.
To find the height, they reasoned that triangle ABC was isosceles, making the perpendicular bisector AY also be the height of the parallelogram. They wanted to find the area of the parallelogram first. The one student who got it also struggled, but he was good about our rule of "never tell an answer." What I gathered from seeing their boards and listening to them explain:
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