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Area Ratio

On Friday we wanted the kids to learn about area and perimeter ratios. We had them make rectangles that took up a fourth of their desk. They saw that even those the sides were half the size of the sides of the desk, the area of the rectangle they drew was 1/4 the area of the desk. They did the same thing with a rectangle who's sides were 1/3 the size of the sides of the desk. They saw that the area was actually 1/9 of the actual desk. Then we gave them two rectangles and had them calculate the perimeter and area for both. Then we had them find the similarity ratio, perimeter ratio, and area ratio. Then they had to make a prediction about what the area ratio would be of two shapes with a similarity ratio of 5/6. It was AWESOME to see the kids work together throughout this activity to make guesses, question each other, and think logically for a good 15 minutes. This was really encouraging, but then I had to have them spend 5-10 minutes putting an example in their notes and writing the 2 important points down... I literally saw the learning stop for a lot of kids as soon as I started "teaching"...

What I saw was that they are perfectly capable of making discoveries and making arguments about simple elegant mathematical questions. Here is what needs to change.

1. We have not enabled students to take it further to test their own arguments, their peers arguments, come up with counterarguments, counterexamples, or further questions.

2. We don't have time to have kids discover very often. Although when I think about how much they'll actually remember and value in the future compared to how much they might value and remember if they had the opportunity to discover things and question things all the time, it makes me reconsider everything I do in the classroom...

3. Kids won't find patterns and test them and have the same ideas at the same pace... If we could truly allow students to discover more of the curriculum, everyone would be in different places. At the same time, it would be valuable for the students to see the different ways their peers came to similar or different conclusions and then work together to figure out what makes the most sense.

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