This has been a very interesting week in my math classes. I've had a couple students who came up with interesting points in two different math classes.

This week, I've spent time helping the geometry class do some basic vocabulary work using the Freyer model which requires a definition, fact, example and non-example for each work.

I started them with point, ray, line, line segment, midpoint and intersection. So of course when the non-example came up, the kids thought they would be silly and called out apple or pine cone which I said were great. When we got to intersection, one child called out "A mountain!" for a non-example of an intersection. Another child disagreed.

Her argument was beautiful. She said it did represent an intersection because it was the bottom part of the intersection and the top of the mountain was at the actual point of intersection. This is out of a 9th grader. This lead to a really interesting discussion and the final conclusion from the class boiled down to the two lines did not extend all the way after crossing so it was a non-example. After all, an intersection of roads continued after they crossed. I let them draw their own conclusion on this argument.

The other interesting conclusion came from the Algebra I class with their review of exponents. I asked for examples of exponents in real life and most students came up with scientific notation but one student provided a variation.

He pointed out that we use exponents when we label the answer to area or volume problems. A couple students disagreed because the number itself did not repeat but the student stated the unit was being repeated.

To finish of this exponent question of can units be repeated, I'm having students create a piece of art where they choose an item like a piece of candy, draw it with an exponent and state that is equal to the item times itself.

For instance, a ice cream cone squared equals a ice cream cone times an ice cream cone. I hope this reinforces the basic concept of exponents. Let me know what you think of my student's arguments.