David and the "Important .5"

David is being home-schooled. He is learning math from an on-line program called, "ALEKS." It's a pretty decent program with a lot of interactive / response code built. For example, if he fails to get a problem right, he will have to practice the problems associated with the concept more times; if he gets it quickly, he will only be required to do 2 exercises. There is an "Explain" option that explains the concept and walks through how to solve the problem.

The issue with David is that he learns the steps and does not always get the "concept." This is where Grandma comes in.

On this particular day, the program wanted David to calculate the surface area of a prism. He wanted to jump right in.

"David, how many sides does this figure have?"
"I don't know," was the immediate, frustrated response.
"Well, let's figure it out."
"One, two, three ..." he started counting, "... four, five, six, seven, eight, nine ..."
"David, don't count the edges [at least, I think that's what he *might* have been doing], count the sides. What shape is the back?"
"A square." [It was a rectangle but he knew that.]
"What shape are the sides?"
"Triangle." Etc.
"So, how many sides are there?"
"Five."
"So to get the area, we add up ..."
"The sides."

Ready to start the problem. He uses the calculator way too much for my liking.

Open paren. '9' '*' '5'
"David, HOW MUCH is 9*5?"
A glance. A momentary lull. "45!" he announced. Backspace, backspace, backspace ... "45" gets typed.

Perhaps I should have let that one go. As he started entering information about the five sides, he had to stop and figure out which sides he's already done. Had he left the "9 * 5" it might have been more obvious.

As he got to the triangular sides, he announced, "... and the .5, that's important."
"What?"
"The .5. You have to use it."
"Why?
"I don't know."
"OK, David, do you remember we talked about this before ... [drawing paper] Suppose we had a triangle with 3", 4" and 5" sides. If we complete a square using the two shorter sides, what is the area of the square?"

(3" * 4") = 12"

"And if we cut it in half with a diagonal line, that forms the traingle. See? That's where the .5 comes from."

Had to repeat it a couple of times, and may have to again. But he gets it. Score one for David (and for Grandma) today.