How do we get from one-half to one-and-a-half?

Teaching JD to count by halves made me want to throw a chair through our sliding glass doors. Let me explain.

The assignment: to measure area by square inches.
The format: there were four problems. Each problem had a picture using polygons to depict an area to measure. There were squares which equaled one square inch. There were also half-squares that were either depicted as a triangle or as a rectangle. To get the answer, we counted by halves and wholes.

The problem in our case: JD did not know how to count by halves. He barely understands what a half is. I get it - a piece is a piece. Cutting a pizza or a pie apart just makes smaller whole pieces, from a certain perspective.

I drew a number line to try to help him understand how to see that halves come between the numbers he already counts:

0          1          2          3          4          5          6          7          8          9          10

We read those. Then, I put halves in between so it looked like this:

0     1/2    1      1 1/2        2       2 1/2       3      3 1/2        4       4 1/2       5      5 1/2        6      6 1/2...

He just didn't get it. The connection wasn't there. I tried so hard and so patiently. On top of trying to teach this, he's looking out the window at the snow falling under the street light in the alley. He's yawning because it's around 7pm and we had been at it for 25 minutes and were on our second problem. (There were four problems to complete.)

"Okay, JD, let's count. Is this a square or a half?"

"Half," he would say.

"That's right!" I replied enthusiastically. Using a pen, I pointed to the next one, a whole square. "How about this one? Is it a half or a whole?"

"One-and-a-half?" he guessed.

AAAAAAAAAAAAAAAAAAAAAAAAAHHHHHHHHHHHHHHHHHHHHHHHHHHHH!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Chair through window.

Just kidding.

I don't know if it was more frustrating for him or for me, though.

And the reality is this: that's MY problem, not his. Failing to grasp the concept is not something he is actively seeking. Being autistic, he probably doesn't know why he needs to learn this. More specifically, he is apathetic to the need to learn it.

It is my false expectation that he should be able to pick this up easily. I was able to easily understand math concepts like this at his age. Projecting that ability on to him is terribly unfair. He will likely excel at many things with which I struggle in life. I need to chill out and accept that counting by halves will take a little while longer.

And, if it's so hard for him and so important to me that he learns it, I need to spend more time with him on this subject.

Or do I choose nuanced language?
Or do I work on communicating using Wh- questions?
Or do we read more books?
Or art projects?
Or physical fitness?
Or give the kid a break and let him play with his action figures because he doesn't get home until after 5pm three out of five school days?

Hey, at least teenage years will be easier!

To get the answers faster, I had him count the total number of half-squares, then find how many wholes that equaled on the number line. Then he counted the total number of whole squares. Then we found the sum. It got us through the four problems in 55 minutes. Maybe he'll be able to use that strategy on the test. Sigh.