Does "center" always mean "half"?

I'm spending the next couple months trying out some new non-dance, moving- and body-scale math lessons for my new book.  There's an activity in Mathematics Their Way (from the '70s) that I want to try out.

The lesson is ostensibly about using your whole, moving body to find the "middle" or the "center" of a space that you can't fold. I was thinking I'd start my version of the lesson by asking first and second graders "What is half?" I planned to give them a piece of 8 1/2" x 11" paper and see what they come up with.

Except, as I was getting the materials ready today I noticed some paper cut into big triangles. I took one up to my 10yo who was doing her homework. I asked her to fold it in half. This is what she did.

 Then I asked her to fold it into half again. The image below illustrates what she was doing for a split second before she said..."ah...no..."

At which point she said, "I can't fold it into half that way. But I can fold it into fourths."

I tried to get her to tell me how she knew when she had folded it in half, but she wasn't interested in talking. But it did make me wonder how I might fold this triangle into eighths.

Then I thought: Wait a minute! We've folded this triangle into eight one-eighths and still haven't found its center/middle. Or, well, maybe we did on the first fold ... but maybe there's another center we haven't yet found?

I love it when what I think I know is challenged. We need to do this for our students on a regular basis.

And, I'm rethinking my lesson for tomorrow. I'll focus on the language of "middle" or "center" and use rectangular and triangular paper. But "half"?  Half I'll save for another time.

Half A Cake

"Hey, tomorrow's your half birthday!  I know, let's make a cake to celebrate!"

Turns out the recipe called for 1 1/2 cups of flour.  Since our cake was ingredient free due to multiple food allergies, we used even parts of two gluten free flours.

"Okay so we need one and one half cups of flour.  How many halves in a whole?"

"Two!"

"Okay, so here is the 1/2 cup measuring cup.  One scoop of the buckwheat, one of the rice... Now, we need 1/2 cup more flour, using both our flours.  What's half of 1/2?"

"One-fourth!"

"How'd you know that?"

"Oh, I just knew..."
..............
Later in the morning, at the library.

"Hey look Mama!  The [really big, red tiled] square is made up of smaller squares!"

"Very cool.  How many small squares make up the big square?"

"Three, six...nine!"

"So, you wanna hear something interesting?  When you hear someone call a number 'square' that's what it means.  Look, nine squares literally make a 'square' number!  Isn't that cool?"

"Mmmm hmmm..."
.............
After lunch.

"Hey, it's time for cake!"

"YAY!!"

"But we can only eat half.  How would you cut it in half?"

[Girl gestures a line from the middle of one side to the middle of the opposite/parallel side.]

"Well, that's one way.  How else could you do it?"

[Girl pauses and then gestures diagonally from corner to corner.]

"Excellent!  But I have one more idea.  How many squares made up the big square at the library today?"

"Nine."

"So, I"m going to divide the cake into nine parts.  Here's one for you, and one for me. We'll have more later, after dinner."
...........
After dinner, and after one more piece each for the child and myself.  (Papa doesn't like chocolate, so no cake for him!)

"Mama, can I have a little more cake?"

"Hmmm, maybe.  We're only going to eat half of the cake -- let's see if there's any left....  How many pieces did we have to start?"

"Nine."

"What's half of nine?"

"Four and a half."

"So, together you and I have eaten four pieces.  There's half a piece left and you can have it!"

Ladies and gentlemen, I present to you: half a cake!