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Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.

“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed each of my two hands, opening up all my fingers as I did.

“10!”

“OK, so five times two is ten.  That’s five groups of two.  Six times two is one more group of two right?”

“Right.”

“So, when you add that extra group of two in what do you get?”

“12?”

“Yeah.  Is twelve the same thing as eighteen?”

“Snort!” she chortled.

“OK, I have to hit the potty, I’ll be right back out.”

When I returned, No. 1 looked at me, beamed a little, and said, “It’s nine!  Eighteen divided by two is 9!”

“Awesome job,” I beamed back.  “How’d you figure it out?”

“Well, I did ten divided by two, and eight divided by two, and then I added up all the groups of two.”

“Why 10 and 8?”

“Well they’re two parts of 18.”

“Why’d you do it that way?”

“It was easier.”

“Why was it easier?”

“Once I took apart eighteen into ten and eight, then I could do all the division on my fingers to figure out how many groups of two there were!

And, that’s how No. 1 reinvented the distributive property... of division, of all things.  Because she needed it!

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