Woke up this morning andsaw the kids watching Saturday morning cartoons. Jack was eating cereal on a little tv table, and Lilly was upside down in the Lay-z-boy. I wiped some of the eye crust from my eyes and stumbled into the kitchen where I made some coffee, and loaded the dishwasher when I saw that Lilly's homework on the counter.

"Lilly, you were supposta do your homework yesterday!" I reminded her.

She bounced into the kitchen and said in her trademark fast speak "Ohyeah iforgottodothat." She gathered up the papers and went back to the tv room and started setting herself up.

After a few minutes she called out, "Daaaaaaad, I need your help!"

I looked over her shoulder and looked at the page of additions and subtractions she was working on. About 120 problems were lined out like an anal retentive's M&Ms on the sheet and she was right about in the middle of them. Simple problems of adding and subtracting single digit numbers.

"What's 6 take away 4?" she asked as she bounced up and down in the chair. She had written a 2 for the answer. "Is this right?"

I asked her if she was taught how to check her work by adding her answer with the bottom number to get the top number. She is more confident at adding than subtracting. "2 plus 4 is... 6!" she squealed and she waved her arms in the air. She almost stabbed me with her pencil.

"Yes," I smiled "that's how you check your work."

"Butsomeoftheseareaddingandsomearesubtractingandthey'reallmixedup,daddy"

"You know how to check your adding?" She nodded enthusiastically. "Do you need me to help you any more?" She shook here head without transisitioning and with just as much energy. "Okay, bobble-head, you're doing great!" She giggled to the pet name as I went back to the kitchen to finish the dishes.

With the coffee poured, the dishwasher on, the girls cratching her pencil on paper and cartoon voices emitting from the television, I went outside, I went outside for my morning smoke and I thought about that interaction with her. I'm no great mathematics theorist, like my brother is, but I like to play with ideas in my head. I started wondering about checking math. Isn't it interesting that the way you check Addition is the same way that you check Subtraction. Add the sum to the subtractor and you get the origin number in Subtraction. Minus the difference from the second second number to get the first number in Addition. Pretty easy stuff, really, for an adult. but the interesting part comes in when you try to do the checking with the opposite numbers. In Addition, you can sutract either either or the numbers from the sum to get the other number, but in Subtraction, this doesn't happen. This is easier to illustrate than to explain.

I'll use these simple problems: 4+5=9 and 8-6=2.

In checking addition, you can say either 9-4=5 or 9-5=4. Either way the sum is the same amount. In checking subtraction it doesn't work the same way. 2+6=8, which is correct checking, but 2+8 does not equal 8, it equals 10.

So that led me to this pondering: In Addition, does it really matter in what order the addition numbers are. 4+5=5+4=9, correct? Yes, of course it's correct. but are they the same numbers? is the 4 in the first equation the same as the 4 in the second equation? I say they are not. If you have 4 apple and your friend loans you 5 more apples, you are holding 9 apples. If you have 5 apples and your friend loans you 4 apples, you are holding 9 apples. but when it comes time to returning your apples to your friend, you have a real dilemma. do you give him 4 apples or 5 apples? Breaking it down to numbers, 4+5 does not equal 5+4, because what you start out with must be the same as you end with when checking. If you were holding 8 apples and you loaned 6 of them to your friend, you would be left hold 2 apples, and when your friend returned those to you, you would have 8 again, not 10.

Hmm. So what am I getting at? I'm not really sure. I guess what I'm saying is that by checking your addition it is important to know that the problem 4+5 is not the same as 5+4, even though the sums might both be 9, they are different 9s with different qualities. And THAT I think is what was interesting to me. That 9 does not always equal 9 fully. Each number has different qualities depending on its own unique history.

That's enough of that. Lilly came in wants a snack.

"What do you want for a snack, Lilly?"

She smiles and wiggles around "Can I have an apple, Daddy?"

"Absolutely"

"Can you cut it up?"

"ok"

She looks at me seriously. "Only cut it into six pieces this time, please, not eight like last time," is her explicit request.

"Oh?"

"Yeah. I can't eat eight slices."