A: When you're seven years old and you've just finished counting all the dimes in the change jar (91). You've successfully made an array of ten dimes in nine rows. There's one dime left over. You count each row of ten: "Ten, twenty...ninety..." and then you look at the lonely little dime and say, "And ten more! 100."

After a hugely frustrating

**and**amounts, your mother ends the lesson (that was, honestly, a little too long but she pushed it a little since it

*had*been going well up until that point) and goes off to wash dishes or sweep the floors or do laundry which is what she does when she needs a little space and time to think something over.

It's at that point when your mother thinks back to something she read about fractions and units in this article from the MAA. And to a fabulous video lesson by Christopher Danielson and produced by TED-ED on units called One is one...or is it? She also thinks back to a post by Julie on using Cuisenaire rods for a factoring activity, and, finally, to a book called 1+1=5 that she read about in this Moebius Noodles post.

She thinks and she thinks, for days, actually, and then comes up with a very simple worksheet she titles "Unit Equations!" The exclamation point is intentional because, really, the world of numbers is an exciting mystery to be unlocked, whether you, the seven year old, know it or not.

The worksheet poses questions like:

1 carton = 12 _____

2 hands = _____ fingers

4 seasons = _____ year

_____ hour = 60 minutes

1 triangle + 1 square = _____sides

...and, of course:

10 pennies = 1 _____

What you don't know is that your mother was judging for frustration tolerance level during that activity and has decided to see if you can go a little further. She worked up a little activity on Saturday night which included finding the factors of each factor of 3 up to 15. The basic instruction is that, no, we're not adding, we're finding multiples. Each row of blocks must be exactly the same length (thanks Julie!) and each row must be a different color.

This activity, overall, is in the 'just-right challenge' category, but there is still some confusion. Two light greens (2 x 3) is the same as all whites (6 x 1) is the same as reds (3 x 2) is the same as dark green (6 x 1) which looks blue in the picture.

"What does that mean exactly?" you ask.

Your mother stands the dark green rod (6) on its end and says, "This rod is 6. There is one rod to make six. So, that's 6

*one*time. The white rods are each 1, and there are six of them. So, that's 1

*six*times. The funny thing is," and here your mother pauses, "...can I give you a super big word? It's called the commutative process. That's just a fancy word that means that no matter what order you multiply or add numbers, the result is the same."

And, as your eyes do not convey complete understanding, your mother continues, "What that means is that even if there are two 3-rods or three 2-rods, they are both the same total length."

From there, the activity proceeds wonderfully, all the way up to five 3s, and when it is all over, your mother asks hopefully, "So, what did you notice while you were doing this activity?

And the seven year old replies succinctly, in her sweet voice that resembles nothing of the frustrated girl from last week, "That one thing can also be something else with many parts."

YES!!!

You're hilarious. ;) Thanks for posting at Math Monday. This reminds me of our recent lesson--using egg cartons for fractions--and talking about how 12 is really one. Or...how can 12 mean 1? ;)

ReplyDeleteThis is great! Yes, the frustration is there and, at least with M and myself, on both sides. Love the idea of a worksheet. We did play a similar Cuisinaire game some time ago, but somehow it didn't translate into understanding money. So for now trying to exchange 5 pennies for 1 nickel is still a problem here (somehow M feels that 5 pennies contain more value than 1 nickel).

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