More patterns on the multiplication machine

When we left off last week, I had challenged Tabitha to find a pattern on the multiplication machine so that there would be the same number of buttons up as down. This challenge followed up on her sophisticated argument that her down-up-down-up pattern yielded more downs than ups.

ups.and.downs

There are 81 buttons, so the task of evening out the ups and downs is not possible.

But Tabitha is 6 years old. She knows little about even and odd numbers. Searching for a way to share 81 things equally (between up and down in this case) is a good way to get her thinking about the idea.

You may recall that I had shooed Tabitha off to her bath on giving her this challenge. This is where our story picks up.

At the end of the bath, she puts on her jammies and announces…

Tabitha (6 years old): I know how!

She runs into the room to get the machine.

T: Now Daddy, I don’t know if this is going to work, so just keep your ideas to yourself.

This line is awesome, is it not?

I do as I am told.

She produces this:

Photo Nov 20, 9 41 52 PMT: Oh no.

Me: What?

T: These [she points to top and bottom rows] are both up.

She tries again, producing this:

Photo Nov 20, 9 42 20 PMT: Oh no. Still too many up.

At this point she gives me a look which I take to mean that I can have a try. So I go back to her first pattern.

Photo Nov 20, 9 41 52 PMAnd I start to share out the bottom row—half up, half down.

patterns.6Tabitha: But Daddy! That’s not a pattern!

So what do we learn?

The raw beauty of Tabitha’s line, “I don’t know if this is going to work, so just keep your ideas to yourself!” strikes every time I think about this conversation.

Children enjoy investigating their ideas. I have to work very hard to get many of my college students back to a mental place where they trust that they have mathematical ideas worth investigating.

The best thing a parent or teacher can do in this situation is be quiet and let the kid work it out.

Starting the conversation

As all interesting conversations do, this one had a trajectory. We started in one place (making fun patterns), focused our attention on one part of what we were doing (comparing the number of ups and downs) and finished off with a “what if” question (what if ups and downs were equal in number, what would that pattern look like?)

You can practice that with your child. It doesn’t matter whether any particular conversation goes anywhere (many of ours do not), eventually you’ll hit on something interesting to both of you and pretty soon you’ll notice that 10 or 15 minutes have gone by.

And then the next time will be easier. Soon it will be a habit.

 

4 thoughts on “More patterns on the multiplication machine

  1. YES! I meet with 5th graders for an hour before school starts to play with math. I showed them what the word conjecture means for mathematicians. Now we always mark off a portion of the white board for CONJECTURES. Everyone and anyone is encouraged to write any and all conjectures in the box. Then we all go through and see if we can convince each other that a conjecture is true. Or find a reason why a conjecture is false. My dream is for the CONJECTURES box to be the entire board and every student has a conjecture up there.

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