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.

 

Patterns on the multiplication machine

Tabitha (6 years old) has been playing with the multiplication machine off and on for a few months now. We have never once used it for learning multiplication facts, but we have had a ton of math-learning fun with it.

Recently, she carefully did an up-down-up-down pattern that she continued along all of the columns.

ups.and.downsWe discussed her patterning a bit before I sent her upstairs for her bath.

I followed behind and, when I knew the machine was out of reach, asked her a question.

Me: You know how some buttons are up and some are down right now? Are there more ups or more downs?

Tabitha (6 years old): There are the same amount.

This was the answer I expected. As we will see, her reason for the answer surprised me.

Me: How do you know?

T: I did two patterns. Up down up down… and Down up down up. Let me show you.

We head downstairs together.

Me: OK. In this one column [I point to the first column on the left], are there more ups or downs?

T: [She counts each type] Downs. But in this one [the next column to the right] there’s more ups.

Me: So how about the whole thing?

T: Same!

Wait.

I’m gonna figure this out.

1,1,2,2,3,3,4,4,5

There’s more downs.

Me: How do you know?

T: Watch. [She points to columns as she counts, one column for each number word] 1, 1, 2, 2, 3, 3, 4, 4, 5.

Me: Right. But what does that mean?

T: One. There’s more downs. Then one. There’s more ups. Then two, and two, like that. But five is more downs.

Me: Wait. One has more downs, then the other one has more ups. So together the 1s have the same number of ups as downs?

T: Yeah.

Me: So then what?

T: Same for the 2s and 3s and 4s. But 5 has more downs.

Me: So there are more downs total.

T: Yeah.

Me: I have a challenge for you. If you can, try to make a pattern that has the same number of ups as downs. But not now. Now you have to get in the bath.

(to be continued)

So what do we learn?

We have to keep an open mind when our kids are telling us what they think.

I knew there were more downs than ups because we started with a down in the upper left. Moving across the row, there is an up for each down. The last down at the end of the row has an up at the beginning of the next row.

Every down has an up.

Except for the bottom right corner. That down has no up to pair with. So there are more downs than ups.

This is not at all how Tabitha saw it. She made groups—each column has an extra up or an extra down. Then she grouped these groups—each pair of columns has equal downs and ups.

By keeping an open mind, I was able to listen to her thinking. This let me ask follow up questions, which helped Tabitha make her own thinking better and more clear. I try as much as possible to have explain your answer be about convincing each other of something. You think they’re the same? Convince me. You think there are more downs? Convince me.

Starting the conversation

Patterns. Notice them. Play with them. Ask about them.

Any repeating pattern will do.

Even if the pattern had been Down down up, down down up (i.e. two downs for every up), we would have had a lovely conversation about which there were more of. The keys to the thinking in this pattern were:

  1. There were too many things for counting to be a convenient solution, and
  2. We started thinking about it when we didn’t have the machine in front of us.

I would have been content if Tabitha had counted 41 downs and 40 ups. But I was very much hoping to push her to use the pattern she had created to reason rather than to count.

Multiplication Machine [Product review]

We have in our house a thing called the “Multiplication Machine”. It is a flash-cardy sort of thing. The multiplication facts are written on top of a 9×9 array of spring-loaded buttons. You press one and the button pops up, revealing the product. You can buy such a thing at your nearest teacher supply store. Ours came from Lakeshore Learning.

Talking Math with Your Kids is dedicated to helping parents and other caregivers to identify the mathematical opportunities afforded by everyday life, so we will not discuss here the traditional, intended use of this product (which is drilling and reviewing multiplication facts).

No, I want to talk about this thing as a toy.

Tabitha was hard at work pressing buttons on the Multiplication Machine the other evening. When I peeked in on her, I saw a scene that looked an awful lot like the one below.

patterns.3

She was playing with the arrangement of up and down buttons, not with the multiplication facts written on them. Patterns are tons of fun. So I went with it.

We developed the up, up, up, down pattern. We went across each row from left to right, top row to bottom row, as you would read a book.

Up Up Up Down, etc.

We developed its opposite—Down, down, down, up.

Down down down up.

We developed the Up, up, up, up, down pattern. This proved much more difficult for Tabitha, as she could not subitize the four ups. She counted them on her fingers, which she also needed for pressing buttons. She worked it out, though.

Up up up up down.

Before executing this last one, we noticed the right-to-left diagonals we had gotten from the Up, up, up, down pattern and predicted what we thought would happen when three ups became four. She correctly predicted the left-to-right diagonals, but I do not know why she predicted this.

We have not yet investigated the down-down-up pattern together, but I suspect she will get a kick out of it.

Down down up.

There are many more cool patterns to play with here. A few ideas that I am sure we’ll explore in the coming weeks:

  • What will happen with lots of different combinations of ups and downs?
  • What if we do columns instead of rows?
  • Are there any patterns where you cannot tell whether the person did rows or columns?
  • What if we follow a path back and forth across the rows, instead of starting at the left-hand side of each row?
  • What if go right-to-left? Or bottom-to-top?