Make an Organized List
Yesterday, I taught a “Make an Organized List” lesson to the intermediate math group (approx. grades 3-5).
I asked them, “If I want to make exactly 16 cents, using only nickels and pennies, what are all the different ways I could do that?”
I made a table and had them suggest ways, which I recorded. We ended up with a table that looked something like this:
|Number of Nickels||Number of Pennies|
Before putting in that crossed out 4-nickel row, they spent some time trying to come up with additional solutions. When they appeared stuck, I asked if they thought we had all the answers and they said they did. I asked how they knew, and they were able to explain that we had used all the numbers of nickels less than 4, and that 4 was too many nickels, since that would be 20 cents. (Then we had a discussion, initiated by the kids, about how it would be ok if we could have a negative number of pennies, but that that didn’t make sense in this problem.)
I asked them to look for patterns. They found that the numbers in the pennies column all differed by 5. They also found that if the number of nickels was even, so was the number of pennies, and likewise for odds.
I asked how we might have organized our list better from the start, so that we’d have known right away when we had found all the answers. They suggested organizing by number of nickels, starting from zero. So, I sent them off on their own to do the same problem with 22 cents. Easy!
Then I asked them to find all the possible ways of making 22 cents using only dimes, nickels and/or pennies. Predictably, this list was harder to organize. It took them a while to realize that they could have several rows for a given number of dimes. But most of them soon realized that the zero dime rows would look just like their results from the nickels and pennies only problem. And in the end, most of them did end up with organized lists, and being confident that they had all the solutions.
As they finished, I had them start playing “21” with the other kids who had finished. More about that in the linked post.
I liked this lesson because doing the example together at the beginning clearly showed them the value of organizing the list. And they basically got it, and were able to organize a more complicated list (some with more support than others).
On Wednesday, I did an “Act It Out” lesson with the youngest group (K-2 level) where one of the questions was “If there are cows and/or chickens in the yard, and I count 14 legs, what could I have”. These kids used blocks to represent the legs, and grouped them into piles of 4 (representing cows) or 2 (representing chickens). Although I didn’t expect them to organize their lists of answers, I did ask them to try to come up with all the answers they could. At the end I asked them to share their answers, and asked them how I might organize them on the board to make sure I didn’t repeat any. So we did a little intro to organized lists there too. We sorted by number of cows. These guys also noticed some patterns — like the fact that the number of legs on the cow side increases by 4 each time, that all the numbers of legs were even, etc.
It takes a long time for the idea of making an organized list really sinks in. But I think we’re well on our way with both of these groups of kids.