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 1 11 3 1 0 16 2 6 4 ???

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.

1. September 24, 2008 10:16 pm

making an organized list is easy you just have to practise it

February 19, 2009 5:14 pm

Could you tell me what the definition of Organized list is?
I need it for math class by tomorrow.

3. February 19, 2009 5:39 pm

Laura,

An organized list is just a list that’s organized. More specifically, it’s a list of all the possible ways to do something, and organized in such a way that it’s easy to tell when you’ve exhausted all the possibilities. [Put that in your own words before you hand it in as the answer to your homework.]

If I ask you how many “words” you can make with two vowels (A, E, I, O our U) if both letters are allowed to be the same, you could just start listing valid “words” such as “IE, OA, UU, …” but you might end up with duplicates, or not be sure if you got everything. If you start with all the words that start with A, then E, etc. and go in a predictable order each time, you’ll be sure you got them all. Also, you may notice a pattern before you finish that will allow you to figure out the correct answer without listing all the possibilities. So you would start with “AA AE AI AO AU EA EE EI…” and making the list in this *organized* way helps you solve the problem more quickly and accurately.

Hope that helps.

4. February 10, 2010 3:18 pm

If i was playing a game of rock paper scissors with a friend with 20 rounds, and an outcome for each game either results from win,lose or tie, how many possibilities of the game would there be?

WIN LOOSE TIE
1 0 19
2 0 18
3 0 17
4 0 16

IM CONFUZED!

5. February 10, 2010 4:00 pm

P, one thing you have to decide, is do you care about the order of the games (W, L, T, W, W, ….) or just the totals (won 4, lost 6, tied 10)?

If you care about the order, then you have 3 choices of what can happen for each game (W, L or T) and you multiply those all together for 3^20 possible ordered outcomes.

If you don’t care about the order then think about it like this:

If you win only 1 game, how many different outcomes are there?
W1, L0, T19
W1, L1, T18
W1, L2, T17

W1, L18, T1
W1, L19, T0

So there are 20 possible cases where you won 1 game (the number of losses goes from 0 to 19, with one row for each of those)

What about if you win 2?

W2, L0, T18
W2, L1, T17
W2, L2, T16

W2, L17, T1
W2, L18, T0

So there are 19 cases where you win 2 games. See a pattern? Work from there. Don’t forget about the cases where you win 0, and be sure to go up to winning all 20.

• October 8, 2010 3:17 am