Fiona O. at Wilson Middle School began her solution to this week’s Pre-Algebra Problem of the Week by listing all of the possible strategies she thought she might use:

• draw a diagram
• make a table
• list possibilities
• simple computation/logical reasoning

I think Fiona’s list covers almost every single solution submitted. Good call, Fiona! The only other strategy I noticed was that some students used “Solve a Simpler Problem” to help them with their diagrams, tables, possibilities, or logical reasoning.

The problem, Trick-or-Treat Route, is about the Anderson children’s plan for trick or treating in their neighborhood (the students also had information about how many minutes it would take to walk between each pair of houses): Reading the solutions, I was really fascinated by thinking about how the strategies Fiona named (plus Solve a Simpler Problem) are interrelated. Here are some things I’m wondering about:

• Would making organized lists help students who jumped right to logical reasoning, but whose answers didn’t match the story?
• What are different ways to organize lists and tables?
• Do some ways of organizing lists and tables make it easier to see patterns and simplify the problem?
• How can you tell if you’ve found all the possibilities?
• How can you tell if you’ve over-counted?
• How do students learn to check their own work on “find all the possibilities” problems?

The first thing I noticed was that of the students who said, “I kept track of all the possibilities” and then over- or under-counted only two submitted their lists. The rest talked about their lists but didn’t type them up. I wonder… did those students not make written notes? Were they too hard to read to re-type? Is there a correlation between not wanting to write down the possibilities and not finding the right number?

Here’s a typical example of those not-quite-there-yet submissions:

1.15 different routes 2.a-b-e-c-d-a 3a-c/e-b/d-d/b-e/c-a
1.I kept track of all the different ways I found
2.It deosn’t go back and forth it goes in a circle type fashion
3.it kinda goes to one area and then back near the other

Almost all of the students who did find 24 possible routes did submit their list, and talked about how they had organized it. In fact, I only saw a handful of lists associated with a correct submission that didn’t seem to have a clear organizing principle — and maybe they had one and I couldn’t see it. Here are some of them:

For the first question I made a list I only put the initials of the first letter of each house: 1.A-B-E-C-D-A=26 min. 2.A-D-B-E-C-A=35 min. 3.A-E-B-D-C-A=41 min. 5.A-E-C-D-B-A=36 min. 6.A-B-D-E-C-A=37 min. 7.A-B-D-C-E-A=36 min. 8.A-E-D-B-C-A=50 min. 9.A-C-E-D-B-A=37 min. 10.A-B-C-D-E-A=41 min. 11.A-D-E-C-B-A=35 min. 12.A-D-C-E-B-A=26 min. 13.A-C-B-D-E-A=50 min. 14.A-B-E-D-C-A=33 min. 15.A-E-D-C-B-A=41 min. 16.A-D-C-B-E-A=39 min. 17.A-C-B-E-D-A=40 min. 18.A-C-E-B-D-A=35 min. 19.A-E-C-D-B-A=43 min. 20.A-E-B-C-D-A=39 min. 21.A-D-B-C-E-A=43 min. 22.A-C-D-E-B-A=33 min. 23.A-B-C-E-D-A=35 min. 24.A-D-E-B-C-A=40 min.

Maybe it’s just hard for me to see patterns because the different paths aren’t listed vertically? But I’m having a hard time seeing if there was a system to generating the paths. But they did get them all!

1. Anderson to Bader to Elliot to Campbell to Daily and back.
2. Anderson to Elliot to Daily to Bader to Campbell and back.
3. Anderson to Campbell to Daily to Bader to Elliot and back.
4. Anderson to Daily to Campbell to Elliot to Bader and back.
5. Anderson to Campbell to Bader to Elliot to Daily and back.
6. Anderson to Campbell to Elliot to Bader to Daily and back.
7. Anderson to Bader to Daily to Elliot to Campbell and back.
8. Anderson to Bader to Campbell to Elliot to Daily and back.
9. Anderson to Daily to Bader to Elliot to Campbell and back.
10. Anderson to Bader to Daily to Campbell to Elliot and back.
11. Anderson to Campbell to Daily to Elliot to Bader and back.
12. Anderson to Campbell to Bader to Daily to Elliot and back.
13. Anderson to Campbell to Bader to Elliot to Daily and back.
14. Anderson to Bader to Elliot to Daily to Campbell and back.
15. Anderson to Bader to Campbell to Daily to Elliot and back.
16. Anderson to Elliot to Campbell to Bader to Daily and back.
17. Anderson to Elliot to Bader to Campbell to Daily and back.
18. Anderson to Elliot to Bader to Daily to Campbell and back.
19. Anderson to Elliot to Campbell to Daily to Bader and back.
20. Anderson to Elliot to Bader to Campbell to Daily and back.
21. Anderson to Daily to Campbell to Bader to Elliot and back.
22. Anderson to Daily to Elliot to Bader to Campbell and back.
23. Anderson to Daily to Bader to Campbell to Elliot and back.
24. Anderson to Daily to Elliot to Campbell to Bader and back.

It seemed like the students got more organized as they went along, which was cool to see! I would have found it a bit easier to see the patterns if the students had used the initials A, B, C, D, and E. I wonder if they would have found it easier to type as well?

The most common organization of all possible paths began like this:

ABCDEA
ABCEDA
ABDCEA
ABDECA
ABECDA
ABEDCA

Then, some students continued to use their organization scheme to generate another 18 paths. But some students (maybe a third of all the organized listers?) noticed something like, “Since there are 6 ways of starting with Anderson to Bader, I thought if I did Anderson to Elliot or Anderson to Campbell or Anderson to Daily, there would also be 6 ways.” [-Cat D., MDES]
I wonder if the students who listed all 24 possibilities figured, “hey, I can generate them quickly, it’s worth it just to list them all?” Sometime being “lazy” can be a good way to generate new math ideas, because you look for shortcuts and simpler calculations. Did students who listed all possibilities take time to notice patterns and see ways they could have generated the number of possible outcomes more quickly?

Student Thirty at Caughlin Ranch Elementary School found an even simpler problem to solve and build their ideas off of (or at least a simpler way of representing their work):
I got 24 routes by figuring out that there are 6 routes from Anderson to either Bader, Campbell, Daily, or Elliot.  An example is if I am going from Anderson to Bader, I can get 6 different combinations for Campbell, Daily, and Elliot (C, D, and E).
The combinations of C, D, and E are:
C D E
C E D
D E C
D C E
E D C
E C D
These are all the different routes the children can take from A to B (Anderson to Bader). I did the same strategy for A to C, A to D, and A to E.
My hypothesis coming out of reading all this student work is that students need to move from trying to find possible routes (through listing, drawing, or acting out), to finding ways to write the routes in more organized ways, to noticing patterns in their writing. I also think that students as a group probably go through those stages (some kids, most likely, will skip stages or go through them in a different order) in terms of what strategies they can use. A class might not start out being able to get to an organized list for a problem like this, but by the end of a unit on Make a Table they might all be able to. But I also think that on each individual problem, you might have to go through all these stages, depending on the complexity of the problem. Like, “hey, I don’t get this, let me try some possible paths. Ok, that one worked. Will this one? Yeah, I seem to be getting some. Let me try to write them down. Is that organized enough? Are there any patterns or ways to be systematic? Hey wait, am I going to have to generate all the possibilities or can I take advantage of patterns or symmetry?”
When students get through the making and organized list and looking for patterns, they might be able to jump straight to the patterns and not have to make lists at all, which is what the rest of the successful solvers did (using the logical reasoning/simple computation strategy Fiona named). Let’s let Fiona take it home:
To solve the question, how many different routes could the children take, I first wrote what I knew. Then, I brainstormed different strategies I could use to solve this problem. I came up with drawing a diagram, making a table, making a list, or thinking logically/using basic computation. I decided that thinking logically and using basic computation would be the best way to solve this problem. To solve the problem this way, I thought that the pre-recorded times that were listed would be very helpful. I noticed that they began with the possible paths that could be taken starting at the Anderson household. There were 4 possibilities. Then, it listed the possible routes leaving the Bader household. There were 3 possible routes. Then, it listed the Campbell household, with 2 possibilities. Lastly, the Daily household was listed with 1 possibility. In previous years, I learned that if you want to find the maximum possibilities from a list of options, that you should multiply the options. From the strategy I learned, I multiplied the options together (4x3x2x1) and got the answer of 24 possible routes. I thought about the option of the route being the same if they walk in the reverse direction, but then i thought that it wasn’t the same because they would be going in a different direction.
My final question to leave you all with is this: which Common Core Math Practices are being evidenced/not evidenced by students at the different stages of solving? Certainly persistence is coming into play, but are they at different stages in “looking for regularity?” “making use of structure?” Is there a practice around organization in particular? Around solving simpler problems? Looking for patterns? How does looking at a snapshot of students at different developmental levels help us think about how the practice can be intentionally developed?