Problem-Solving Activities that Engage Low-Level Students?

I'm going to be starting teaching a course called algebra COE, which is for students who didn't pass the required state algebra exam to graduate and are now seniors, to do spaced-out exam-like extended problems after extensive support.

I don't want to start the class out with "getting down to business" because I want the students to feel comfortable in the class, with me and with each other. The "getting down to business" will happen during the second week of class. Therefore, I'd like to start out with a class collaboration to solve a "fun" problem. (There are 5 students in the class)

At the same time, I don't want to start out with a problem that feels too contrived, or too much like "school math" problems. They have clearly been turned off from "school math." I want one or some that feel more like they are doing a puzzle, yet still engage algebra-related skills and open up a discussion about problem-solving as a process and skill that can be honed.

Some problems I have considered, yet I believe are too "math-feeling":

• The exponential chessboard and rice problem
• How many squares are there on the chessboard? (note: more than 64)
• The "lockers" problem
• The ooops game

Any suggestions?

• To clarify: You are asking about initially engaging students in non-"school math," and then — once the class of 5 and you feel comfortable around doing a "fun" problem — you will "get down to business" and have them doing "spaced-out exam-like extended problems"? Aug 29 '15 at 0:45
• Yes, somewhat. The general structure of the class before I came here was "review algebra, do practice problems, have students do state-mandated tasks, do more practice problems, have students do next set of state-mandated task" and so on. Each state-mandated task is essentially an extended algebra word problem. Aug 31 '15 at 16:41
• What does "COE" stand for? Sep 1 '15 at 7:03
• "Collection of Evidence" -- the students create documented evidence of their algebra learning through these tasks instead of an end-of-course assessment (which they failed). Sep 1 '15 at 16:42

The article titled "Math Teachers’ Circles: Partnerships between Mathematicians and Teachers" by B. Donaldson, M. Nakamaye, K. Umland, and D. White in the December 2014 Notices of the AMS (pp. 1335-1341) (pdf version) mentions a nice problem called "Dividing squares."

Shown below are some pictures of a square divided into smaller squares, not necessarily of the same size. (Image taken from the linked file.)

A few possible questions you can ask:

1. For which whole numbers $$n$$ is it possible to subdivide a square into $$n$$ smaller squares?
2. For which whole numbers $$n$$ is it not possible to subdivide a square into $$n$$ smaller squares?
3. Is it possible to perform the division so that there is a unique square of smallest size?
4. Is it possible to perform the division in such a way that no two squares have the same size?

The problem can lead to discussions on arithmetic sequences.

• I explored this problem once with a mathematical proof / proving class (directed at undergraduate math education majors). As far as number 2: Proving (in some rigorous way) that this is impossible for $n = 5$ is not so easy! (It should be possible for $n = 1$ and $4$, and for $n \geq 6$...) Sep 1 '15 at 22:14

The frog jumping puzzle is nice, in that it is very simple but hides some surprising but manageable complexity. Here's a "video game" version. Have the students play with this, and then try to generalize to n male frogs and m female frogs.

For another favorite, see the map folding puzzle of Martin Gardner as problem 29 here. This is a beautiful way to show the surprisingly delicate complexity arising in mathematics.

I've broken all sorts of posting rules here, but perhaps someone will find better links. Please feel free to modify this post accordingly!

What about the Monty Hall Problem?