# Mathematical Task with Various Solutions

I need to come up with a mathematical task for middle school (9th grade), which involves either algebra, functions, probability or statistics (anything but geometry actually). My problem is, that the idea behind the task is to allow students to work together and learn from ideas of each other. Therefore, the task should be such that there are 5 different ways of solving it.

I will give you an example. In the picture below, you see a sequence of hexagons. If I asked students to calculate the perimeter of the hexagons, and to predict the perimeter of 20 hexagons, they would solve it in various ways. One student would count the perimeter (each edge is 1), find out an arithmetic sequence and derive the formula (4n+2). Another student could count the upper edges, multiply by 2 and reduce 2, leading to the same solution. A third student could multiply the number of hexagons by 6 and reduce the mutual edges. What I am saying, that different approaches lead to the same solution (4n+2).

I need to come up with another idea where students can come up with different solutions, different ways of thinking, and end up with the same final solution. The problem should not be too hard, middle school, and not concerning geometry (for algebra lesson, which includes the topics I specified above).

The example below doesn't count as geometry since it involves sequences. No geometry means no Euclidean geometry. I would appreciate it if you could give me some ideas because I am pretty much lost. Thank you in advance.

• (May try to answer when not pressed for time but...) Check out Leikin's writing on multiple-solution tasks to see if there are helpful examples. – Benjamin Dickman Nov 11 '16 at 12:32
• Thank you Benjamin, I will do that. I have a few days to perform this, so if you come up with an idea, I would appreciate it if you share. :-) – user2899944 Nov 11 '16 at 13:34
• I had a look. Leikin gives an example of a system of equations. One of the method of solving is the use of matrices, which is way too far for middle school. I was thinking about 2 equations, but how many ways do you have of solving it? I can think of 3 (isolating one variable, subtracting one equation from another, and graphically). – user2899944 Nov 11 '16 at 13:37
• I was thinking, maybe something with sequences, or probability (counting , formula, venn,...) – user2899944 Nov 12 '16 at 9:23

Here is one I enjoyed from middle school. This was a project: I think we had a whole week to experiment, and discuss, and come up with a solution.

Consider a rectangle a 231 by 84 rectangle which is tiled with 1 by 1 squares. How many squares does a diagonal of this rectangle pass through? I think this was phrased in terms of a mouse running from one hole to another along a tiled floor (for cuteness points).

We did a lot of experiments with smaller rectangles. Students noticed that sometimes the line would pass through a lattice point, and asked if this counted as passing through all 4 adjacent squares or not. We had a discussion about this and decided, as a group, that we would define "passing through a square" to exclude only touching one of the corners: you had to pass through the "interior" of the square.

People had lots of different approaches. The solution involves the common factors of the edge lengths, making this more "number theory" than "geometry".

How can we transform the colored octagon into the octagon C? Name the transformation(s).

There are at least 5 options for just one transformation!

Edit: My bad. I didn't notice that geometry is not disired!

• Hexagon? Square? Octagon? – mickep Nov 13 '16 at 7:11

One possibility: Counting the number of unlabled trees:

No closed formula is known, but the number for each small $n$ is known. You'd have to explain when two trees are structurally different. Aiming for $a(7)=11$ would generate a lot of variety.

Counting the number of labeled trees leads to Cayley's formula, $n^{n-2}$. It may be more satisfying to have a formula, but I think it is more difficult to manage the labeling, as the count grows so quickly.

• One could also count rooted binary trees. – Steven Gubkin Nov 15 '16 at 13:35
• @StevenGubkin: Good point. Leads to the Catalan numbers. – Joseph O'Rourke Nov 15 '16 at 13:38