The Garden Problem

For the beginning of the school year I wanted a short module to review and extend ideas about linear functions.  Simultaneously the module should involve students in working together and looking for patterns.  Well I found something that worked pretty well!  So let’s do the internet thing and share what works.

It might be helpful to know that my class is quite heterogeneous in terms of math background, and all of them are ELLs.  It’s an Algebra 2 class so they’ve “seen” linear functions before.

First, the Files

The Garden Problem

I found the Garden Problem while searching for components of algebraic thinking.  In her aptly named webpage, “Just WHAT IS ALGEBRAIC THINKING?“, Dr. Shelley Krieger offers a fine introduction and a beautiful problem that I swiped for my own use:

I introduced the pattern like this…

And then I changed the questions slightly…


  1. How many border tiles are required for a garden of length 12?
  2. How many border tiles are required for a garden of length 30?
  3. How many border tiles are required for the very long garden of length 1000?  Show and explain how you get your answer.
  4. If you know the garden length, how can you find the number of border tiles?
  5. Show how to find the length of the garden if 152 tiles are used for the border.
  6. Can there be a garden that uses 2011 tiles? Explain your reasoning.  (When you’re done with this question let me know!)

Question 3 is something that I’m starting to see as the bridge between “Length 12” and “Length x” from the original problem statement.  If you can do it with 1000, then you can do it with x.  Question 6 was also nice because some students used their formula to show that the garden length would not be a whole number if it used 2011 tiles, while other students made the reasonable observation that every garden tile number is even.

Extension Problem

Knowing that I have some students who would breeze through these questions and appreciate something a little trickier, I devised an extension problem for this scenario, as follows… (I’ll leave the solution to the reader)

Group Poster Problems

To build on their work with the Garden Problem, I designed a group task for students to solve and present solutions for a new patterns.  The patterns were copied over from an activity written by the nice people at the Math Learning Center, who appear to be having some website problems.  The activity that was formally available at can be downloaded from my dropbox here.

Here’s a sample of what a group might encounter…

You might notice that this pattern is not linear!  So we start the discussion for comparing linear and nonlinear functions.

Finally, if you paid attention to the files at the beginning, you might have noticed some sort of “Audience Participation Activity.”  I wanted something that helped the students get to know each other (in this case writing down names), and to help them get engaged in the presentations.  I made the “write an interesting question” part optional because it can be difficult to come up with such a question, and I didn’t want to stress out the students about this non-critical part.

If I get my act together I’ll take some pictures of the posters — there were some nice ones.  This whole introduction to linear problem-solving occupied the greater part of four days at the beginning of this school year.  As I start looking at exponential functions, I’m wondering how I can engage in the same sort of patterns-based problem-solving and algebraic thinking.  Time will tell, and I hope to share it if it works.

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