Math = Love: Evaluating Functions Puzzles - Open Middle Style

Saturday, June 4, 2016

Evaluating Functions Puzzles - Open Middle Style

After making this activity to practice classifying relations as a function or not a function, I was eager to make a similar activity to practice a different skill.  I settled on creating a practice activity for evaluating functions and better understanding function notation.

I wanted to make this activity use the same integer cards (-4 to +4) that the previous activity used to save me time cutting and laminating.

I grabbed a notepad, wrote a few functions, and fiddled around with the numbers until they worked.  

This is what I came up with: 

Place the integers between -4 and +4, inclusive, in the missing boxes to make each function evaluate properly.  Each number may only be used once.  

I printed off the puzzle and gave it to my husband/lovely beta tester to give it a shot.  After trying what felt like a zillion combinations, he came up with a solution.  I took a look at it and realized it was a slightly different solution than I had in mind when I originally created the problem.  For one of the functions, he used the same three integers I had; however, he had the integers placed in different locations within the puzzle.

We discussed the problem and decided we didn't know how long our students would stick with it.  The overall idea was good, but it seemed like there was just too much guessing and checking needed.  The presence of a square root symbol did mean there could only be a 0, 1, or 4 in that slot.  But there were no other similar clues throughout the puzzle.

I dropped the idea for a few days until picking it back up today.  I stared at the puzzle for a while before deciding maybe it would be more logically solve-able if I provided the solver with some of the input and output values.

This led to a different version.  My functions were longer and more complex, so I had to change the paper orientation.  That was annoying.


Place the integers between -4 and +4, inclusive, in the missing boxes to make each function evaluate properly.  Each number may only be used once.

My husband found a solution to this puzzle in about five minutes.  Given that he's one of the mathiest people I know, I'm expecting it would take students at least 15 minutes to solve the puzzle.

One exciting thing was that the solution I intended when making the puzzle and the solution Shaun came up with were totally different!  For one of the functions, we used the same three integers, but we placed them in different places.  For the other two functions, we used completely different sets of integers to solve each.  This makes me wonder if there are even more solutions to this puzzle?

I'm not as thrilled with these as I am with the function/not a function puzzle.  I just feel like these puzzles are missing something that I can't put my finger on.

If you're interested in using either of these puzzles in your classroom or trying them yourself, I've uploaded them here as editable Publisher files and non-editable PDF files.

Each file comes with a puzzle sheet and a sheet of integer squares to cut out.


  1. Hi Sarah, I actually like the first version of your problem better but I agree that is a bit challenging. I only found 6 possible solutions. You could change the functions a to allow more possible solutions. If you remove the squared in the g(x) function you can increase it to 10 possible solutions and if you change it to g(x) = __x-5 you can increase it to 18 possible solutions. Anyway, just my 2 cents worth. Thanks for the great ideas.

    1. Thanks for the feedback. I like the change you suggested!

  2. I did a problem like this for function notation, but called it "drag and drop" to mirror the question type of our state's EOC. You have inspired me to write a blog entry about it. Check it out
    My problem is easier than yours but it may be a good way to scaffold up to your problem. Keep the good ideas coming! :)

  3. Thank you for sharing!
    It takes time to create any activity and I appreciate your time and effort to put it out there to share.

  4. I did this activity today with my kids. THey LOVED it. LOVED. Great way to review function notation. There was excellent "number talk" happening. Thanks for sharing

    1. Yay! Looking forward to trying it with my kids now!