Welcome to Variable Analysis!

Variable Analysis Games are terrific to encourage algebraic thinking in your students, and they are expandable to many different levels — elementary, middle, and secondary.  And the best part is that all of your students can win — the game is not over when the first person wins, nor will that first person be bored once they “get it”.  I’ll give you strategies for keeping all students engaged for the entire time each round.

I’m going to focus on middle school problems that my students can use, but if this site catches hot fire in the MATH TWITTER BLOGOSPHERE (#MTBoS) I’ll open up new authors for elementary, secondary, or even college level Variable Analysis Games.


Each variable analysis game (see the ARCHIVES on the right for the current collection!)  is a set of three rows of numbers, with column headings labeled a, b, c, etc. (the number of columns can vary depending on how difficult you want to make the game).

The students’ job is to find the relationship between the numbers, and that relationship is consistent in each row.  They must communicate that understanding not by explaining the relationship, but instead by creating a new row of numbers underneath the original three that correctly fits the relationship.  This way everybody gets to play — the first person doesn’t give away the answer but instead provides more evidence to help the rest of the class.

Let’s do a very simple example just to see how it’s played:

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So, you would instruct your students to create an equation with the number 2, 6, and 3.  No extra numbers.  Just those three.  That same structure should work for 4, 24, and 6 as well as 5, 20, and 4

Most students are going to immediately see that ac = b.  And that’s correct!  But that’s not their answer.  The students must create a new row of numbers that fits the relationship, go up to the front of the class and write the new row in alphabetical order (so as to not give their thinking away to their classmates).

So, for example, Bobby might go up to the board and write 5, 25, 5 while Melissa writes 8, 72, 9 and Isaiah writes 10, 90, 9, etc.

Students who do not immediately see the relationship are supported by these extra examples that don’t give away the answer.

After enough time has passed (I usually decide on a percentage of students that will be successful before announcing that there’s a minute or two left in the round), I’ll have one of the first students tell me the formula using the column headings (variables) instead of the numbers.  Inevitably, if one student says “a times c equals b”, another students hand rockets into the air and says “But I did b divided by a equals c”.  This leads to some rich discussions involving equations and the properties of equality.

The Official Best Friend of Variable Analysis, Dr. David Butler at the University of Adelaide, did a lovely Desmos-ified version of the game…try it out by clicking on the image below:

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  •  students that finish first can become “judges” along with the teacher.  I often have three of four spots on the board where kids can come put a response.  I can’t keep my eye on all of them at once.  So with a “thumbs up”/”thumbs down” method, I have my successful students watch the board and help me determine who else has been successful.
  • students that finish first might need to stay more busy than that.  So I also have tasks ready like “Can you now make an example using all ones and zeroes?” or “…using only negative numbers”, or “…using only the same number.”  Depending on the Variable Analysis game those tasks can be quite challenging. (Just plan ahead to make sure you don’t give the student an impossible task).
  • students who struggle to be successful can be helped with small hints — sometimes you might come up with an example besides the three given ones that sort of “telegraphs” the answer a little more.  Or you might limit the operations for them (tell them ahead of time “This one only uses multiplication and subtraction”, for example).

UPDATE!  Thanks to Glenn for capturing my TMC 16 presentation about Variable Analysis.  Check it out:


I’m Joel Bezaire, Pre-Algebra teacher at University School of Nashville.  Check out my curriculum at Pre-Algebra.info

A mentor teacher (PAEMST winner Glyn Burton) introduced me to Variable Analysis Games, I refined the rules a bit and here they are.