Walk Like a Mathematician

Over the weekend, I attended the CMC-North conference at the Asilomar conference grounds. It’s my favorite math teacher event of the year.

Photo by Captain Albert E. Theberge, NOAA Corps (retired)

When I was a teacher, the combination of the beautiful, coastal setting and being among a sea (ha ha) of math educators, all spending their weekends learning and sharing, always provided a much-needed boost. Now, I look forward to running into my former colleagues and catching up with them.

One of the teachers I ran into this weekend was Mr. F., who has stuck with teaching through many cycles of reform. He mentioned that even though his (and my former) school has gone back to traditional textbooks, he still likes to do something different and fun every month to keep the students engaged, like the “Algebra Walk.”

I first did the Algebra Walk, a.k.a. Human Graphing, when I was a student in the credential program at U.C. Berkeley. The activity goes like this:

You need two long ropes to use as x– and y-axes, with evenly spaced markers, and a large open area. (Good weather helps as well.) Lay out the ropes at 90˚ angles to form a coordinate system. Assign eleven students an integer from –5 to 5 to use as their starting point on the x-axis. Assign the rest of students to be recorders, and switch roles periodically.

If possible, you and the recorders should stand in an elevated spot for better viewing. You call out a rule for the students to apply to their value. For example, “Add one to your number.” Give students a moment to do the calculation, then call out, “Walk to your output value.” Each student walks parallel to the y-axis to their output value. In this case, the students will walk to points along the line y = x + 1. The recorders sketch the graph and write the rule. Then the students return to the x-axis.

A video of a dynamic sketch is worth at least 1,000 words, so here’s a visual demonstration of my last two paragraphs:

How does this activity relate to the Common Core State Standards?

Well, the 8th Grade standards are the first time the term “function” comes up in the Common Core State Standards:

  1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
  2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Both of these standards are addressed in the “Algebra Walk” activity. Students directly experience applying the function rule to generate an output value and then observe the graph of ordered pairs. They also have to translate the teacher’s verbal rule into a mathematical rule and relate the three representations: verbal, symbolic, and graphical. As a follow-up to the activity, you might ask students to make a table, and then apply the rule to input values that weren’t included, like decimal values.

When I did this activity as a credential student, walking the linear rules was fun but not very eye-opening. But the last rule we were given was “Square your number.” Now, I knew intellectually that the rule would generate a parabola. But it still made an impact on me when the person at 0 stood still, the people at –1 and 1 took a single step, and those folks with input numbers 4 and 5 kept walking, and walking, and walking. Check it out:

The symmetry of the graph is beautifully revealed as the students at opposite input values walk to the same output value.

The best thing about this activity is that it becomes a touchstone experience to refer back to. Instead of saying, “Remember that day we graphed a lot of lines?” you can say, “Remember that day we created human graphs?” You can videotape the activity and use it for review and extension. You may not be able to get outside in December, but you can follow up with an activity using Dynamic Geometry® software that gets students to work backwards. (Example prompt: “What rule generates this graph?“)

Thanks, Mr. F., for that reminder of an engaging activity that might just leave a bit of residue with your students.

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