[Karen Arth] Making Functions in Algebra Active and Interesting

Karen Arth is a strong communicator, full stop, and for that reason alone I’ll be sure to look for her byline in future conference schedules.

She also works for CPM, so if that’s curricula you use, it’s likely you’re already familiar with the function activities she described in her talk. I’ll explain them briefly and share her resource [PDF]:

Function Machines

You give student four functions on cards. Ours were:

  • f(x) = sqrt(x)
  • k(x) = -x/2-1
  • g(x) = -(x-2)^2
  • h(x) = 2^x-7

Then you give them an input and tell them what the output should be after you compose all those functions.

For example, Arth gave us 6 as the input and told us that 11 had to be the output of the last function.

We shuffled cards around and finally determined that f(h(k(g(6)))) = 11.

These functions aren’t chosen arbitrarily, though, and that’s where you take your small-group instruction and whole-group debrief. k(x) changes the sign of the input and creates non-integer outputs if the input is odd, for instance. So Arth asked us why we made the choices we made, and our responses revealed a lot of mathematical understanding.

Silent Board Games

“Teachers love these,” Arth said. “Because students love them, because they’re silent, and because the hand that goes up is not necessarily what you consider the A student.”

Arth drew a horizontal x-y table on the board. There were two complete x-y pairs. The other pairs were only half complete.

Students were to raise a hand when they thought they knew a missing part of the table. Arth would then invite the student up to write the missing part. She’d do nothing if the number was correct and she’d erase it if it was incorrect.

Beneath that table on the board was a blank for “English Sentence” and then “Math Sentence.” It felt a bit like Wheel of Fortune, where you know the missing pieces and then decide to solve the puzzle.

Here, again, Arth emphasized process. “What was your strategy? What were key moments for you?”

One participant said that once someone else completed the x-y pair (0, -1) she knew an important fact.

Personally, I was confused throughout much of this activity. If Arth said the function was linear, I didn’t hear it, which meant I had to figure out what function family we were even looking at all while my fellow participants were having no trouble completing the puzzle.

I’m interested in hearing from people who have tried this activity in a CPM class. Do struggling students find it as easy to participate as Arth suggested?

Number Walks

Arth took us outside and laid out a large x-y axis on the ground. Several of us stood from -4 to +4 on the x-axis. Arth then said, “Take your number and add two.” As we all formed the line y = x + 2, she had us turn and link shoulders, creating a visual of the line.

Other people took their turn. f(x) = 2^x was a real treat to watch, as the people standing at negative numbers struggled, first, to figure out if they should step backwards, after which they did their best to shuffle forward just an inch.

Treasure Hunt

You can find this resource on page ten of the handout but my group never managed to make sense of the instructions, which probably means they’ll need to be clarified or simplified for students.

Resources

Featured Comment

Matt Vaudrey:

I’ve done the Silent Function Game with my students (though I didn’t have the English Phrase and Math Phrase, which I will add). I added a few components to make it work well for any class:

–Each person can only get up once.
–You can erase or write, but not both.
–The last person writes the function rule (y=2x-7).

Then I sit in the back and make menacing looks at students who make noise.

3 thoughts on “[Karen Arth] Making Functions in Algebra Active and Interesting

  1. Pingback: [Karen Arth] Mathematical Modeling | MathRecap

  2. andrew says:

    I used the CPM sequence for several years and have done several variations of these activities at different age levels.

    For middle school Alg students, the “silent board game” activity creates a very low barrier for guesses. In that context, there’s only so many function families that are in play. Students quickly learn that two points gives them a pattern to try, but they need another bit of data to confirm that pattern.

    What I found helpful when I first started with these exercises a decade ago, was the CPM focus on non-reactive “less helpful” judging. It extended to the point where I didn’t even remove incorrect guesses, but would let the next student either add a number or amend a current guess.

    In classes with a broader range of functions in play, construction of prompts that are unique but not trivial is much more challenging.

  3. Matt Vaudrey says:

    I’ve done the Silent Function Game with my students (though I didn’t have the English Phrase and Math Phrase, which I will add). I added a few components to make it work well for any class:
    –Each person can only get up once.
    –You can erase or write, but not both.
    –The last person writes the function rule (y=2x-7).

    Then I sit in the back and make menacing looks at students who make noise.

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