[Karen Arth] Mathematical Modeling

This is the second session I attended from Karen Arth at this conference and I’ll re-iterate again that I appreciate her communication skills. She also has a lot of good ideas, even though I disagreed with one of her premises in this particular session.

She started by asking for our definitions of “modeling” and I thought the audience’s suggestions were astonishingly perceptive. Two well-turned phrases:

  1. “Start with life, do the math, check back with life.”
  2. “Approximating the world.”

“So what I’m hearing from you guys is ‘application’,” Arth said, and I made a note to ask her a question about that later.

She gave the following Algebra II problem, which she called “traditional”:

Write the equation of a parabola going through the points (0,0), (20,50), and (40,0).

a) Use the form y = a(x-h)^2 + k.
b) Solve for y when x = 25.

Then she gave us the following revised problem, which she implied was the same task only with modeling:

McDougal’s Restaurant has a play area for children under and around their giant arch (in the shape of a parabola with negative orientation). They plan to set up a new activity that allows children to bungee jump from the arch. The manager, upon hearing of your team’s expertise, hires you to calculate the maximum stretch of the rope that will keep the kids safe. The arch is 50 feet high and 40 feet wide at the base. The jumping location will be 5 horizontal feet away from the axis of symmetry of the arch.

a) Write an equation to model the shape of the arch.
b) What’s the maximum length to which the cord could stretch to keep McDougal’s safe from lawsuits?

She had us create a graphic organizer for our work that included sections for “Notes,” “Labeled Picture,” “Table,” “Estimates,” “Assumptions,” “Calculations,” and “Recommendations.”

“If I look at your picture, if I look at your table, if I look at your notes, I want to be able to tell the whole story,” she said.

We worked on the task in groups and presented it. Then I asked my question. “Earlier you equated modeling with applications. Is there ever an applied task that doesn’t involve modeling?”

She told us about several SBAC performance tasks in the modeling strand. Bruce Grip asked if my question had been answered. I said I was still unclear on what features of those performance tasks made them modeling. Is it enough to have “real world” objects in your problem?

Our own definitions of “modeling” are interesting, of course, but the CCSS and other documents have offered their own, very specific definition, which includes, among other skills:

  • Identifying essential variables.
  • Formulating a model that uses those essential variables.

Arth had us talk about our model’s assumptions, which is a core component of modeling, but her task also gave us the essential variables (the height of the arch, the width of the base) and it gave us the model also (“the arch is in the shape of a parabola with negative orientation”). This seemed to make the task something other than modeling.

I said all of that and she replied that, given her student population, she had to offer them some of that information because the task might be too foreign and unmanageable otherwise. I’m sure she knows her students’ capabilities better than I do. Whatever our students’ capabilities right now, though, I hope we’re all moving in the same direction, towards the same definition of modeling.

[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.


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.