Next day, I’d start with one of those graphs, and maybe say, “Every point on a line communicates information. For example, THIS VERY POINT says that …” Have students explain another point, work on interpreting the graph, maybe throw up the second student graph I chose to reinforce the ideas, then move on to another activity to build on these same ideas. Time permitting, of course. (ha ha)

In brief, I’d want to return to the meaning of a point (in context) for a student unable to sketch a graph.

*Unless the student didn’t sketch merely for fear of being wrong, in which case you’re looking at a different kind of problem.

This can even work with the Opt Out learner – especially if you unpack and prompt with Polya question stems such as “do you understand the problem?” If that fails to elicit a response, maybe try asking them to talk through someone else’s graph.

It’s important for me to get in the habit of using the same script whether the solution is correct or not – that develops rigour in thinking and learners don’t come to hear “Talk me through that..” as “you’re wrong, but I’ll humor you.”

BTW we *did* ask learners to sketch them on a mini whiteboard *before* the group work or class discussion didn’t we? [Lemov, Take a Stand]

I’d translate a student’s graph back to them in words and ask them if that’s what they wanted to represent.

ie. For A:

“So you’re saying here with one rubber band, the Barbie has fallen a little bit. Is that right?”

https://www.dropbox.com/s/zsofsyyb71h7j1s/BarbieBungee.ggb?dl=0

Part 1

Have students: 1) create a slider in GeoGebra going from 0 to 12 counting by 1’s; 2) create a vertical line segment of length 2*r+3; 3) slide the slider and describe what is happening; 4) Answer the question: what could this be a model of? Take answers looking for a variety.

Part 2

Have students: 1) create another vertical line segment of length 2*r+6 right next to the original one, noting the change of the constant term from 3 to 6; 2) Slide the slider (which will animate both segments) and answer the question: what effect did changing the 3 to a 6 have?; 3) Answer the question: if this were modeling (pick a phenomena they brought up in step 4 of Part 1), what would the change of 3 to 6 mean in that context?

Part 3

Repeat Part 2, but with a third vertical line segment of length 4*r+3, noting that only the multiplier is different from the original.

Part 4

Introduce the Barbie context and have students do the drop, complete with fall height measurement, with 2 rubber bands so they have the opportunity to see and handle the props in context. Then explain the connection between the model and the situation. Specifically tell them two things: 1) The “r” in the expression for the length of the vertical line segment will stand for the number of rubber bands (so right now, r=2); and 2) the length of the line segment represents the distance from the point where the rubber band chain is attached to the support at the top to the tip of Barbie’s head at the bottom of the fall (in whatever units you want, let’s say inches).

Part 5

Have the students slide the slider to r=2 and determine which of the three line segment models predicts the length of the actual fall the closest (none will be very close, likely, especially if you work in millimeters! The purpose of this part is simply to get them to perceive the fact that different values for the multiplier and constant produce models of better or worse value for predicting the fall height.) Then challenge them to create a fourth segment, by changing the multiplier and constant, that comes as close as they can. There will be many strategies here from “playing with different numbers” all the way to making very accurate measurements of Barbie’s height and the length of single stretched rubber bands.

Part 6

Have them use their new and improved models to predict the fall height for 4 rubber bands. Then have them check their model’s effectiveness by actually doing it and seeing. Have them re-alter their parameters to get their model to make a more accurate prediction for 4 rubber bands without losing much accuracy for 2 bands.

Part 7

Have them repeat this “use your model to predict, then actually do the Barbie drop, then improve your model” procedure with 6, 8, 10, and 12 rubber bands. By the end, they should have a fairly fine-tuned model.

Part 8

Discuss strategies. In the discussion, bring out the connection between the multiplier and the length of a single rubber band, as well as the constant and the height of Barbie.

Part 9

Do a linear regression fit on the data and compare the student parameters with the ones generated by the fit.

Boom. There it is. Oh, and by the way, during the data gathering and function tailoring stages play Hideaway by Kiesza, then Black and White by Parquet Courts then Control by Olympic Ayres (at 70 decibels, cuz that’s what the research says is loud enough to stimulate creativity but not loud enough to distract). People will be bobbing their heads a little feeling like they’re doing something cool and inventing stuff.

Notice this version still uses Dan’s wisely proposed “sketch first, measure later” strategy, but there are a few key differences here: 1) The “sketch” is a line segment instead of a height vs. time graph. This is a much less abstract visual representation of the fall and easier to process; 2) The visual properties of the sketch are controlled simultaneously by your hands (sliding the slider with your finger on an iPad) and by mathematical processes (slider values entering into functions and coming out as line segment lengths). It’s hard to describe how satisfying this is. 3) The strategy is used multiple times throughout the activity instead of once, with immediate feedback following each instance. This allows students to “carve out” the linear model, slowly fusing their intuitive, informal knowledge about the situation with the formal mathematical notation and measurements.

As a teacher, convergence can be easier and quicker but if we can get twice as many miles out of the same lesson by making our jobs a tad more difficult, seems like a worthy trade off (not that I think this is possible with all classes).