Present your students with a linear modeling challenge in a familiar context:
- When will the phone be fully charged?
One evening, with my phone battery nearly depleted, I plugged in and took a series of screenshots to track the percent battery charge (as a function of time). I limited my use of the phone to screenshots and nothing else during my data collection.
Display the following image:
Ask students to write down what they notice and what they wonder. Here are some student responses from earlier this week:
There may be some other interesting questions to ask, but I’m unapologetically driving toward the “how long until fully charged” question here.
With the question established, ask students to write down a too low, too high, and just right (or “Goldilocks”) guess. The too low and too high guesses (borrowed from Dan Meyer’s approach to Three Act lessons as well as Andrew Stadel’s framework for Estimation 180 challenges) give students an easy path to making their actual guess.
With a guess written down on their paper, I find students are anywhere from slightly to dramatically more invested in the problem (and the pursuit of an answer).
Depending on your preference, you can ask students to describe the additional information that would help them, or you can cut to the chase and show them this image:
From there, students should have enough information to create a linear model for extrapolation. I recommend having students use a combination of low-tech (paper and pencil, to organize the data into a table, calculate slope, identify the y-intercept, etc.) and high-tech tools (Desmos is an ideal pairing for this task) to build and use their model.
If using Desmos, students have quite a few options:
- Calculate the rate of change, identify the y-intercept, and simply use Desmos to confirm that the model fits the data
- Enter the data in a table, add an equation with sliders (e.g., y = ax + b), and adjust the sliders to find a good (albeit informal) line of fit
- Enter the data in a table and use regression to find an equation (learn regression here)
Whichever path they take, students should end up with something like this:
A little rounding and a bit of arithmetic leads to:
Now for the big reveal:
Cue student disbelief.
Depending on the parameters of their model, most students will be off by about an hour. One hour!
So what happened? This is where the lesson turns from run-of-the-mill linear modeling task to a great opportunity for discussing assumptions, pitfalls of extrapolation (using a model to predict beyond where we gathered data from), and (if working with older students) piecewise functions.
To facilitate those discussions, I use the following Desmos graph:
Click on each folder icon, one at a time, to show how the data and our model compare.
Then click the play button for slider a to see where our model falls apart (the red line is my linear model; the black dots represent the actual data):
- Find a piecewise function that models the percent battery charge as a function of time for the entire 140 minutes.
- Discuss the meaning of the linear model parameters in context. Consider using this image to facilitate the discussion:
- CCSS 8.F.A
Define, evaluate, and compare functions.
- CCSS 8.F.B
Use functions to model relationships between quantities.
- CCSS 8.SP.A
Investigate patterns of association in bivariate data.
- CCSS 7.RP.A
Analyze proportional relationships and use them to solve real-world and mathematical problems.
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