Present your students with a linear modeling challenge in a familiar context:

### Challenges

- When will the phone be fully charged?

### Setting

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.

### Lesson Notes

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:

And…

**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:

https://www.desmos.com/calculator/hzlch8fx0x

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):

### Extensions

- 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:

### Resources

- All images (.zip)
- Slide deck (Keynote)
- Slide deck (.pdf)
- Desmos Graph

### Key Standards

- 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.

### Support Standards

- CCSS 7.RP.A

Analyze proportional relationships and use them to solve real-world and mathematical problems.

### Shout Outs

### More Lessons

Looking for more lessons? Click here.

## Comments 7

Something crazy going on there at that 80 percent mark… it would be interesting to also test the battery life claims as well. Great work here.

With my new phone last chistmas I got an “adaptive fast charger”. It’s amazing how quick it charges up my nearly dead device. Would love a follow-up, which might allow a comparison of the rate of change in a very relate able context.

Leeanne, that sounds like a great idea! Let me see what I can put together… 🙂

Thank you so much for this rich investigation, Michael. We used it with a group of Grade 8 & 9 math teachers today and it worked really well. Our focus was to elicit what we call up here mathematical competencies, something akin to the mathematical practices.

I made a couple of changes that seemed to play out well. First, I replaced an earlier data point with one further along (though still in the linear domain). Specifically, the data we used were (0, 5), (8, 14), (24, 33), (56, 70). Notice the pattern in the differences (8, 16, 32), which gives a nice way to notice the linearity even without plotting the points. Second, rather than jumping to the Act 3 reveal, after they were pretty confident with their linear answer I gave them two new data points, (80, 89) and (100, 95). This had the same ‘what happened?’ impact, plus it gave them a chance to revise their model before the final reveal.

Marc, glad to hear the activity went well. Thanks for sharing your tweaks!

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