Here are three gas station pumps. Which ones are trying to rip you off? Can you tell just by looking?

After your students have that debate and share their reasons (expected: “the third is a ripoff because it’s moving faster”) invite your students to collect data for each pump and enter it at Desmos. Here we’re establishing a *need* for a graphical representation. It may reveal patterns that our eyes can’t detect.

The third act helps clarify the underlying trends. The third pump is spinning faster, but the price and the gas still exist in a proportional relationship. The first pump, meanwhile, pumps less gas per dollar the longer it runs.

I am indebted to William G. McGowan and Sean Berg, whose NCTM 2016 session description included the words “gas pumps have been hacked,” and there went my weekend.

Their description reminded me how important it is to expose students to counter-examples of the relationships they’re studying, protecting against over-generalization. (ie. “*Everything* is proportional. That’s the *chapter* we’re in!”) I’m becoming fascinated, in general, by problems that ask students to prove that a mathematical model is *broken* rather than just apply a model that *works*.

[Download the goods.]

**Featured Comment**

I’ve written before about expanding teaching to the “neighborhood” of the special case. If we always show the highlight reel, students never get appreciation for how special and how powerful the ideas are. So I like that this lesson is about finding the non-proportional “ripoff” as it stands out in contrast to the “normal/expected” proportional relationships. (Ironically I would have said that proportionality is the special case, and nonproportionality is the ‘normal’— before thinking about what we expect as consumers. ‘normal’ is all subjective!)

## 20 Comments

## chase orton

March 21, 2016 - 4:02 pm -I need to analyze the video more, but it has me wondering about the other proportional relationship standards for 6th and 7th grade (ratio tables and double number lines in particular). It’d be more pencil and paper but an excellent investigation for them to do, especially in 6th grade when they’re discovering a need for the coordinate plane. The graph is quite a spectacular representation. I worry that the double number lines might bog down the learning flow/energy, and I’m wondering how best to avoid that. Thoughts? How low could this go in grade levels?

## Scott

March 21, 2016 - 5:40 pm -I like this thing you say at the end:

>how important it is to expose students to counter-examples of the relationships they’re studying, protecting against over-generalization. (ie. “Everything is proportional. That’s the chapter we’re in!”)

I’ve written before about expanding teaching to the “neighborhood” of the special case. If we always show the highlight reel, students never get appreciation for how special and how powerful the ideas are. So I like that this lesson is about finding the non-proportional “ripoff” as it stands out in contrast to the “normal/expected” proportional relationships. (Ironically I would have said that proportionality is the special case, and nonproportionality is the ‘normal’– before thinking about what we expect as consumers. ‘normal’ is all subjective!)

Also, worth noting in the video/digitalgraphing mode, we have dozens of data points to work with communicated in a 30 second video. That’s probably quicker than a student reading/interpreting a paper multiple choice question, “Here are five ratios… which one does not belong?”

I’m confused by your own desmos graph– you force the linear relationship for pumps 2 and 3? Is the data imperfect? (or did you run out of weekend? haha)

I do wonder if the act 2 mode (the students graphing) can be improved/varied. Perhaps they will graph pump one meticulously, notice its not a line, and then develop shortcut ideas for pumps 2 and 3 to avoid doing all the data work. McGowan’s and Berg’s session will probably dig into some ideas!

## Karim

March 21, 2016 - 5:40 pm -Wasn’t this an Amplify project?

## Jere Confrey

March 22, 2016 - 3:28 am -Glad you liked that problem. I came up with it pumping gas one day and thinking about whether you could tell if the rate changed. Jere Confrey

## Karim

March 22, 2016 - 5:16 am -@Jere, I recall attending your presentation at NCTM a few years ago and really liking the task. I agree with Scott that substituting video for receipts (if I recall correctly) communicates more data more quickly. Of course, the underlying problem is still very elegant. Nice work!

## Dan Meyer

March 22, 2016 - 7:20 am -Jere:Oh nice. Please bring along your treatment of the problem when we catch up in April. Let’s trade notes.

@

Scott, lots of meat in this paragraph. I’m floating it up into the post.Do you have a reference on your writing on the “neighborhood”? (Incidentally my Google Search for “Scott Farrar math blog” didn’t turn up your math blog! Check your SEO, man!)

I suppose I assumed those pumps were functioning as expected, since I didn’t monkey with their footage. I’ll have to go back and make sure I wasn’t ripped off in actual fact.

## Kevin Hall

March 22, 2016 - 10:50 am -I think this is great, but it targets the easier aspect of detecting proportionality: checking that the rate is truly constant. I think it’s more important for students to generate some kind of intellectual need to check whether y=0 when x=0.

For example, suppose you put 2 candies in a cup and weigh the cup The scale says the total weight is 14 grams. Then you ask students, how much would it weigh with 4 candies? Most students immediately recognize that your prediction is meaningless if the additional candies aren’t the same (in weight, at least) as the originals. That’s the aspect that your lesson reinforces: does this scenario have a constant rate? What students tend to overlook, however, is that even if the new candies are the same, the new total weight will not be 28 grams, since although you’re doubling the weight of the candies, you’re not doubling the weight of the cup. The cup is the y-intercept, and the y-intercept isn’t 0.

I have some ways of trying to surface this point for students, but nothing as slick as what you usually produce. If you ever get around to making a sequel, I’d love if it targeted that aspect

## Scott

March 22, 2016 - 2:44 pm -@dan you can click on my name for my blog I think. Or http://www.scottfarrar.com

As for SEO I do need to work on that. I’m fighting a war for namespace with the Hollywood special effects guy Scott Farrar.

Here’s the specific posts I referenced:

http://scottfarrar.com/blog/neighborhoodofthespecialcase/

http://scottfarrar.com/blog/continuummath-join-the-movement/

## Lisa Schall

March 22, 2016 - 5:25 pm -Did this activity today with my 7th & 8th graders & it was great! I had them graph the data in Excel & then explain how they knew pump #1 was ripping them off. They also calculated the approximate price per gallon & wrote a linear model for each pump. All the students knew that pump #1 was the answer because the graph was different. Getting them to explain how it was a rip off involved some thought. How did they know, for example, that the pump wasn’t working to their advantage? This activity was a fun & valuable way to tie together the concepts of slope, rate of change, unit cost, proportionality, & linear equations.

## Dan Meyer

March 23, 2016 - 2:10 am -Lisa:Oh that’s slick. I hadn’t thought of that question.

I’m curious: were students ever able to see pump #1’s bustedness in the

videoor were they relying entirely on thegraph?## Will McGowan

March 23, 2016 - 10:26 am -@Karim Yes, the “Gas Pump Mystery” is an Amplify math project, and was created by Jere Confrey. At the 2016 session, we were going to have teachers do the entire project. As a result of Amplify’s reorganization, we will no longer be running the session.

## l hodge

March 23, 2016 - 3:02 pm -Instead of gallons & cost maybe the same setup with income and income tax shown. The marginal tax rates would result in something that is not proportional. Plotting points would allow for speculation on where the changes take place.

Otherwise, I am wondering if the lesson goes something like this: Me: Which gas station is ripping you off? Sally: What are the prices? Me: I don’t know. Sally: Doesn’t it say on the pump? Isn’t there a huge sign on the corner with the prices? Me: Shut up Sally. Obviously we need to gather data and make a graph to figure it out.

## chase orton

March 23, 2016 - 3:13 pm -I agree with your larger point that sometimes we can find answers to some of these problems using common sense. We can google “how many pennies in the pyramid of pennies” and arrive at the answer pretty quick. Or, let’s just each walk to the taco truck by different routes each way and find out instead of doing all this math. But it’s only by calculating a unit rate after a purchase that we can feel more certain about the validity of the pump (or the sign on the corner). This assumes that the pump isn’t lying about the amount of fuel it’s dispensing. And gas pumps are highly regulated by inspectors and anti-tampering measures (like security tape). Which is probably the best argument a student could make that gas pumps don’t rip you off.

I may be missing something to your point however so let know if I am. I think though that Sally’s question shows a need to verify the sign on the corner with our experience at the pump. I suppose that finding the unit rate procedurally through division makes the graph less necessary. Hmmm. You have me wondering more. Thank you.

## Nathan Kraft

March 24, 2016 - 2:38 am -My class worked on Fenton’s Charge the other day. Great example of how not everything is proportional.

## Bowen Kerins

April 1, 2016 - 6:46 am -It would be interesting if one of the pumps was a ripoff not because of its changing rate, but because of a hidden offset charge that happened before the part of the video you can see.

This one was certainly a freeze-framer!

## Dan Meyer

April 1, 2016 - 7:28 am -@

Bowen, I thought for awhile about that one. Decided to keep it all in the proportional family. Might have been the wrong call.## Sue Hellman

May 31, 2016 - 7:56 am -@Bowen Your idea reminded me of an experience buying mushrooms at my local grocery store. Here in Canada paper bags are supplied to keep them fresh longer. I’d always assumed the scales at the checkout were automatically ‘zeroed’ for this kind of purchase, but asked one day and found they are not. So the y-int when buying mushrooms > 0. Mushrooms here are usually $4- $5 per pound, and at that rate I don’t pay much for one bag. The more interesting question is how much profit the store makes on these bags in a week or a year of mushroom sales. Have they already factored bag cost into the price of the mushrooms so we’ee paying twice? How much would it affect their bottom line to be honest? Is this an industry-wide practice? How different is from merchants in historical times putting a thumb on the scale? By the way, his happened at a big chain store.

## Mark Schlatter

July 28, 2016 - 10:17 am -I was thinking about 3act problems (having only seen the pool table example) and was wondering how I could do something similar with decimal arithmetic and gas pumps. Specifically, I don’t know if gas stations always round up on purchases of gas or if they use typical rounding. In my only gas purchase since thinking of this, you would have rounded up anyway.

## Dan Meyer

July 28, 2016 - 4:04 pm -Nice idea,

Mark. I like the idea of showing students (through video magic) the first missing digit in the price. Maybe that’s the ten-thousandths place. They see it ticking along furiously, and then you see some trickery. The price is 29.3281, with the 1 being unseen to the customer. Then we see the price round up to 29.329. What? Over a day, a year, how much do those shenanigans add up to?