“The Cup Is the Y-Intercept”

Are your students overgeneralizing their models? After working exclusively with proportional relationships for the last month, are they describing every new relationship as proportional?

This isn’t a task, or a lesson, or anything of that scope. It’s a resource, a provocation, one that gives students the chance to check their assumptions about what’s going on.

Play this video and pause it periodically, asking students to decide for themselves, and then tell a neighbor, what’s coming next.

10 marbles weigh 350 grams. So 20 marbles should weigh how much? I’m curious which students will say the answer is less than, exactly, or more than 700 grams. I’m curious which students will say it’s impossible to know.

That will be surprising for some. Now invite them to speculate about 30 marbles. 40 marbles. And 0 marbles.

Let me end with three notes.

First, my thanks to Kevin Hall who had the fine idea for the video and encouraged me to make it. I’ve never met Kevin. That’s the kind of internet collaboration that makes my week.

Second, the stacking cups lesson offers a similar moment of dissonance. Can you find it?

Third, here’s Hans Freudenthal on technology in 1981:

What I seek is neither calculators and computers as educational technology nor as technological education but as a powerful to arouse and increase mathematical understanding.

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I always like creating a proportional reasoning speed bump by giving these types of questions.

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Hey! Nice idea for helping kids make the turn from proportional to linear relationships. There were two things I wanted to change:

• the discrete nature of the domain
• the way it’s not clear in the still images whether we are being shown the mass of just the marbles or the mass of the marbles + the glass together (the brief shot of the balance scale with the glass on it at the beginning of the video wasn’t doing it for me).

So I made a video! Here! It was shot on my phone using a jar of cumin to stabilize, so it could certainly be professionalized.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. Xavier

May 31, 2016 - 8:22 am -

It remembers me the Tawny ports problem [https://blog.mrmeyer.com/2013/real-world-math-that-isnt-real-to-students/]. When I ask for the price to my students, most of them see \$16. But when I ask for price of Tawny of 137 years, then most of them use proportional and get “wrong” value. They do not consider the “fixed” cost of the service: 7\$.

I used it as model of afine functions.

2. hunter

May 31, 2016 - 8:39 am -

I find the Domino’s lesson by Mathalicious offers that same moment to my math 8 students. “if 4 toppings cost \$___, then what will 8 toppings cost?” I use these as one of those “break their tools” moments as they transition from 7th grade proportional relationships, to 8th grade non-proportional relationships. I will add this to my resources. well done.

3. Kate Nowak

May 31, 2016 - 8:42 am -

Hey! Nice idea for helping kids make the turn from proportional to linear relationships. There were two things I wanted to change:
– the discrete nature of the domain
– the way it’s not clear in the still images whether we are being shown the mass of just the marbles or the mass of the marbles + the glass together (the brief shot of the balance scale with the glass on it at the beginning of the video wasn’t doing it for me).

So I made a video! Here! It was shot on my phone using a jar of cumin to stabilize, so it could certainly be professionalized.

4. David Cox

May 31, 2016 - 10:18 am -

I keep imagining the kid who says, “Why didn’t he just tare the scale?”

5. naw

May 31, 2016 - 6:56 pm -

Is there a purpose to showing this instead of kids doing it?
I think the question misleads the action.

6. Jessica Breur

June 1, 2016 - 4:53 am -

Kate and Dan – I think both videos would work well together. You could take the kids to discussing, what would be similar and what would be different about the graphs. This could highlight the discrete vs. continuous domain and range.

7. Kevin Hall

June 2, 2016 - 11:57 am -

Dan, thank you for making this. I’ll definitely use it next year. (And perhaps yours, too, Kate). Also grateful for the recognition, and for the chance collaborate with you. Cheers!

8. Karim

June 4, 2016 - 10:58 am -

Agree with Kate about having the scale be ever-visible; though perhaps unintentional, the shift in focus feels like a bit of a sleight of hand which may obscure exactly the “aha” moment the task is trying to inspire.

Disagree, though, about changing it to be continuous. If the activity serves as the turn from proportional relationships to linear functions, it seems reasonable to keep things easily countable. I wasn’t able to access Kate’s video, but I assume cumin is measured in something like tablespoons. Students may find a slope of “y grams per x tablespoons” a bit confusing. Not only is the weight-weight thing a bit tough, but I’d worry that — especially if the video shows someone scooping cumin into a glass — they might just treat the spoon as a discrete object…which would be problematic unless the scooping was extraordinarily precise. Marbles are easy to count, and lend themselves to a pretty intuitive “grams per marble” discussion.

Oh, and I agree with Hunter. Of course.

9. Kate Nowak

June 5, 2016 - 6:00 am -

My bad. I think I messed up sharing settings. I don’t think discrete vs continuous is a huge deal. And since this is an intro type thing, discrete may indeed be preferable. Students should get plenty of experience modeling all types of quantities. https://goo.gl/photos/Gb8had3ijmfzQb7V7

10. Karim

June 5, 2016 - 12:22 pm -

@Kate: I think you’re putting your finger on an important issue. If students graph weight vs. marbles, it could be a nice opportunity for a teacher to ask whether it should be a solid line, a dashed line, a series of dots, etc. This may preserve the mathematical understanding that you want to elicit, and in a very explicit way. (It’s been a while since we wrote Domino, but I suspect we probably had a similar convo about half-toppings.)

Watched your milk video. Nice. What if you shot it from the top (so that you saw the weight but only a 2D circle of milk) so that students had to predict number of cups based on weight? Seems like we usually change x and predict y but rarely vice-versa. If this doesn’t muddle too much the independent/dependent relationship, maybe this could be a cool alternate angle…pardon the pun.

11. Kevin Hall

June 10, 2016 - 6:36 pm -

One more thought: I didn’t realize until this year that there are 2 levels of difficulty in identifying non-proportional situations. In easier situations, the y-intercept object (e.g., the glass jar) is visibly different from the slope object (e.g., the marble). They are clearly objects of different types, and even when lots of marbles have accumulated, it’s always easy to visually separate them from the jar.

In harder situations, the y-intercept and the slope are not objects of different types. For example, suppose a puppy weighs 3 lbs when it is born and then gains 1 lb per week. The 3 lbs of birth-weight and the pound-per-week of gained weight are not visually distinguishable. They mix together. You can’t look at a picture of a 5-week old puppy and point to the 3 lbs that were there at birth. Similarly, when you turn on an oven and let it heat up to 400 degrees, you can’t point to the 71 degrees that were there from ambient temperature before you turned the oven on.

Harder non-proportional situations can’t really be understood by students until they really get that the y-intercept represents the amount when time = 0. Or, more simply, until they get that in Algebra we always decompose a single quantity into its original value plus its added value, even though that may feel like an unnatural thing to do.

As a result, you can teach identifying simpler non-proportional situations in your first introduction to linear functions (e.g., Stacking Cups or Domino Effect). But you can’t graduate to identifying harder ones until you’ve talked about graphing stories and really nailed the idea that the y-intercept is the (temporal) starting value.

12. Heresolong

June 14, 2016 - 11:39 am -

In re your eggs problem. The correct answer would, of course, be that it still takes 3 minutes because you just put five eggs in the pot instead of one.

13. Dan Meyer

June 15, 2016 - 7:06 am -

Yep. Pretty sure that’s the author’s intent. Mix up the non-stop proportionality with some un-proportional relationships.