# How Do You Turn Something Interesting Into Something Challenging?

[Correction: an oil barrel contains 158,987.295 ml.]

Nat Torkington writes the Four Short Links column for O'Reilly's Radar, highlighting interesting articles around the web on a daily (or near-daily) basis. Recently, he's pitched me a few links via e-mail under the heading "WCYDWT?" which, due to my fallen nature, I have taken as a challenge to my sacred honor.

Here's one: the relative price of different liquids which illustrates the disturbing fact that HP printer ink is several orders of magnitude more expensive than crude oil.

So I opened our first day back from winter break with a learning moment built around Nat's link and then recorded video of the moment which you'll find below. My apologies in advance for the pitiful production value. Initially, I was going to forward this only to Nat as some kind of retort but I found the experience so difficult, messy, and exhilarating, I had to debrief myself here. Notwithstanding the video quality, you're welcome to pummel me for anything you see.

Classroom Video

Color Commentary

Synonymous with "What Can You Do With This?" is "How Do You Turn Something Interesting Into Something Challenging?" I have asked educators that question on this blog, in online classes, and in several conference presentations over several years. Here is — by far — the most common answer:

"I'd put it on the wall and we'd talk about it."

Which is a weak start. A certain kind of student inevitably dominates these pseudo-Socratic discussions and then invites another kind of student to disengage. But Nat has dealt us a strong hand. If we play those cards right, we can retain and empower a lot of those (mathematically and conversationally) reticent students.

1. Calm down with the math for a moment. Invite their intuition.

At one point in my career, I would have led this off by giving them all the data and asking them to compute the ratio of cost to volume. but my blue students are poorly-served by that approach. So many of them have been burned so badly by math that if I open the conversation with terms like "ratio" and "volume," pushing numbers and structure right at them, I'll lose the students I want to keep. Moreover, this confuses master with slave. We use math to make sense of the world around us more often than the reverse.

So I put seven liquids on the wall and asked them to rank them from most expensive to least. Simple speculation. Nothing more mathematical than that. Please imagine, here, how much more fun it is to walk around and talk about the question, "Which do you think is the most expensive?" rather than the lead balloon "Which has the highest ratio of cost to volume?"

Ask a student to come up and share her ranking with the class. Argue a bit. Entertain opposing opinions. Ask a student if he'd trade a can of Red Bull for a can of his own blood. Student investment at this point is very nearly 100%. It's mine to lose.

2. Slowly lower mathematical structure onto their intuition.

"Here's the answer," I told them, but students know at this point to triple-check me. Several went straight for Red Bull, which totes does not cost \$51.15.

"So you're saying that how much you get matters as much as how much it costs."

Fine.

We used cell phones to text Google and ask for unit conversion. This always strikes my students as magical and suspicious.

And here, finally, we talked about the ratio of the cost of blood to how much blood you get. I asked them to visualize one milliliter of blood. "What does .40 mean?" We talked about the cost of one milliliter and how it's useful to compare that cost across liquids.

The rest (hopefully) writes itself, though, for the record, I kind of hate how explain-y I get in the last third of the video.

The Virtuous Cycle Specific To Our Line Of Work

1. Find an interesting thing1.
2. Transform that interesting thing into a classroom challenge.
5. Repeat.

The feeds in your reader then spiral upwards and out of your control. WCYDWT ideas begin to pile up faster than you can capture them. It'll freak you out and you'll wish you could turn it off for just a few hours while you're watching TV but you realize this a rare ancillary benefit in an occasionally tortuous job and you accept it gratefully.

[BTW: Mr. K rightly points out that this problem is of a piece with the nickel thieves from a few years back.]

[BTW: You should read Burt's commentary on the lack of real-world meaning of these statistics.]

[BTW: Great list of liquids and prices here.]

1. It's sad how often the conversation with other teachers ends here, after it becomes obvious that they just aren't interested in all that much.
1. Dan,

The ideas and stuff you’re putting out there are absolutely mindblowing! This is my third year teaching math, here in Iceland, and your methods, and line-of-thought, plus all the material you are sharing with the world are helping me immensely on the way to become a better teacher.

Thanks so much for all this and keep it coming please!

Best regards,
Björn Karlsson
Iceland

2. Great work Mr. Meyer!

I just graduated from high school, and I have to say I wish I had had more math teachers that taught like you. Algebra one and two are definitely the groundwork for advanced math classes, and any of your students that do choose to move on and take accelerated courses will actually be prepared.

3. I conducted this exercise as an introduction to the use of data and information with a group of 25 nurse managers. I had them divide into groups of 5 to rank the list of liquids and then we came back together to walk through the process.

Our conclusions as a group:
(1) Every group will have a different opinion given the same information.
(2) The way things actually are may be different than what you believe them to be.
(3) When analyzing a situation, you need to establish a framework in order to transform data into information.
(4) Data is not helpful – information is.

Thanks for the great exercise!

4. I love you. This is brilliant.

5. Dan,

I love this idea! I did a variation of this lesson today and thought you might like to hear how it went. First off, I took off the vodka because I felt that my students particularly (in a juvenile detention center) might try to figure out the cost of making a Red Bull and Vodka drink. I replaced it with chocolate syrup. I also had my students calculate the cost per cup of the liquid in stead of cost per ml. I thought that a cup size was a bit more accessible and I actually had a measuring cup lying around in class. My students sounded exactly like yours did, “Misssss… Red Bull does not cost \$51.15″ was the first thing that popped in their heads. Overall I have to say this was a really successful lesson. I had students who would rather “take a zero” than do any math engaged and willing to ask questions. I also learned the joy of simply stating “that’s a really great question,” rather then saying that and giving them the answer immediately.

Thanks!

-Serafina

6. Bam! Thanks for the recap.

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8. That’s why printers are so cheap, the companies make their money because we have to continually buy ink (you buy one printer but many ink cartridges!)

This style of activity also is possible with (lemonade/soda) cans: I have various sizes (airplane =250mL), standard Australian size = 375mL, European (?) size = 330mL and the “special” that they occasionally have which is 500mL. Which can is the most economical to produce (amount of material (tsa) v volume ratio) ? Why are the last three sizes all the same radius but the variable is the height (why choose that radius in the first place? eg suitable for adult male hand to grab) and the questions go on….

9. Best Maths lesson ever!!!!!!!!!!!!!!!!! This worked so well. I really left them hanging to calculate the unit volume on their own. Kids who I have rarely seen use a calculator ended up getting one and giving it a go. Brilliant!

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13. Looks awesome Dan,

I’m gonna use this to kick off the work we do on ratio and best buys for my GCSE resit students in a couple of weeks.