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This clip is from the movie *Holes*, which is inexplicably billed as a movie for kids. (Sue Van Hattum kindly brought it to my attention.) The grim premise is a penal colony of children, each digging one hole per day in the desert for the duration of their sentences. On our hero Stanley Yelnats’ first day, he accidentally takes another kid’s shovel which is slightly shorter than the rest. Drama!

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**Step One**:

- Play the video.
- Ask the students for questions that perplex them.

**Aaron**: “does the shorter shovel really matter?” +4 others

**Peter**: “how long does it take to dig a standard hole?” +2 others

**Other questions**. +5

This data is invaluable to my curriculum development. *Invaluable*. Consider last week’s responses to the boat in the river video.

**Steve**: “How long will it take Dan to go up the down escalator?” +15 others

**Other questions**. +2

It’s obvious to me which problem has the stronger current. Maybe I can do something about that; maybe I can’t. Regardless, I had to make a more authoritative call on the problem than I prefer. I said, “Okay, let’s talk about the first question. ‘Does it really matter that X-Ray’s shovel is a couple inches shorter?’

**“How many pounds of extra dirt is Stanley going to dig at the end of a full year?”**

**Step Two**:

- Ask the students to guess at the answer to our question.
- Ask the students to set an upper bound on an acceptable answer.
- Ask the students to set a lower bound on an acceptable answer.

Our median [lower bound, guess, upper bound] was [100 pounds, 1,000 pounds, 10,000 pounds]

**Step Three**:

- Ask the students to define the information they’ll need to solve our question.

**VoijaRisa**: Weight of dirt and extra length of shovel

**Frank**: how many days/ week?

**Mr_K**: Density of dirt

**Steve G.**: Density of dirt, dimensions

**Aaron**: mass of dirt, “shorter” shovel length

The movie doesn’t define the shorter shovel’s length, which leads to an awesome moment where the students and the teacher can basically *make something up*, some number that has no material effect whatsoever on the mathematics they’re practicing but which gives everyone the sense that “this is *our* problem.” Big win.

Director Andrew Davis didn’t think it fit the narrative of his film to mention the weight of a cubic foot of desert dirt so we faced a similar dilemma w/r/t density.

**Step Four**:

- Give them a pile of information to use as they see fit.

**Step Five**:

I put twenty minutes on the clock and asked everyone to email me either a scan or a camera photo of their work when they finished.

Example #1

Example #2

Example #3

Example #4

**Step Six**:

- After they compute their final answer, ask them to compare it to their error bounds from step two.
~~Play the answer video~~.
- Compare the answer to our guesses from step two. Determine who guessed closest.
- Discuss sources of error.
- Discuss follow-up questions.

**Stacy**: I love this problem, but we still don’t have a good way to check our answer.

So this problem receives a certain demerit for not allowing us to *observe* the answer. I accept that demerit.

These problems require some kind of plan for challenging students who finish early. The attendees offered two approaches I want to highlight here:

**Aaron**: change the size of the original shovel.

**Justin & Anna**: how much shorter would the next shovel have to be for the difference to be the same?

**Aaron** has changed the input quantities and asked his students to find another output. His students will use the same operations on different numbers. From my experience, this leaves the teacher vulnerable to charges of assigning busy work.

**Justin and Anna**, by contrast, have made the old output quantity the new input. Before, their students were solving for the total quantity of dirt. Now, Justin and Anna have given their students the quantity of dirt and asked them to work backwards to new inputs. This is a great, versatile way to quickly create a new problem for students who finish early.

**Open Questions**

**ft**^{3} or pounds? In a workshop recently, we ended this problem by calling a local composting company and asking them how many cubic feet of compost they brought in on each dumptruck. The units on our final answer, then, were “dumptrucks.” There are different, subtle ways to frame the same question. Do you ask for mass, weight, volume, or dumptrucks? **Mr. K** and I went back and forth over the difference between these two questions. “How many extra *pounds* of dirt will Stanley dig after a year?” vs. “How many extra *cubic feet* of dirt will Stanley dig after a year?” What are their advantages and disadvantages? I’m still going over this in my head but my sense is that our students’ early estimates will be more accurate in ft^{3}, but the answer will be more tangible to them in pounds. (See also: How Big Really?.)
- I’m curious, if you were in that session,
**what does it do for your engagement with the problem to see your classmates’ guesses?** (I welcome any other comments from the participants, of course.)
**What app will let people email files to a public folder?** I planned to use Dropbox but, last minute, I realized it didn’t support that function so I had participants email *me* their files, which I uploaded to my domain after updating an HTML file, and the whole thing was ridiculous. How do I cut out the middleman (me) here?

**Miscellaneous**:

- Here is the session transcript.
- I had DimDim’s whiteboard up at the start of the workshop, which turned out to be accidentally awesome. Participants started doodling as they waited for the session to start. One participant drew a map of the US and asked everyone to identify their location with a star.
- I need to type questions after I speak them so the responses in the transcript make a little more sense to me afterwards.
- I’m crazy enough to look up the shooting locations for
*Holes*. It remains to be seen if I’m crazy enough to drive down to the Mojave Desert with a scale and weigh a cubic foot of dirt, which is clearly what needs to happen here.