I'm teaching College Algebra for the first time this summer, as apart of trying to revise the class. We're focusing the syllabus, and shifting some emphasis to practices from just content. I thought this problem was a good one. We're not doing proportional reasoning as an individual focus, but rate of change and difference quotient is a part of the class.

Then over the weekend, Marty & Burkard, the Australian math popularizers who collect math in the movies clips among other things, sent a link to this video from Burkard's new YouTube channel, Mathologer. It features a great math clip from Little Big League. If Joe takes 3 hours to paint a house and Sam takes 5 hours...

So much good in there, you wonder if the writer had some teaching experience.

- Number mashups - check.
- My uncle was a painter - check.
- Trick question trope - check.

I told the story Glenda Lappan tells, about the shepherd with 132 sheep, who delivers them evenly to four fields, how old is the shepherd? They laughed, then one person gave the common student answer: 33 years old.

So then they tried to solve the problem. Some people decided 8 hours, some 4. I asked what's the most it could be: we got to less than 3. That gave one person the courage to share their answer of 2.

We showed the rest of the movie clip, but stopped at 3:22 before the math explanation started. Were they satisfied with 3x5/(3+5)? No. Earlier talking about the course (doing the Piece of Me activity) someone had asked what my teaching style was. I didn't know how to describe it, but had said I would not be up front telling them stuff, they would be working and discussing. I used this part of the clip to support that, him telling the answer did nothing for us.

I did a bit of a demonstration (more like an early 'with' in gradual release terms): what do we know for sure? Set up a timeline: at 3 hours, 1 house plus a part. (Should be more than half, students say) At 5 hours, two houses plus more than half again. At 6 hours, 3 houses... what time would make sense to think about here? 30 hours, a student said. "It's like finding a common denominator." So at 30 hours, Joe's painted 10 houses and Sam has 6. 16 houses in 30 hours. Does that tell us anything about one house? A student suggested 30/16. Why? "Because that's their average. Per house." Awesome! They invented unit rate and in hours per house not houses per hour hour. Then someone pointed out the average was the same as the answer from the clip.

Jot down what you're thinking about after solving that. Share it with your table.

So onto our next problem. I warned them that it's not the same as the painter problem because it's solved the same way, but it is the same in that we want to make sense of it. I adapted the NRICH problem for less obvious units and less information.

Work in Progress

A job needs three people to work for two weeks (10 working days).

Andi works for all 10 days.

Burt works for the first week and Claire works for the second week.

Dave works for 6 days, but then is too sick to work.

Edie takes his place for 3 days, then Fred does the last day.

When the job is finished they are all paid the same amount. At first they could not work out how much each man should have, but then Fred says: “If Edie gives $150 to Dave, then at least Dave’s got the right amount.”

If we have enough information, how much was paid for the whole job, and how much does each person get? If we don’t, what’s a little bit of information that would let you figure it out?

They really gave it a go. A couple people got to a quick answer with the numbers involved, and all their tablemates couldn't dissuade them. I recorded their strategies as I heard them:

What they got from making sense of what the problem asked.

Asked a student to record her chart on the wall.

After they worked for a while, most students felt like they did not have enough information. Those who felt they had a solution could not convince the class of it.

They consolidated the chart into the center information on days worked, and grouped them into three ideal workers, who worked all ten days.

In discussion I pointed out that they hadn't used the information that $150 made Dave's pay fair. They kept losing track of the idea that they were all paid the same, and wanted to know how much Edie had left. Once they decided for sure they didn't have enough information I gave them more. (I considered just saying 'yes, you do' but they were stuck. ) So I said Fred had another idea. If he gave all his money to Andi, she would be set.

Then they were off and running. Several solutions popped up. One was more algebraic, which killed the interest that many people had.

This was revived when someone presented a just logical idea. If Andi was set, so were Burt and Claire. Because they worked half of Andi's work, and had half of Andi's money. If they were fair, then Dave had pay for 5 days, so then the $150 was one day's pay.

Once they knew one day's pay, they backward engineered the pay per person and the total.

Someone asked about the original information, and I supplied that we knew how many days of pay (person-days, like man-hours, in my head) there were - 30 - and that was split among 6 people. This didn't have a lot of traction, as I think man-hours is a weird unit. Some people connected it with work and got it.

In general, use of symbols was a barrier, not a help, which means we have our work cut out for us. On the other hand, this lesson with these two problems was packed full of the values of the class culture I want to establish, and got them discussing math for the first or one of the few times in their life.