Since each class only worked through one scatterplot problem yesterday, I thought another one would be useful to reinforce the ideas from yesterday, and to provide some practice. So I gave my morning class a similar problem to yesterday's: an agricultural context (from the MathWorks12 textbook), a table of values, and two questions (one interpolation, one extrapolation).

Well that didn't work well....there was virtually no engagement with this problem. Each group delegated the work to one member, sometimes with the argument of "I did the last one". I take from this that the standing, working in groups approach really needs novel problems in order for there to be good engagement. It's not a method that is useful for a practice scenario, especially if there is nothing new in the problem.

So I got creative with my afternoon class and gave them this problem instead: The Vitruvian Man is a drawing that Leonardo da Vinci made in about 1487. It shows what da Vinci believed to be the proportions of all humans.

He thought that the ratio of arm span to height is the same for everyone. Is he correct?

(I got the idea for this from an "Activity" in the MathWorks12 textbook.)

Students measured each other, then began listing ratios--a reasonable approach given the wording of the problem. So I said that, given what we worked on yesterday, what approach might be reasonable?

This is a small class, and attendance was so embarrassingly low, that there ended up being one large group working at the front of the class. Engagement was definitely waning. This problem didn't work as well as I had hoped, perhaps because:

- The small class meant we didn't have enough data points to definitely see a trend (so I had to talk about what we would have seen had we measured more people)
- It was the second last block on a Friday
- The question, although more novel than what I gave my morning class, was still not novel enough to really draw students in

We eventually had a fully labeled scatterplot with line of best fit, outliers, an interpolated point, and an extrapolated point (we used the height of a staff member who says he's 5 foot 17).

After renewing some of my notes on creating a culture of thinking, I was reminded that there isn't enough time to make good thinking problems for every class, and that traditional teaching will still be required.

So now the question becomes, how to balance the following:

- using novel thinking problems that truly engage students (with students in groups, working on whiteboards)
- using traditional instruction when novel problems can't be found or created
- giving students time to practice their new-found skills (and what format should that take?)