Near the end of class in physics, I challenged my students to solve a word problem where a faster runner catches a slower runner, when the slower runner starts sooner.

Most groups approached the correct answer (they were bit rushed), but not using the method I anticipated. As shown above, most graphs were drawn incorrectly. Well, the graph leads to the correct answer, but it doesn’t represent what the runners are doing. The graph shows that one runner starts ahead of the other and they both start at the same time. Strictly speaking, the graph is wrong if the independent variable represents a *clock reading*.

In any event, the general problem solving showed that many kids have transferred their knowledge from qualitative understandings to quantitative analysis.