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From: Building on student mathematical thinking in whole-class discourse: exploring teachers’ in-the-moment decision-making, interpretation, and underlying conceptions
| Lesson | Finding the distance between a point and a line | Discovering the complement rule |
| Problem | “Calculate the distance between point \(P(\mathrm{6,2})\) and line \(l:y=1/2 x+4\)” | “In a television show, the finalist is placed in front of 20 doors. Behind 5 doors is a valuable price, and behind the other 15 doors is an empty envelope. The finalist is allowed to open six doors. Calculate the probability that at least one price is won”. |
| Context | It was the first time the students had been asked to calculate the distance between a point and a line \(l\), given by an equation in a coordinate system. Instead of providing the students with a step-by-step solution method, as the textbook does, Jesse wanted the students to find their own solution methods | In previous lessons, the class had worked on calculating probabilities, including the sum and product rule for probabilities, and Denise had explained the urn model |
| Mathematical goals | “Students should know the distance must be perpendicular” “Slopes \(-1\), students should know that product is perpendicular” In our words, Jesse wanted the students to learn that the shortest distance from a point to a line is in the direction that is perpendicular to the line, and that two lines are perpendicular if and only if the product of their slopes equals \(-1\) | “Students should know the complement rule and be able to apply it” In our words, Denise wanted the students to think of and compare different solution methods to decide collectively that the one that uses the complement of the required probability is the most elegant method and saves effort |
| Preparation, student solution methods, and whole-class discourse | Jesse had used the five practices to prepare for the lesson. He had found several solution methods. While his students worked on the problem, Jesse had monitored their work and selected three solution methods to be presented to the class. Only the first two methods were discussed in whole-class discourse, and this took up the final 20 min of the 50-min lesson. The first solution method was presented by Thijs. He calculated angle \(\alpha\) using the slope of line \(l\) (see Fig. 2)*. Then he calculated \(\beta\) using \(\alpha\), calculated \(PB\), and used the sine rule in right-angled triangle \(\Delta BCP\) to calculate \(PC\). The second solution method was presented by Sarah. She drew line \(k\) through \(P\), perpendicular to \(l\). Then she counted “squares** ”—two horizontal squares and four vertical squares—and she calculated the requested distance using Pythagoras’ theorem | Denise had thought of at least three different solution methods: a) calculating and adding the probabilities of winning one, two, three, four, and five prices by using urn models, b) calculating and adding those probabilities by using products and binomial coefficients, or c) using the complement rule. Denise expected that these three methods would be used by her students, and indeed they were. Students worked on the problem in groups, and every group wrote their solution method on a large piece of paper. These nine papers were attached to the whiteboard before whole-class discourse started. Whole-class discourse took up 13 min out of the 50-min lesson |
- *The students were not provided with an illustration. Most students produced their own illustration. We provide one here to complement the textual explanation of the students’ solution methods.
- **The students were equipped with notebooks with grid paper.