Lesson 1.2
Distance and Midpoint in the Plane
Focus: taxicab geometry, the distance formula, the midpoint formula, proportional reasoning
This lesson earns its place in this unit by doing three things at once: it gives students a genuine thinking task that produces the distance formula through their own reasoning, it introduces the Pythagorean theorem connection that will recur throughout the course, and it closes with a proportional reasoning application that previews one of the most important tools in the similarity unit.
The Google Maps activator works because it shows two different answers to the question “how far is it?” — and most students immediately conjecture that one is as-the-crow-flies and the other is as-the-car-drives. The question of which is which, and how you could test it, is exactly the kind of question this class is building the habit of asking.
Save the resolution of the activator for the end of class. Students will be more motivated to do the proportional reasoning calculation if they have something to check their answer against, and the wait builds genuine curiosity. Most students will conjecture the answer correctly before calculating — which gives you the opportunity to revisit Guess My Rule: a correct conjecture is not a proof.
The taxicab geometry task has a pleasant side effect: students find it genuinely fun and want to explore it. One thing to watch for is students who start using the taxicab formula — adding the horizontal and vertical distances — in later problems where the Euclidean formula is needed. Name this directly during consolidation: taxicab geometry was our entry point for thinking about distance, but the formula we want to remember is the Pythagorean one. The hidden right triangle is the key.
Timing guidance: approximately 45 minutes for the tasks, 20–25 minutes for making notes on the three formulas (distance, its relationship to the Pythagorean theorem, and midpoint), then the activator resolution with proportional reasoning to close. When groups finish the main task, challenge them to write their own notes on the formulas and then pick two random points and find both distance and midpoint. This begins the practice of making meaningful notes from their own work rather than copying from the board.