Lesson 1.3
Polygons and Area in the Plane
Focus: polygon definitions, concavity, area of rectangles and triangles, perimeter with the distance formula
This lesson brings the definition-building work of Lesson 1.1 back into play. The security camera task produces a wide variety of polygons on the boards around the room, and the consolidation is organized around noticing what all those shapes have in common and how they differ — exactly the classification and differentiation framework students developed with the shoe task.
There are two things worth being deliberate about in this lesson. The first is the connection between perimeter and the distance formula. It is early in the year and students are not yet accustomed to tying ideas together across lessons. Some will try to find the length of a slanted side by counting grid squares — which works for horizontal and vertical sides but not for diagonals. A compass demonstration is a vivid and memorable way to show why: place the compass at two endpoints of a slanted segment and at two endpoints of a horizontal segment that appears to be the same length visually, and students will see immediately that the arcs do not match.
This is also the first time in the course a compass appears. Worth noting briefly: this tool will return in Unit 3 when we begin constructions in earnest. Students do not need to know how to use it yet — they just need to see it do something that counting cannot.
The second thing to be deliberate about is the area of a triangle. The triangle-in-rectangle task is designed to lead students to conjecture that the triangle takes up exactly half the rectangle — and the visual argument (draw a vertical from the top point to the base, see two rectangles each divided in half) makes the why transparent. Push for this argument. Students who can see why A = ½bh rather than just applying it are students who will not forget it.
Timing: approximately 45 minutes on the security camera task and polygon consolidation, 15 minutes on notes, then CYU problems and homework. Do not go past 45 minutes on the task — the consolidation and practice time are genuinely necessary.