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**Standard(s): **
[MA2019] GEO-19 (9-12) 18 :

18. Given the coordinates of the vertices of a polygon, compute its perimeter and area using a variety of methods, including the distance formula and dynamic geometry software, and evaluate the accuracy of the results.

The module opens with a modeling challenge (G-MG.A.1, G-MG.A.3) that reoccurs throughout the lessons. Students use coordinate geometry to program the motion of a robot bound in a polygonal region (a room) of the plane. MP.4 is highlighted throughout this module as students transition from the verbal tasks to determining how to use coordinate geometry, algebra, and graphical thinking to complete the task. The modeling task varies in each lesson as students define regions, constrain motion along with segments, rotating motion, and move through a real-world task of programming a robot. While this robot moves at a constant speed and its motion are very basic, it allows students to see the usefulness of the concepts taught in this module and put them in context.

In Lesson 1, students use the distance formula and previous knowledge of angles to program a robot to search a plane. Students impose a coordinate system and describe the movement of the robot in terms of line segments and points. In Lesson 2, students graph inequalities and discover that a rectangular or triangular region (G-GPE.B.7) in the plane can be defined by a system of algebraic inequalities (A-REI.D.12). In Lesson 3, students study lines that cut through these previously described regions. Students are given two points in the plane and a region and determine whether a line through those points meets the region. If it does, they describe the intersection as a segment and name the endpoints.

18. Given the coordinates of the vertices of a polygon, compute its perimeter and area using a variety of methods, including the distance formula and dynamic geometry software, and evaluate the accuracy of the results.

Lesson 9 begins Module 4, Topic C with students finding the perimeter of triangular regions using the distance formula and deriving the formula for the area of a triangle with vertices (G-GPE.B.7). Students are introduced to the “shoelace” formula for area and understand that this formula is useful because only the coordinates of the vertices of a triangle are needed. In Lesson 10, students extend the “shoelace” formula to quadrilaterals, showing that the traditional formulas are verified with general cases of the “shoelace” formula and even extend this work to other polygons (pentagons and hexagons). Students compare the traditional formula for area and area by decomposition of figures and see that the “shoelace” formula is much more efficient in some cases. This work with the “shoelace” formula is the high school Geometry version of Green’s theorem and subtly exposes students to elementary ideas of vector and integral calculus.

Lesson 11 concludes this work as the regions are described by a system of inequalities. Students sketch the regions, determine points of intersection (vertices), and use the distance formula to calculate perimeter and the “shoelace” formula to determine the area of these regions. Students return to the real-world application of programming a robot and extend this work to robots not just confined to straight-line motion but also motion bound by regions described by inequalities and defined areas.