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Geometry, Module 4, Topic C: Perimeters and Areas of Polygonal Regions in the Cartesian Plane

Classroom Resource Information

Title:

Geometry, Module 4, Topic C: Perimeters and Areas of Polygonal Regions in the Cartesian Plane

URL:

https://www.engageny.org/resource/geometry-module-4-topic-c-overview

Content Source:

EngageNY

Type:

Lesson/Unit Plan

Overview:

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.

Content Standard(s):

Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis

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.

Unpacked Content

Evidence Of Student Attainment:

Students:
Given the coordinates of the vertices of a polygon,

Use the distance formula to find the measure of the sides to aide in computing the perimeter and area using a variety of methods.

Methods may include:

Using the distance formula to find the length of each side.
Using slopes of perpendicular lines to determine when a side of a triangle or quadrilateral may be used as the height.
Using a system of equations and the distance formula to find the height of a non right triangle.
Using geometry software such as geogebra.org to find the perimeter and area of polygons to aide in the accuracy of results.

Teacher Vocabulary:

Coordinates
vertices
perimeter
Area
Distance formula
Evaluate
Accuracy

Knowledge:

Students know:

The distance formula and its applications.
Techniques for coordinate graphing.
Techniques for using geometric software for coordinate graphing and to find the perimeter and area.

Skills:

Students are able to:

Create geometric figures on a coordinate system from a contextual situation.
Accurately find the perimeter of polygons and the area of polygons such as triangles and rectangles from the coordinates of the shapes.
Explain and justify solutions in the original context of the situation.

Understanding:

Students understand that:

Contextual situations may be modeled in a Cartesian coordinate system.
Coordinate modeling is frequently useful to visualize a situation and to aid in solving contextual problems.

Diverse Learning Needs:

Essential Skills:

Learning Objectives:
GEO.18.1: Define area, perimeter, regular polygons, inscribed polygons, circumscribed polygons, and vertices.
GEO.18.2: Analyze the given information to develop a logical process to calculate area or perimeter.
GEO.18.3: Create equations for area and perimeter based on given information.
GEO.18.4: Illustrate graphically an inscribed or circumscribed polygon.
GEO.18.5: Solve equations given the area and perimeter.
GEO.18.6: Plot given coordinates on the Cartesian plane.

Prior Knowledge Skills:

Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
Demonstrate an understanding of an extended coordinate plane.
Draw and label a 4 quadrant coordinate plane.
Draw and extend vertical and horizontal number lines.
Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
Recall how to graph points in all four quadrants of the coordinate plane.
Define area.
Analyze the area of other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.
Apply area formulas to solve real-world mathematical problems.
Demonstrate how the area of a rectangle is equal to the sum of the area of two equal right triangles.
Explain how to find the area for rectangles.
Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
Recognize and demonstrate that two right triangles make a rectangle.
Define vertices.
Apply absolute value to find the length of a side joining points with the same first coordinate or the same second coordinate.
Plot points on a coordinate plane., then connect points for the vertices to sketch a polygon.
Identify ordered pairs.
Recognize polygons.
Define perimeter and area.
Identify the length between vertices on a coordinate plane.
Calculate the perimeter and area using the distance between the vertices.

Alabama Alternate Achievement Standards

AAS Standard:
M.G.AAS.10.18 Find the perimeter or area of a square, rectangle, or equilateral triangle to solve real-world problems when given the length of at least one side.

Tags:

area, coordinates, distance, formula, perimeter, polygons, rectangles, triangles

License Type:

Custom Permission Type
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For full descriptions of license types and a guide to usage, visit :
https://creativecommons.org/licenses

Accessibility

Text Resources: Content is organized under headings and subheadings

Comments

There are three lessons on this topic.

This resource is free for teachers to access and use. All resources required for the lessons are available to print from the site.

This resource provided by:

Author:

Hannah Bradley

ALEX Classroom Resource

Geometry, Module 4, Topic C: Perimeters and Areas of Polygonal Regions in the Cartesian Plane