Lesson 9 | Polygons and Algebraic Relationships | 10th Grade Mathematics

Objective

Calculate and justify the area and perimeter of polygons and composite shapes off the coordinate plane.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.GPE.B.7 — Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Expressing Geometric Properties with Equations

G.GPE.B.7 — Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

The foundational standards covered in this lesson

6.G.A.1

Geometry

6.G.A.1 — Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
7.G.B.6

Geometry

7.G.B.6 — Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Describe that the area of a region is the sum of each individual area minus the overlapping section using expressions and verbal descriptions.
Describe the area of a region as the sum of each individual area when there is no overlap using expressions and verbal descriptions.
Use (or derive) the formulas for area necessary to find the area of composite shapes.
Identify equivalent expressions or equations that describe the area of composite shapes.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson will prepare students to access G-GPE.7 standard by discussing and reviewing the area and perimeter of composite shapes off the coordinate plane.
Have students use or develop a reference sheet as needed. Here is an example: Massachusetts Comprehensive Assessment System Grade 10 Mathematics Reference Sheet
EngageNY, Geometry, Module 3, Lesson 2 uses set notation of ∪,∩,and ⊆ to describe the union, intersection, or subset, respectively, of a figure. This is not very important according to the Common Core Standards but is an additional topic that you could introduce as an extension.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Calculate the area of the shaded figure below.

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 3 > Topic A > Lesson 2 — Exploratory Challenge #2a

Geometry > Module 3 > Topic A > Lesson 2 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem 2

Two triangles $${ABC}$$ and $${DEF}$$ are shown below. The two triangles overlap forming $${\triangle DGC}$$.

What is the area of the shaded region?

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 3 > Topic A > Lesson 2 — Exploratory Challenge #3

Modified by Fishtank Learning, Inc.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Wood pieces in the following shapes and sizes are nailed together to create a sign in the shape of an arrow. The pieces are nailed together so that the rectangular piece overlaps with the triangular piece by 4 inches. What is the area of the region in the shape of the arrow?

References

EngageNY Mathematics Geometry > Module 3 > Topic A > Lesson 2 — Exit Ticket, Question #1

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

EXTENSION: Find the area of regular polygons by composing or decomposing polygons into other shapes to find the area. For example, find the area of a regular hexagon by decomposing it into two congruent trapezoids.

Illustrative Mathematics Overlapping Squares — (Include questions that will align with the Criteria for Success)
EngageNY Mathematics Geometry > Module 3 > Topic A > Lesson 2
CK-12 Area and Perimeter of Composite Shapes

Lesson 8

Lesson 10

Lesson Map

Topic A: Distance on the Coordinate Plane

Use the Pythagorean Theorem to calculate distance on a coordinate plane. Develop a formula for calculating distances.

G.GPE.B.7

Partition horizontal and vertical line segments into equal proportions on a number line.

G.GPE.B.6

Divide a line segment on a coordinate plane proportionally and identify the point that divides the segment proportionally.

G.GPE.B.6

Topic B: Classify Polygons using Slope Criteria and Proportional Line Segments

Algebraically and using the Pythagorean Theorem, determine the slope criteria for perpendicular lines.

G.GPE.B.5 G.SRT.C.8

Describe and apply the slope criteria for parallel lines.

G.GPE.B.5

Identify and create parallelograms, rectangles, rhombuses, and squares on the coordinate plane.

G.GPE.B.4

Algebraically verify midsegment, median, and parallel line relationships in triangles.

G.CO.C.10 G.SRT.B.4

Algebraically verify diagonal relationships in quadrilaterals and parallelograms.

G.CO.C.11 G.GPE.B.4

Topic C: Area and Perimeter On and Off the Coordinate Plane

Calculate and justify the area and perimeter of polygons and composite shapes off the coordinate plane.

G.GPE.B.7

Calculate and justify the area and perimeter of parallelograms and triangles on the coordinate plane.

G.GPE.B.7

Calculate and justify composite and irregular areas on the coordinate plane.

G.GPE.B.7 N.Q.A.3

Describe how the area changes when a figure is scaled.

G.GPE.B.7

Solve area applications by creating and solving quadratic equations.

A.CED.A.1

Describe a polygon on the coordinate plane using a system of inequalities.

A.CED.A.3 G.CO.A.2

Calculate and justify the area and perimeter of polygons on the coordinate plane given a system of inequalities.

A.REI.D.12 G.GPE.B.4 G.GPE.B.7

Polygons and Algebraic Relationships

Objective

Common Core Standards

Core Standards

Expressing Geometric Properties with Equations

Foundational Standards

Geometry

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Target Task

References

Additional Practice

Lesson Map

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