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

Objective

Calculate and justify the area and perimeter of parallelograms and triangles on 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

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

Find the area of a figure on the coordinate plane by either:
1. Identifying the area of a rectangle that “inscribes” the shape and subtracting the shape outside the figure (decomposition).
2. Identifying the lengths required by the formula to calculate.
Find the perimeter of a figure on the coordinate plane by determining all the side lengths and finding the sum.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This is the first lesson out of two that focus solely on calculating and justifying the area and perimeter of polygons on the coordinate plane.
The Massachusetts Comprehensive Assessment System Grade 10 Mathematics Reference Sheet is useful for this lesson.

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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

Find the area and perimeter of the following figure:

Guiding Questions

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Problem 2

Find the perimeter and area of the following figure:

Guiding Questions

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Problem 3

Given $${A(2,3)}$$,

What two other vertices create a triangle that can have an area of 8 square units?
What is the perimeter of the triangle you drew?

Guiding Questions

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Target Task

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

On graph paper, draw a line segment with endpoints $${ A(0,2)}$$ and $${B(0,6)}$$.

Plot point $${C(x,y)}$$ such that $${\triangle ABC}$$ has an area of $$6$$ square units.
Is there more than one point for $$C$$ that satisfies these conditions? Explain your reasoning.
If point $$C$$ were placed at $${ (2.75,0.8)}$$, would $${\triangle ABC}$$ have an area of $$6$$ square units? Show your reasoning.

References

Illustrative Mathematics Equal Area Triangles on the Same Base I

Equal Area Triangles on the Same Base I, accessed on May 31, 2018, 2:39 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

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.

EngageNY Mathematics Geometry > Module 4 > Topic C > Lesson 10 — (Focus on quadrilaterals only)
Illustrative Mathematics Squares on a Coordinate Grid
Illustrative Mathematics Triangle Perimeters
Mathematics Vision Project: Secondary Mathematics One Module 7: Congruence, Construction, and Proof — Lesson 7.1 "Go" (p. 8)
Mathematics Vision Project: Secondary Mathematics One Module 7: Congruence, Construction, and Proof — Lesson 7.3, "Go" (p. 20)
EngageNY Mathematics Geometry > Module 4 > Topic C > Lesson 9 — (Focus on triangles only)

Lesson 9

Lesson 11

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

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Target Task

References

Additional Practice

Lesson Map

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