Polygons and Algebraic Relationships

Students connect algebra to geometric concepts with polygons as they explore the distance formula, slope criteria for parallel and perpendicular lines, and learn to calculate and justify the area and perimeter of polygons.

Unit Summary

In Unit 5, Polygons & Algebraic Relationships, students connect algebra to geometric concepts with polygons through distance on the coordinate plane, partitioning line segments, slope criteria for perpendicular and parallel lines, area (with composition and decomposition), and perimeter. Students then use this knowledge, in part, to describe properties and prove theorems of triangles and parallelograms.

In this unit, students will draw on previous understanding of elementary geometry standards as well as many middle school standards. The primary foundational content students will need to have prior to beginning this unit are application of the Pythagorean Theorem from eighth grade; areas of polygons from sixth grade; and algebraic skills with square roots and factoring from eighth grade, Algebra 1, and Unit 4.

The Unit begins with Topic A, Distance on the Coordinate Plane. Students develop the distance formula using the Pythagorean Theorem to partition line segments proportionally. In Topic B, Classify Polygons using Slope Criteria and Proportional Line Segments, students describe and apply the slope criteria for parallel and perpendicular lines in order to algebraically identify characteristics of triangles and quadrilaterals, with a particular focus on midsegments, medians, and diagonals. Extending these skills, Topic C, Area and Perimeter On and Off the Coordinate Plane, focuses on calculating and justifying the area and perimeter of polygons. In addition, students identify scale factors of dilated polygons and use quadratic equations and systems of inequalities to describe polygons on and off the coordinate plane.

The material from this unit is foundational to the next unit, Three-Dimensional Measurement and Application, where students will need to use the composite area concepts and the Pythagorean Theorem, and focus on measurement units to solve application problems.

Assessment

This assessment accompanies Unit 5 and should be given on the suggested assessment day or after completing the unit.

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Unit Prep

Intellectual Prep

Internalization of Standards via the Unit Assessment

Take unit assessment. Annotate for:
- Standards that each question aligns to
- Purpose of each question: spiral, foundational, mastery, developing
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) that assessment points to

Internalization of Trajectory of Unit

Read and annotate “Unit Summary.”
Notice the progression of concepts through the unit using “Unit at a Glance.”
Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions

Essential Understandings

Build on previous understanding of the Pythagorean Theorem and similar triangles to define distance on the coordinate plane, partition line segments proportionally, develop slope criteria for perpendicular and parallel lines, and apply these criteria to the classification, area, and perimeter of polygons.
Describe and algebraically verify properties of polygons involving their sides, diagonals, medians, and midsegments on and off the coordinate plane.
Solve area and perimeter problems through algebraic reasoning (both on and off the coordinate plane), estimation, and composition and decomposition of shapes.
Use quadratic equations and systems of inequalities to describe polygons.

Vocabulary

Distance formula	Pythagorean Theorem
Polygon	Partition proportionally
Ratio	Directed line segment
Slope	Perpendicular line
Parallel line	Parallelogram
Rectangle	Rhombus
Square	Midsegment
Median	Point of concurrency
Diagonal	Trapezoid
Area	Perimeter
Composite shape	Irregular shape
Scale factor	Quadratic equation
System of inequalities

Unit Materials, Representations and Tools

Massachusetts Comprehensive Assessment System Grade 10 Mathematics Reference Sheet

Lesson Map

Topic A: Distance on the Coordinate Plane

1

G.GPE.B.7

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

2

G.GPE.B.6

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

3

G.GPE.B.6

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

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

4

G.SRT.C.8

G.GPE.B.5

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

5

G.GPE.B.5

Describe and apply the slope criteria for parallel lines.

6

G.GPE.B.4

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

7

G.CO.C.10

G.SRT.B.4

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

8

G.CO.C.11

G.GPE.B.4

Algebraically verify diagonal relationships in quadrilaterals and parallelograms.

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

9

G.GPE.B.7

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

10

G.GPE.B.7

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

11

N.Q.A.3

G.GPE.B.7

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

12

G.GPE.B.7

Describe how the area changes when a figure is scaled.

13

A.CED.A.1

Solve area applications by creating and solving quadratic equations.

14

G.CO.A.2

A.CED.A.3

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

15

A.REI.D.12

G.GPE.B.4

G.GPE.B.7

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

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

Congruence

G.CO.A.2 — Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Congruence

G.CO.A.2 — Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Congruence

G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G.CO.C.11 — Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Congruence

G.CO.C.11 — Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Creating Equations

A.CED.A.1 — Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Creating Equations

A.CED.A.1 — Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Creating Equations

A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Expressing Geometric Properties with Equations

G.GPE.B.4 — Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

Expressing Geometric Properties with Equations

G.GPE.B.4 — Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
G.GPE.B.5 — Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Expressing Geometric Properties with Equations

G.GPE.B.5 — Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
G.GPE.B.6 — Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Expressing Geometric Properties with Equations

G.GPE.B.6 — Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
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.

High School — Number and Quantity

N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

High School — Number and Quantity

N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Reasoning with Equations and Inequalities

A.REI.D.12 — Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Reasoning with Equations and Inequalities

A.REI.D.12 — Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Similarity, Right Triangles, and Trigonometry

G.SRT.B.4 — Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

Similarity, Right Triangles, and Trigonometry

G.SRT.B.4 — Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 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.

Similarity, Right Triangles, and Trigonometry

G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 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

Congruence

G.CO.C.9

Congruence

G.CO.C.9 — Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Creating Equations

A.CED.A.2

Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Expressions and Equations

8.EE.A.2

Expressions and Equations

8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.EE.B.6

Expressions and Equations

8.EE.B.6 — Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
8.EE.C.8

Expressions and Equations

8.EE.C.8 — Analyze and solve pairs of simultaneous linear equations.

Geometry

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.
6.G.A.3

Geometry

6.G.A.3 — Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. 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.
8.G.A.1

Geometry

8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:
8.G.B.6

Geometry

8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.8

Geometry

8.G.B.8 — Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Ratios and Proportional Relationships

6.RP.A.1

Ratios and Proportional Relationships

6.RP.A.1 — Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."
6.RP.A.2

Ratios and Proportional Relationships

6.RP.A.2 — Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Expectations for unit rates in this grade are limited to non-complex fractions. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."
6.RP.A.3

Ratios and Proportional Relationships

6.RP.A.3 — Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
7.RP.A.2

Ratios and Proportional Relationships

7.RP.A.2 — Recognize and represent proportional relationships between quantities.

Reasoning with Equations and Inequalities

A.REI.B.4

Reasoning with Equations and Inequalities

A.REI.B.4 — Solve quadratic equations in one variable.
A.REI.C.6

Reasoning with Equations and Inequalities

A.REI.C.6 — Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Future Standards

Building Functions

F.BF.B.3

Building Functions

F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Circles

G.C.A.3

Circles

G.C.A.3 — Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Congruence

G.CO.D.13

Congruence

G.CO.D.13 — Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Geometric Measurement and Dimension

G.GMD.A.1

Geometric Measurement and Dimension

G.GMD.A.1 — Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
G.GMD.A.3

Geometric Measurement and Dimension

G.GMD.A.3 — Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.