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.

Unit Prep

Intellectual Prep

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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

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  • 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

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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  

Lesson Map

Topic A: Distance on the Coordinate Plane

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

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

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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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.

  • 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.

  • 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).

  • 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.

  • 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.

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.

  • 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

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Congruence
  • G.CO.C.9

Creating Equations
  • A.CED.A.2

Expressions and Equations
  • 8.EE.A.2

  • 8.EE.B.6

  • 8.EE.C.8

Geometry
  • 6.G.A.1

  • 6.G.A.3

  • 7.G.B.6

  • 8.G.A.1

  • 8.G.B.6

  • 8.G.B.8

Ratios and Proportional Relationships
  • 6.RP.A.1

  • 6.RP.A.2

  • 6.RP.A.3

  • 7.RP.A.2

Reasoning with Equations and Inequalities
  • A.REI.B.4

  • A.REI.C.6

Future Standards

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Building Functions
  • F.BF.B.3

Circles
  • G.C.A.3

Congruence
  • G.CO.D.13

Geometric Measurement and Dimension
  • G.GMD.A.1

  • G.GMD.A.3

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.