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.

Math

Unit 5

10th Grade

Unit Summary

In Unit 5, 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 8th Grade Math; areas of polygons from 6th Grade Math; and algebraic skills with square roots and factoring from 8th Grade Math, Algebra 1, and the previous unit. 

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.

Pacing: 17 instructional days (15 lessons, 1 flex day, 1 assessment day)

Assessment

The following assessments accompany Unit 5.

Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

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  

Materials

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

Congruence

Creating Equations

Expressing Geometric Properties with Equations

High School — Number and Quantity

Reasoning with Equations and Inequalities

Similarity, Right Triangles, and Trigonometry

Foundational Standards

Congruence

Creating Equations

Expressions and Equations

Geometry

Ratios and Proportional Relationships

Reasoning with Equations and Inequalities

Future Standards

Building Functions

Circles

Congruence

Geometric Measurement and Dimension

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.

New Math Materials Now Live!

New Math Materials Now Live!

Now Live: All new Grade 1 units and enhanced Grade 7-8 materials. Coming soon: Brand new Kindergarten course and enhanced Grade 6!

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