Polygons and Algebraic Relationships

Lesson 6

Math

Unit 5

10th Grade

Lesson 6 of 15

Objective


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

Common Core Standards


Core Standards

  • 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).

Foundational Standards

  • 6.G.A.1
  • 6.G.A.3
  • 8.G.B.8

Criteria for Success


  1. Use slope criteria to identify and verify parallelograms on a coordinate plane. 
  2. Use slope criteria to identify and verify rectangles on a coordinate plane. 
  3. Use the theorem that opposite sides of a parallelogram are congruent to identify and verify parallelograms on the coordinate plane. 
  4. Given coordinate points that form a polygon and no coordinate plane, determine the classification of the polygon based on slope criteria and/or congruent sides. 
  5. Given partial information about a polygon on a coordinate plane, define the rest of the polygon using equations of the line and/or coordinate points.

Tips for Teachers


  • Students may need to review basic definitions of parallelograms, rectangles, rhombuses, and squares before they can fully access this lesson. It is recommended to spend time outside of class building this specific skill.
  • Properties of parallelograms and diagonals of quadrilaterals will be discussed in future lessons. 
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Anchor Problems


Problem 1

Is the shape below a parallelogram? A rectangle? A rhombus? A square?

Guiding Questions

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References

Mathematics Vision Project: Secondary Mathematics One Module 8: Connecting Algebra and GeometryLesson 8.3 "Set"

Module 8: Connecting Algebra and Geometry from Secondary Mathematics One: An Integrated Approach made available by Mathematics Vision Project under the CC BY 4.0 license. © 2016 Mathematics Vision Project. Accessed March 12, 2017, 5:09 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

A quadrilateral with vertices $${(-6,2),\space(-3,6),\space(9,-3),}$$ and $${(6,-7)}$$ is drawn.

  1. Is this quadrilateral a rectangle? Explain.
  2. Jason states that the opposite sides are congruent in this quadrilateral. Show the work that Jason needs to do to make this claim.
  3. Is the work Jason does in part (b) enough to prove that this shape is a rectangle? Explain your reasoning. 

Guiding Questions

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References

Illustrative Mathematics Is this a Rectangle?

Is this a Rectangle?, accessed on March 12, 2017, 5:11 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.

Problem 3

Ashanti is surveying for a new parking lot shaped like a parallelogram. She knows that three of the vertices of parallelogram $${ABCD}$$ are $${A(0,0)}$$, $${B(5,2)}$$, and $${C(6,5)}$$. Find the coordinates of point $$D$$. Justify mathematically that the figure you have drawn is a parallelogram.

Guiding Questions

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References

JMAP G.GPE.B.4: Quadrilaterals in the Coordinate Plane 2Question #12

G.GPE.B.4: Quadrilaterals in the Coordinate Plane 2 is made available on JMAP by Steve Sibol and Steve Watson. Copyright © 2017  JMAP, Inc. - All rights reserved. Accessed June 1, 2018, 4:38 p.m..

Target Task


The following rectangle has one side that is contained in the line $${y=2x+3}$$.

a) What are three possible equations that will contain the other three sides of the rectangles?
b) What are the vertices of the rectangle you have defined?

References

MARS Formative Assessment Lessons for High School Classifying Equations of Parallel and Perpendicular LinesSlide P-11 "Lines and Rectangles"

Classifying Equations of Parallel and Perpendicular Lines from the Classroom Challenges by the MARS Shell Center team at the University of Nottingham is made available by the Mathematics Assessment Project under the CC BY-NC-ND 3.0 license. Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham. Accessed March 12, 2017, 5:05 p.m..

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.

  • Include problems where students are given partial information about a polygon on a coordinate plane and they must define the rest of the polygon using equations of the line and/or coordinate points.
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Lesson 5

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Lesson 7

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

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

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