Download Geometry Unit 4 Quadrilaterals

Geometry Unit 4 – Quadrilaterals
Enduring understanding (Big Idea): Students will understand that the properties of quadrilaterals are evident and useful in
real world applications such as: architectural designs, building structures, production of industrial or business equipment,
geometric art, etc.
Essential Questions:
1. What is the Polygon Angle-Sum Theorem?
2. What conjecture can be made about the sum of the exterior angles of any convex polygon?
3. How can you learn to recognize, classify, and apply the properties of quadrilaterals?
4. How can coordinate geometry be used to prove relationships among quadrilaterals and their properties?
BY THE END OF THIS UNIT:
Students will know…
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The interior and exterior angle sum theorems for polygons
The properties and classification of quadrilaterals (parallelogram,
rectangle, square, rhombus, trapezoid, kite)
How to use coordinate geometry to prove relationships in quadrilaterals
(by applying distance, midpoint, and slope formulas)
Students will be able to…
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Vocabulary: regular polygon, irregular polygon, diagonal, equilateral
polygon, equiangular polygon, consecutive angles, opposite angles, opposite
sides, parallelogram, rectangle, square, rhombus, trapezoid, midsegment of a
trapezoid, isosceles trapezoid, kite, coordinate geometry, coordinate proof
Unit Resources
Learning Tasks:

Interior Angle Sum Theorem (6.1 Dynamic Activity) online
teacher resources: www.pearsonsuccessnet.com
 Exterior Angle Sum Theorem (Concept Byte: Textbook p. 352)
Performance Task: Poster – Use Teacher Online Resources (6-5
Activities, Games, and Puzzles) Title – Puzzle: Shape Sort
Mini Project: My Organizer: Properties of Quadrilaterals
Unit Review Game: Quad Property Competition – Select a quad
name from a hat and name as many properties as possible.
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Compute angle measures as well as the sums of interior and exterior
angles of a polygon
Classify quadrilaterals
Apply properties of quadrilaterals and their diagonals
Use algebra to compute angle measures and side measures of a polygon
Use coordinate geometry to classify and prove relationships (i.e.
theorems) among quadrilaterals
Write variable coordinates to figures in the coordinate plane
Compute area of triangles and quadrilaterals
Mathematical Practices in Focus:
1.
2.
3.
4.
7.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Look for and make use of structure.
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 4 – Quadrilaterals
CORE CONTENT
Cluster Title: Polygon Angle Sum Theorems
Standard: G-CO.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.
Concepts and Skills to Master
 Find interior angle and side measures of convex regular using the Polygon Angle Sum Theorem.
 Find exterior angle measures of a regular polygon using the Polygon Exterior Angle Sum Theorem.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Triangle Angle-Sum Theorem (especially since the Polygon Angle sum Theorem is an extension of the Triangle Angle-Sum Theorem) Note: The
Common Core introduces and teaches the Triangle Angle-Sum Theorem in 8th grade.
Academic Vocabulary
equiangular polygon, equilateral polygon, regular polygon, irregular polygon, n-gon, diagonal of a polygon, convex polygon,
overlapping triangles, interior angles, exterior angles, consecutive angles
Suggested Instructional Strategies
Resources
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Have students sketch regular polygons (3 sided to 8 sided
shapes). Then have students make a conjecture about the
number of overlapping triangles in each after drawing
diagonals that connect all vertices – remember: triangles
cannot overlap. See if students can discover the Polygon
Angle Sum Theorem.
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Extension: Have students label all interior angles of the
sketched polygons, extend the vertices, and label the
measures of each exterior angle formed. Ask students
Essential Question #2 – What conjecture can be made about
the sum of the exterior angles of any convex polygon?
Interactive Learning: 6-1 Solve It (Dynamic Activity – Online Teacher
Resources –Interactive Digital Path)
www.pearsonsuccessnet.com
 Wiki Exploration: Exterior Angle Sum Theorem
http://www.geogebra.org/en/wiki/index.php/Angles
(Click on Polygon Exterior Angle Sum Theorem – explore – click next
in the top right hand corner to continue exploration)
 Concept Byte Exploration Activity: p.352 Exterior Angles of Polygons
 Cluster Review – Use Links Below:
http://freedownload.is/ppt/3-4-the-polygon-angle-sum-theorems-ppt
Click on 3.4 (also labeled 3.5 in description) The Polygon Angle Sum Theorems
http://www.mathwarehouse.com/geometry/polygon/
-Scroll entire web page to see questions.-
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 4 – Quadrilaterals
Sample Formative Assessment Tasks
Skill-based task
Problem Task
Question I: A polygon has n sides. An interior angle of the
polygon and an adjacent exterior angle form a straight angle.
a. Use an algebraic expression to represent the sum of the
measures of the n straight angles?
b. Use an algebraic expression to represent the sum of the
measures of the n interior angles?
c. Using your answers above, what is the sum of the
measures of the n exterior angles?
d. What theorem do the steps above prove?
1. For each regular polygon, state the sum of the measures of the
interior angles and give the measure of an interior angle.
2. For each regular polygon, state the sum of the measures of the
exterior angles and give the measure of an exterior angle.
Question II: A triangle has two congruent interior angles and
an exterior angle that measures 100. Find two possible sets of
interior angle measures for the triangle?
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 4 – Quadrilaterals
CORE CONTENT
Cluster Title: Properties of Quadrilaterals
Standard: G-CO.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.
Concepts and Skills to Master
 Identify, verify, and classify properties of quadrilaterals.
 Define and classify special types of parallelograms
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Prior knowledge of common quadrilateral properties (grades K– 4); especially square and rectangle.
 Prior knowledge of congruent triangle theorems as well as properties of parallel and perpendicular lines.
 Prior knowledge of the definition and properties of the isosceles, equilateral, and right triangles.
Academic Vocabulary
quadrilateral types: parallelogram, rectangle, square, rhombus, trapezoid, isosceles trapezoid, kite, and midsegment of trapezoid
Suggested Instructional Strategies
Resources
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Have students create an organizer (foldable, Venn diagram,
mapping, table grid, concept map, chart, poster, etc.) to categorize
the studied quadrilaterals and their properties. The finished
product may be used as a study tool and can count as a mini
project. (Note: Design a rubric to give to students if you grade the
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product as a mini project.)
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Figure 4-1 is a Venn Diagram
sample of a Quadrilateral
Graphic Organizer. If students
use a Venn Diagram, they must
come up with a creative way to
include properties as well.
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Quadrilateral Family Tree
http://www.mathwarehouse.com/geometry/quadrilaterals/
Quadrilateral Properties: Online Game
http://www.onlinemathlearning.com/quadrilateralproperties.html
Interactive Learning
Quadrilateral Quest: Do you know their Properties?
http://teams.lacoe.edu/documentation/classrooms/amy/geometry/
6-8/activities/quad_quest/quad_quest.html
Online Teacher Resource Center
www.pearsonsuccessnet.com
Parallelogram Scramble Puzzle (Activities, Games, and Puzzles
6-2)
Puzzle Shape Sort (Activities, Games, and Puzzles 6-5)
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 4 – Quadrilaterals
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Sample Formative Assessment Tasks
Skill-based task
What value of x makes each figure the given special parallelogram?
1. Rhombus
Quadrilateral Graphic Organizer – Word Document
Problem-based task
1. Find EF in the trapezoid.
2. Rectangle
2. Use the information in the figure. Explain how you know that ABCD is a
rectangle.
Classify each figure as precisely as possible. Explain your
reasoning.
3.
3.
ABCD is a rhombus. What is the relationship between 1 and 2?
4.
Explain.
Teacher Created Argumentation Task (W1-MP3&6)
Error Analysis: Since a parallelogram has two pairs of parallel sides, it certainly has one pair of parallel sides. Therefore, a
parallelogram must also be a trapezoid. What is the error in this reasoning? How would you explain this error as well as the
correct reasoning to another student who is struggling with the property? Give a thorough explanation. Also provide a sketch
to support your answer.
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 4 – Quadrilaterals
CORE CONTENT
Cluster Title: Proving Relationships Among Quadrilaterals with Coordinate Geometry
Standard: G-GPE.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…
Concepts and Skills to Master
 Use of formulas for slope, distance, and midpoint to classify quadrilaterals and to prove geometric relationships for
quadrilaterals in the coordinate plane.
 Use of variables to name the coordinates of a figure; allowing relationships among quadrilaterals to be shown true for a
general case.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Know how to compute slope, distance, and midpoint when given two coordinate points.
 Recall relationships among parallel and perpendicular lines that is determined by slope
 Understand how to graph points on a coordinate plane (graph paper) – Covered in Grade 5 with CCS
 Recall classification of triangles: scalene, isosceles, or equilateral.
Academic Vocabulary
coordinate plane, coordinate proofs, (Also recall prior vocabulary: quadrants, origin, x-axis, y-axis, variables)
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.
Geometry Unit 4 – Quadrilaterals
Suggested Instructional Strategies
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Allow students to use graphing paper when teaching this
cluster.
Review the slope, midpoint, and distance formulas with
students if needed.
Discuss the importance of being thorough when writing a
coordinate proof.
Students will need assistance understanding the logical
placement of quadrilaterals on the coordinate plane for
general cases. Show students when it makes sense to place
a segment on an axis.
Resources

Preparing Proofs in Coordinate Geometry
– Online examples for modeling
http://regentsprep.org/Regents/math/geometry/GCG4/Coordinatelesson.htm
 Coordinate Geometry Challenge
http://regentsprep.org/Regents/math/geometry/GCG4/Coordinateresource.htm
 Floor Pattern
Square
 Tutorials – Demonstrations on How to Use Coordinate Geometry to
Prove that you have a Parallelogram, Rectangle, Rhombus, and Square
Use the link below:
http://www.sophia.org/coordinate-geometry-of-quadrilaterals--2tutorial
Sample Formative Assessment Tasks
Skill-based task
1. W (-4, -2)
X (5, -2)
Y (8, 4)
Problem Task
Z (-1, 4)
1. Describe two ways you can show whether a quadrilateral in
the coordinate plane is a square. Which of the two ways you
1a. Is WXYZ a parallelogram? Verify using slope to show that
described is a more efficient method for determining if the
opposite sides are parallel - or not.
figure is a square? Explain.
1b. Is WXYZ a rectangle? Verify using slope to show that
2. A vertex of a quadrilateral has coordinates (a, b). The xadjacent sides are perpendicular - or not.
coordinates of the other three vertices are a or –a, and the ycoordinates are b or –b. What kind of quadrilateral is the
1c. Is WXYZ a rhombus? Verify using slope to show that the
figure? Draw a sketch to support your answer.
diagonals are perpendicular - or not.
3. Describe a good strategy for placing the vertices of a
1d. Is WXYZ a square? Explain why or why not.
rhombus when beginning a coordinate proof.
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are
listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.