Geometry Unit 4 – Quadrilaterals
BY THE END OF THIS UNIT:
CORE CONTENT
Cluster Title: Polygon Angle Sum TheoremsStandard: 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
· 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.
· 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? / Resources
· 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.-
Sample Formative Assessment Tasks
Skill-based task
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. / 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?
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?
CORE CONTENT
Cluster Title: Properties of QuadrilateralsStandard: 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
· 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 product as a mini project.)
· 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. / Resources
· Quadrilateral Family Tree
http://www.mathwarehouse.com/geometry/quadrilaterals/
· Quadrilateral Properties: Online Game
http://www.onlinemathlearning.com/quadrilateral-properties.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)
· Quadrilateral Graphic Organizer – Word Document
Sample Formative Assessment Tasks
Skill-based task
What value of x makes each figure the given special parallelogram?
1. Rhombus 2. Rectangle
Classify each figure as precisely as possible. Explain your reasoning.
3. 4. / Problem-based task
1. Find EF in the trapezoid.
2. Use the information in the figure. Explain how you know that ABCD is a rectangle.
3. ABCD is a rhombus. What is the relationship between Ð1 and Ð2?
Explain.
CORE CONTENT
Cluster Title: Proving Relationships Among Quadrilaterals with Coordinate GeometryStandard: 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)
Suggested Instructional Strategies
· 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) Z (-1, 4)
1a. Is WXYZ a parallelogram? Verify using slope to show that opposite sides are parallel - or not.
1b. Is WXYZ a rectangle? Verify using slope to show that adjacent sides are perpendicular - or not.
1c. Is WXYZ a rhombus? Verify using slope to show that the diagonals are perpendicular - or not.
1d. Is WXYZ a square? Explain why or why not. / Problem Task
1. Describe two ways you can show whether a quadrilateral in the coordinate plane is a square. Which of the two ways you described is a more efficient method for determining if the figure is a square? Explain.
2. A vertex of a quadrilateral has coordinates (a, b). The x-coordinates of the other three vertices are a or –a, and the y-coordinates are b or –b. What kind of quadrilateral is the figure? Draw a sketch to support your answer.
3. Describe a good strategy for placing the vertices of a 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.