Lessons

Perplexing Parallelograms

Discipline

Mathematics

Grades

9, 10, 11, 12

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Professional Commentary

A surprising result occurs when two line segments are drawn through a point on the diagonal of a parallelogram and parallel to the sides. From multiple instances of this construction, students can arrive at various conjectures. The basis of this lesson is considering strategies for proving (or disproving) one of those conjectures. This lesson is a nice example of a result that might not be arrived at without the use of technology. An applet, activity sheet, questions for students, suggestions for assessment, lesson extensions, and prompts for teacher reflection are included. (author/sw)

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Common Core State Standards for Mathematics

Standards for Mathematical Practice

CCSS.Math.Practice.MP3

Construct viable arguments and critique the reasoning of others.

High School - Geometry

Congruence

Prove geometric theorems

HSG-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.*Similarity, Right Triangles, and Trigonometry

Prove theorems involving similarity

HSG-SRT.B.5

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Ohio Mathematics Academic Content Standards (2001)

Geometry and Spatial Sense Standard

Benchmarks (8–10)

B.

Describe and apply the properties of similar and congruent figures; and justify conjectures involving similarity and congruence.

E.

Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools, such as straightedge, compass and technology.

H.

Establish the validity of conjectures about geometric objects, their properties and relationships by counter-example, inductive and deductive reasoning, and critiquing arguments made by others.

Grade Level Indicators (Grade 8)

1.

Make and test conjectures about characteristics and properties (e.g., sides, angles, symmetry) of two-dimensional figures and three-dimensional objects.

Grade Level Indicators (Grade 10)

3.

Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including:
a. prove the Pythagorean Theorem;
b. prove theorems involving triangle similarity and congruence;
c. prove theorems involving properties of lines, angles, triangles and quadrilaterals; and
d. test a conjecture using basic constructions made with a compass and straightedge or technology.

4.

Construct right triangles, equilateral triangles, parallelograms, trapezoids, rectangles, rhombuses, squares and kites, using compass and straightedge or dynamic geometry software.

Principles and Standards for School Mathematics

Geometry Standard

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Expectations (9–12)

analyze properties and determine attributes of two- and three-dimensional objects;

explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;

establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;

Use visualization, spatial reasoning, and geometric modeling to solve problems

Expectations (9–12)

analyze properties and determine attributes of two- and three-dimensional objects;

explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;

establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;

draw and construct representations of two- and three-dimensional geometric objects using a variety of tools;