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Teaching Math: A Video Library, 9-12

Properties of Parallelograms

Process Standards
Communication, Problem Solving
School: Arlington High School; 970 students
Location: Arlington, Massachusetts
Teacher: Carol Martignette Boswell
Years Teaching: 21
Students in Classroom: 16
Grade: 10 – 12
Properties of Parallelograms
Video Overview
Students present homework on changing a right triangle into a rectangle. They evaluate one another's presentations with peer assessment cards numbered 1, 2, 3, or 4 (4 as highest), and explain the scores they give. Then students work in small groups to cut a triangle into pieces to form a parallelogram. During the activity, students explain to Ms. Martignette Boswell how their work illustrates the properties of a parallelogram, and she encourages them to use mathematical vocabulary. At the end of class, students are told they will present their work the following day.

Course and Curriculum
This required Geometry course focuses on basic geometry concepts. Prior to this course, students took algebra. The goal of this lessonÐwhich is part of a unit on area, properties of quadrilaterals and triangles, and informal proofÐwas to discover properties of parallelograms. In earlier lessons, students worked with tangrams to learn properties of triangles and quadrilaterals. Students also learned how a shape can be cut and made into another shape and investigated the concept of area by covering one shape with another shape.
During the lesson, Ms. Martignette Boswell assessed how well students learned the geometry concepts and worked cooperatively with one another. Each day, she has students assess one another in their groups using peer assessment cards, and discusses with students their scores from these cards. She also has students assess themselves according to five categories: group work; preparation for class; participation in class; performance on homework, tests, and quizzes; and portfolios. In addition, she conducts peer and self–evaluation through periodic student self–assessment questionnaires.
Following this lesson, students reviewed and summarized class presentations. They continued to explore area by covering one shape with another and investigated how formulas for parallelograms and triangles are derived. Students presented arguments to generalize the covering relationships and eventually developed theorems. Following this unit, students studied triangle congruency. They also offered suggestions for improvement after they gave peer assessment scores.

A Pre–Viewing Exploration for Teacher Workshops
Using Triangles to Form Other Polygons
Objective: To explore the relationships between properties of triangles and parallelograms.
Materials: Paper, scissors, and straightedges for each pair.
Activity
Do this activity in pairs.

Draw any large scalene triangle and list as many of its properties as you can. Include relationships between angles, sides, and special segments of the triangle.
Visualize how the triangle might be cut to form a parallelogram. Draw a line on the scalene triangle, cut out the shapes, and construct a parallelogram. List as many of the parallelogram's properties as you can. Present your construction to the group and explain your reasoning.

Exploring Content

What properties did you use to form the parallelogram?
What questions could you ask to help students understand the relationships between the triangle and the parallelogram formed from the triangle?
How could this activity help students explore the area formulas for these two shapes?
What knowledge do students need to complete this discovery?

Exploring Teaching Issues

How do informal explorations of geometric relationships prepare students to construct formal arguments? How can you encourage them to search for several relationships?
How would you address students' misconceptions or inaccurate conclusions about geometric relationships?

Exploration Answers
Activity

Properties of scalene triangles: Angles total 180°; no two sides are congruent; no two angles are congruent; the midline, which connects the midpoints of two sides, is parallel to the third side and half its length.
Cut in triangle to form parallelogram. Properties of parallelograms: Opposite sides are parallel and congruent; opposite angles are congruent; consecutive angles are supplementary; each diagonal forms two congruent triangles; diagonals bisect each other; total measure of the four angles is 360°.

Exploring Content

Properties used: Midline of triangle is parallel to third side; midline cuts side of triangle into congruent segments; segments in new
Answers will vary. Possible questions include: In triangle ABC, what is the relationship between angle B and the sum of angles A and C? What is the relationship between the base angles of a parallelogram? If you were to draw a line through the triangle, parallel to the base, how would it divide the sides it intersects? Where would it make sense to draw this line?
Students must show that the midline cuts the altitude in half.

Triangle Parallelogram

A = (bh₁)/2 A = bh₂

(bh₁)/2 = bh₂ when the height of the triangle is twice the height of the parallelogram. For example, ((6)(8))/2 = (6)(4), h₁ = 8 and h₂ = 4.
Relationships of parts of triangles and parallelograms, supplementary angles, property of midline in a triangle, congruent segments, and parallel line relationships.

Topics for Discussion
What is the value of having students paraphrase their understanding in mathematics?

What intuitive understandings did students have about the geometric relationships? How did they use these understandings and their prior knowledge?
How did students' understanding and misunderstanding of terminology affect their work? How important do you think the relationship is between knowing mathematical vocabulary and understanding mathematical concepts? Explain.
What criteria would you use to evaluate student work and the oral presentation of that work? Discuss how sharing that criteria with students before they begin an assignment affects their work.
Discuss Ms. Martignette Boswell's interventions and their impact on student discourse.

How does Ms. Martignette Boswell promote discourse with students and between students?
What strategies did Ms. Martignette Boswell use in response to students' misconceptions?
Cite several strategies for responding to students' misconceptions and discuss the advantages and disadvantages of each.
How do informal explorations prepare students to build formal arguments?

Create a list of other possible approaches for engaging students in geometry.
Cite lessons, contexts, or projects that might motivate students to seek understanding and to justify their thinking.
Describe specific informal explorations that have helped your students build formal arguments and gain a better understanding of the structure of mathematical principles.
How can computer software be used for student exploration in geometry?