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Intersecting and Perpendicular Lines: Activities to Prevent Misconceptions

Using a Journal Article as a Professional Development Experience
Geometry

Title:          Intersecting and Perpendicular Lines: Activities to Prevent Misconceptions
Author:      Regina Mistretta
Journal:     Mathematics Teaching in the Middle School
Issue:        October 2003, Volume 9, Issue 2, pp. 84-91

Rationale/Suggestions for Use

Principles and Standards for School Mathematics (PSSM), NCTM (2000), states that students in the middle grades should "…understand relationships among types of two- and three-dimensional objects using their defining properties” (p. 232).  This article describes a series of activities of how to help students understand the relationship between interesting and perpendicular lines.

The article offers an example of how to use concept cards as a teaching tool that helps students to visualize, explore, and communicate their ideas.

Procedures/Questions

1. Engage teachers by having them complete the pre-assessment questions shown in Fig. 1.
• Teachers need to record their responses.
• Lead a group discussion regarding each question.  Teachers may well have the same misconceptions as the students' misconceptions described in the article.
• Discuss the role of pre-assessment in guiding instruction.
2. Have teachers read about the first activity that uses concept cards to investigate the relationship between intersecting and perpendicular lines. (p. 86-89).
• Ask teachers if they have used concept cards in their classes and for what specific concepts.
• Refer to the guided dialogue questions (p. 88) and ask small groups to work together in answering the questions.
• Discuss the role of the concept cards in focusing on the important characteristics of intersecting and perpendicular lines so that definitions of each contain necessary and sufficient information. Point out that the examples/non-examples need to help students sort, classify, and form a definition for the concept that is to be taught.
3. Ask teachers to design a concept card for parallelograms.
• First discuss the properties of a parallelogram you want students to understand: a four-sided polygon with both pair of opposite sides parallel.  Discuss the additional properties of special parallelograms (rhombus, rectangle, square).
• Distribute a concept card template for each teacher to use.  Give ample time for teachers to draw their examples/non-examples of parallelograms.
• Have teachers share their examples/non-examples along with the rationale for their samples.

Did teachers mention orientation? (fig. a-b) Did any examples exclude vertical and horizontal sides? (fig. c-d) Did non-examples include non-quadrilaterals? (fig. e-f)

• Ask pairs (or groups) of teachers to design a concept card (on chart paper) for other special quadrilaterals: rhombus, rectangle, square, kite, and trapezoid.
• Use a technique (i.e. gallery walk) to allow for participant feedback on each figure drawn as examples/non-examples.  Facilitate a whole group discussion of one or more of the special quadrilaterals.  As participants are discussing the concepts, use chart paper to record potential dialogue questions.  The dialogue questions should focus on the properties that help sort quadrilaterals.  Example: Are all rectangles squares? Why or why not? Justify your answer with illustrations from the concept cards.
• Close this activity by drawing a flow chart or Venn diagram that summarizes the relationship between types of quadrilaterals. (see PSSM p.233)
• Extension: Suggest to teachers that the collection of quadrilateral concept cards could be used as a bulletin board display or poster.

Next Steps

1. Have teachers complete a journal entry by responding to one or both of the following writing prompts.
• In this activity I learned ...
• Using concept cards will help students …
2. Ask participants to use a concept card to teach a concept in the next few weeks. Bring the concept card (s) and guided dialogue questions to next session.
3. Have teachers explore the diagonals of a quadrilateral using dynamic software, bendable straws and elastic thread, coffee stirrers, or pencil and paper. There is a discussion of this activity in PSSM (p. 233).

Connections to Other NCTM Publications