Quadrilaterals: Transformations for Developing Student

Illinois Mathematics Teacher – Fall 2012 .....................................................................................36

Quadrilaterals: Transformations for Developing Student Thinking Colleen Eddy – University of North Texas, [email protected] Kevin Hughes – University of North Texas, [email protected] Vincent Kieftenbeld – Southern Illinois University Edwardsville, [email protected] Carole Hayata – University of North Texas, [email protected]

Students begin learning about transformations in the early elementary grades. They might create simple tessellations by cutting and pasting cardboard pieces, or investigate symmetry by folding figures on patty paper (Common Core State Standards [CCSS] 4.G.3). Their knowledge and ease with transformations continues to grow through middle school, where the everyday language of turn, flip, slide, bigger and smaller is connected with the corresponding mathematical terminology of rotation, reflection, translation, and dilation (National Council of Teachers of Mathematics [NCTM] 2000). By the end of middle school, students should be able to describe the effect of transformations on two-dimensional figures (CCSS 8.G.3).

In high school, however, students often study transformations in isolation, establishing few connections with other parts of geometry. This disconnects the informal thinking of elementary and middle school, and the deductive reasoning required in a high school geometry course. Coxford and Usiskin (1971, 1975) originally proposed the use of transformations to build conceptual understanding based on prior experiences before deriving proofs. Transformations also form the basis of geometric understanding in the vision of the Common Core. For instance, for high school geometry “[t]he concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation” (National Governors Association for Best Practices & Council of

Chief State School Officers, 2010, Geometry: Introduction section, para. 4).

Using transformations to strengthen your students’ understanding of geometry sounds like a good idea, but how can you actually do this in the classroom? This article describes geometry activities that incorporate transformations to discover and justify definitions and properties of quadrilaterals, and the relationships between the different quadrilaterals. We provide examples how you can help students use transformations in their reasoning. The lesson described in this article incorporates CCSS geometry content from across the grades. Specifically, it addresses the following standards in the Congruence domain (G-CO):

Experiment with transformations

in the plane G-CO.3 Given a rectangle, parallelogram,

trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

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Prove geometric theorems 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.

Completing the activities presented in this article will help your students develop the following mathematical practices [CCSS MP]: 1. Make sense of problems and persevere in

solving them; 3. Construct viable arguments and critique

reasoning of others; 5. Use appropriate tools strategically. Blackline masters for all activity pages are included at the end of this article. Examples of student work and suggestions on how to deliver the material can be found throughout the text.

Discovering and developing an understanding of quadrilateral properties

As mathematics teachers, we know

that simply providing students with a list of definitions and properties of quadrilaterals to memorize is an ineffective practice. This approach leads to minimal understanding of the quadrilateral properties and how they are derived. In addition, little or no connections are made between the properties and how the quadrilaterals relate to each other. Transformations give students the opportunity to develop the definitions and properties of quadrilaterals through discovery and to build connections among the family of quadrilaterals. In the activity we describe, students create quadrilaterals

using transformations of triangles and discuss how to derive the properties of the resulting quadrilateral. With teacher guidance, students synthesize and summarize their findings and investigate how these properties relate the different quadrilaterals to each other. Students then develop definitions to classify each quadrilateral shape accordingly (CCSS MP 3). The outline for this activity is as follows:

1. Form heterogeneous groups of 3 - 4

students. 2. Assign each group one or two

quadrilaterals to investigate parallel-ogram, rhombus, rectangle, square, trapezoid, isosceles trapezoid, or kite. a. Give each student in the group an

investigation sheet to record their findings. Blackline masters for these six investigations can be found at the end of the article.

b. In addition, give each group a sheet of graph paper, a ruler, and a sheet of patty paper, which students can use to complete the prescribed rotations in some of the investigations (CCSS MP 5).

3. During the investigation phase of the lesson, each group creates a quadrilateral by completing a set of transformations on a given triangle. In the process, the students discuss the various properties of the quadrilateral shape they create with respect to the sides, the vertex angles, the diagonals, and the symmetry of the quadrilateral.

4. Students are asked to go back and discuss how they could justify the various properties using their prior knowledge of transformations (CCSS MP 3). In particular, students should have developed the concept that rotations, reflections, and translations preserve congruency of segment lengths

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Advancing geometric thought through the use of transformations

The van Hiele levels of geometric

thought (1986) provide a sound justification of why transformations are a good bridge between initial student thinking and the abstract nature of high school geometry. The model includes five levels of geometric reasoning: visual, descriptive, informal deductive, (formal) deductive, and rigor. Although a student progresses through the levels linearly, there is no strict dependence on age. For example, a fourth grader and a high school geometry student could be at the same level. The levels overlap as a student transitions from one level to the next. Most students entering a high school geometry course operate at or below the van Hiele visual level of understanding (Shaughnessy and Burger 1985). These students are able to identify different shapes, but they may find it hard to recognize and reason with specific characteristics of shapes. For example, doing transformations with everyday language as described in the introduction is at the visual level. Students at this level are often not yet ready for the abstract reasoning required in high school geometry. Students using the mathematical terminology for transformations are at the overlap of the visual and descriptive levels. Students discovering and deriving quadrilateral properties are at the informal deductive level.

Transformations provide a bridge between the initial visual intuition of the students and the more formal reasoning of the higher van Hiele levels. The quadrilateral activities provided entry to students at the descriptive level. Students drew conclusions about the sides and angles of their original triangle and then rotated the triangles to create the specified quadrilateral. This provided students the opportunity to apply prior experiences of transformations

and the properties of triangles. With the discussions, the teacher guided students to the informal deductive level by having them discover the properties of quadrilaterals and create informal arguments deriving the quadrilateral properties. Finally, students unify their understanding of quadrilaterals by creating the family tree.

Conclusion

Incorporating transformations to develop understanding in high school geometry helps students move from the visual and descriptive van Hiele levels to the informal deductive and formal deductive levels. All students, regardless of their van Hiele level, stand to benefit from an integration of transformation geometry in a high school geometry course (Usiskin 1972; Okolica and Macrina 1992). In the quadrilateral activities described above, the teacher used this approach to guide the students in building their understanding of quadrilateral properties. When students can connect geometric concepts to the prior knowledge and experiences that they bring to class, they can formulate deductive arguments. The study of transformations in the early grades and the continuing development of these concepts in middle and high school can provide the bridge geometry students need to reach the informal and formal deductive levels of geometric reasoning.

References Coxford, Arthur F., and Zalman P. Usiskin.

Geometry: A Transformation Approach. River Forest, IL: Laidlaw, 1971, 1975.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.

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National Governors Association for Best Practices & Council of Chief State School Officers (2010) Common Core State Standards for Mathematics. http://www.corestandards.org/the-standards

Okolica, Steve, and Georgette Macrina. “Integrating Transformational Geometry Into Traditional High School Geometry.” Mathematics Teacher 85, no. 9 (December 1992): 716–19.

Shaughnessy, J. Michael, and William F. Burger. “Spadework Prior to Deduction in Geometry.” Mathematics Teacher 78, no. 6 (September 1985): 419–28.

Usiskin, Zalman P., and Arthur F. Coxford. “A Transformation Approach to Tenth-Grade Geometry.” Mathematics Teacher 65, no. 1 (January 1972): 21–30.

Usiskin, Zalman P. “The Effects of Teaching Euclidean Geometry via Transformations on Student Achievement and Attitudes in Tenth-Grade Geometry.” Journal of Research in Mathematics Education 3, no. 4 (November 1972): 249–59.

Van Hiele, Pierre M. Structure and Insight: A Theory of Mathematics Education. Orlando, FL: Academic Press, 1986.

Blackline masters for the activity pages are included below and on subsequent pages.

Building Quadrilaterals: Parallelogram 1. On a sheet of graph paper, draw obtuse οܥܤܣ.

x Draw one of the sides of οܥܤܣ along one of the grid lines. x Be sure all vertices are placed at the intersection of grid lines.

2. Locate the midpoint, ܯ, of ܤܣതതതത. 3. Draw the median to side ܤܣതതതത.

4. Label the figure you have drawn by indicating congruent sides, angles, and measures using

appropriate markings.

5. Rotate οܥܤܣ by ͳͺͲι around point ܯ. x Label the new image with the correct markings to indicate congruent sides, angles, and measures.

6. Discuss with your group the properties of the parallelogram that you created using the transformations above. Pay attention to the properties of the sides, the vertex angles, the diagonals, and the symmetry of the figure.

x Summarize your findings under the headings on the chart.

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Building Quadrilaterals: Rectangle 1. On a sheet of graph paper, draw scalene right οܥܤܣ.

x Draw both legs of οܥܤܣ along grid lines. x Draw the right angle at vertex ܥ. x Be sure all vertices are placed at the intersection of grid lines.

2. Locate the midpoint, ܯ, of ܤܣതതതത.

3. Draw the median to hypotenuse ܤܣതതതത.



5. Rotate οܥܤܣ by ͳͺͲι around point ܯ. x Label the new image with the correct markings to indicate congruent sides, angles, and measures.

6. Discuss with your group the properties of the parallelogram that you created using the transformations above. Pay attention to the properties of the sides, the vertex angles, the diagonals, and the symmetry of the figure.


Building Quadrilaterals: Rhombus 1. On a sheet of graph paper, draw scalene right οܥܤܣ.

x Draw both legs of οܥܤܣ along grid lines. x Draw the right angle at vertex ܥ. x Be sure all vertices are placed at the intersection of grid lines.



3. Reflect οܥܤܣ across the line containing ܥܤതതതത. x Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use

prime marks for the image vertices. x What kind of figure do you have now? Justify your answer.

4. Reflect οܣܤܣԢ across the line containing ܣܣԢതതതതത. Be sure to also reflect ܥܤതതതത.

x Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use prime marks for the image vertices.

5. Discuss with your group the properties of the parallelogram ܣܤܣԢܤԢ that you created using

the transformations above. Pay attention to the properties of the sides, the vertex angles, the diagonals, and the symmetry of the figure.


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Building Quadrilaterals: Square 1. On a sheet of graph paper, draw isosceles right οܥܤܣ.

x Draw both legs of οܥܤܣ along grid lines. x Draw the right angle at vertex ܤ. x Be sure all vertices are placed at the intersection of grid lines.



3. Reflect οܥܤܣ across the line containing ܥܤതതതത. x Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use

prime marks for the image vertices. x What kind of figure do you have now? Justify your answer.

4. Reflect οܣܥܣԢ across the line containing ܣܣԢതതതതത. Be sure to also reflect ܥܤതതതത.


5. Discuss with your group the properties of the square ܣܥܣԢܥԢ that you created using the

transformations above. Pay attention to the properties of the sides, the vertex angles, the diagonals, and the symmetry of the figure.


Building Quadrilaterals: Kite 1. On a sheet of graph paper, draw scalene acute οܥܤܣ.

x Draw side ܥܤതതതത of οܥܤܣ along a grid line. x Be sure all vertices are placed at the intersection of grid lines.

2. Draw an altitude, ܦܣതതതത, of οܥܤܣ from point ܣ to ܥܤതതതത.

3. Label the figure you have drawn by indicating congruent sides, angles, and measures.

4. Reflect οܥܤܣ across the line containing ܥܤതതതത.


5. On a separate sheet of graph paper, repeat steps 1-4 with a scalene obtuse οܥܤܣ.

6. Discuss with your group the properties of the kites ܣܥܣԢܤ that you created using the transformations above.


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Building Quadrilaterals: Trapezoid 1 1. On a sheet of graph paper, draw a large scalene οܥܤܣ. 2. Label the figure you have drawn by indicating congruent sides, angles, and measures using


3. Locate a point anywhere on ܤܣതതതത. x Construct a line parallel to ܥܤതതതത through point . x Construct the intersection of this line with ܥܣതതതത at point .

4. Discuss with your group the properties of the trapezoid ܥܤ that you created using the

transformations above. x Summarize your findings under the headings on the chart. x Hint: What connections can be made between the measures of the angles in this figure and your prior

experiences with parallel lines cut by a transversal?

Building Quadrilaterals: Trapezoid 2 1. Now draw a large isosceles ο.

x Draw the vertex angle at .

2. Locate a point ܣ anywhere on തതതത. x Construct a line parallel to തതതത through point ܣ. x Construct the intersection of this line with തതതത at point ܤ.

3. Discuss within your group the properties of the trapezoid ܤܣ that you created using the

transformations above. x Summarize your findings under the headings on the chart. x Hint: What connections can be made between the measures of the angles in this figure and your prior

experiences with parallel lines cut by a transversal?

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VERTEX ANGLES:

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Quadrilaterals: Transformations for Developing Student