PRoPoRtional

Copyright © 2011 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Reasoning with a

PYRaMid A three-dimensional model and geometry software can help develop students’ spatial reasoning and visualization skills.

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ami Mamolo, Margaret Sinclair, and Walter J. Whiteley

Proportional reasoning pops up in math class in a variety of places, such as while making scaled drawings; finding equivalent fractions; converting units of measurement; comparing speeds, prices, and rates; and comparing lengths, areas, and volume. Students need to be exposed to a variety of representations to develop a sound understanding of this concept, particularly in the middle grades. Some meaningful explorations

of proportion occur in geometry, in particular, with respect to dilation, or scaling. Dilation is a similarity map in which a shape is enlarged or reduced by a nonzero factor while keeping the edges of the new shape parallel to the original shape’s edges. Through hands-on geometry experiences, students can develop proportional reasoning that can be extended and connected to other contexts, such as those in the number-sense strand. Vol. 16, No. 9, May 2011

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a tWo-PaRt geoMetRY actiVitY Described here is an activity designed to foster learners’ ability to use visualization, spatial reasoning, and geometric modeling to solve problems. Research shows that increased inclusion of visual and kinesthetic approaches helps students make connections among various representations of underlying mathematical concepts (Clements and Battista 1992). Filling

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Fig. 2 When the base of the inverted

pyramid is parallel to the floor, the exposed water surface creates a shape that is geometrically similar to the base.

pyramid is tilted and not parallel to the floor, the exposed water surface creates a new shape.

Stewart Craven

PROPORTIONAL REASONING AND DILATION IN THREE DIMENSIONS Lanius and Williams (2003) state: “Proportionality is so far-reaching in scope that it has connections to most, if not all, of the other foundational middle-grades topics and can provide a context for their study” (p. 395). The first part of Filling the Pyramid uses physical models to investigate proportionality. Depending on the class, the activity can focus on linear relationships, or it can be broadened to include proportionality in area or volume. The materials used are clear plastic pyramids (with triangular or square bases, with a portion of the base removed) and a watering can. Filling the Pyramid involves pouring varying amounts of water and observing and reflecting on the shapes that are formed by the surface of the water enclosed by sides of the pyramid. When the base of the pyramid is parallel to the floor, the exposed water surface creates a shape that is geometrically similar to the base (e.g., a similar square if the pyramid is square-based and a similar triangle if the pyramid is triangular-based). (See fig. 1.) Pouring more water in the receptacle scales the surface shape, giving a visual and kinesthetic sense of dilation, or geometric proportionality. Consider the solid shape that is

Fig. 1 When the base of the inverted

Stewart Craven

the Pyramid is a hands-on investigation that incorporates dynamic geometry software. It explores three-dimensional models and two-dimensional geometric representations to illustrate properties of proportion and dilation. Over the past several years, we have tested the activity with various groups, including math-anxious undergraduates interested in teaching as well as practicing K−8 teachers. They provided feedback on both their learning experiences and how to make the activity accessible to middle school students.

created by the water in the pyramid: As its volume increases, the lengths of its sides increase proportionally to the height of the water, and the area of its surface shape increases proportionally to the square of the height. The (bottom) vertex of the pyramid acts as the center of this geometric dilation. Students can detect the ratios and proportions of two surface polygons by comparing the side lengths for the two surfaces. They can also investigate the relationship between these measurements and the distance from the center of dilation to the vertices of the different surface shapes. Plastic pyramids can be anchored using PlayDoh® so that (1) the base of the pyramid is parallel to the floor and (2) the pyramid is tilted (see fig. 2). Figure 3’s prompts can be distributed in activitysheet form, with appropriate spacing for student notes and diagrams. (Note: The effects of surface tension, which make it difficult to take accurate measurements, can be lessened by adding a few drops of detergent to the water to reduce but not completely eliminate the meniscus effect. When such errors are noticed by the students, they should be addressed and the reasoning behind them discussed.)

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Figure 3’s prompts offer students an opportunity to recreate on paper the images they see in the pyramids. This will help them gain a visual and spatial sense of proportional changes. Teachers may want to stimulate further discussion or extended explorations by encouraging students to reflect on their observations of the similarities and differences between the two cases (i.e., when the pyramid base is parallel to the floor and when it is tilted). For example, the teacher might ask: • In what ways, if any, did the shape of the surface area change when you tilted the pyramid? • In what ways did the shapes’ measurements change? We found that these prompts helped draw attention to the fact that the surface shape could change while the volume of water remained constant. One pair of students remarked: “When we tilted the first pyramid, the surface area actually increased, even if the same amount of water was poured in. Two sides of the shape became longer, making it no longer a square, so there is more surface area.” Both the undergraduate students and

Fig. 3 these prompts on Filling the Pyramid can be produced in activity sheet form for students.

1. Pour a small amount of water into the pyramid. 2. examine the shape that the water surface takes. 3. Make a sketch of the shape in your notebook. Use your estimation skills to make your sketch as accurate as possible. 4. Pour a bit more water into the pyramid. examine and sketch the shape that the water surface takes. 5. Pour again and again. What shapes do you notice? Sketch them. 6. Compare the shapes you have drawn, including the lengths of the sides, angles, and areas. 7. repeat this activity with the same pyramid tilted at a different angle. What do you notice?

the practicing teachers returned to these guiding prompts throughout the lesson. Observations that were at first vague (e.g., “The shape changed when we added water because the lengths increased”) developed into more specific descriptions about angle sizes and scale factors (e.g., “The surface area changed by a constant factor” and “When increasing and decreasing the water levels, the angles will be the same from the previous one”). To help focus attention on the changes in dimension of the surface shape as more water was added and as some was poured out, we found it helpful to ask participants to describe, in their own words, what was happening to each dimension of the water’s shape and to approximate how the changes in dimension were related.

To encourage their spatial reckoning, we further asked them to predict whether (and why) these observations would hold true for new shapes. These prompts stimulated a fruitful discussion while students sought to identify or recall proper terminology for unfamiliar shapes. The discussions provided a launching pad for participants to generalize their reasoning and draw connections, using measurement tools or formulae, to numerical approaches to proportion. One way to enhance the threedimensional exploration; encourage deeper understanding; and support visual, spatial, and numerical reasoning with proportion is by using The Geometer’s Sketchpad©. We describe a two-dimensional representation of dilation, which was completed using Sketchpad.

dilation and PRoPoRtional Reasoning We designed the initial Sketchpad exploration using a triangle to complement what students would do with the triangular-based pyramid models. By dilating a triangle about a center, we created a skeleton for the investigation. We wanted to design a sketch that would help students see the connections between this two-dimensional representation and the three-dimensional model. We also wanted them to move flexibly between the two representations (either in their minds or with the visual models). It was important that students be comfortable moving, dragging, and exploring in ways that were not accessible in the previous activity. To encourage them to notice the connections between the sketch and the three-dimensional model, we colored the triangle and its dilated image and used broken lines to link the center, original vertices, and dilated images’ vertices. Figure 4 shows the sketches the students used. Vol. 16, No. 9, May 2011

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The Sketchpad drawing can keep track of area measurements and their ratios, so that when students drag and change the shape of the triangles, the numerical values change accordingly. Thus, multiple representations (e.g., numerical and geometric) can provide different perspectives on the effects of proportional change and help students make connections across mathematical strands. As one student explained about Sketchpad: [It] gave me a visual/spatial look on the movements of shapes. It helps explain thoughts better and definitely gave me more of an understanding of a shape’s proportions.

We recommend that after students complete initial explorations with the three-dimensional models and Sketchpad drawing they be given time to work with and access both resources. This can help them make connections between the two-dimensional and threedimensional representations of dilation. Once students are able to see the threedimensional model in the Sketchpad drawing, they can explore different avenues, conjecturing about “what would happen if ” and making measurements. For example, students might use a Sketchpad drawing to model filling a pentagonal-based pyramid to various levels or to tilt a pyramid beyond what would be possible with the physical materials in Filling the Pyramid. One pitfall of this activity is its length. We recommend setting aside at least two days for this lesson, with additional time dedicated to introducing Sketchpad, if students are unfamiliar with the software. Guiding prompts may be necessary to keep students focused on the intended goals of the activity. For example, teachers may need to explicitly direct students’ attention to how ratio or area measurements change when students drag a point or line on one of the triangles.

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PROPORTION AND DILATION IN TWO AND THREE DIMENSIONS By comparing what students notice with the three-dimensional models and the Sketchpad drawing, they can exercise their visual and spatial skills and can discover several key ideas about proportion. For instance, both tasks provide evidence of the following: • Scaling preserves all angles and all ratios of distances within any given shape. • There is a special relationship between the original and scaled image; specifically, all lengths are scaled by a constant factor, and areas are scaled by the square of this constant factor. The measurements offered by Sketchpad can validate the visual sense of size changes developed with the three-dimensional models. (The idea that volume is scaled by the cube of a constant factor can be explored by further experiments with pouring and measuring.) • When scaled up or down, reflective symmetries are preserved. For example, in the Sketchpad activity, if we dilate an isosceles triangle (which has a line of symmetry, or mirror), the new triangle will also be isosceles. • If shape A scales to shape B by a ratio of m:1, then shape B scales to shape A by a ratio of 1:m. If a design is scaled by a ratio of 5:1, the new design can be rescaled to the original using a ratio of 1:5. This idea is related to adding and removing water in the three-dimensional models. • Scaling shape A to shape B by a ratio of m:1 and shape B to shape C by a ratio of n:1 means that shape A scales to shape C by a factor of m × n. In particular, if a design is enlarged using a ratio of 2:1, and the new design is enlarged using a ratio of 3:1, then the ratio of the final design to the original will be 6:1. This property is related to pouring water into the pyramid multiple times. 548

Fig 4. The dilation sketch was prepared in The Geometer’s Sketchpad. We suggest that teachers provide a preconstructed sketch that students can explore by dragging various parts of the sketch. As an extension or homework assignment, teachers can encourage students to explore properties of dilation further by creating and dilating their own polygons.

(a)

(b)

• Dilating a shape preserves important geometric features of the original shape. In building an understanding of some of these ideas, it is important that students talk about what they notice (e.g., the surprising increase in area when the water is only a little higher) and what puzzles them (e.g., why the surface shape is different from the base shape when the pyramid is tilted). To engage in meaningful conversation, students should be encouraged to use correct terminol-

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ogy, such as vertex, base, pyramid, similarity of shape, surface shape, height, area, volume, proportion, ratio, dilation, and center of dilation. The Sketchpad exploration can help in this regard; as students navigate the menus, they can familiarize themselves with some of the terminology. Commercial or student-made posters can also help. The key is to provide sufficient time and opportunity for students to become comfortable with the new words and the new ideas. Working in teams during the

explorations can help students develop both their reasoning skills and their ability to communicate their ideas about proportion. Follow-up class discussions around the three-dimensional model and Sketchpad investigations are also important. Throughout the lessons, students should be encouraged to talk about what they notice; use gestures and incorporate new vocabulary; and relate their experiences to different areas, problems, and applications of mathematics that involve proportional reasoning.

conclUsion Proportional reasoning, a central concept in middle school mathematics, connects to a wide range of mathematical disciplines—in particular, important geometric content around length, area, and volume scale. It is important that teachers and curriculum designers continue to develop and refine their pedagogical approaches to this concept. As Clements (1999) observes, it is important for educators to help students build and connect multiple representations of mathematical ideas. Filling the Pyramid investigates scaling transformations in two dimensions and three dimensions. It provides a multirepresentational approach that offers students an opportunity to strengthen their— • visualization skills by seeing similarities in the dilated and original shapes (e.g., the preserved reflective symmetries, angles, and shape) and by seeing the (larger) growth in area compared with the growth in side length; • spatial reasoning skills by predicting and recreating shapes and changes in those shapes, by estimating lengths and angles in Sketchpad, and by comparing changes in area and volume when tilting the pyramid; and • kinesthetic skills through handling

the three-dimensional model, tilting it, pouring and removing water, and manipulating the Sketchpad drawing. This activity also offers students a meaningful and authentic experience with proportional reasoning—a concept that has been hailed as both “a capstone of elementary mathematics” and “a cornerstone of advanced mathematics” (Lesh, Post, and Behr 1988, p. 116).

ReFeRences Clements, Douglas. “Concrete Manipulatives, Concrete Ideas.” Contemporary Issues in Early Childhood 1 (1999): 45–60. Clements, Douglas H., and Michael T. Battista. “Geometry and Spatial Reasoning.” In Handbook of Research on Mathematics Teaching and Learning, edited by Douglas A. Grouws, pp. 420–64. New York: Macmillan, 1992. Lanius, Cynthia, and Susan Williams. “Proportionality: A Unifying Theme for the Middle Grades.” Mathematics Teaching in the Middle School 8 (April 2003): 392–96. Lesh, Richard, Thomas Post, and Merlyn Behr. “Proportional Reasoning.” In Number Concepts and Operations in the Middle Grades, edited by James Hiebert and Merlyn Behr, pp. 93–118. Reston, VA: National Council of Teachers of Mathematics, 1988. National Council of Teachers of Math-

ematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000. ami Mamolo, amamolo@ yorku.ca, is a postdoctoral fellow in mathematics education at York University. Specializing in university students’ learning and understanding of mathematics, she is currently involved in a research program to (re)design the teaching of large first-year mathematics courses. Margaret sinclair, [email protected], is associate professor of education at York University and codirector of the York-Seneca Institute for Mathematics, Science, and technology education, who specializes in the use of technology in teaching elementary and secondary mathematics. Walter J. Whiteley, whiteley@mathstat .yorku.ca, is professor of mathematics and statistics at York University and a member of the graduate program in education. He specializes in research in discrete applied geometry and in geometry education. the steps for creating the sketches in this article, including snapshots of the screen, are appended to the online version of this article at www .nctm.org/mtms.

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Instructions for Creating the Sketchpad Dilation Drawing 1. To create the triangle, select the Point Tool on the tool bar at the left-hand side of the sketch, and draw three points on the page. 2. Using the Arrow Tool, select all three points. 3. Go to the Construct tab in the menu, and select “Triangle Interior.” 4. Specify colors of the triangle or the corner points using the “Color” item in the Display tab of the menu.

Creating a colored triangle 5. O  nce the triangle interior is established, draw a fourth point on the page (not on the triangle). 6. Using the Arrow Tool, select this point (if it is not already selected). 7. Go to the Transform tab in the menu, and select “Mark Center.” This establishes the fourth point as the center of dilation.

Constructing a center for dilation 8. C  onnect the center of dilation to the points of the triangle as follows: A. Use the Arrow Tool to select the center point and one of the corner points. B. Go to the Construct tab, and select “Segment.” C. Repeat for the other two vertices. 9. Specify the line segment style or color by selecting the segment and using the “Line Width” or “Color” items, respectively, in the Display tab. Connecting vertices to the center 10. U  sing the Arrow Tool, select the triangle, including its interior and vertices. 11. Go to the Transform tab in the display menu, and select “Dilate.” 12. Input an appropriate ratio and click “Dilate.” This will create a reduced or enlarged triangle (depending on the ratio), which is a dilation of the original.

Dilating the triangle by a fixed ratio

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Instructions on Including Measurements in the Sketchpad Drawing 1. T  o add distance measurements, select two vertices on one of the triangles, go to the Measure tab in the menu, and select “Distance.” 2. Labels can be tailored through the Edit tab by selecting and adjusting the “Properties.” 3. To add ratio measurements, select the two distance measurements, go to the Measure tab, and select “Calculate.” 4. The lengths can be input into the calculation by selecting them from the Values menu. Dividing the lengths will yield the ratio. Keeping track of lengths and ratios 5. T  o add area measurements, select both of the triangles, go to the Measure tab in the menu, and select “Area.” 6. Labels can be tailored through the Edit tab by selecting and adjusting the “Properties.” 7. To add ratio measurements, select the center point, a point on the dilated triangle, and its corresponding point on the original triangle. The order in which the points are selected will impact the ratio measured. 8. Go to the Measure tab and select “Ratio.”

Keeping track of areas and ratios

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