Aiming a Basketball for a Rebound: Student Solutions Using Dynamic Geometry Software

Transcription

1 Aiming a Basketball for a Rebound: Student Solutions Using Dynamic Geometry Software Diana Cheng, Tetyana Berezovski, Asli Sezen-Barrie Abstract Sports can provide interesting contexts for mathematical and scientific problems. In this article, we present a high school level geometry problem situated in a basketball context. We present a variety of student solutions to the problem using dynamic geometry software and offer possible extensions. Keywords: geometry, sports, dynamic geometry software, problem solving strategies Published online by Illinois Mathematics Teacher on January 21, How can we get a basketball into a hoop? This is a simply stated, open-ended, and complex problem faced by athletes, but it is also a problem that can be motivating for high school students in a geometry or physics class. We created a two-dimensional model of a basketball hoop with dynamic geometry software, Geometer s Sketchpad (Jackiw, 1991). To increase accessibility options, we wanted students to be able to model the problem using a pencil-and-paper printout as well. Thus, we chose the scale factor of the model to be such that a National Basketball Association (National Basketball Association, 2006) regulation-sized basketball with a 29 inch circumference (9.23 inch diameter) could be represented by a US penny coin with 0.75 inch diameter. Using this scale factor, we represented an 18 inch diameter NBA hoop as the circle with its center at point B and radius BD = 1.85 cm (see figure 1). The 24 inch backboard was represented by line segment IL = 4.95 cm. 1. Basketball Aiming Problem We posed this problem to students: You are a basketball player standing at point Q. Your coach tells you to always aim for the center of the hoop, point B. Place point F on the backboard IL such that you will be using the backboard to Corresponding author Figure 1: Basketball hoop model created using Geometer s Sketchpad help you make your shot. Explain how you found point F. Find two different ways to determine point F s location. This is a cognitively challenging problem (Stein et al., 1996) as there is no explicit solution pathway given. The problem involves a modeling situation, as it is realistic that basketball players would aim the basketball towards the center of the hoop, and many shots use a rebound from the backboard. Learning goals for this activity include having students determine which quantities (such as coordinate points, distances, angles) are relevant and how these quantities should be used to solve the problem. There are multiple solutions to this problem, some of which Illinois Mathematics Teacher 1

2 Diana Cheng, Tetyana Berezovski, Asli Sezen-Barrie are discussed in this article. 2. Method The participants in this study were seven in-service middle and high school mathematics teachers taking a semester-long problem solving graduate course in spring All of the participants were enrolled in a master s degree program in mathematics education offered by the mathematics department of a public university. The instructor of the course is the first author of this paper. The participants were given this problem to solve independently as part of their final exams taken in class in a computer laboratory. The participants were provided access to the Geometer s Sketchpad file with the diagram shown in figure 1 and a protractor and ruler to solve the problem. Because participants were asked to provide two solution methods to the problem, a total of fourteen solution strategies were collected, with some used by more than one participant. Eight solutions provided by the participants are reported in this article and are categorized into six distinct categories of strategies. 3. Solution Strategies In our problem, we simplified basketball to two dimensions. A mathematically correct solution to this problem involves recognizing that (a) the trajectory of the ball is not a single straight line because the directions state that the ball must bounce off the backboard, and (b) the angle at which the ball hits the backboard will be the same angle it reflects off of the backboard. That is, the angle of incidence is congruent to the angle of reflection. Connections can also be made from physical observations of Descartes-Snell s Law, used in geometrical optics. This law states that you will always get a constant number when you divide the sines of the angles of incidence and of refraction. This constant gives the reflective indices of the boundary objects (e.g., water, glass) on which a light wave hits; each object has a different reflective index (The Physics Classroom, Figure 2: Productive strategy 1: Slopes 2014). The way a light wave bounces off a surface is similar to the way a basketball hits the backboard. While we did not specifically mention Descartes-Snell s Law in class, it is possible that the teachers heard of it through their high school or undergraduate coursework. 4. Productive Strategies Three strategies used by students were considered to be productive in the sense that applying them accurately would yield a correct answer to the problem. The productive strategies each took into account that the angle at which the basketball hits the backboard is the same angle at which the basketball bounces off the backboard. Participants started out by finding point F on backboard IL. Auxiliary lines were then created as described below. If the strategy involved a measure of two relevant quantities that needed to be equivalent, then the measures of these two quantities had to be made explicit in the participant s work. Because of the low level of precision that is possible using Geometer s Sketchpad, if angles were measured, they only had to be within one degree of each other in order for answers to be considered sufficiently accurate. Strategy 1: Slopes Participants using this strategy created line segments F Q and F B. They then measured the slopes of F Q and F B. Point F was then dragged along IL until the magnitudes of these two slopes were as close as possible (see figure 2). 2 Illinois Mathematics Teacher

3 Aiming a Basketball for a Rebound Figure 3: Productive strategy 2: Congruent angles This strategy is effective because it presumes that the slopes of F B and F Q need to have the same magnitude but be opposite of each other. This strategy is equivalent to making IF B and LF Q congruent, as these angles are the inverse tangents of the slopes. Strategy 2: Congruent Angles Participants using this strategy created line segments F Q and F B and measured IF B and LF Q. Point F was dragged until these two angles were as close as possible. In the work shown in figure 3, the measures of IF B and LF Q only differ by 0.1 degree. This strategy is productive because it creates congruent angles IF B and LF Q such that the basketball bounces off the backboard at the same angle at which it bounces onto the backboard. Strategy 3: Reflection Participants using this strategy created line segment F Q. Then they then created a line perpendicular to IL through point F. This line was used as a line of reflection for F Q. Point F was dragged until the image of F Q passed through point B (see figure 4). This strategy is effective because it creates congruent angles IF B and LF Q. Figure 4: Productive strategy 3: Reflection of F Q through point Q. Line IL will intersect this perpendicular line at point A with coordinates (5.03, 1.85). The other two vertices of the larger triangle are Q and F. The smaller triangle is formed in the following manner: Draw the radius of the hoop that is perpendicular to the backboard, intersecting IL at point E with coordinates (0, 1.85). The other two vertices of the smaller triangle are B and F (see figure 5). Triangle F EB and triangle F AQ are similar by angle-angle similarity. Angles F EB and F AQ are both right angles by construction of the perpendicular lines, and EF B and AF Q are as close to congruent as possible because the angle of incidence is assumed to be congruent to the angle of reflection. Since corresponding sides in similar triangles are proportional, EB/EF = AQ/AF. Since AE is a straight line of length 5.03 cm and AE = AF + EF, we can find that EF = 1.22 cm and the location of F is (1.22, 1.85). Strategy 4: Similar Triangles Another way to solve the problem, not mentioned by the participants of this study, involves setting up a proportion to compare corresponding lengths of two similar triangles. The larger triangle is formed in the following manner: Draw the line perpendicular to IL Figure 5: Productive strategy 4: Similar triangles Illinois Mathematics Teacher 3

4 Diana Cheng, Tetyana Berezovski, Asli Sezen-Barrie 5. Unproductive Strategies There were three unproductive strategies attempted by participants: the use of a fixed angle, the use of a straight line, and the use of irrelevant congruent angles. These strategies are considered unproductive because they do not involve the use of relevant quantities or a relevant trajectory of the ball. The strategies are further described below. Strategy 5: Fixed Angle While the use of angles can be a productive strategy, in the work shown below it is unclear how participants decided which angle to measure and what the measure of that angle should be. Two different angles were identified by different participants, with different angle measures. In the first example (see the top image in figure 6), the participant found point F on IL such that m F BQ = 90. In the second example (see the bottom image in figure 6), the participant found point F on IL such that m F BD = 45. No justification of these angle measures was provided. We speculate that these angle measures were chosen because many diagrams involving triangles in high school geometry textbooks are right triangles or isosceles triangles. Strategy 6: Straight Line This was the most common strategy used by participants. Participants using this strategy thought that the ball would travel in a straight line through B to the backboard. Thus, they drew in line segment QB and extended it until it intersected IL at point F. Participants using this strategy did not take into consideration LF Q (see figure 7). We speculate that participants using this strategy misunderstood the prompt whereby the coach tells you to aim for point B. Participants may have thought that the only way to aim for point B was by going in a straight line from Q to B. Figure 6: Two examples of unproductive strategy 5: Fixed angle Figure 7: Unproductive strategy 6: Straight line 4 Illinois Mathematics Teacher

5 Aiming a Basketball for a Rebound 6. Discussion Figure 8: Two examples of unproductive strategy 7: Irrelevant congruent angles Strategy 7: Irrelevant Congruent Angles Participants using this strategy recognized that two angles needed to be congruent, but used irrelevant angles. In the first example (see the top image in figure 8), the participant found the angle bisector of LQB, and thought that point F would be located at the intersection of the angle bisector and IL. In the second example (see the bottom image in figure 8), the participant drew in the midpoint V of IL and created point F along V L. The participant then drew in V B, F B, F Q, and QL. The participant dragged point F along V L until the measures of V BF and F QL were almost the same. We hypothesize that participants using this strategy may have encountered Descartes-Snell s law and were trying to apply it, but did not remember which angles needed to be taken into account. A rich set of responses to this basketball modeling problem was obtained and reported in this article. Strategies that yielded correct solutions as well as unproductive strategies were described. In order to solve this problem correctly, participants needed to synthesize knowledge across disciplines and recognize that a physical principle about angles should be applied in this geometry problem. Taken as a whole, the solution paths can apply a variety of mathematical content standards from the Common Core State Standards (Common Core State Standards Initiative and others, 2010). For example, all of the strategies involved using geometric shapes, their measures, and their properties to describe objects on the basketball court (HSG.MG.A.1) and applying geometric methods to solve design problems of aiming the basketball to bounce off the backboard (HSG.MG.A.3). The strategies all involved making formal geometric constructions with the dynamic geometry software (HSG.CO.D.12). Strategy 1 involves representing constraints by equations... and interpret[ing] solutions as viable or nonviable options in a modeling context (HAS.CED.A.3) since the trajectory of the basketball is being modeled by slopes of lines. Strategy 3 involves representing transformations in the plane, particularly reflections (HSG.CO.A.2). Strategy 4 involves using congruence and similarity criteria for triangles to solve problems and prove relationships in similar triangles (HSG.SRT.B.4), and establishing the angle-angle criterion for two triangles to be similar (HSG.SRT.A.3). Integrating this basketball modeling problem as a science activity is consistent with recommendations by the Next Generation Science Standards (NGSS Lead States, 2013). Descartes-Snell s Law could be discussed as a follow-up to this activity, focusing on the core scientific idea that when light shines on an object, it is reflected, absorbed, or transmitted through the object, depending on the object s material (MS-PS4-2) and the related scientific and engineering practice of using math- Illinois Mathematics Teacher 5

6 Diana Cheng, Tetyana Berezovski, Asli Sezen-Barrie ematical representations to describe and/or support scientific conclusions and design solutions (MS-PS4-1). Understanding the basketball model as an instance of Descartes-Snell s Law involves students using analogic reasoning, treating the path of light against a surface similarly to the path of the basketball as it bounces off the backboard (English, 2013). Once students have successfully made the basketball shot, we can point out that reflecting the hoop across the backboard transforms the shots into a straight line. Multiple additional problems can be posed concerning this two-dimensional model. Two extension problems are proposed here. They both involve finding a locus of points within the hoop under different constraints. Your coach does not need you to aim for point B. Ignoring the backboard IL, shade in a locus of points in the hoop at which the ball may be centered while still remaining entirely in the hoop. Find the locus of points on the backboard IL for which the ball would rebound into the hoop (taking into consideration the requirement that the ball must end entirely within the hoop). NGSS Lead States (2013). Next Generation Science Standards: For States, By States. Washington, DC: National Academies Press. Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33, The Physics Classroom (2014). Snell s law. URL: Lesson-2/Snell-s-Law. Diana Cheng MATHEMATICS DEPARTMENT TOWSON UNIVERSITY TOWSON, MD dcheng@towson.edu Tetyana Berezovski MATHEMATICS DEPARTMENT ST. JOSEPH S UNIVERSITY PHILADELPHIA, PA tberezov@sju.edu Asli Sezen-Barrie PHYSICS, ASTRONOMY & GEOSCIENCES DEPARTMENT TOWSON UNIVERSITY TOWSON, MD asezen@towson.edu Readers who want to further extend this model can consider a three-dimensional representation of the basketball hoop and/or take into consideration the amount of spin that the basketball may have. References Common Core State Standards Initiative and others (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. English, L. D. (Ed.) (2013). Mathematical reasoning: Analogies, metaphors, and images. Routledge. Jackiw, N. (1991). The geometers sketchpad. computer software. National Basketball Association (2006). Rule no. 1 court dimensions equipment. URL: analysis/rules_1.html. 6 Illinois Mathematics Teacher