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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, 2016.
“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
∗
34
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
Aiming a Basketball for a Rebound
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 2014. 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,
Illinois Mathematics Teacher
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).
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Diana Cheng, Tetyana Berezovski, Asli Sezen-Barrie
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.
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
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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).
Figure 5: Productive strategy 4: Similar triangles
Illinois Mathematics Teacher
Aiming a Basketball for a Rebound
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.
Illinois Mathematics Teacher
Figure 6: Two examples of unproductive strategy 5: Fixed
angle
Figure 7: Unproductive strategy 6: Straight line
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Diana Cheng, Tetyana Berezovski, Asli Sezen-Barrie
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.
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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 “mathIllinois Mathematics Teacher
Aiming a Basketball for a Rebound
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.
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 (2), 455–488.
The Physics Classroom (2014). Snell’s law. URL:
http://www.physicsclassroom.com/class/refrn/
Lesson-2/Snell-s-Law.
Diana Cheng
MATHEMATICS DEPARTMENT
TOWSON UNIVERSITY
TOWSON, MD
E-mail : [email protected]
Tetyana Berezovski
• 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.
MATHEMATICS DEPARTMENT
ST. JOSEPH’S UNIVERSITY
PHILADELPHIA, PA
• 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).
PHYSICS, ASTRONOMY & GEOSCIENCES DEPARTMENT
E-mail : [email protected]
Asli Sezen-Barrie
TOWSON UNIVERSITY
TOWSON, MD
E-mail : [email protected]
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: http://www.nba.com/
analysis/rules_1.html.
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