TCCRI College Readiness Assignments
Instructor Task Information
Bull’s Eye Math
Overview
Description
Students are asked to use their knowledge of circles, area of circles, and sectors to
calculate probabilities found in games that involve circular targets and to design a
target.
Final Product: Students will first solve problems and answer questions relating
probability and geometry. Students will then design a target whose part and/or whole
areas will fit given ratios. Students will write a report describing their thought process
and how their final product fits the given criteria.
Subject
Geometry or Mathematical Models with Applications
Task Level
Grades 9-11
Objectives
Students will:
•
Use the ratios of the areas of parts of a figure and the whole figure to determine
geometric probability.
•
Create a target whose part and/or whole areas fit given ratios.
Preparation
•
Read the Instructor Task Information and Student Notes.
•
For each student, make a copy of the Student Notes and handouts.
•
Collect a checkerboard and pennies to demonstrate Target Game. Optionally,
gather other game boards with colored areas of different shapes.
•
Students will need access to calculators.
Prior Knowledge
Students should understand the concept of geometric probability. Students should be
able to find the area of a circle and parts of a circle such as regions created by
concentric circles and sectors.
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Bull’s Eye Math
Instructor Task Information
Key Concepts and Terms
•
Area
•
Concentric circles
•
Diameter
•
Geometric probability
•
Radius
•
Sector
Time Frame
This assignment will take from three to five class periods, depending on whether
students work in class or outside of class on the design of targets.
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Bull’s Eye Math
Instructor Task Information
Instructional Plan
Getting Started
Learning Objectives
Students will:
•
Use the ratios of the areas of parts of a figure and the whole figure to
determine geometric probability.
Procedure
1. Distribute the Student Notes and introduce the task.
2. Provide students with the Square Target handout. Have students work in
pairs to answer questions 1–4. This activity is intended to refresh students’
prior knowledge of geometric probability. Facilitate a class discussion of the
checkerboard coin toss game and the geometric probabilities involved.
3. Optionally, display and discuss other game boards you have collected.
4. Provide students with the Circular Target handout. Have students work in
pairs on Parts A and B. Monitor students’ work and facilitate a class
discussion, if needed, before students proceed to design their own targets.
Investigating
Learning Objectives
Students will:
•
Use the ratios of the areas of parts of a figure and the whole figure to
determine geometric probability.
•
Apply understanding of geometric probability to design a target whose part
and/or whole areas fit given ratios.
Procedure
1. Students now begin work individually to design a target. You may want to set
a deadline for students to present their preliminary ideas for a target shape
before they finalize their designs. Students should be able to explain how
their design meets the given criteria.
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Bull’s Eye Math
Instructor Task Information
Drawing Conclusions
Learning Objectives
Students will:
•
Solve a design problem by creating a game target that adheres to specified
geometric probabilities.
•
Verify mathematically that the design meets criteria by calculating areas and
probabilities.
•
Evaluate the process used to solve the design problem.
•
Communicate the design process and product in a written report that
incorporates a diagram, calculations, and mathematical language.
Procedure
1. Students finalize their designs and create a sketch with specifications
indicated.
2. Students then write brief reports on designing their targets.
3. You may wish to have students share their final sketched designs without
specifications indicated, and then have other students check that the designs
meet the criteria.
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Bull’s Eye Math
Instructor Task Information
Scaffolding/Instructional Support
The goal of scaffolding is to provide support to encourage student success,
independence, and self-management. Instructors can use these suggestions, in part or
all together, to meet diverse student needs. The more skilled the student, however, the
less scaffolding that he or she will need. Some examples of scaffolding that could apply
to this assignment include:
•
Work through the Getting Started handouts with students.
•
To help with the archery target, show students how to do multiple calculations
nd
that are similar on a graphing calculator by using 2 enter to repeat the previous
entry.
•
Pair students with more advanced understanding with students whose
understanding is less advanced.
•
Check in with students at regular intervals during their work designing a target to
ensure they are progressing in a productive manner. Redirect if necessary.
•
Have students who struggle with the written portion of the assignment describe
their process verbally, and assist them in making the translation from verbal to
written expression.
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Bull’s Eye Math
Instructor Task Information
Solutions
The solution provided in this section is intended to clarify the problem for instructors. This
solution may not represent all possible strategies for approaching the problem or all
possible solutions. Solutions are provided for reference only.
Square Target
1.
5
1
= or 20%
25 5
2.
8
or 32%
25
3. Make 14 of the 25 squares white.
4. You would have to calculate the areas and use the ratios of areas to find
probabilities.
Circular Target
A. Archery Target
Ring #
#1 ring
#2 ring
#3 ring
#4 ring
#5 ring
#6 ring
#7 ring
#8 ring
#9 ring
#10 ring
1.
Color
white
white
black
black
blue
blue
red
red
yellow
yellow
Point value
1 point
2 points
3 points
4 points
5 points
6 points
7 points
8 points
9 points
10 points
2
Area (cm )
1491.5
1334.5
1177.5
1020.5
863.5
706.5
549.5
392.5
235.5
78.5
Probability
19%
17%
15%
13%
11%
9%
7%
5%
3%
1%
π (502 ) ≈ 7850 cm2
2. See chart.
3. See chart and answers below.
a.
235.5
= 3%
7850
b. From the chart 11% + 9% = 20%
c.
100 – (3% + 1%) = 96%
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Bull’s Eye Math
Instructor Task Information
B. Dartboard
2
1.
⎛ 17.7 ⎞
2
π ⎜
⎟ ≈ 245.93 in
2
⎝
⎠
2.
1
(245.93 ) ≈ 12.30 in2
20
3. Area of one double ring section = π 6.65 2 − 6.35 2 ≈ 0.61 in 2
20
(
P(double section) =
)
0.61
≈ 0.002 ≈ 0.2%
245.93
4. Area of one triple ring section = π 4.15 2 − 3.85 2 ≈ 0.38 in 2
20
(
P(triple section) =
)
0.38
≈ 0.002 ≈ 0.2%
245.93
Target Design
Designs will vary. A simple one is given here.
A circle is divided into 10 equal sectors.
The diameter is 16 in.
( )
Area of target = π 82 ≈ 201 in2
Area of each sector = 201 = 20.1 in 2
10
P(white) = 5 × 20.1 = 100.5 = 0.5 = 50%
201
201
P(gray) = 3 × 20.1 = 60.3 = 0.3 = 30%
201
201
P(black) = 2 × 20.1 = 40.2 = 0.2 = 20%
201
201
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Bull’s Eye Math
Instructor Task Information
TCCRS Cross-Disciplinary Standards Addressed
Performance Expectation
Getting
Started
Investigating
Drawing
Conclusions
P
P
P
I. Key Cognitive Skills
B.1. Consider arguments and conclusions of self and
others
B.2. Construct well-reasoned arguments to explain
phenomena, validate conjectures, or support
positions.
P
C.1. Analyze a situation to identify a problem to be
solved.
P
P
C.2. Develop and apply multiple strategies to solve
problems.
P
P
P
D.1. Self-monitor learning needs and seek assistance
when needed.
P
P
P
D.2. Use study habits necessary to manage academic
pursuits and requirements.
P
P
P
P
P
P
D.3. Strive for accuracy and precision.
P
D.4. Persevere to complete and master tasks.
E.1. Work independently.
P
P
E.2. Work collaboratively.
P
P
P
II. Foundational Skills
B.1. Write clearly and coherently using standard
writing conventions.
P
B.3. Compose and revise drafts.
P
C.6. Design and present an effective product.
P
E.1. Use technology to gather information.
P
TCCRS Mathematics Standards Addressed
Performance Expectation
Getting
Started
Investigating
Drawing
Conclusions
P
P
P
III. Geometric Reasoning
C.2. Make connections between geometry, statistics,
and probability.
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Bull’s Eye Math
Instructor Task Information
C.3. Make connections between geometry and
measurement.
P
P
P
P
P
P
A.3. Determine a solution.
P
P
P
A.4. Justify the solution.
P
V. Probabilistic Reasoning
B.1. Computation and interpretation of probabilities.
VIII. Problem Solving and Reasoning
P
A.5. Evaluate the problem-solving process.
P
B.1. Develop and evaluate convincing arguments.
P
IX. Communication and Representation
B.2. Summarize and interpret mathematical
information provided orally, visually, or in written
form within the given context.
C.3. Explain, display, or justify mathematical ideas and
arguments using precise mathematical language
in written or oral communications.
P
P
P
P
TEKS Standards Addressed
Bull’s Eye Math - Texas Essential Knowledge and Skills (TEKS): Math
111.34.b.8. Congruence and the geometry of size. The student uses tools to determine measurements
of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem
situations. The student is expected to:
111.34.b.8.E. use area models to connect geometry to probability and statistics.
111.36.c.1. The student uses a variety of strategies and approaches to solve both routine and nonroutine problems. The student is expected to:
111.36.c.1.A. compare and analyze various methods for solving a real-life problem.
111.36.c.1.B. use multiple approaches (algebraic, graphical, and geometric methods) to solve
problems from a variety of disciplines. 111.36.c.1.C. select a method to solve a problem, defend
the method, and justify the reasonableness of the results.
111.36.c.4. The student uses probability models to describe everyday situations involving chance. The
student is expected to:
111.36.c.4.A. compare theoretical and empirical probability.
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TCCRI College Readiness Assignments
Student Notes
Bull’s Eye Math
Introduction
Many games involve geometric boards that can teach us about more than just
“playing fair.” In this activity, you will explore geometric probability by looking at
two common games. You will use your knowledge of geometry and probability to
predict the likelihood of certain outcomes. Then you will design a game target
that meets given criteria and report on your design.
Directions
Getting Started
1. With a partner, work through the Square Target handout. This will provide an
opportunity to refresh your probability skills.
2. With your partner, work through parts A and B of the Circular Target handout.
You will need a calculator.
3. Ask questions, if you need to, to clarify your understanding of finding the
geometric probabilities before you begin to make a target of your own design.
Investigating
1. Now you will design a game target. It can be any shape, but it must meet the
following criteria:
•
Have a least two sections each of white, gray and black
•
The sections that match in color should be congruent shapes
•
The probabilities of randomly hitting the colors are:
Color
Probability
White
50%
Gray
30%
Black
20%
2. Experiment with designs for your game target. Keep a record of your work.
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Bull’s Eye Math
Student Notes
Drawing Conclusions
1. Decide on a final design. Verify that it meets the criteria.
2. Make a final sketch of your design to submit with your report. Provide the
specifications so a reviewer can verify that the design meets the criteria:
indicate the area of each section and the probability for each color.
3. Write a report describing the process you used to create your target.
•
State the problem to be solved. Include the constraints.
•
Tell about the process of designing the target.
•
Explain how you know your target meets the criteria.
•
What was most interesting? What were the biggest challenges? What
would you do differently if you were to design a target again?
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TCCRI College Readiness Assignments
Student Handouts
Square Target: Reviewing Geometric Probability
A game played at carnivals gives a player points as they toss a penny onto a square
target. Points are awarded based on the color of the square in which the greater part of
the penny lands.
A sample scoring scenario is shown:
Black square = 10 points
Gray square = 5 points
White square = 2 points
The player who tossed the three pennies
shown on the target was awarded 17 points.
The probability that a coin will land on a particular color is determined by the part of the
target board that is that particular color. For example, this board is made up of 25
squares, and 12 of the squares are white. Therefore, the probability of landing on a white
square is 12 or 48%.
25
Another way to determine the probability is to compare the total areas of the white
squares to the area of the entire board. Each of the squares on this target is 1.6 inches
by 1.6 inches, so the total area of all white squares is (1.6 × 1.6) × 12 = 30.72 square
inches. The entire board measures 8 inches by 8 inches, so the area is 64 square
inches. Using the ratio 30.72 to 64 the probability is 0.48 or 48%.
1. What is the probability that a penny will land on a black square?
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Bull’s Eye Math
Student Handouts
2. What is the probability that a penny will land on a gray square?
3. How could the board be changed so the probability of landing on a white
square is 56%?
4. What if the colored squares were not all the same size and/or shape? How would
different sizes and shapes effect how you would determine the scoring probabilities?
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Bull’s Eye Math
Student Handouts
Circular Target
A. Archery Target
Now consider a target that is not square. The outdoor archery target shown here
measures 100 centimeters in diameter and has concentric circles that form rings of equal
width. The radius of the center circle (10) is the same as the width of each ring. Each
ring is assigned a point value as indicated in the figure and chart here.
10
9
87
65
43
21
5 cm
http://en.wikipedia.org/wiki/Target_archery
Ring #
#1 ring
#2 ring
#3 ring
#4 ring
#5 ring
#6 ring
#7 ring
#8 ring
#9 ring
#10 ring
Color
white
white
black
black
blue
blue
red
red
yellow
yellow
Point value
1 point
2 points
3 points
4 points
5 points
6 points
7 points
8 points
9 points
10 points
Area (cm2)
Probability
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Student Handouts
1. What is the area of the entire target? Use π ≈ 3.14 .
2. Calculate the area of each ring to the nearest tenth of a square centimeter and
complete the chart.
3. Assume a beginning archer, and not a professional archer, shoots the arrow and that
the arrow hits the target. This implies that each point on the target is equally likely to
be hit. Calculate the geometric probability of an arrow hitting each of the rings
indicated and add the probabilities to the chart. Then, determine the following
probabilities.
a. The probability of hitting yellow ring #9:
b. The probability of hitting the blue region:
c. The probability of hitting a part of the target that is not yellow:
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Bull’s Eye Math
Student Handouts
B. Dartboard
Like the archery target, this target is also round and contains concentric circles.
Dartboards come in a variety of measurements. This one is 17.7 inches in diameter.
Each of the 20 sectors is intersected by rings that form double and triple point sections.
There is also a center ring called the bull, and within that, the bull’s eye.
http://en.wikipedia.org/wiki/Darts
The diagram below shows the measurements of one sector.
0.3"
0.3"
3.25"
2.2"
0.3"
2.2"
0.3"
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Bull’s Eye Math
Student Handouts
1. What is the area of the entire dartboard? Use π ≈ 3.14 and round your answer to the
nearest hundredth of an inch.
2. What is the area of one entire sector? Hint: Each sector is 1 of the circle.
20
3. What is the area of the double ring section within one sector? If you throw a dart at
the dartboard, what is the probability the dart will land within the double ring section
of the 13-point sector?
4. What is the area of the triple ring section within one sector? If you throw a dart at the
dartboard, what is the probability the dart will land within the triple ring section of the
13-point sector?
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SH-6