Mathematics Senior Level Capstone Course Unit Overview
Title of Unit: PEEPS! Unit Designers:
Mikhail Balachov (Arlington) Su Chuang (Loudoun) Mirela Geagla (Arlington) Hunter Hagerty (Loudoun) Kiera Poplawski (Loudoun) Debora Strickler (Loudoun)
Edited by Diane Leighty, UVA-SCPS Office of Mathematics Outreach Context:
Summary of the issue, challenge, investigation, or problem.
Design one level of the video game, Angry Birds, and create mathematical models to simulate the most effective and efficient method of completing this level of the video game and hitting all your targets.
Number of Class Hours:
Estimated 10 hours Unit Design:
___Task Based _X_Project Based Other Subject
Areas/Disciplines Addressed:
Physics, Career and Technical Education, Writing
Driving Question: How can you design a mathematical model to maximize the chances of hitting a target? Mathematics
Content Addressed:
Quadratic Functions, Distance, Projectile Motion, Pythagorean Theorem, Right Triangle Trigonometry
MPE Addressed:
Problem Solving, Decision Making, and Integration; Understanding and Applying Functions Assumption of Prior
Knowledge:
Understanding Quadratic Functions, basic right triangle trigonometry,
College and Career Readiness/21st
Century Skills to be taught (T) during this unit or
expectation (E) for student use during this unit and assessed (A): BIE Page 35-37
Communication (Oral and/or Written) Technology: Students will use Texas Instruments’ CBR: Calculator Based Rangers to model quadratics.
T
Critical Thinking/Decision Making Other: (Describe)
Major Products and/or
Performances:
Group – Design of a level in Angry Birds, Efficient Mathematical Model simulating the completion of one level of Angry Birds, Presentation of a real world application of quadratics.
Presentation Audience: Class School Individual – Mathematician’s Journals – prompts about
quadratics, applications, misconceptions of parabolas representing path of objects.
Expert Community Other:
Launch: Event or experience used to engage the students interest and inquiry:
Students will play several levels of the Angry Birds game (if available) or a free version of a similar game, Angry Animals. http://hoodamath.com/games/angryanimals.php
The objective of the game is to hit various “targets” using a slingshot and an animal as a projectile. Students can identify the “targets” on each level before releasing the projectiles in order to complete each level.
Evaluation: Formative Assessments (During the Unit)
Interview X Practice Presentations
Mathematicians Journal X Notes
Preliminary
Plans/Outlines/Prototypes
Checklists
Rough Drafts X Concept maps
Field Tests X Other:
Summative Assessment (End of Project)
Written Products, with a rubric X Peer Evaluation, with a rubric
X Oral Presentation with a rubric X Self Evaluation, with a
rubric
X Other Product(s) or Performance(s),
with a rubric
Resources Needed: On-site people, facilities: Facilitator/Teacher
Equipment/Technology: Computers with Internet access, Calculator Based Rangers (CBRs), graphing calculators, Logger pro(See note below), which may be available through the science department. If CBRs are not available, a loaner set can be requested for free from Texas Instruments if requested one month ahead of time. For more information, visit the following website: http://education.ti.com/educationportal/sites/US/nonProductSingle/global_forms_loan.html
Logger pro is a data collection and analysis software. For this unit, Logger Pro will be used to capture video of a projectile in motion and the data collected for analysis. If the software is not available, a free 30-day trial is available for download at:
http://www.vernier.com/products/software/lp/
Materials: Grid Chart Paper, graph paper, Ball (for CBR Activity) Community Resources: None
Reflection Methods: Individual, Group, and/or Whole Class
Mathematicians Journal X Small/Focus Groups
Whole Class Discussions X Fishbowl Discussions
Survey Other:
Material Adapted From:
NASA: http://search.nasa.gov/search/edFilterSearch.jsp?empty=true
Texas Instruments: http://education.ti.com/calculators/downloads/US/Activities/ Hooda Math: http://hoodamath.com/games/angryanimals.php
Virginia’s Senior Level Capstone Course Instructional Plan
Unit Title: PEEPS!
Driving Question: How can you design a mathematical model to maximize the chances of hitting a target?
Project: Design one level of the video game, Angry Birds, and create mathematical models to simulate the most effective and efficient method of completing this level of the video game and hitting all of the targets.
ENGAGE How will student’s interested be peaked so they want to engage in the inquiry in this unit? Number of hours _1___
Begin this project by familiarizing students with the Angry Birds game. This game can be downloaded onto iPods or iPad devices. If students do not have access to these devices, a free version of a similar game, Angry Animals, can be played online at
HoodaMath.
http://hoodamath.com/games/angryanimals.php
Have students play several levels of the Angry Birds game or the Angry Animals game.
Teacher Note: Teachers who are not familiar with the Angry Birds or Angry Animals games may want to play the games to prepare for this unit.
If game play is not available, students can watch a video of Real World Angry Birds -
http://www.youtube.com/watch?v=s9TxM3Jpo8o
This video clip is of a real life simulation of the Angry Birds game. The player uses a red ball to simulate the bird/animal as the projectile and builds his own “targets”.
Discussion Points:
What kind of function does the path of the projectile follow? Predict the path of the projectile given a starting point and the
location of the target.
Teacher Note: At this point, all classroom discussions about parabolas can be general, without getting into the quadratic equations or transformational graphing of quadratics.
Mathematician Journal Prompts EXPLORE Teacher provides
Title: Rolling the Ball Activity – (Introduction to the CBR) Goal of the activity:
The Rolling the Ball Activity is an optional activity introducing
the
explorations to prepare students with the knowledge and skills to engage in the task. Students will self-assess on the prior knowledge and skills assumed for the unit
Number of hours: 3 - 4
reviewing quadratics. Description of the Activity: HO #1: Rolling the Ball Activity
In this activity, students create a ramp at different angles and roll a ball down the ramp. The CBR device collects data on the distance of the ball to the CBR over time.
Materials Needed:
Plank of wood to serve as a ramp Large ball (basketball or dodge ball) Calculator Based Ranger (CBR) TI-83/84 graphing calculator Protractor
Directions for Instructors:
If CBR devices or TI graphing calculators are not available, Texas Instruments has a product loaner program. The devices can be requested for free on loan if requested one month ahead of time. For more information, visit the following website:
http://education.ti.com/educationportal/sites/US/nonProductSingle /global_forms_loan.html
HO #1a: Rolling the Ball: Teacher Notes offers step by step directions to set up the CBR activity. Screen shots of the TI graphing calculator screen are available to guide this activity. HO#1b: Rolling the Ball: Student Notes is the student
recording sheet for the activity. Teachers may want to prompt students to justify their responses and predictions in their mathematician’s journals.
Anticipated Reactions:
Depending on the results collected during the activity, the data may appear to be linear. Teachers may want to use a longer ramp to collect additional data and discuss why the previous trials resulted in graphs that appeared to be linear.
Title: Bouncing Ball Activity Goal of the Activity:
It is a common misconception that parabolic graphs always represent the path of a trajectory. This CBR (Calculator Based Ranger) activity can be used to address this common
misconception. In this activity, students are examining the graph of the distance of a bouncing ball from the CBR over time.
What are your predictions if the incline of the ramp is increased? What would your graph look like and why?
What are your predictions if the CBR device is placed at the bottom of the ramp? What would your graph look like and why?
Before
Description of the Activity: HO #2: Bouncing Balls Activity
Students explore the rate of change at various points on the graph and describe these points in the context of the bouncing ball. Students also find the curve of best fit for their data.
Materials:
Large ball (basketball or dodge ball) Calculator Based Ranger (CBR) TI-83/84 graphing calculator Directions for Instructors:
HO #2a: Bouncing Balls Instructions provides step by step instructions for setting up this activity.
HO#2b: Bouncing Balls Data Collection is the student recording sheet for the activity. Teachers may want to prompt students to justify their responses and predictions in their mathematician’s journals.
Mathematicians Journal Prompt: Teachers may want to offer students a different type of ball and have them predict what the graph might look like. Have students bounce two types of balls and visually observe the height and frequency of
bounces. From those observations, have students predict and justify their graph and the curve of best fit.
Anticipated Reactions:
Students may need to conduct this experiment several times to obtain the best data. Teachers may want to conduct a whole class demonstration of the bouncing ball activity prior to student groups conducting the activity.
Exploring Transformational Graphing
The goal of this activity is to explore quadratic equations for various “Angry Animals/Birds” from the Angry Birds/Angry Animals games. Using transformation graphing, students explore how variations to the path of trajectory may affect the quadratic equation. (See attached HO #3, Angry Animals Worksheet) What is the equation of a parabola that models the path of the
projectile? How accurate is your equation to the actual path of the projectile needed to hit the target? (This may be done on a white board, drawing the predicted path on the white board and projecting the Angry Birds game over the predicted path.) Teachers may choose to “zoom out” so that the entire Angry Birds level is viewable without scrolling left and right. This best allows for students to see the entire path of the projectile and match that to their predicted path.
Teacher Notes:
Potential Issue: Website may run slow, which could be an issue with having a whole class of students on it at one time. Also, on Level 4, it’s hard to see the change that’s happen at first. Let the students replay the level a few times in order to figure it out. Optional additional preliminary task: Refer to HO #4
It is suggested that you and your students play the game “Angry Birds” before attempting the main project in order to understand how it works. Or, teachers may implement some other real world activities involving quadratics in various ways. These are some ideas:
Activity involving shooting a basketball or paper into trash can and using technology to help write equations of trajectory Activity involving shuttle launch. (Check the NASA site).
EXPLAIN Teacher introduces the main task of the unit and prepares students to in small group independent work...
Number of Hours: 1
Introduce the project: Create your own level of angry birds using three birds to potentially “destroy” the pigs.
Description of final product and/or presentation: Final product is a blueprint of the level of play, one blueprint just of the level, and a second blueprint that includes the quadratic equations that model successfully “destroying” the pigs.
Prepare students for working independently in their groups. Describe the expectations for how students will work in their group, including a discussion of the peer/self-evaluation form, HO #8.
Project: Student HO #5
Skills or knowledge needed: Ability to determine “best fit” for data; how to create a blueprint
Materials/Equipment/Resources Needed: Computers, smartphones or iPhone Directions for Instructor:
For struggling students, you may want to provide them with the three heights they are to use for the target pigs for their “level”. Allow students to research the requirements for a blueprint prior to creating one of their own.
ELABORATE The student groups are working independently with teacher consultations. Number of Hours: 2 - 3
Design and construct the Angry Birds level.
The teacher interviews students to make sure they understand the task and are on the right path toward completing the project. Students are to write a paper explaining their process in creating their blueprint, including any problems they had along the way. They should discuss what worked, what didn’t work, and any improvements they are able to make to complete the project. Student handout #5
Mathematician Journal Prompts
EVALUATE Working groups submit products or make
presentations Number of Hours: 1
Students will submit their blueprints along with their paper describing the process they went through to find the correct equations. Rubric attaches as HO #6
Students will “solve” the level created by another group to the best of their ability. Rubric attached as HO #7
Peer/Self-Evaluation Form – HO #8
Map the Unit
What do students need to know and be able to do to complete the task/project/problem
successfully? How and when will they assess their own necessary knowledge and skills? How will they remediate their own gaps or weaknesses in knowledge and skills? Look at each major task for the unit and analyze the tasks necessary to produce a high-quality product.
Task: Design one level of the video game, Angry Birds, and create mathematical models to simulate the most effective and efficient method of completing this level of the video game and hitting all your targets
KNOWLEDGE AND SKILLS NEEDED Assumed already learned
Students will self-assess
Will be taught during the unit
1. Quadratic Functions X X
2. Distance Formula X X
3. Projectile Motion X X
4. Pythagorean Theorem X X
5. Right Triangle Trigonometry X X
6. Create a Blueprint X
7. Using a CBR properly X
8. 9.
10. 11.
What project tools will student’s use? Know/need to know lists Daily goal sheet
X Mathematician’s Journals Briefs/Memos
Task lists
Planning Calendar
□ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________
HO #1a
Rolling the Ball: Teacher Notes
I. Data Collection
1. Answer question 1 on the Activity sheet. Use the protractor to set the ramp at a 15° incline. Lay the CBR2 motion detector on the ramp and flip the sensor head so it is perpendicular to the ramp.
Mark a spot on the ramp 15 cm from the CBR2 motion detector. Hold the ball at this mark, while your partner holds calculator and CBR2 motion detector.
HINT: Aim the sensor directly at the ball and make sure that there is nothing in its path. 2. Run the EasyData App.
3. To set up the calculator for data collection:
a. Select Setup (press WINDOW) to open Setup menu.
b. Press 2 to select 2: Time Graph to open the Time Graph Settings screen.
d. Enter 0.1 to set the time between samples in seconds.
e. Select NEXT (press ZOOM) to advance to the Number of Samples dialog window. f. Enter 30 to set the number of samples. Data collection will last for 3 seconds.
g. Select Next (press ZOOM) to display a summary of the new settings.
h. Select OK (press GRAPH) to return to the main screen.
4. When the settings are correct, choose Start (press ZOOM) to begin sampling. 5. When the clicking begin, release the ball (don’t push) and step back.
6. When the clicking stops, the collected data is transferred to the calculator and a plot of distance vs. Time is displayed.
Answer questions 2, 3, 4, and 5.
II. Explorations
1. Predict what will happen if the incline increases. Answer question 6.
2. Adjust the incline to 30°. Repeat steps 2 through 6. Add this plot to the drawing in question 6, labeled 30°.
HO: #1b
Rolling The Ball: Student Handout
Data collection
1. Which of these plots do you think best matches the Distance-Time plot of a ball rolling down a ramp?
2. What physical property is represented along the x-axis? ___________________
What are the units? ________________________________________________
What physical property is represented along the y-axis? ___________________
3. Sketch what the plot really looks like. Label the axis. Label the plot at the points when the ball was released and when it reached the end of the ramp.
4. What type of function or functions does this plot, between the two points you identified, represent?
Explorations
6. Sketch what you think the plot will look like with a greater incline. (Label it prediction, and be sure to label your axes appropriately.)
Degree of incline: ___________
HO #2a
Name __________________________ Bouncing Balls Instructions
1. Run the RANGER program on your calculator. It can be accessed using the APPS menu, and selecting CBL/CBR.
2. From the MAIN MENU of the RANGER program, select 3:APPLICATIONS.
3. Select 1:METERS, then select 3:BALL BOUNCE.
4. Follow the directions on the screen of your calculator. Release the ball. Press the TRIGGER key on the CBR as the ball strikes the ground.
5. Your graph should have at least two bounces. If you are not satisfied with the results of your experiment, press ENTER, select 5:REPEAT SAMPLE, and try again.
6. When you are satisfied with your data, sketch a Distance-Time plot. On the grid below. Label the axes.
HO #2b
Name __________________________ Bouncing Ball Data Collection
1. The goal here is to “capture” one complete curve. Choose the best curve that your bouncing ball created. Press ENTER and go to 4: PLOT TOOLS. Choose 1: SELECT DOMAIN. Use the right arrow to trace to a point near the lower left side of the parabola that you chose and press ENTER. Continue tracing until you reach a point near the lower right side of this parabola and press ENTER.
2. Now you are going to clean-up and perfect your graph. Trace on the graph to find the maximum and use it and a few other points that you get from tracing to draw an accurate graph below.
4. In what interval(s) is the ball traveling the slowest? Explain.
5. Why is the graph curved? What is the ball doing that makes the graph curve?
6. Write a paragraph describing what the graph tells us about the motion of the ball. Be sure to interpret all the important features of this type of graph.
7. Using mathematical language, describe the type of graph and function that seems to fit this motion.
Function type: _____________________Graph type: _________________________
9. Graph your equation along with the data from the ball drop. How well do they match?
HO #3
Angry Animals (adapted from Hooda Math)
Level 1:
Using the base of the slingshot as the origin, think about the Cartesian coordinate system. In
which quadrant, are you pulling the sheep back? In which quadrant, does the sheep fly out of the
sling shot?
Level 2:
Launch the first pig and click in mid-air to divide it into three pigs. Use the arrows
at the bottom of the game to scroll back to where the green pig magically turns into
three pigs. Assuming the parabolic formula for the original pig is y = -0.1x
2+ x + 2, and
the parabolic formula for the highest pig is y = -0.1x
2+ x + 3. What is the parabolic
formula for the lowest pig?
Level 3:
Launch a bull and click in mid-air to drop a milk bottle. Using the base of the slingshot as the
origin, estimate the equation for the line created by the dropping bottle to destroy the first alien
and the second alien assuming that each small bull shown in the large bulls’ path is one unit?
Level 4:
After you launch the chicken it follows along a parabola, but when you click in
mid-air the path of the chicken changes into a different parabola. Looking at the
parabolic equation,
y
ax
2
bx c
, what is the meaning behind each of the constants a, b, and
HO #4
Writing Quadratic Equations From Angry Birds Possible Tasks:
Students can use smartphones to take a screen shot of a particular Angry Bird level. They will want to launch a bird and take a screen shot of the path the bird made. Save this image as a JPEG. They can then import the image into a Smartboard screen or project the image with a LCD projector. If they are using the Smartboard, they can place a grid over the picture which will allow them to plot actual points that they can use to write the equation of the path. If they are projecting the image using a LCD projector, they can project onto a graph and again plot points. This could be done as a whole group or in smaller groups depending on size of class.
I. Students are to plot at least 3 different paths and write the corresponding quadratic for each. They will need to draw a simulation of the graph and list the points they are using for the quadratic. II. Students will design a level of Angry Birds on paper and show the optimum point of impact for
the most damage. They will write the equation of the quadratic necessary to hit this point using an appropriate scale.
III. Extension after this lesson: Students will research possible real-world applications for quadratic equations. Projectile motion is used in many sports for success, hunting has a form of projectile motion that incorporates parts of quadratic understanding-there are scopes and range finders that calculate this information now for hunters and golfers. How are they designed? What things are needed for them to work?
http://www.real-world-physics-problems.com/physics-of-sports.html
There are also projectile motion simulators where students can explore quadratics. They allow students to take out the physics part and have only gravity as a force affecting the projectile. They allow students to change angle and velocity to show how they affect the projectile. They can get points by tracking time and height as they shot the simulator. Some possible simulators:
http://www.squadron13.com/games/projectile/projectile.htm
http://jersey.uoregon.edu/vlab/Cannon/
http://www.livephysics.com/simulations/mechanics/projectile-motion.html
I. Students will draw a simulation of what their projectile did-listing angle and velocity with included points for use in regression models. They will need to do several of these simulations to get a complete idea of what is going on with the quadratics.
Real world simulation: If your school has a pitching machine, you can control velocity and angle so students can see how these affect the projectile. The shots can be recorded and then used in programs such as Logger Pro (http://www.vernier.com/products/software/lp/) or Tracker Video Analysis (http://www.cabrillo.edu/~dbrown/tracker/) so that students can then write the equations for the quadratics. One feature of using the pitching machine is that you can drop 4 or 5 balls in at the same time and they will fly together so students can see the path better.
I. Students can place targets at specified distances and calculate the needed angle and velocity to hit the target. Then run the launch and write about what happened and what they need to change to hit target.
HO #5
“Angry Birds” Project Directions
Your task is to create your own level of angry birds. You will have three pigs to “destroy” that
must be at three different heights. You will be given three red birds to “destroy” the pigs. For
the purposes of this activity, you must hit the pigs with a bird to “destroy” it.
Part 1: The Blueprint
Create a blueprint of your level. Then create a second blueprint which includes the trajectories
of the birds and the quadratic equations that go with them. Your blueprint without the
trajectories and equations will later be given to another group to “complete.”
Part 2: The Paper
Write a paper explaining how you chose your trajectories. Use the available technology
(graphing software) to simulate your proposed trajectories to “complete” the level of Angry
Birds. Did your proposed equations “destroy” each target? How would you change your
trajectories to hit all three targets? What are your new quadratic equations?
Part 3: Completing Another Level of Angry Birds
HO #6
Rubric for Blueprint and Paper
CATEGORY 4 3 2 1
Mathematical Concepts Explanation shows complete understanding of the mathematical concepts used to determine most economical use of solar energy.
Explanation shows substantial
understanding of the mathematical concepts used to determine most economical use of solar energy
Explanation shows some
understanding of the mathematical concepts used to determine most economical use of solar energy
Explanation shows very limited understanding of the mathematical concepts used to determine most economical use of solar energy Mathematical
Reasoning
Uses complex and refined
mathematical reasoning to choose the best energy source.
Uses effective mathematical reasoning to choose the best energy source.
Some evidence of mathematical reasoning to choose the best energy source.
Little evidence of mathematical reasoning to choose the best energy source.
Explanation Explanation is detailed and clear to justify their conclusion mathematically.
Explanation is clear to justify their conclusion mathematically.
Explanation is a little difficult to understand, but includes critical components to justify conclusion. Explanation is difficult to understand and is missing several components OR was not included. Strategy/Procedures Uses an efficient
and effective strategy to solve the problem(s).
Typically, uses an effective strategy to solve the
problem(s).
Sometimes uses an effective strategy to solve problems, but does not do it consistently.
Rarely uses an effective strategy to solve problems.
Neatness and Organization
The work is presented in a neat, clear, organized fashion that is easy to read.
The work is presented in a neat and organized fashion that is usually easy to read.
The work is presented in an organized fashion but may be hard to read at times.
The work appears sloppy and unorganized. It is hard to know what information goes together.
Diagrams and Sketches
Blueprints are clear and greatly add to the reader's understanding of the procedure(s).
Blueprints are clear and easy to
HO #7
Rubric for Solving Other Created Level
1-Insufficient Understanding
2-Fair 3-Good 4-Excellent
Understanding of Concept (Writing and Solving Quadratic Equations) Demonstrates no understanding of the main concept.
Demonstrates little understanding of the main concept.
Demonstrates partial understanding of the main concept. Demonstrates mastery knowledge of the main concept. Accuracy Quadratic Models
Equations do not represent
successful “hits”.
Most equations do not represent successful hits.
One equation does not represent a successful hit, while others are accurate.
All equations represent successful hits.
Summary No organization of work or support for final solution.
Very weak evidence of organization or support for final solution.
