Basketball on the moon? Football on Mars? How would the rules of the game change for different sports on the moon or a planet other then earth? Using mathematical modeling and quadratic equations, you will re-write the rules for playing a sport on another planet. Quadratic equations are used to describe many real-world patterns. You will learn several methods for solving quadratic equations including graphic representation and analysis.
This unit will take sixteen fifty minute class periods. The initial investigations and research for the project require Internet access. The bouncing ball activity can be done in the classroom, in the hall or in the gym. I have the students conduct the Barbie Bungee Jumping data collection in the hallway and culminating jump in the gym. (Our gym has a balcony overlooking the gym floor.) All other activities and tasks can be accomplished in the regular classroom. Students may need to conduct initial velocity experiments in the gym or outdoors.
Rules of the Game project: Students will work in pairs. The end product of the project will be a PowerPoint presentation to the Intergalactic Athletic Rules Committee suggesting rule changes for a sport on the moon or a planet other than earth. (Jupiter should not be used because of the high gravitational force.) The teacher will provide the students with Internet book marks for information about the moon and planets as well as Earth Rules for the different athletic events.
Web site for planet data:
The students choose a sport that involves vertical motion of a ball. (football, basketball, soccer, volleyball) For the purposes of this project, we will only consider the effects of gravitational force on the vertical component of the balls motion, not the horizontal component. Changes in the athlete's motion but not in other characteristics (eyesight for example) should also be considered. The student's presentation will demonstrate mathematical evidence supporting the rule changes. The only rule changes that should be addressed concern changes to the physical playing field and to the equipment used. Students can use Internet research to find initial velocity values or conduct an experiment using the CBR to find an initial velocity value. The project will be presented to the class and a Rules Committee made up of other faculty members, administrators and/or parents. The main focus of the assessment rubric is the mathematics in the presentation but will include assessment on oral presentation and the PowerPoint product. Students should conduct preliminary work on the project throughout the unit outside of class.
Key terms for guiding discussions: quadratic function, x-intercept, y-intercept, domain, range, coefficient, factor, trinomial, binomial, parabola, vertex, axis of symmetry, roots, zeroes, maximum, minimum, independent variable, and dependent variable.
Day 1 & 2: Introduce the project portion of the unit. Give a brief overview of what will be happening for the next three weeks of class. Review how to solve equations of the form ax^2 = b; a, b constants. Review the definition of a quadratic expression, a quadratic equation in one variable, and a quadratic function. Demonstrate the geometrical representation of several different quadratic expressions using algebra tiles. Have students model several expressions using algebra tiles. While modeling the quadratic expressions, review factoring of quadratics and point out the geometrical representation of the factors. Use several expressions that are perfect square trinomials and several expressions that are not perfect square trinomials, pointing out to the students the difference both algebraically and geometrically. Students work in pairs to investigate Internet sites that have been book marked and use the information to complete a practice worksheet on solving quadratic equations by factoring.
Web Sites for factoring and solving quadratic equations by factoring:
Day 3 & 4: Go over the practice worksheets from the previous days and answer any questions students may have. Conduct a short group discussion concerning the students' mastery of solving quadratic equations by factoring. Write a quadratic expression on the board that is not factorable. Ask the students to try to model it with the algebra tiles. Ask what needs to be added to the geometrical representation to make it factorable. Demonstrate solving a quadratic equation by completing the square using the algebra tiles. Students use algebra tiles to solve several quadratic equations by completing the square. Students work in pairs to investigate Internet sites that have been book marked and use the information to complete a practice worksheet on solving quadratic equations by completing the square.
Web Sites for Completing the Square:
Day 5: Discuss the level of mastery of students for solving quadratic equations by completing the square. Write a quadratic equation in the general form on the board
(ax^2 + bx + c = 0) and ask students to work with their partners to complete the square solving for x. Emphasize that their answer is called the quadratic formula and can be used for solving any quadratic equation. Students work in pairs to investigate Internet sites that have been book marked and use the information to complete a practice worksheet on solving quadratic equations using the quadratic formula.
Web Sites for solving quadratic equations using the quadratic formula:
Day 6 & 7: With the aid of a graphing calculator (TI-83+), transition from quadratic equations in one variable to quadratic functions. Demonstrate the procedure for entering and graphing quadratic functions on the calculator. Emphasize the relationship between the window on the calculator and a piece of graph paper. Also emphasize the correlation between an ordered pair chart used for graphing on paper and a table generated by an equation on the calculator. Have students work in pairs to investigate the affect of changing the values of the constants a, b, and c in the function f(x) = ax^2 + bx + c. Have students sketch the graphs of the equations that they investigated and write a paragraph summarizing their findings. Discuss each group's findings during class to be sure that all groups came to the correct conclusions.
Web sites for graphing quadratic equations:
Day 8: Conduct a group discussion comparing linear motion and projectile motion. Mathematically, discuss the difference between a linear function and a quadratic function. Introduce the students to the CBR units and have them practice capturing data.
Day 9: Conduct the Velocity and the Bouncing Ball experiment using the CBR units. After conducting the experiment, students answer the questions about the activity. The Bouncing Ball activity is found in the Texas Instruments Explorations manual Modeling Motion: High School Math Activities With The CBR.
Day 10 & 11: Introduce the Barbie bungee jumping activity with a short presentation showing different bungee jumping sites. Students work in groups of three to conduct the Barbie bungee jumping experiment. Problem: Create an algebraic model that will allow you to predict the number of rubber bands required for Barbie's bungee jump to span a given distance. The final test of your algebraic model will be a bungee jump with the objective of Barbie coming as close to the ground as possible without actually touching the ground. Procedure: Work in a team of three or four students. Using linked rubber bands as the bungee cord, create test data that will allow you to create an algebraic model. The independent variable will be the number of rubber bands linked and the dependent variable will be a distance. Start by linking a couple of rubber bands together and measuring the length of Barbie's fall. Continue with different numbers of rubber bands linked together. You should conduct at least ten test drops. Decide if your data fits a linear model or a quadratic model the best. Explain why your test data doesn't fit perfectly. One paper per group should be handed in explaining the experiment, the algebraic model used (and why), an accurate graph of the test data with the algebraic model superimposed. Bungee Jumping web sites:
Day 12: Tie it all together with a group discussion relating the unit activities to quadratic equations and to the unit project Rules of the Game. Review methods of solving quadratic equations and graphing quadratic functions.
Day 13: Written test.
Day 14, 15: Go over tests with the students. Students work on projects.
Day 16: Project presentations.
The teacher's role will be to provide book-marked sites for the Internet investigations, to explain expectations for the unit, and to review assessment guidelines. The teacher will be the facilitator of class discussions guiding students to use correct mathematical terminology. The teacher will also act as an aide to students as they conduct their investigations and experiments. The teacher must provide feedback to the students concerning their mathematical abilities relating to the unit.
Students will work in pairs and individually on daily investigations and on the culminating presentation. Students will work in groups of three to conduct the experiments. Students will take on the traditional role as the teacher while working in pairs by teaching each other and assessing each other's success. Students will access their partner in the Internet investigation using the collaboration rubric. Students will evaluate other PowerPoint presentations using the rules of the Game Rubric.
Other faculty members, administrators, and/or parents can act as the Rules Committee for the presentations. It is important to use the same committee for all of the presentations.
Standards with * will be mastered during this unit. Other standards will be introduced or reviewed, but will not be completed in this unit.
The student will:
*select, justify, and apply a technique to solve quadratic equations over the set of complex numbers and interpret the results graphically.
*solve problems using the quadratic formula including graphic representation and analysis.
*create algebraic models to represent problem situations.
*use various representations of functions. (example: graphs, tables, symbolic forms)
*demonstrate and explain the effects that changing coefficients and/or constants has on the graph of a function.
*determine which type of function best models a situation, write an equation, and use this equation to answer questions about the situation.
analyze the relationships among the coefficients, factors, and roots of polynomials
compare quadratic growth with linear growth and exponential growth.
determine the domain, range, zeros, y-intercepts, end behavior, relative maximum and minimum points, and symmetry of functions.
create tables or graphs to interpret relations and/or functions.
create geometric and numerical patterns that model relations and/or functions.
demonstrate the relationships between force and motion in Newton's laws.
relate gravitational or centripetal force to projectile or uniform motion.
apply quantitative relationships among mass, velocity, force, and momentum.
Internet based investigations will be assessed with a rubric self-assessment, peer assessment and teacher assessment using a collaboration rubric. The worksheets will be assessed using a homework rubric. The bouncing ball experiment will be assessed using the homework rubric. The Barbie Bungee Jumping activity will be assessed with a rubric. Mastery of the standards addressed in this unit will be assessed by a written test (traditional assessment) and a rubric assessment of the project. View copies of Worksheet 5-1, Worksheet 5-2, Worksheet 5-3, and the Unit 5 Test.
Computers with Internet access and PowerPoint software, TI-83+ calculators, TI Graph Link, CBRs, basketballs, footballs, soccer balls, volleyballs, racquet balls, rubber bands, meter sticks, Barbie dolls, book marked Internet sites, algebra tiles, worksheets, test, graph paper, Texas Instruments Explorations manual Modeling Motion: High School Math Activities With The CBR with copies of the Velocity and the Bouncing Ball Activity, rules for various sports (I use information from Rules of the Game by the Diagram Group.