Shoot for Your Grade with Parametric Equations - Physics
Additional Resources:
Bonus: 2D Motion Test
Graphing Calculator
Motion in 2D
Parametric Equations - Physics
Notes: Day 1
Notes: Day 2
Notes: Day 3
Notes: Day 4
Notes: Day 5
Day 7 Class Notes
Notes: Day 7
Projectile Motion 1 - Problems
Projectile Motion 2
Projectile Motion Quiz 2
Shoot for your grade
2D HW Calculator
Michelle Bresett - Regents Physics Grade 12
Troy Creurer – Precalculus Grade 12
Madrid-Waddington Central
Technology Intergration
Technology Hardware: Promethean Board, TI-83 Graphing Calculator, Digital Video Camera
Technology Software: Activstudio, TI-83 Presentor, Adobe Premier
- Learning Standards
- Physics
- Mathematical Analysis Standard 1: Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions.
- Key Idea 1: Abstraction and symbolic representation are used to communicate mathematically
- M1.1: Use algebraic and geometric representations to describe and compare data.
- Standard 4 The Physical Setting: Students will understand and apply scientific concepts, principles, and theories pertaining to the physical setting and living environment and recognize the historical development of ideas in science.
- Key Idea 5: Energy and matter interact through forces that result in changes in motion.
- Performance Indicator 5.1: Students can explain and predict different patterns of motion of objects (e.g., linear and uniform circular motion, velocity and acceleration, momentum and inertia).
- Major Understanding 5.1f: The path of a projectile is the result of the simultaneous effect of the horizontal and vertical components of its motion; these components act independently.
- 5.1g A projectile’s time of flight is dependent upon the vertical component of its motion.
- 5.1h The horizontal displacement of a projectile is dependent upon the horizontal component of its motion and its time of flight.
- Standard 4 – Process Skills
- Key Idea 5: Energy and matter interact through forces that result in changes in motion.
- 5.1vii. Sketch the theoretical path of a projectile.
- Precalculus
- NCTM Standards
- Algebra:
- Represent and analyze mathematical situations and structures using algebraic symbols. Use a variety of symbolic representations including parametric equations for functions and relations.
- Use mathematical models to represent and understand quantitative relationships. Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships. Draw reasonable conclusions about a situation being modeled.
- Measurement:
- Understand measurable attributes of objects and the units, systems, and processes of measurement.
- Problem Solving:
- Apply and adapt a variety of appropriate strategies to solve problems
- Communication:
- Organize and consolidate their mathematical thinking through communication.
- Use the language of mathematics to express mathematical ideas
- Connections:
- Recognize and apply mathematics in contexts outside of mathematics
- Problem Statement
- Determine the initial velocity of a horizontally launched projectile and predict its landing spot in order to “shoot for your grade”. Part two is to record the time is takes for a projectile launched at an angle to land on the floor and use this information to calculate the initial launch speed.
- Essential Question
- Essential Questions:
- What is two dimensional motion?
- What is a parametric equation?
- How can two dimensional motion be modeled using parametric equations?
- Learning Objectives
- Learning Objectives:
- Express a function in parametric equations
- Use parametric equations to model horizontal, vertical and two-dimensional motion with a graphing calculator
- Understand the relationship between physics and mathematical notation
- Appreciate the connection between the horizontal and vertical motion as it relates to the independent variable of time
- Demonstrate algebraic and graphical understanding of parametric equations through written work
- Apply vector analysis to the path of a projectile
- Manipulate parameters to meet specific result criteria – can students work backwards through a problem
- Predict the horizontal distance traveled by a projectile
- Determine a projectile’s time of flight
- Understand the shape of a projectile’s trajectory is parabolic
- Determine optimal launch angle for a projectile with an initial position at ground level
- Resources
- For the teacher:
- Precalculus with Trigonometry by Paul Foerster
- Reviewing Regents Physics by Prentice Hall
- Conceptual Physics by Paul G. Hewitt
- For the student:
- Precalculus with Trigonometry by Paul Foerster
- Conceptual Physics by Paul G. Hewitt
- Problem-Solving Exercises in Physics by Jennifer Bond Hickman
- Teacher generated worksheets
- Steps for Implementing Learning Experience
- Physics and Precalculus classes must have a significant overlap of students
- Introduce parametric equations
- Model vertical motion
- Model horizontal motion
- Combine vertical and horizontal motion to represent two-dimensional motion
- Model behavior that approximates real world conditions
- Introduce launch angle
- Practice
- Shoot for Grade lab Part 1
- Shoot for Grade lab Part 2
- Instructional/Environmental Modifications
- No modifications were necessary for this lesson in our classrooms this year.
- Time Required Planning
- Planning (Duad and Individual)
- 4 hours meeting formally to discuss topic for integration and planning for implementation of the learning unit
- 2 hour of informal discussions
- Precalculus – Twenty hours creating lesson plans and assessments
- Physics – Twenty hours creating lessons plans and assessments and learning graphing calculator
- Time Required Implementation and Assessment
- Implementation:
- Precalculus – 5 class periods (41 min)
- Physics – 10 class periods and 4 lab period (41 min)
- Assessment
- Precalculus – 2 class periods for 1 quiz and 1 test
- Physics - 3 class periods for 2 quizzes and 1 test
- Assessment Tools
- Precalculus Unit test
- Homework rubric
- Whiteboards and class discussion
- Lab report rubric
- Precalculus Quiz
- Physics Quiz 1
- Physics Quiz 2
- Physics Unit Test
- Reflection
- Precalculus
- Physics
- Student Work
- Rigor/Relevance Framework
The topics of 2-D motion and parametric equations are completely new to my students and require instruction for algebraic, graphic and calculator understanding. The presumption that students will understand how to translate function equations into parametric equations in one day overestimated their ability and the amount of practice necessary for mastery of this skill. I should have also built in more time for learning how to master using the calculator to represent the equations graphically. There should have been more time allotted for practice with more complicated two-dimensional problem solving. It seems that a common theme has emerged and the amount of time for this unit needs to be expanded for more practice and more explanation. The assignment sheet and notes have already been adapted for use next year.
This unit should have been restricted to parametric equations instead of including topics on periodic functions. The periodic functions were included because you can use parametric functions to graph circles. Initially I felt that the combination of the 2-D motion and periodic functions might be a good fit, but upon completion the periodic functions should be incorporated into its own unit. There is enough material in working with 2-D motion to complete an entire unit.
The evaluations in the Precalculus class need to be redone to better space the two quizzes throughout the unit. The homework worksheets that were used by both the Physics and Precalculus classes were a useful exercise and helped reduce the amount of homework that a student in both classes might have to complete. I might even advocate for more collaborative homework to allow for there to be time to complete quality work.
The intention of this unit is to help students who are in Physics with their understanding of the equations that represent 2-D motion and introduce parametric functions to their mathematical vocabulary. I believe the collaboration and timing of the instruction is helpful in establishing the link between the two subject areas. In this regard I felt that the planning and effort put forth was time very well spent.
Two dimensional motion can be challenging for many physics students. I was looking for a way to make this concept easier for students to really understand. One of my former students went to the Precalculus teacher for help with a problem and she came to me and said these problems could be solved in parametric mode on the graphing calculator. I was unfamiliar with this, but willing to learn to help my students integrate math and physics more easily. I spent time with the Precalculus teacher in the summer learning how to use the graphing calculator in parametric mode and writing the equations necessary to model two-dimensional motion with the calculator. This gave us the opportunity to discuss the similarity and difference in notation between the two classes.
I came up with a series of lessons by first modeling free fall and then linear motion and then asking the students to launch a horizontal projectile from some initial height. There was constant communication with the Precalculus teacher and I was able to sit in on many of his classes. Most of the students followed Precalculus with Physics so I was able to pick up were he left off and utilize the students from his class to assist those in Physics who were not in the Precalculus class. The students in both classes certainly adapted to the modeling of two-dimensional motion very quickly and were very helpful to the few students not in both classes.
I do think there was a higher level of mastery with the two-dimensional motion concept this year. There is a better understanding that time of flight links the horizontal and vertical components of the motion together.
However, it did take me longer to get through this material and I added an additional challenge piece to my lab which also extended the time spent in the lab setting as well. In the challenge part of the lab students had to use two step equations to solve for the initial velocity of the projectile launched at an angle. This required quite a bit of algebraic work for the students and some were frustrated with this. I do think it is important to challenge the students and I would continue with this second piece of the lab again. It would not be necessary, however, or essential to a student’s understanding of two-dimensional motion.
On a personal note I found this to be a very rewarding learning experience. I had personal success with using the graphing calculator to graph model the motion in the summer and thought it was very cool. I kept trying to think of the next set of equations I could input to model the next type of motion I would want the students to see. It definitely requires time spent with the calculator if you are not already familiar with this parametric mode. The cross-over opportunity is great. I believe it is so important for the students to see their teachers working together and being excited about something. I was even able to step in and teach Troy’s Precalculus one day when he had to be away. This certainly let the students know I was comfortable with the topic outside of the realm of the Physics classroom. I would certainly complete this again and again with the hope of making it better and better.
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