Presentation Transcript

1. PREPARING PRECALCULUS STUDENTS FOR PHYSICS

   Mike Thayer, Summit High School, Summit, NJ Association of Mathematics Teachers of New Jersey Annual Conference – October 16, 2009
2. Synopsis Why should we care? What are some typical curricula for physics courses? What are some common topics that we precalculus teachers can address? What are some topics limited to specific physics courses (calculus-based)? How do math & physics teachers “think differently”? What should we change about our teaching? What is the role of technology? Where else does physics “hide out” in our precalculus curricula? Conclusions
3. Why should we care? In our precalculus courses, many of our juniors will be taking physics concurrently or as seniors Adding physics topics can add context to topics in our courses Adds “interdisciplinary” aspect to our course Gives an excuse to add some of the neat things we don’t often always talk about in precalculus (vectors, etc.) To help our students!!!
4. What do some curricula look like? Many high schools offer several levels of physics for junior or senior students: “Conceptual” physics courses, a la Hewitt (typically not for precalculus students) Traditional high school physics (“standard physics”) Honors physics (non-AP) AP Physics, B level (non-calculus-based) AP Physics, C level (calculus-based): Mechanics Electricity and Magnetism The precalculus course a student takes often correlates with the level of physics course they will take. Curricular resources available through AAPT and AP (www.aapt.org, www.apcentral.com)
5. Conceptual Physics From the American Association of Physics Teachers: “AAPT believes that all students should take a physics course in high school. Physics is the basic science … Thus every high school has an obligation to offer at least one broadly appealing physics course that is open to all students … It is not necessary for this physics course to be based upon advanced mathematics.” http://www.aapt.org/Resources/upload/hsguidelines-pdf.pdf
6. Standard Physics Traditional physics course, including topics in: Mechanics Heat and Thermodynamics Waves and Sound Light and Optics Electricity & Magnetism Modern Physics Typical textbooks: Heath Physics (Martindale et al.), Holt Physics (Serway and Faughn), Glencoe Physics (Zitzewitz), PSSC Physics
7. Honors Physics Typically, what separates the “honors” (non-AP) physics course from the standard one is: Degree of mathematical sophistication Pace of course Difficulty of homework questions Depth of discussion of topics Lab experiments may be “deeper”
8. Advanced Placement Physics B From the AP Physics course description: “It is assumed that students are familiar with algebra and trigonometry, although some theoretical developments may use basic concepts of calculus…{it} is not the usual preparation for more advanced physics and engineering courses.” Designed to be equivalent to a one-year college physics course typically taken by students in life sciences, pharmacy, etc. Very lengthy list of topics covered
9. Advanced Placement Physics C 2 distinct parts, Mechanics and E & M Some schools offer both parts, others only offer one (usually Mechanics) Much more intense, calculus-based course Most schools require students to take (concurrently or previously) AP Calculus – BC level preferred
10. What mathematical topics are common to physics courses? Vectors and Trigonometry Solving Equations for a variable Scientific Notation Parametric equations (“kinematic equations”) Graphical analysis skills Word problem skills Dimensional Analysis (“cancelling units”) Proportional thinking Logarithms and exponentials NUMBER SENSE/ESTIMATION Data analysis and statistics (basic)
11. What mathematical topics are only required in Physics C? Derivatives & Integration Some familiarity with polar coordinates is helpful Regardless of the AP course level (B or C): “Students need two things to be successful in AP Physics classes: commitment and sufficient math background.” (From the AP Physics Teacher’s Guide: http://apcentral.collegeboard.com/apc/members/repository/ap07_phys_teachersguide_11-72.pdf)
12. How do math and physics teachers think differently? 2 major areas: Their approaches to graphs and graphical analysis MUCH MORE DATA-DRIVEN THAN MATH TEACHERS! Their attitudes towards numbers MUCH MORE FLEXIBLE! Being aware of the ways we approach these areas differently can only help!
13. A note on calculators and graphical analysis: From the AP Physics “Graphical Analysis” document: “Students need to be able to think about the material in their physics courses in terms of conceptual, verbal, graphical, and mathematical [i.e., numerical] ideas. As part of these comprehensive skills for understanding the physical world around them, students must be able to perform graphical analysis in its many forms. … With the use of graphing calculators, students appear to be losing the ability to draw, interpret, and understand graphs. ... The AP Physics courses should provide an opportunity to bridge the gap between physics and math for these students.”http://apcentral.collegeboard.com/apc/public/repository/AP_Physics_Graphical_Analysis.pdf
14. Graphical analysis in a physics class: A thought experiment Imagine the following experiment: A car is on a flat table. A rope is attached to its front, and the other end of the rope goes over a pulley at the end of the table. A weight is tied to the other end. The car is released from rest, and moves with a constant acceleration toward the pulley. The students collect data on the time elapsed and the distance traveled by the car. Formula relating distance x and time t: So what happens? Picture source: http://www.batesville.k12.in.us/physics/APPhyNet/lab/experiments/kinematics/constant_acc_2.htm
15. Our thought experiment (2) The students collect data that, for example, looks like this:
16. Our thought experiment (3) And they make a lovely graph like this: And then the teacher asks: “Are quantity and distance proportional?” And the students say…
17. Our thought experiment (4) “Why, yes! The distance goes up as the time goes up!” So the physics teacher has them look at the data another way:
18. The punchline So what quantities are proportional? Remember: We would ask this as “Does distance vary directly with time squared”? A physics teacher would say “Distance and time squared are directly proportional – verifying the formula we discussed earlier!” Nice website for this discussion of proportionality: http://www.batesville.k12.in.us/physics/APPhyNet/Measurement/is_this_proportional.htm
19. The moral (For now) When math teachers look at our students’ graphs, we see curves When physics teachers look at their students’ graphs, they see data The goals the two groups have for their students are COMPLETELY DIFFERENT Math teachers – we want the students to see the “big picture” Physics teachers – we want the students to draw “physics conclusions” Words like “proportional” are used differently by the two camps!
20. How do math and physics teachers handle numbers differently? Math teachers: Care about rationalizing denominators Care about having “exact” answers (in terms of π, fractions, etc.) Care about simplifying radicals Use radians Physics teachers: Want answers that can be worked with intuitively 3√2 less good than √18; Θ=π/3 less good than Θ=60° Want answers that are sensible (significant figures appropriate to the problem)
21. Does this change how we have to teach precalculus? The short answer: “NOT REALLY”. We may want to change our emphasis on some topics, delete some and add others We may want to refocus our thinking about the types of questions we ask Calculator usage remains a concern – negative influence on students’ number sense and graphing skills Much research on physics education – some applicable to teaching of precalculus topics (example resource: “Some misconceptions and things often not done well in the MEI mechanics papers”, by David Holland: http://www.mei.org.uk/files/conference06/D1.pdf)
22. So, what specifically should we do? Many places where we could connect physics to existing precalculus curricula: Within parametric equations: Free-fall and projectile motion problems Conic sections (non-rotated) Change parametrization to get idea of “speed” across (e.g., x = t, y = t vs. x = 2t, y = 2t) Within vectors: Brief reminder of what forces are; introduce terms like resultant, equilibrant Use boldface or “hat” notation for vectors; physicists don’t use bracket notation (2i + 3j, not <2, 3>) GRAPHING SKILLS – Emphasize the generation and interpretation of graphs
23. What to do (continued)? Solving equations and systems for a variable: Solving physical formulas for particular variables (example: express velocity in terms of mass and kinetic energy) Calculus without calculus: “optimization” problems (example: cylinder with height h and radius r, fixed surface area, find maximum possible volume) “rate” problems Estimation and number sense problems Answers in physics problems range over much wider magnitudes than those in typical precalculus problems Also important to try to get a sense of scale – example: how big is a Newton? (weight of a small apple) Power laws: Proportional thinking What happens to quantity X when quantity Y is doubled? Halved? Depends on relationship between them! Statistics: lines of best fit, regression, etc.
24. Technology: Does it play a role? Simulations available for “demos” in a precalculus class: http://phet.colorado.edu/index.php Applets for vector addition, function/derivative/integral simultaneous graphs, estimation, curve fitting, etc. Handheld data-collection units or computer-interfaced probes for “actual” demos PASCO, Vernier, etc. Your science department may have these! Use of computer spreadsheets to perform data analysis
25. Other places physics can show up in our (math) classrooms We often bring in topics that can connect to physics without trying: Newton’s Law of Cooling (exponential decay) Sound intensity – decibels (logarithms) Snell’s Law for refraction of light (right-triangle trigonometry) Ohm’s Law relating voltage, current, resistance (can lead to systems of linear equations with loop laws) The phenomenon of “beats” in sound (sums of sinusoidal functions)
26. An example: Newton’s Law of Cooling Using a PASCO or Vernier temperature probe and a cup of coffee, you can generate a graph like this in a couple of hours:
27. Conclusions We have an opportunity (that we’re often not aware of) to help our students prepare for their physics courses even more than we do now Knowledge of what they will be dealing with in their physics course can help us choose more relevant topics in our precalculus courses We need to be mindful that we are not there to teach the students physics, but to give them some tools that will be useful to them when they take the course