2. Chapter 2 The description of motion* in one dimension.

3. Chapter 2 The description of motion* in one dimension. What sort of motion is referred to? Describe the motion of the cart? the cart’s front right wheel? A point on the edge of that wheel?

4. Chapter 2 The description of motion* in one dimension. *translational motion of objects (as the motion of a specified point**) or the motion of any point **such a point exists

5. Introduction: Our approach Preliminaries –definitional, historical, cognitive Lessons from math experience –guidance, caution Representations of translational motion –visual, mathematical Mathematical language in physics Exploring uncertainty Modeling motion Wrap-up

6. Preliminaries Definition: What is/isn’t translational motion? History: Galileo’s revolutionary description of motion –continuous, without a center of the universe –And the next revolutions? Cognition: What we will “know” and the shifting ground of knowing (more later) –Received knowledge, Subjective knowledge, Procedural knowledge, Constructed knowledge –Reflections on the First Day Exercise

7. Lessons from math experience Two math problems –providing lessons, observations, insights –Problem #1#1 –Problem #2#2 The essential role of representation –visual and mathematical (good quote)good quote A caution on generalizing all math “problem solving” experience to Physics and problem solving guidance (go)go

8. Representation of motion - 1 Visual representations of motion –different positions at different times (always with respect to a frame of reference) [next] –strobe view ranking tasks –motion diagrams –motion graphs (conceptual, not a picture) Visual language can provide a starting point for understanding

10. Representation of motion - 1 Visual representations of motion –different positions at different times (always with respect to a frame of reference) –strobe view ranking tasks ranking tasks –motion diagrams? –motion graphs? (conceptual, not a picture) Visual language can provide a starting point for understanding and representing, and sometimes is an end itself. –in kinematics, especially motion graphs!

11. Representation of motion - 2 Mathematical representations provide clarity and precision about position, velocity, acceleration in one dimension. –vectors in general (a bit) –kinematic vector quantities in one dimension –instantaneous and average values –slope function = derivative –questions (brain storm/prioritize/ask) Real motion –prediction and observation ILDs

12. Representation of motion - 3 Constant velocity motion –examples, graphs, equations Constant acceleration motion –examples, graphs, equations Exercises –(+,-,0 ) x, v, a (home exercise) –shapes of motion graphs (home exercise) –Same data run? (homework exercise) Looking at the mathematical functions –http://www.shodor.org/interactivate/activities/FunctionFlyer/http://www.shodor.org/interactivate/activities/FunctionFlyer/

13. Representation of motion - 4 The use of mathematical language in Physics – start of a dialog (begin)begin Representation of problem solving (!) –And the stages of learning problem solving –Recognizing where we commonly begin (see)see Class activity: Time to second bounce –Begin, then continue in following days –Take notes for assignment on learned skills Changing acceleration motion –auto performance (homework exercise)

14. 1-D kinematics problem solving Finish time to second bounce activity Sequence of learning important problem solving elements (see)see Final diagram activity (go to) (handout)go to Self-assessment assignment (handout)

15. Modeling motion (numerical integration) Modeling language (Stella) Working backwards from “rate of change” –numerical integration –like skipped part of chapter

16. Wrap up What questions do we have? –questions (brain storm/prioritize/ask)

17. the end

18. Note on problem-solving “Representation entails more than a direct or literal translation of a problematic situation into a mathematical model such as a formula or a diagram. When engaging in representing, problem solvers imagine a visual story – one that is not always or necessarily implied by the problem formulation. They impose that story on the problem, and, acting on this representation, they derive from it the sought solution (Arcavi 2003).” from Mathematics Teacher vol. 101, no.5. backback

19. 1-D kinematics x(t), v(t), a(t) relations solution problem Below we recognize a common student view of kinematic problem solving before a challenging engagement with a real problem. back

20. 1-D kinematics x(t), v(t), a(t) relations solution problem Below we recognize a common student view of kinematic problem solving before a challenging engagement with a real problem.

21. 1-D kinematics Physical situation mathematical representation x(t), v(t), a(t) relations solution problem

22. 1-D kinematics Physical situation visual representation motion graphs mathematical representation x(t), v(t), a(t) relations solution problem

23. 1-D kinematics Physical situation visual representation motion graphs mathematical representation x(t), v(t), a(t) relations solution problem physical and conceptual assumptions position as continuously varying point

24. 1-D kinematics Physical situation visual representation motion graphs mathematical representation x(t), v(t), a(t) relations solution problem physical and conceptual assumptions position as continuously varying point

25. 1-D kinematics Physical situation visual representation motion graphs mathematical representation x(t), v(t), a(t) relations solution problem physical and conceptual assumptions position as continuously varying point

26. 1-D kinematics Physical situation visual representation motion graphs mathematical representation x(t), v(t), a(t) relations solution problem physical and conceptual assumptions position as continuously varying point back

27. 1-D kinematics Physical situation visual representation motion graphs mathematical representation x(t), v(t), a(t) relations solution problem physical and conceptual assumptions position as continuously varying point 1 2 3 4 5 6 7 8 back

28. math problem #1 Start with two different non-negative real numbers. What is the relation among their sum, difference (>0), product, and quotient (<1)? Think about this yourself for 2 minutes and then work in groups to solve the problem. You may ask me and each other questions. I will report interesting questions, and may make comments. Note.

29. note on problem #1 One needs to make a mathematical representation of what the problem refers to. The solution comes from this representation. How do we know this rather complicated relationship in a series of equations is the best we can do? Other comments? Concerns? back

30. math problem #2 Start with three separate points on a plane, given by (x 1,y 1 ), (x 2,y 2 ), (x 3,y 3 ), with 0 < x 1 < x 2 < x 3, 0 < y 1 < y 2 < y 3. What is the area of the triangle formed by the lines between these three points? Think about this yourself for 2 minutes and then work in groups to solve the problem. You may ask me and each other questions. I will report interesting questions, and may make comments. Note.

31. notes on problem #2 Starting with a visual representation is a good idea, and often necessary. From the visual representation, a mathematical representation of the situation the problem refers to can be created. From the mathematical representation, one gets the solution. Other comments? Concerns? back

32. problem solving caution Math class exercises often ask for deducing a solution from a given set of equations, so creating a mathematical representation isn’t needed. Generalizing from these exercises, one begins by work backwards to create what is needed for a sufficiently robust mathematical representation. Novice physics problem solvers start this way with an “ends-means” strategy; whereas expert Physics problem solvers do the opposite and start with the physical situation and use its characteristics to move forwards toward the creation of a mathematical representation.

33. the moral to this story Appreciate the limits of the successful ends-means strategy in math classes, and the necessity and central skill of creating representations that is the more common goal of physics problems. To do this, schemas for moving toward the mathematical representation starting from the physical situation are required. Hypothesis: shifting strategic approaches goes deeper and more efficiently by being mindful about it. back

34. mathematical language in Physics Start of a dialog Consider the following equations (handout) Assignment: Write a short (<250 words) essay on your initial thoughts on the mathematical meanings of these equations. Submit e-copies, bring hard copy for group discussion for the next class.