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 Representations of motion –visual –mathematical Exploring uncertainty Modeling motion Wrap-up

6. Preliminaries What is and isn’t translational motion? Galileo’s revolutionary description of motion (without a center of the universe) –And the next revolution? What we will “know” per the Study Guide and a closer look at knowing (more later) –Received knowledge –Subjective knowledge –Procedural knowledge –Constructed knowledge

7. 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

9. 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 it is an end itself.

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

11. 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/

12. Representation of motion - 4 Problem solving and representation –problem solving/representation quotequote 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)

13. 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)

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

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

16. the end

17. 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

18. 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

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.

20. 1-D kinematics Physical situation mathematical representation x(t), v(t), a(t) relations solution problem

21. 1-D kinematics Physical situation visual representation motion graphs 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 physical and conceptual assumptions position as continuously varying point

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 back

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 1 2 3 4 5 6 7 8 back