Chapter 2

The description of motion* in one dimension.

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?

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

Introduction: Our approach• Preliminaries

– definitional– historical– cognitive

• Representations of motion– visual– mathematical

• Exploring uncertainty

• Modeling motion

• Wrap-up

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

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

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

– 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.– in kinematics especially motion graphs!

Representation of motion - 2• Mathematical representations provide

clarity and precision about position, velocity, acceleration in one dimension.– vectors in general (a bit)– vector quantities in one dimension– instantaneous and average values– slope function = derivative– questions (brain storm/prioritize/ask)

• Real motion– prediction and observation ILDs

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/

Representation of motion - 4• Problem solving and representation

– problem solving/representation quote

• Representation of problem solving (!)– And the stages of learning problem solving– Recognizing where we commonly begin (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)

1-D kinematics problem solving

• Finish time to second bounce activity

• Sequence of learning important problem solving elements (see)

• Final diagram activity (go to) (handout)

• Self-assessment assignment (handout)

Modeling motion (numerical integration)

• Modeling language (Stella)

• Working backwards from “rate of change”– numerical integration– like skipped part of chapter

Wrap up

• What questions do we have?– questions (brain storm/prioritize/ask)

the end

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. back

1-D kinematics

x(t), v(t), a(t) relations

solutionproblem

Below we recognize a common student view of kinematic problem solving before a challenging engagement with a real problem.

back

1-D kinematics

x(t), v(t), a(t) relations

solutionproblem

Below we recognize a common student view of kinematic problem solving before a challenging engagement with a real problem.

1-D kinematicsPhysical situation

mathematical representationx(t), v(t), a(t) relations

solutionproblem

1-D kinematicsPhysical situation

visual representationmotion graphs

mathematical representationx(t), v(t), a(t) relations

solutionproblem

1-D kinematicsPhysical situation

visual representationmotion graphs

mathematical representationx(t), v(t), a(t) relations

solutionproblem

physical and conceptual assumptionsposition as continuously varying point

1-D kinematicsPhysical situation

visual representationmotion graphs

mathematical representationx(t), v(t), a(t) relations

solutionproblem

physical and conceptual assumptionsposition as continuously varying point

1-D kinematicsPhysical situation

visual representationmotion graphs

mathematical representationx(t), v(t), a(t) relations

solutionproblem

physical and conceptual assumptionsposition as continuously varying point

1-D kinematicsPhysical situation

visual representationmotion graphs

mathematical representationx(t), v(t), a(t) relations

solutionproblem

physical and conceptual assumptionsposition as continuously varying point

back

1-D kinematicsPhysical situation

visual representationmotion graphs

mathematical representationx(t), v(t), a(t) relations

solutionproblem

physical and conceptual assumptionsposition as continuously varying point

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