##
* *Reflection: Connection to Prior Knowledge
Creating Velocity Graphs from Position Graphs - Section 3: Pride Points Problem: Non-linear Position Graphs

It is clear from the Pride Points work that students are right on target with the idea of using slopes to create velocity profiles. In addition, every team of students hit upon the strategy of approximating small curved sections of the position graph with short, straight segments whose slopes could be assembled. This is new to me - in previous years, students were not as ready to do this. Before considering why this might be, I want to examining two sample student products.

In the first image, the velocity graph is a staircase function - constant for a segment with quick transitions from interval to interval.

Most teams produced graphs like the one above with only minor variations. One team produced this graph which looks continuous:

While this is closer to the truth, their thinking (not shown here) reveals that they followed exactly the same strategy as others but decided to simply plot the midpoint of the the velocity value for each interval!

I think there are two reasons why students were so ready to move to straight line approximations. I overheard many students refer to earlier approximation methods (like estimating the area under nonlinear force functions to get work) as a way to convince teammates that "getting close" was preferable to struggling, perhaps in vain, for a perfect answer. This is very rewarding to me as it indicates a level of sophisticated thinking that we've been striving for and a comfort with uncertainty that is admirable. In addition, students engaged with a simulation in the last lesson that previewed this kind of nonlinear position graph. Both influences likely played a huge role in their comfort with approximating in this problem.

*Examining Student Thinking*

*Connection to Prior Knowledge: Examining Student Thinking*

# Creating Velocity Graphs from Position Graphs

Lesson 3 of 10

## Objective: Students will incorporate the idea of slope as an essential idea in studying motion, applying it to create velocity graphs based on position graphs.

*80 minutes*

#### Warmup

*15 min*

Though today's lesson is a continuation of examining position versus time graphs to produce velocity graphs, I start with a warmup problem that is focused on some ideas from our previous unit. Students have an upcoming quiz on that material and I want to provide some time each day, in the interim, to keep those ideas fresh.

The essence of the problem is to use the given wave information to wave write functions of time and space. I remind students to access their resource sheets (part of the Electromagnetics Mock Quiz handout) which has any formula needed and other helpful information. Students may work individually or in small groups. I circulate to assist and affirm. After just five minutes or so, I give a two-minute warning, and begin to show solutions on the board.

First, I show the solutions for finding wavelengths. After a short discussion, I use this information and the other given information to produce solutions for the wave functions. I end the segment asking students to indicate their comfort level with these topics via a "fist-to-five" vote (where a fist means complete lack of understanding and five fingers means complete comfort). Most students indicate a high degree of comfort with mainly four and five fingers showing.

The balance of this lesson is, indeed, a continuation of our study of objects in motion, focusing today on the development of velocity graphs from position graphs. Students employ mathematical reasoning and model the physical properties of position and velocity with linear equations, two of the Science and Engineering Practices. The concepts of position, velocity, and acceleration are essential to achieving NGSS Performance Expectation HS-PS2-1.

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#### Creating Velocity Graphs

*30 min*

To begin this short lecture, I show students a multi-part position graph.

I ask students to make observations about this graph. Comments include that the object is moving faster in the first three seconds than in the last five seconds and that the object has stopped during the interval from three to six seconds. When I press for a rationale, students comments are variations on the interpretation of the slope. Given that idea, I add a "mantra" to our graph:

I then ask students to create, with me, a velocity graph that is based on the slopes of the segments of the position graph. We create the following:

Based on this idea, I provide my students a set of practice problems for in-class formative assessment. Students work on the problems while I circulate get a sense of understanding. While I don't show solutions at the board, I do provide a significant amount of immediate feedback to students as they work. After about ten minutes or so, I hand out an assignment that is very similar to the practice and announce a due date. Students store the assignment and we proceed to the next activity.

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As we have done in previous classes, I ask my students to assemble into their "Pride Points" teams. I provide each team with two copies of the challenge problem and give them about 25 minutes to generate a response.

The intent of this activity is to shift anxiety away from grades while still providing just enough incentive to put forth a genuine effort. The competitive aspect of the Pride Points standings seems to be the right incentive - each event is scored out of 100 points and I keep a separate spreadsheet with the teams' accumulated points for the year.

Teams generate a scheme for turning a nonlinear position graph into a velocity graph. The image below captures a common strategy: to approximate short curved sections with straight lines (one line below even looks like it might be tangent - this will ultimately be our strategy) and use the lines to get slopes and, hence, velocities.

The strategy shown above leads to staircase velocity graphs - constant segments followed by abrupt changes between intervals.

Students work on their schemes and, with about ten minutes left in class, they submit their best work as we transition to our final activity: sharing strategies at the board.

#### Resources

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In the final few minutes, I ask the Pride Points teams to come to the board to share some strategies for solving the previous problem. I want to provide a chance for students to see and hear an important idea that has just emerged: short segments of a curved position graph can be approximated as straight-line segments whose slopes give the velocity in that short interval of time. Here are two examples of these short presentations:

We take the remaining time in class to show as many of these short presentations as possible. This sets up the next idea: creating acceleration graphs from position and velocity graphs.

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- LESSON 1: Introduction to Objects in Motion
- LESSON 2: Virtual Exploration of Motion
- LESSON 3: Creating Velocity Graphs from Position Graphs
- LESSON 4: Nonlinear Position Graphs
- LESSON 5: Electromagnetics Test & Motion Task
- LESSON 6: Graphical Summary of Motion
- LESSON 7: Acceleration Lab - Day 1
- LESSON 8: Acceleration Lab - Day 2
- LESSON 9: Acceleration Lab - Day 3
- LESSON 10: Wrapping up the Acceleration Lab