# Finding the Slope of a Position vs Time Graph

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## Objective

Students will determine the relationship between the slope of a position vs time graph and the average speed and velocity of an object.

#### Big Idea

The slope of a position vs time graph is a way to calculate the velocity of an object.

## Overview

The goal of this lesson is to help students use multiple models for the motion of objects in terms of the slope of a position vs time graph. This lesson addresses the HSF-IF.C.7 and HSS-ID.C.7 standards because it asks students to use mathematical reasoning to interpret the meaning of one graph to generate average velocity and average speed for an object. It aligns with the NGSS Practices of Analyzing and Interpreting Data (SP4), Obtaining and Communicating Information (SP8) and Developing and Using Models (SP2) because students complete activities that demonstrate their understanding relationship between the slope of a kinematic graph and the motion of an object.

Within this lesson, students obtain additional information from their peers, a whole class discussion and a credible website on the meaning behind the slope of a position vs time graph. Students then use their understanding of mathematical models to complete a set of modeling activities. The activity asks students to make connections between the slope of a position vs time graph and the determination of average speed and velocity of an object.  I assess student understanding throughout the lesson using informal check-ins and assess each student's work at the end of the school day.

## Bell-ringer

5 minutes

At the beginning of each lesson, I have a quick bell ringer activity to get students focused on the tasks for today's lesson. There is a slide with the date, the objective and an additional prompt projected on the interactive whiteboard with a red label that says "COPY THIS" in the top left-hand corner. Students write this information in their notebooks. Sometimes the additional prompt is a BIG IDEA for the lesson, or the Quote of the Day or a Quick Fact from current events that is related to the lesson. The red label helps my students easily interact with the information as soon as they enter the room and avoids losing transition time as students enter the classroom.

Today's additional prompt is the BIG IDEA that the slope of a position vs time graph helps us calculate the velocity of an object. I use this BIG IDEA because I want students to learn that patterns help us quickly analyze data.

## Teaching Slope Types through Physical Activities

10 minutes

I want students to become familiar with the process of making connections between physics concepts and the world around them. With this in mind, after the bell-ringer, I have a kinesthetic activity where students stand up and open their arms to their full wingspan to represent a zero slope. Then I have students make a diagonal line with their right hands highest in the air and their left hands pointed toward the ground to represent a constant positive slope.

Next, I have students make a diagonal line with their left hands highest in the air and their right hands pointed toward the ground to represent a constant negative slope. After students get the hang of these motions, I have students stand up and then I call slopes out in a random order three times for each type and have students demonstrate the appropriate pose. This is a simple activity that works well when implemented on the last block of the day.

When physics is taught with kinesthetic activities, students find interacting with the material to be more fun and enjoyable than running drills and plugging numbers in known equations to solve for unknown quantities.  With this in mind, this part of the lesson uses a physical activity to emphasize three types of slopes that helps to cement patterns of slope behavior for students. I think this type of activity is effective because I believe that shared positive experiences like an entire class moving their arms to meet the verbal cues of their teacher help to create a community of learners who are engaged in learning together.

## Identifying the Meaning of Slope for Position vs Time Graphs

45 minutes

After introducing the differences between slopes using the physical activity in the previous section, I distribute an Activity for students to identify the meaning of the slope of a position vs time graph. For the first fifteen minutes of this section, students work independently on this worksheet. After fifteen minutes have elapsed, I ask students to turn to their elbow neighbors and discuss their solutions to the worksheet for ten minutes.

After students discuss their solutions with their elbow neighbors, I ask students to work in pairs to construct the solution to a single problem of their choice and to take turns explaining their solution to the whole class using the interactive whiteboard at the front of the room. I create a flipchart of each of the problems on the modeling handout and ask students teach small sections of the handout to the rest of the class. Click here to see an example of student work. In the next section of the lesson, students write their perspective on the meaning of the slope for a position vs time graph.

## Student Perspectives on the Meaning of Slope for Position vs Time Graphs

15 minutes

In this section of the lesson, students choose the work product that I assess their understanding of the slope of a position vs time graph. I distribute a choice board with three writing options.

The options include:

• Writing a letter to another physics student describing the meaning of the slope of a position vs time graph
• Crafting a poem that expresses the meaning of the slope of a position vs time graph.
• Creating a public service announcement about the meaning of position vs time graph.

Each option asks students to give a step by step solution to a problem involving position and time similar to the ones from today's lesson. Students work individually on a written work product for 10 minutes that their peers edit using this Peer Edit Form. Students must have a peer edit their work before the beginning of the next class and turn the form in by the beginning of the next lesson.  I use choice boards to help get students engaged in the process of learning and doing physics. I emphasize that written communication is an essential skill for scientists. While students work on their work products, I circulate the room ensuring that everyone is on task and is clear on the expectations of the task.

## Closure

10 minutes

I provide students with an Exit Slip where students are tasked with both identifying their personal level of understanding of three different types of slopes and their meanings for position vs time graphs and the underlying reasons behind their understanding.  Some student responses include, "Positive slope of a position vs time graph means there is a positive velocity", "A graph with a zero slope means that an object has stopped moving", and "A negative slope means the skater moves to the left."

To wrap up the lesson, I remind students that I will return the exit slips at the beginning of the next lesson and we will go over the feedback from their exit slips during the beginning of our next class.