# Modeling Motion in Terms of Velocity vs Time Graphs, Part 1

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## Objective

Students will compare and contrast the shapes of velocity vs time graphs.

#### Big Idea

The shape of a velocity vs time graph depicts how quickly and in which direction an object is accelerating.

## Overview

The goal of this lesson is to help students use multiple models for the motion of objects in terms of velocity vs time. This lesson addresses the HSF-IF.C.7 standard because it asks students to use mathematical reasoning to interpret the meaning of one graph to generate a related graph. It aligns with the NGSS Practices of Analyzing and Interpreting Data (SP4) and Developing and Using Models (SP2) for Science because students complete activities that demonstrate their understanding relationship between the graphical models of two kinematic graphs.

Within this lesson, students obtain additional information from their peers, a whole class discussion and a credible website on the meaning behind the shape of a velocity vs time graph. Students then use their understanding of mathematical models to complete a pair of graphing activities. The first activity asks students to make connections between motion maps and velocity vs time graphs. The second activity asks students to generate velocity vs time graphs given the graph of an object's position vs time. I assess student understanding throughout the lesson using informal check-ins and assess each student's work at the end of the school day.

## Warm Up

5 minutes

Within this class, I want students to recognize the significance of patterns and to make connections between concepts they are learning and those they are more familiar with. For example, many students can readily define slope as rise over run, but have little to no explanation as to the meaning of the slope. This lesson follows introductory lessons on Galileo's scientific legacy, modeling motion using position vs time graphs and modeling motion using motion maps. In this lesson, I want students to begin to bridge the gap between linear fits of graphs, using units to keep consistency within a mathematical model and the ability to connect the slope of one kinematics graph to the creation of a related graph.

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.

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. This lesson's Big Idea relates that the shape of a velocity vs time graph communicates information about that object's acceleration.

## Revisiting Motion Maps

35 minutes

During the next five minutes of class, I give student groups each a different scenario and ask them to create in their notebooks a corresponding motion map with a short sentence or two to explain how to construct the motion map. After five minutes have elapsed I have each team create a whiteboard summary of their scenarios, this time adding the corresponding position vs time graph. I circulate asking clarifying questions and ensuring that students correctly label each part of their graphs. In this section of the lesson, one of the activities asks students to tie concepts from the last few lessons together because integrating similar ideas is an important skill for science students to learn and use. During the next activity, each student will be given all of the scenarios to analyze.

As a class, we spend five additional minutes discussing the connections between motion maps and position vs time graphs. Then we discuss the connection between motion maps and velocity vs time graphs. I spend about ten minutes sharing ideas with students in a whole class discussion about velocity and its relationship to displacement and ask questions about how to jump from a motion map to a velocity vs time graph. Some students find it easier to sketch position vs time graphs from the motion maps first and then transform the slope data from one graph into the point data on the other graph.

I encourage students to spend the next five minutes asking their peers about the easiest steps to use while converting a motion map into a velocity vs time graph. After we discuss the connection between motion maps and velocity vs time graphs, I distribute this worksheet that brings it all together. Students spend about twenty minutes working on this activity. After students work on the activity individually they are free to discuss their work with their table mates. I use this Velocity vs Time Graph worksheet because I want students to construct ideas about the slope of the line graphs and the area between the line graphs they generate using motion maps and the horizontal axis. Click here to see an example of student work.

## Sketching Velocity vs Time Graphs from Position vs Time Graphs

25 minutes

During the first ten minutes of this section, I ask students to use the Chromebooks to go to this physics classroom tutorial to gather more information about what the shape of velocity vs time graphs mean. Students spend fifteen minutes using this sheet to work independently. I circulate the room as students complete the activity to answer student questions.

After fifteen minutes pass, I hand out this activity that asks students to construct ideas about the slope of the position vs time graphs and the velocity vs time graphs they generate. I also want students to use units to help them construct ideas about the meaning of the area under the velocity vs time graph in particular. This activity asks students to use mathematical reasoning to compare the shapes of different kinematic curves. Most students are able to recognize that the slope of a position vs time graph corresponds to the value of velocity vs time graph for the same time interval. Click here to see an example of student work.

While the main focus of this section of the lesson is the slope of kinematic graphs of position vs time and velocity vs time, there is a question on the activity that asks students to determine the area under a velocity vs time graph that confuses some students. Some students struggle with the units of the area under the curve and suggest that the area under a velocity vs time graph has the units of acceleration instead of displacement. The student misconception comes from an error in mathematical reasoning where students equate (m/s) * (s/1) as (m/s^2) instead of (m/s) * (s/1) as m. I address this misconception by showing the calculation on the interactive whiteboard at the front of the room.

## Closure

10 minutes

I provide students with an Exit Slip with a set of writing prompts for a routine called connect extend challenge, where students are tasked with making connections between what they already knew and the concepts they discussed in today's lesson. Some of the student responses include: "Motion maps make graphing velocity easier to understand", "Once you calculate the slope of your position vs. time graph, graphing velocity vs. time is simple" and "This makes me want to pay more attention in math since we use it so much."

I use this type of closure because I want students to use the mindset that connections are valuable ways to extend our current understanding of a topic. 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.