Modeling Motion in Terms of Velocity vs Time Graphs, Part 2

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Students will determine the area under a set of velocity vs time graphs.

Big Idea

The area under a velocity vs time graph is equal to the displacement of an object during a specific duration of time.


This is the second installment of a pair of lessons where student 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 a velocity vs time graph. This lesson also addressed the HS-PS2-1 standard because it asks students to describe the relationships between position and velocity graphs for objects subject to an unbalanced force.  It aligns with the NGSS Practices of Analyzing and Interpreting Data (SP4) and Using Mathematical and Computational Thinking (SP5)because students complete activities that demonstrate their understanding by interpreting the slope and area of a velocity vs time graph.

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.

Quick Summary: Connection between Mathematics and Physics

15 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. 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 BIG IDEA is that the area under a velocity vs time graph corresponds to the displacement of an object from its original location. I chose this type of Warm-Up activity to add structure to my lessons. I begin and end each lesson with the same type of opening and closure routines so that students have a sense of order within our learning environment, which some people may generally describe as organized chaos.  

After students write today's objective and BIG idea in their notebooks, I project a writing prompt on the interactive whiteboard for to complete in their notebooks. In this section of the lesson, I have chosen an introductory activity where students share their answers to the question "How does physics connect to Mathematics?" as a class. First the students write their answers in their notebooks and then they share their answers with their station partners. They can write their answers down as word webs, bullet points, a pair of short paragraphs or as a mind-map. At the end of five minutes, I call on a representative from each table to share their answers with the class.  

I circulate, read and stamp student notebooks, adding comments to help students stretch their understanding as they revisit these notes. I have three stamps, a smiley face for competent, a check mark for proficient, and a butterfly for highly proficient. I turn the smiley face upside down for incomplete work. As I circulate, I look for examples of correctly labeled graphs, mathematical expressions for physical quantities or maybe even the general formula for conversions from earlier this semester. Later on this week, I read and add comments to these summaries in more detail.  Some of the comments I use include, "Add details to your graph to go from competent to proficient", "What ideas that you have learned before were useful in solving problems that relate to motion?", and "Can you give me an example of the connections between mathematical expressions and motion?"

Finding the Area Under a Velocity vs Time Graph

35 minutes

During the previous section, many students suggest that algebra and maybe trigonometry are branches of mathematics that are directly connected to physics. Within this section of the lesson, I tell students that ideas from basic geometry can also be useful when studying the physics. With this in mind I ask students to complete an activity that connects geometry to physics.  I spend a minute distributing a review activity that students use to revisit basic geometry of regular polygons. Students use this activity along with this tutorial from the physics classroom as a lens to understand the area under a velocity vs time graph.

Students spend 15 minutes completing the geometry review individually. After 15 minutes pass I ask students to switch papers with their elbow neighbors and to assess their peers understanding of basic geometry. Students spend 5 minutes assessing each others' work as I circulate and discuss the geometry review with students at each lab station. One of the common errors that students make is to calculate the area of every velocity vs time graph as though it were a rectangular area.  I think that adding a screen-cast of helpful geometry tips to our class Edmodo wall to practice using a similar activity may help students correct this misconception.

After circulating the entire room, I distribute a modeling activity on velocity vs time graphs and displacement for a set of scenarios and ask students to work in groups of 2-4 using the concepts from the geometry review to determine the area under each curve. Click here and here to see example work for this activity. Students spend 20 minutes working on the different scenarios I have distributed. I ask students to write their solutions and corresponding explanations in their lab notebooks as they work on analyzing each scenario. 

Student Perspectives on the Area under a Velocity vs Time Graph

15 minutes

After students complete the activities in the previous section, I ask students to spend fifteen minutes creating a summary of useful information from today's lesson. I give students the option to create an open letter to new physics students, or design an information cube or a blog entry that summarizes the ideas each student identifies as "Key Information" from the previous section of the lesson. At this point in the semester, students have worked with most of their classmates and have created visuals on best practices for learning physics, so this activity aligns with skills students have experience leveraging within our classroom.

I remind students that materials like markers, rulers, chart paper, and scissors as well as Chrome books are available at the resource area at the front of the room. Students spend the next fifteen minutes creating a summary of the meaning of the area under a velocity vs time graph. Students work in pairs to create a visual representation that demonstrates the connection between geometry and velocity vs time graphs. Click here to see an example of student work.


10 minutes

I provide students with an Exit Slip with a set of writing prompts for a routine called a 3-2-1. In this writing routine, students identify three areas they feel proficient in understanding, two areas where they have extended or shifted their understanding based on new information and one area they still feel they need to grow in understanding. I ask students to identify their personal level of understanding of key ideas, including areas of success and places for growth within the unit. Some student responses include, "I understand how to find displacement from a v vs t graph", "I know how to use the shape of a v vs t graph to find an object's displacement" , "I know how to use a position vs time graph to create a velocity vs time graph", "I don't know how to calculate displacement when the graph has a negative slope". 

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.