# Distance vs Displacement

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## Objective

Students will use their understanding of concepts related to distance and displacement to model motion in terms of changes in position.

#### Big Idea

An object can travel a large distance and have zero displacement.

## Overview

The goal of this lesson is to help students use their shared experience of commuting to explore the idea that a person can travel a large distance and have a zero displacement. This lesson addresses the HSN-VM.A.3 standard because it asks students to represent displacement as a change in position vectors using scaled arrows. Scaled arrows allow for representation of different values through a scale. For example, a 7 cm arrow can represent a distance of 70 m with a scale of 1 cm = 10 m. This aligns with the NGSS Practices of Using Mathematical Reasoning (SP5) because students will use mathematical logic to create scaled arrows to represent their position for each leg of their commute. This lesson also is aligned to the NGSS Cross Cutting Idea of Patterns because students must recognize that when the final position and initial position are equivalent then an object has a zero displacement.

Within this lesson, students will begin constructing a model of walking along a path using arrows to describe their position on each leg of their journey. It is important in physics that students can identify the difference between how far an object has traveled and how far an object has traveled from its starting position and final position in a given direction. In physics distance takes into account the total amount of ground an object has traversed and displacement only concerns the directional change in position between the starting and final location of an object regardless of the path the object travels. This idea of a starting position as a point of reference is important in physics because, reference frames introduce the concept of context to understanding the physics at work within a system. Students work in teams of 2-4 to complete a set of practice problems that illustrate an object's motion in terms of position. Finally, students will then use their understanding of proportional reasoning to construct a visual that represents the position of an object over time. I assess student understanding throughout the lesson using informal check-ins, and will assess each student's work at the end of the school day.

## Warm-Up

5 minutes

During this portion of the lesson, I ask students to write the objective and big idea in their notebooks. I ask my students that, instead of relying on a set of instructions, they focus on constructing an explanation using tools that are familiar to them and their understanding of conversions. This relates to (SP5) because students have to combine skills from their daily lives along with those that they have learned from mathematics to extend their physical understanding of the system at hand: a student's commuting to and from school via train.

Today's additional piece of information is a BIG IDEA which states that an object can travel a large distance and have a zero displacement.  Later on within this lesson I ask students make several choices which lead them down unique paths where they create ways to meet the objectives of the lesson. I want students to learn that leveraging skills from their daily lives is an important way to raise their level of understanding of physics.

## Practice Problems

30 minutes

I transition to a set of practice problems to leverage skills and knowledge that students already employ while traveling through New York City (NYC) and introduce students to the idea that there is a difference between total distance and the linear distance from a starting position. I also want students to have a clear example of traveling a large distance and having a zero displacement to drive home the importance of precise explanations of scientific vocabulary. During this portion of the lesson, I introduce a problem that includes distance, displacement and a conversion. I include a conversion to extend students' current understanding of distances from city blocks to meters.

I introduce a set of Problems where I ask students "How far Terry travels during her journey?" and then ask students to determine "How far away and in which direction did Terry travel from her starting position?” Students copy the first problem into their lab notebooks and spend about 5 minutes working on it independently. After 5 minutes elapse I ask students to discuss their solution with their elbow partners. After students have discussed the problem with their partners I ask for a volunteer to lead the class through the problem at the interactive whiteboard at the front of the room. A student comes to the front of the room, writes and explains the steps of their solution answering a small number of questions at the end of their explanation. After we discuss the first problem, I project the second problem and we repeat the process: students copy the second problem into their lab notebooks and spend 5 to 10 minutes working on the problem and sharing their results with their elbow partners that we then discuss as a class. Click here to see an example of student work for this section of the lesson.

After I have introduced these problems, I tease out the idea that distance is like a pedometer reading that only is concerned with how far Terry walked, while displacement is the difference between Terry's final position and her starting position. I tease out the idea that while distance and displacement can have the same numerical value, they communicate two different types of information.  I walk from one end of my classroom and back. I ask the class: "How far did I travel?" Students give estimates some in terms of tiles, some in terms of feet others in terms of meters. I ask students to think of how far I traveled as the total distance I travel. Then I ask the class how far I traveled away from my starting position. Student replies include you didn't, none and zero. Then ask the entire class of students to think of the difference between my final position and starting position as displacement.  Finally, I ask students how my walk connects to the big idea of the lesson. I get several replies along the lines of if I walk a distance away from a starting position and back then I have traveled a large distance but have zero "displacement".

I take a temperature reading, where students give a thumb up if they are feeling competent in their understanding of the difference between distance and displacement; or a thumb down if they feel as though they lack competence in distinguishing between distance and displacement.. Typically, for a temperature reading, if at least 80% of students feel competent in their understanding, I ask students if they can they demonstrate their understanding by correctly comparing and contrasting between distance and displacement in their notebooks. If less than 80% of the class feels competent in their understanding, I ask students to discuss distance and displacement with their station partners using a whiteboard and dry erase marker to create an explanation of the terms that includes visuals, math models and graphs. Then I circulate to ask questions like "Why do you say that?", "How does your answer change if..." and, "Why do we care?” to probe how deep student understanding of the topic goes. This relates to (SP1) because here I am using questions that students can investigate within our classroom with the resources available to them. The resource area in the front of the room has poster paper, markers, color pencils, rulers, calculators and Chromebooks.

I use a checklist to ensure that I check each student's understanding by the end of the lesson.  I like this checklist because it is easy to customize for lesson goals and student needs. Each column of the checklist corresponds to specific goals of the lesson. The first column is where I note whether a student uses the G.I.R.L.S. protocol to solve the practice problems. The second column is where I note whether a student actively participates in discussions at with their table mates. The third column is where I note if students correctly calculate the distance an object travels. The fourth column is where I note if students calculate displacement correctly. The final column is where I check student work for correct units and mathematical reasoning between steps in a multi-step solution.

## Assessing Student Understanding of Distance and Displacement

35 minutes

After I circulate the entire classroom to check on student progress toward proficiency in distinguishing between distance and displacement, I project a set of choices on the interactive white board for students to work on during this section of the lesson. I remind students that the materials (including a Subway Map and Map Reading Grid), are located in the front resource area in labeled. The map reading grid is used to map changes in position. The grid has a scale that students can set based on the amount of distance they travel during each leg of their trips. The map reading grid also breaks student commutes into legs to make the total distance and displacement easier to visualize. Click here to see an example of a student's work for this section of the lesson.

Students use the next 35 minutes to complete a task of their choice. In this section of the lesson, I introduce task choices because I want to give students multiple access points to improving their knowledge and demonstrating their proficiency. The choices are based to the top three learning styles from the student baseline lesson

Although the final work products will be different, I ask students to compare their displacement and distances traveled on an example commute students create using a grid and a scale. Students use the idea that distance is like a pedometer reading to calculate the the total distance and use arrows to represent changes in position to determine their displacement on the visuals they produce.  Although I want students to think of displacement in terms of the change between final and initial position, many students focus on the convention that treats North as a positive direction while South is a negative direction.

## Closure

10 minutes

I provide students with an Exit Slip with a set of writing prompts for a routine called Can You Prove It? In this routine I ask students to individually identify their personal level of understanding of key ideas within the unit and make connections between the key ideas and an event from their daily lives.

I use this type of exit slip because I want students leverage the experiences they have working in teams and small groups to make connections between basic events in their daily lives and the enduring understandings that unfold during physics class. I want to make sure students are able to make these connections as individuals just as effectively as they do while working in teams. Mainly, I want students to think of physics as a lens to understand the way they interact with the world around them. I want students to benefit from learning in a collaborative classroom environment in a manner that builds their knowledge and improves student communication of that knowledge when working independently on complex problems.

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 that they provide.