# Using Math to Model the Work-Energy Theorem

## Objective

Students will create a foldable that illustrates a mathematical model for the Work-Energy Theorem.

#### Big Idea

Work is equivalent to the change in kinetic energy of a system.

## Overview

This lesson addresses the HS-PS3-1, RI.11-12.7 and HSA-REI.A.1 standards as a way to effectively create a foldable about a step-by-step math model of the work-energy theorem using information from class. Students research concepts related to work using the NGSS Practices of Developing and Using Models (SP2), Using Mathematical and Computational Thinking, and Constructing Explanations (SP6) that illustrates the mathematical model for the work-energy theorem. Students begin today's lesson by revisiting the units for work, energy, force and momentum in a quick activity. I then give students a set of equations including Newton's second law, work, kinetic energy and one of Galileo's equations of motion. I ask students to show and annotate the steps that convert the equation for work into one that uses of mass and velocity using the equations I provide. Next, I ask students to create a foldable that demonstrates that the steps in this derivation of work is equivalent to the change in kinetic energy of an object. At the end of this lesson, I give students some team planning time to modify their roller coaster prototypes from this lesson

I assess student understanding throughout the lesson using informal check-ins and assess each students' work at the end of the school day. I want students to learn to integrate information from various points of this course into a coherent presentation. This relates to (SP6) because students have to leverage skills like solving an equation for an unknown to show that work is equivalent to the change in kinetic energy.

## Bell-ringer

10 minutes

This portion of the lesson begins with a routine where students write the objective and additional piece of information in their notebooks as soon as they enter the classroom. Today's additional piece of information is a Big Idea which states that work is equivalent to the change in kinetic energy. The objective of the bell-ringer is to give students a clear understanding of the focus of today's lesson.

In this lesson, I want students to get ready to leverage information gathered from their understanding of work and energy  to construct an explanation of the work-energy theorem using mathematical logic. I follow this bell-ringer activity with a quick think pair share on the concept of work. I project this slide on the interactive whiteboard at the front of the room. Then I ask students to spend five minutes writing information from the slide in their notebooks. Then they think about each scenario and make a decision as whether each scenario describes work in a physics context.  Students spend three minutes discussing the concept of work with their elbow partners. Finally, we have a whole class share out about each scenario.

Some student responses include, "Studying for a physics test is way more work than pushing or lifting anything.", " Pushing really hard against a wall takes the most amount of work because you are never going to move a wall by pushing someone against this wall.", and "Pushing a box with constant velocity takes the most work because you have to make sure to not change how hard you are pushing." Students are audibly surprised when I point out the scenarios like pushing against a wall and studying for an exam do not meet the physics definition of work.

## Creating A Foldable on The Work-Energy Theorem

30 minutes

I choose this activity because I am a great believer in the idea of implementing hands-on activities that require critical thinking helps students craft a better understanding of physics content. Students love this activity and so do I. Within this section of the lesson students create a foldable that demonstrates the work-energy theorem by incorporating three distinct mathematical models:

• Work
• Newton's second Law
• Galileo's equations of motion

Foldables work best when students have a model to implement, a choice of materials and a clear understanding of the information students want to communicate. I provide these three videos on our Edmodo wall as examples of how to make fold-ables. Some students also use Pinterest as a resource to build a foldable within the time limit of 35 minutes for this section.

I distribute Chromebooks, paper, pairs of scissors, and markers. Students spend about fifteen minutes working on the derivation in pairs by writing the steps in their notebooks. I provide students with the three equations:

• W = F *d (step 1)
• Fnet = m *a (step 2)
• vf^2 = vi^2 + 2ad (step 3)

Students substitute the second equation into the first equation to generate this equation:

• W = m*a*d (step 4)

which we call the work like mad formula.

Then students solve the third equation for "a*d" to generate this equation:

• ad = (vf^2-vi^2)/2 (step 5)

Finally, students substitute this fifth equation into the fourth equation to generate this equation:

• W = 1/2 m(vf^2 -vi^2) (step 6)

This final equation shows that W is mathematically equivalent to the change in kinetic energy. After generating this last equation in their notebooks, students choose a fold-able tutorial to follow, choose materials from the resource area and create a foldable that shows that work is mathematically equivalent to the change in kinetic energy. Click here and here to see examples of student work. I assess student work using a 5-point rubric. In the next section of this lesson, I give students team planning time for a roller coaster exhibition project. This project is a time intensive one and is guided by the essential question, "What does physics have to do with roller coaster design?".

## Team Time on Paper Roller Coaster Design

25 minutes

During the second semester, one of the major exhibitions asks students to create a paper roller coaster that maximizes the contact time a marbles spends during a roller coaster lab. Students work in teams of 2-4 using tutorials on EDpuzzle to create component parts for their roller coaster prototypes that they test.

Students spend the next thirty minutes creating component parts for their roller coaster models. I give student pairs three marbles each with the same diameter but made of different materials. Some of the parts that students create include turns, spirals, loops and drops. At this point in the semester, students have created working roller coaster prototypes and are now tweaking their design to meet the constraints of the slowest, safest ride with at least one loop, one hill and one turn. Two examples of student coasters are below:  ## Closure

10 minutes

The closure activity this section asks students to write down ideas about energy in their notebooks using a Free Write Routine. Student responses include: "It's difficult to build a spiral", "I want a tutorial on how to create steps", and "I like fold-ables and would like to make them again".

To wrap up this section of the lesson, I ask students to look at the upcoming lab report due dates that I post on the class Edmodo wall. I also ask students to share pictures of roller coaster component parts by midnight to receive credit for this portion of the lesson.