##
* *Reflection: Staircase of Complexity
Galileo's Equations of Motion - Section 3: Modeling Motion

Within this lessonI ask students to identify connections between mathematical models and graphical representations of motion. I believe that the layering of these models will allow students multiple ways to access this information and make the information a readily available part of their lexicon or knowledge base.

The errors I see mostly have to do with the step that eliminates the final velocity from the equation. When I reteach this lesson, I will create an edpuzzle screencast with embedded questions that students can follow along with at their own pace. Finally, I tell students at the end of this section that these equations are our "lessons from Galileo" on how to model the motion for objects with constant acceleration.

*Layering Models*

*Staircase of Complexity: Layering Models*

# Galileo's Equations of Motion

Lesson 11 of 11

## Objective: Students will demonstrate an understanding of the connections between a set of kinematic data and Galileo's equations of motion.

## Big Idea: Galileo's equations of motion can be used to model the behavior of objects with constant acceleration.

*85 minutes*

#### Overview

*5 min*

The goal of this lesson is to help students connect graphical data to Galileo's equations of motion. Students model the motion of a dune buggy with graphs for the position, velocity, and acceleration with respect to time. Students also connect Galileo's equations of motion to one another and test the data against the equations that they generate within this lesson. This lesson addresses the HSA-REI.A.1 and HSA-CED.A.2 standards as a way to effectively model the motion of a toy dune buggy car using both graphical and mathematical models. Students investigate and communicate their understanding of the connection between these models of motion using the NGSS Practices of Developing and Using Models (SP2), Analyzing and Interpreting Data (SP4), Constructing Explanations (SP6), and Communicating Information(SP8) that illustrate the Galileo's equation of motion are effective models for an object with constant acceleration.

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 Galileo's equation of motion can be used to model the behavior of an object with constant acceleration.

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. I want students to learn to integrate information from various points of this course into a coherent analysis of the connection between the mathematical models we generate during this class and the data for a toy car's position, velocity and acceleration as a function of time.

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During the first five minutes of this section, I give students a writing prompt 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 "What are the ways we can model the motion of an object?" First, students write their answers in their notebooks. They then share their answers with their station partners. At the end of five minutes, I call on a representative from each table to share their answers with the class. Some student responses include, "Motion maps", "Velocity vs time graphs", and "The slope of a position vs time graph". At this point in the semester, students have used several models including creating motion maps, and kinematics graphs and have investigated motion using both simulations, and a dune buggy lab.

This part of the lesson focuses on synthesizing related information from different lessons and external academic experiences. At the end of this section I ask for volunteers from each station because I want students to summarize their group's feedback and to give candid reflections on the collected information in order to highlight ways that the different models of motion relate to one another. From student feedback, I note that the majority of the class thinks of motion maps as the easiest way to model the motion of an object.

I circulate the class giving both written and verbal feedback as I add stamps to student lab notebooks. Click here to see an example of student work.

#### Resources

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#### Modeling Motion

*45 min*

At the beginning of this section of the lesson, I spend about 1 minute distributing this activity to students. I spend the next 1-2 minutes of this section introducing this task by projecting a set of dune buggy data on the interactive whiteboard at the front of the room. I ask students to derive a set of equations based on this data. Students spend about five minutes graphing the data on the activity sheet. I remind students to label their axes and to include a title and units on their graphical data. Some students ask whether to create best fit lines or to just create an unconnected scatter plot. At this point in the semester, students have worked with several graphs and have used motion maps and mathematical equations to describe the motion of an object.

Next, students spend about twenty minutes using the equations for average velocity and acceleration to generate the remainder of Galileo's Equations of Motion with the assumption of constant acceleration. Students work in self-selected groups to complete this activity in teams of 2-4. Many students remain at their tables and work with their elbow neighbors. However, a few students choose to work at the whiteboard on the center right of our classroom. Click here and here to see examples of student work. Most students are able to generate the equation for final velocity in terms of acceleration, initial velocity and time using basic algebra skills. A few students struggle with generating the equation that connects displacement, acceleration and time.

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In this section, I ask students to write the equations from today's lesson and to then test the data points from the dune buggy lab against them. I distribute this activity for students to complete within the next fifteen minutes of class. Students work on this section with their elbow partners, but each partner turns in an individual copy of the activity.

I ask students to use their six-inch voices and turn the "How are we working Dial" to partner work. At this point in the semester, students have seen the "How are we learning Dial" and have worked in all of the configurations on the Dial. The "How Are We Working?" dial is a poster with the most common student groupings in our classroom with their descriptions in different colors that I display on the whiteboard at the front of the room. It is really helpful for students if they leave the class and re-enter in the middle of a new activity get back into the lesson as soon as they enter the classroom.

To use a six-inch voice means that someone further than six inches from your conversation cannot hear your conversation. I circulate the room and answer any questions that students have about the activity. Some student questions include, "Do we have to write the equation in the standard equation form?", and "Will choosing data points with a large time interval affect my results?" After 15 minutes pass, I collect student work and remind students that I will also post the Challenge Question on our Edmodo wall for them to answer in more detail.

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#### Closure

*10 min*

I provide students with an Exit Slip with a set of writing prompts for a routine called 3-2-1, where I task students with both identifying their personal level of understanding of key ideas within the lesson and identifying the underlying reasons behind their understanding. Some student responses include:"Remembering to cancel units during calculations is important", "Learning how to change one equation into another using substitution is important." While other student responses include suggestions of "Choose data points that are close together or you may not be able to get the math to work", "Add your numbers during the last step in your calculation or your answer will not make sense"

After ten minutes elapse, I collect the slips to grade them. I use this type of closure activity because it encourages students to consider multiple viewpoints on a scientific topic. To wrap up the lesson, I remind students that I will grade and return the exit slips at the beginning of the next lesson.

#### Resources

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- LESSON 1: Lessons from Galileo
- LESSON 2: Investigating Basic Types of Physics Graphs
- LESSON 3: Motion in Terms of Position and Time
- LESSON 4: Introduction to Linear Motion
- LESSON 5: Modeling Motion Using Motion Maps And Position vs Time Graphs
- LESSON 6: Modeling Motion in Terms of Velocity vs Time Graphs, Part 1
- LESSON 7: Modeling Motion in Terms of Velocity vs Time Graphs, Part 2
- LESSON 8: Finding the Slope of a Position vs Time Graph
- LESSON 9: Finding the Slope of a Velocity vs Time Graph
- LESSON 10: Introducing Gravitational Field Strength
- LESSON 11: Galileo's Equations of Motion