Overview:

Welcome to our PBI unit centered around the design of a roller coaster. In this lesson the students will explore topics such as gravitational energy, kinetic energy, conservation of energy, derivatives, relationship between position, velocity, and acceleration, and more. Throughout this lesson the students will be designing a roller coaster along with calculating the kinetic and potential energy at various points on their roller coasters. There is four benchmark lessons incorporated in this unit in order to ensure the students understand concepts they may not have been introduced to before. The lessons include taking derivatives, relationships between position, velocity, and acceleration, drawing free body diagrams, and frictional coefficients. This unit contains two investigations where the students will partake in a two day lab where they explore both conservation of energy and relationships between position, velocity, and acceleration. Throughout each portion of the lesson various formative assessments have been constructed in order to ensure the students are on track and no misconceptions are being formed. To conclude the unit the students will give a presentation, that meets the requirements of the presentation rubric, which will allow for a final assessment of the students knowledge on the various topics.

Welcome to our PBI unit centered around the design of a roller coaster. In this lesson the students will explore topics such as gravitational energy, kinetic energy, conservation of energy, derivatives, relationship between position, velocity, and acceleration, and more. Throughout this lesson the students will be designing a roller coaster along with calculating the kinetic and potential energy at various points on their roller coasters. There is four benchmark lessons incorporated in this unit in order to ensure the students understand concepts they may not have been introduced to before. The lessons include taking derivatives, relationships between position, velocity, and acceleration, drawing free body diagrams, and frictional coefficients. This unit contains two investigations where the students will partake in a two day lab where they explore both conservation of energy and relationships between position, velocity, and acceleration. Throughout each portion of the lesson various formative assessments have been constructed in order to ensure the students are on track and no misconceptions are being formed. To conclude the unit the students will give a presentation, that meets the requirements of the presentation rubric, which will allow for a final assessment of the students knowledge on the various topics.

The 5 essential elements of PBI are that the students engage in inquiry, collaborate to find solutions, use technology to analyze and gather information, create an artifact to demonstrate what has been learned, and that the project is relevant and meaningful to the students (Krajcik, Blumenfeld, 2006). This unit includes all 5 essential elements of PBI as follows

This project will engage the students and should be relevant to the class since it is centered around roller coasters which students enjoy. The launch letter will allow for the students to connect with the project and help keep them driving throughout the unit, since they have a main goal they are working towards. The anchoring experience of the letter will be brought up throughout the unit in order to keep the connection between the students and the project meaningful.

The misconceptions and struggles anticipated with this project based unit in both the mathematics and physics concepts. Differentiating between continuous and not continuous is simple for functions that do not match up at a specific point but can be confusing for other graphs that are connected but have points that are not differential, for example the absolute value function. During the conservation of energy investigation the students may observe a small amount of energy lost to friction between the objects and the surface. If the calculation of potential energy and kinetic energy do not match the students may assume that the either something was wrong or that the remaining energy disappeared without understanding why the energy was lost. Through the kinetic energy investigation the students should draw the conclusion on the first hill in their roller coaster must be the tallest else the cart will not make it to the end. The reasoning behind this can lead to misconceptions about energy being destroyed instead of redistributed due to friction.

- The project is relevant to the students since it is centered around a theme park close to where they live and is related to the students through the launch letter which acts as an anchoring experience.
- The students engage in inquiry throughout the lesson in both the investigations and construction of the final artifact. Both investigations are centered around level 2 and 3 inquiry (Bell, Smetana, Binns, 2005).
- This unit is driven by group work and student collaboration in order to complete the entirety of the unit. the investigations and artifact creates require student to work together to measure and record data along with to create the roller coaster design.
- The students will use technology in the benchmarks and investigations including the use of motion sensors, graphing calculators, and stop watches.
- The students will be creating a presentation as an artifact for this unit. The artifact will include the design of the roller coaster, analysis of various components of the roller coaster, show that it is a continuous function, and other requirements stated in the artifact rubric.

This project will engage the students and should be relevant to the class since it is centered around roller coasters which students enjoy. The launch letter will allow for the students to connect with the project and help keep them driving throughout the unit, since they have a main goal they are working towards. The anchoring experience of the letter will be brought up throughout the unit in order to keep the connection between the students and the project meaningful.

The misconceptions and struggles anticipated with this project based unit in both the mathematics and physics concepts. Differentiating between continuous and not continuous is simple for functions that do not match up at a specific point but can be confusing for other graphs that are connected but have points that are not differential, for example the absolute value function. During the conservation of energy investigation the students may observe a small amount of energy lost to friction between the objects and the surface. If the calculation of potential energy and kinetic energy do not match the students may assume that the either something was wrong or that the remaining energy disappeared without understanding why the energy was lost. Through the kinetic energy investigation the students should draw the conclusion on the first hill in their roller coaster must be the tallest else the cart will not make it to the end. The reasoning behind this can lead to misconceptions about energy being destroyed instead of redistributed due to friction.

Objectives:

The students will be able to:

1. Calculate potential and kinetic energy at various points

2. Determine velocity and acceleration of an object

3. Calculate frictional force using mass and frictional coefficient

4. Compute the derivative of a function

5. Determine the position at various velocities and acceleration for a function of time.

6. Graph the function for speed, velocity, and acceleration of a chosen function.

7. Integrate citations appropriately into presentations with correct formatting.

The students will be able to:

1. Calculate potential and kinetic energy at various points

2. Determine velocity and acceleration of an object

3. Calculate frictional force using mass and frictional coefficient

4. Compute the derivative of a function

5. Determine the position at various velocities and acceleration for a function of time.

6. Graph the function for speed, velocity, and acceleration of a chosen function.

7. Integrate citations appropriately into presentations with correct formatting.

## The standards that will be addressed during this project based lesson include but are not limited to:

**§111.54. Advanced Placement (AP) Calculus AB **

(Taken from College Board Publication Advanced Placement Course Description Mathematics: Calculus AB, Calculus BC)

I. Functions, Graphs, and Limits

Analysis of graphs.

With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.

Continuity as a property of functions

1. An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)

2. Understanding continuity in terms of limits.

3. Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).

II. Derivatives

Concept of the derivative

1. Derivative presented graphically, numerically, and analytically.

2. Derivative interpreted as an instantaneous rate of change.

Derivative at a point

1. Slope of a curve at a point .

2. Tangent line to a curve at a point and local linear approximation

3. Approximate rate of change from graphs and tables of values.

Derivative as a function

1. Corresponding characteristics of graphs of *ƒ *and *ƒ*’.

2. Relationship between the increasing and decreasing behavior of *ƒ *and the sign of *ƒ*’.

Application of Derivatives

1. Analysis of curves, including the notions of monotonicity and concavity.

2. Optimization, both absolute (global) and relative (local) extrema.

3. Modeling rates of change, including related rates problems.

4. Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

** ****§112.39. Physics**

(b) Knowledge and skills.

(7) Linear functions. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to:

(A) analyze situations involving linear functions and formulate linear equations or inequalities to solve problems;

(B) investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities; and

(C) interpret and determine the reasonableness of solutions to linear equations and inequalities

(c) Knowledge and skills.

(4) Science concepts. The student knows and applies the laws governing motion in a variety of situations. The student is expected to:

(D) calculate the effect of forces on objects, including the law of inertia, the relationship between force and acceleration, and the nature of force pairs between objects

(E) develop and interpret free-body force diagrams

**§110.32. English Language Arts and Reading, English II **

(b) Knowledge and skills.

(15) Writing/Expository and Procedural Texts. Students write expository and procedural or work- related texts to communicate ideas and information to specific audiences for specific purposes. Students are expected to:

(B) write procedural or work-related documents (e.g., instructions, e-mails, correspondence, memos, project plans) that include:

(i) organized and accurately conveyed information;

(ii) reader-friendly formatting techniques; and

(iii) anticipation of readers' questions;

(23) Research/Organizing and Presenting Ideas. Students organize and present their ideas and information according to the purpose of the research and their audience. Students are expected to synthesize the research into a written or an oral presentation that:

(A) marshals evidence in support of a clear thesis statement and related claims;

(B) provides an analysis for the audience that reflects a logical progression of ideas and a clearly stated point of view;

(C) uses graphics and illustrations to help explain concepts where appropriate;

(D) uses a variety of evaluative tools (e.g., self-made rubrics, peer reviews, teacher and expert evaluations) to examine the quality of the research; and

(E) uses a style manual (e.g., *Modern Language Association*, *Chicago Manual of Style*) to document sources and format written materials.

**§111.54. Advanced Placement (AP) Calculus AB**(Taken from College Board Publication Advanced Placement Course Description Mathematics: Calculus AB, Calculus BC)

I. Functions, Graphs, and Limits

Analysis of graphs.

With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.

Continuity as a property of functions

1. An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)

2. Understanding continuity in terms of limits.

3. Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).

II. Derivatives

Concept of the derivative

1. Derivative presented graphically, numerically, and analytically.

2. Derivative interpreted as an instantaneous rate of change.

Derivative at a point

1. Slope of a curve at a point .

2. Tangent line to a curve at a point and local linear approximation

3. Approximate rate of change from graphs and tables of values.

Derivative as a function

1. Corresponding characteristics of graphs of

*ƒ*and*ƒ*’.2. Relationship between the increasing and decreasing behavior of

*ƒ*and the sign of*ƒ*’.Application of Derivatives

1. Analysis of curves, including the notions of monotonicity and concavity.

2. Optimization, both absolute (global) and relative (local) extrema.

3. Modeling rates of change, including related rates problems.

4. Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

**§112.39. Physics**(b) Knowledge and skills.

(7) Linear functions. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to:

(A) analyze situations involving linear functions and formulate linear equations or inequalities to solve problems;

(B) investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities; and

(C) interpret and determine the reasonableness of solutions to linear equations and inequalities

(c) Knowledge and skills.

(4) Science concepts. The student knows and applies the laws governing motion in a variety of situations. The student is expected to:

(D) calculate the effect of forces on objects, including the law of inertia, the relationship between force and acceleration, and the nature of force pairs between objects

(E) develop and interpret free-body force diagrams

**§110.32. English Language Arts and Reading, English II**(b) Knowledge and skills.

(15) Writing/Expository and Procedural Texts. Students write expository and procedural or work- related texts to communicate ideas and information to specific audiences for specific purposes. Students are expected to:

(B) write procedural or work-related documents (e.g., instructions, e-mails, correspondence, memos, project plans) that include:

(i) organized and accurately conveyed information;

(ii) reader-friendly formatting techniques; and

(iii) anticipation of readers' questions;

(23) Research/Organizing and Presenting Ideas. Students organize and present their ideas and information according to the purpose of the research and their audience. Students are expected to synthesize the research into a written or an oral presentation that:

(A) marshals evidence in support of a clear thesis statement and related claims;

(B) provides an analysis for the audience that reflects a logical progression of ideas and a clearly stated point of view;

(C) uses graphics and illustrations to help explain concepts where appropriate;

(D) uses a variety of evaluative tools (e.g., self-made rubrics, peer reviews, teacher and expert evaluations) to examine the quality of the research; and

(E) uses a style manual (e.g.,

*Modern Language Association*,*Chicago Manual of Style*) to document sources and format written materials.