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**Definition of a Benchmark Lesson:****Benchmark lessons are used during a PBI unit to introduce new ideas or techniques that students cannot learn through an investigation. Benchmark lessons are used for the purposes to teach, and review important ideas or techniques. These lessons should be student interactive and should help students develop a better understanding of concepts. There are many different forms of benchmark lessons such as mind mapping, discussions of readings, interactive lectures, etc. (Krajcik, Czerniak, 2007). It is important to use an interactive benchmark in order to engage the students to better the chances that the students retain the information presented. The learning cycle stresses the importance of interactive learning to help the students actively learn which has been show to directly relate to retention of information (Gillis, McDougall, 2007).**

__Benchmark Lesson 1:__Overview:

Students will partake in an interactive lecture on taking derivatives of given functions. This will be a refresher for the students, as they should be learning/have learned how to take derivatives in their concurrent calculus/pre-cal classes. The class will go through each example together and the students will present their work to the class. The goal of this lesson is for students to remember/learn how to take derivatives of specific functions. The class will work together on this worksheet with guidance from the teacher. This activity was designed using "Direct Instructional Strategies." This activity is a student driven lecture with teacher guidance.

Objectives:

SWBAT:

- Compute the derivative of a function.
- Explain the relationship between a function and its derivative.

This lesson addresses the following TEKS:

·II. Derivatives

Concept of the derivative

- Derivative presented graphically, numerically, and analytically.
- Derivative interpreted as an instantaneous rate of change.

- Slope of a curve at a point.
- Tangent line to a curve at a point and local linear approximation.
- Approximate rate of change from graphs and tables of values.

- Corresponding characteristics of graphs of ƒ and ƒ’.
- Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’.

- Analysis of curves, including the notions of monotonicity and concavity.
- Optimization, both absolute (global) and relative (local) extrema.
- Modeling rates of change, including related rates problems.
- Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

Download the following file in order to view the benchmark lesson:

benchmark_lesson_1.docx | |

File Size: | 55 kb |

File Type: | docx |

__Benchmark Lesson 2:__Overview: Students will partake in an interactive lecture on the relationship between position, velocity, and acceleration. The teacher will guide the students through examples of how to take the derivative of a position function in order to find the velocity and acceleration of that function. The students and teacher will go through graphing of position, velocity and acceleration and how the graphs relate to one another. The teacher will also introduce how to find the maximum and minimum of both velocity and acceleration. This lecture will be driven by students questions and by students presenting their work to the class. This activity was designed using "Direct Instructional Strategies." This activity is a student driven lecture with teacher guidance.

**Objectives:**Students will be able to:

- Graph position, velocity, and acceleration of a function of time
- Relate the change in slope of graphs to their corresponding derivative.
- Predict the motion and speed of an object based on the graphs given.

**This lesson addresses the following TEKS:**

__Alignment with Texas Essential Knowledge and Skills (TEKS):__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.

- An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)
- Understanding continuity in terms of limits.
- Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).

Download the following file in order to view the benchmark lesson:

benchmark_lesson_2.docx | |

File Size: | 72 kb |

File Type: | docx |

__Benchmark Lesson 3:__Overview:

The teacher will engage the students with a video of a curling stone slowing down on the ice and stopping (or some object with friction acting on it) and probe the students to discuss the various forces acting on the object. After that the teacher will give a brief lecture (7-10min) on forces acting on an object and how those would be draw using a free body diagram. The class will then do a few examples as a class then the students will be given time to finish the worksheet that is based around drawing forces on a free body diagram. The teacher will stress the length of arrows correlate to strength of force and how the sum of forces is properly constructed.

**Objectives:**SWBAT:

- construct and analyze free body diagrams.
- determine the magnitude and direction of forces acting on an object.
- Calculate the corresponding components of forces acting at angles.

__Alignment with Texas Essential Knowledge and Skills (TEKS):__(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

Click the following link for the corresponding worksheet for the benchmark teach. (Only use worksheet 2 for classroom examples)

http://bolesscience.weebly.com/uploads/5/2/3/6/5236980/free_body_worksheets.pdf

__Benchmark Lesson 4:__Overview:

This benchmark lesson is an short lab where students compare the friction of various materials through the velocity of an object sliding down a ramp covered in different materials. This will implement the free body diagrams learned earlier in the unit along with allowing the students to see the need to use material with a smaller frictional coefficient in order to minimize the energy lost in motion.

**Objectives:**SWBAT:

- Calculate the kinetic and static frictional coefficients of different materials.
- Analyze various materials to determine which material has the highest frictional coefficient.
- Apply free body diagrams in order to determine the frictional forces acting on an object.

__Alignment with Texas Essential Knowledge and Skills (TEKS):__(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

Click on the follow link to access the lesson plan based for benchmark 4

http://alex.state.al.us/lesson_view.php?id=23807

__Assessments:____Formative Assessments throughout the Unit:__

Throughout the Roller Coaster Unit we will have a series of warm-ups and exit slips that the students will be required to complete. The warm-ups will be structured similarly to what the students are given as exit slips/homework's. The warm-ups will determine whether or not the students were able to effectively complete the exit slips/homework's. The warm-ups will not be used as a grade but they will give the teacher an understanding of what the students are comprehending and what the students are still struggling with. The exit slips/homework's given will be graded and used as a form of assessment. The exit slips and homework's will be created to match the content that was covered each day. If the student is able to complete the homework with the correct answers then this will give the teacher confidence that the student is understanding the material. If the student is unable to complete the homework and exit slips given then this will let the teacher know that additional days are needed for instruction/learning in that specific area of the unit.

Throughout the lesson the students will be required to calculate both mathematical and scientific equations in order to discover the numerical values corresponding to the created roller coaster. Both kinetic and potential energy will be calculated for various points along the roller coaster, such as the point of maximum kinetic and potential and the points where the potential is equal to the kinetic. The position, velocity, and acceleration throughout the roller coaster will be determined along with the frictional coefficient for the material chosen to construct the roller coaster.

In addition to the exit slips and homework's the students are given, the investigations that the students work on throughout the unit will be used as a form of assessment and will also be graded. The investigations, paired with the benchmark lessons contain the meat of the lesson and how well the students do on these activities is a strong indicator of how well the students are understanding the content. If students are struggling through an investigation, then the teacher will be able to spend more time on the specific investigation or concept that the students are struggling with. While the students are working on either benchmark lessons or investigations it is important for the teacher to be walking around, providing support to the students. The teacher can use this time as formative assessment by asking probing questions to see where student thinking is at.

__Opportunities for Students to critique the products of others:__

During week 4 of the Roller Coaster Unit, students will be given the opportunity to work on their final artifacts which includes the design of their roller coaster. Once the students have a well thought-out rough draft of their final artifacts, the students will switch projects with one another. During this time the students will have the opportunity to give feedback and critique other students work while also receiving feedback and positive criticism of their own work. This time will be beneficial to students because it will give students time to fix or edit any part(s) of their final artifact that received criticism.

__Rubrics:__

Please download the following file to see how you will be graded on your final artifact at the end of the unit:

rubric_for_final_artifact_.docx | |

File Size: | 74 kb |

File Type: | docx |

Please download the following file to see how your roller coaster design will be scored:

roller_coaster_rubric_.docx | |

File Size: | 86 kb |

File Type: | docx |