**An investigation contains all five essential elements of inquiry and must be structured, guided, or open (levels of inquiry). The essential elements of inquiry are as follows: learner engages in scientifically oriented questions, learner gives priority to evidence, learner formulates explanations from evidence, learner connects explanations to scientific/math knowledge, and the learner communicates and justifies explanations.**

__Definition of an Investigation:__An investigation is formed around a specific level of inquiry, which is described below:

Level 1-Confirmation-Students confirm a principle through an activity in which the results are known in advance.

Level 2-Structured Inquiry-Students investigate a teacher-presented question through a prescribed procedure.

Level 3-Guided Inquiry-Students investigate a teacher-presented question using student designed/selected procedures.

Level 4-Open Inquiry-Students investigate topic-related questions that are student formulated through student designed/selected procedures.

A more concise description of the Levels of inquiry can be seen in the picture below:

(Bell, Smetana, Binns, 2005)

__Investigation 1:__Overview:

This investigation incorporates motion sensors to allow for students to create position versus time graphs of their movement. These graphs will then be compared to see the difference in what various motion looks like. The slopes of the lines will then be compared to determine how the graphs change as the students velocity changes. Lastly, the acceleration will be discussed by comparing the curves on a graph and recreating them using the motion sensor. Recreating the graph will allow for the students to see how slowing down and speeding up directly relates to the shape of the graphs. This investigation will be led by the students with them creating the graphs they are using themselves. This investigation is a level 2 to level 3 inquiry based investigation with the main question being provided by the teacher and some possible suggestions of how to perform the lab but the majority of the method along with the entirety of the solutions will be the students responsibility to create and perform.

Objectives:

SWBAT:

- Compute the derivative of a function.
- Determine the position at various velocities and acceleration for a function of time.
- Graph the function for speed, velocity, and acceleration of a chosen function.

Teks:

**§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.

Please click on the following link to view the lesson plan for investigation 1:

http://www2.vernier.com/sample_labs/PSV-35-COMP-graphing_motion.pdf

__Investigation 2:__Overview:

This investigation is centered around kinetic and potential energy, along with the law of conservation of energy. The students will be conducting multiple trials changing the starting height of the marble rolled down the U shaped track. The students will record the velocity of the marble as it rolls along with observing the maximum height the marble rolls up the opposite end. The second part of the investigation is focusing on the conservation of energy by predicting where the marble will land after leaving the ramp.

Objectives:

TSWAT:

- calculate kinetic and potential energy at any point along the path of motion
- predict and analyze the location an object will land after being launched
- measure and record data for different trials throughout the investigation
- implement the law of conservation of energy to solve for and analyze kinetic and potential energy

§112.38. Integrated Physics and Chemistry

(5) Knowledge and skills

(A) recognize and demonstrate that objects and substances in motion have kinetic energy such as vibration of atoms, water flowing down a stream moving pebbles, and bowling balls knocking down pins;

(B) demonstrate common forms of potential energy, including gravitational, elastic, and chemical, such as a ball on an inclined plane, springs, and batteries;

(D) investigate the law of conservation of energy;

Please click on the following link to view the lesson plan for investigation 2:

http://www.energyeducation.tx.gov/pdf/114_inv.pdf