AP Calculus AB Trapezoid, Integration, & Geogebra Rubric

Part 1 – Trapezoid Rule

Pick one of the scenarios outlined, use the trapezoidal rule to solve, and present your findings (in appropriate costume) to the class (maximum 2 groups per topic).

1. Ford Mustang Cobra2. Filling a Swamp3. Running A Marathon4. Coal Gas5. Swimming Pool

Part 2 – Integration using Geogebra

Geogebra Instructions - http://realteachingmeansreallearning.blogspot.ca/2012/06/integration-and-geogebra.html

   Complete a picture using geogebra with at least 5 areas calculated by hand, computer, and/or graphing calculator.

Examples

Grading Scale

30 pts Trapezoid calculations and presentation30 pts Geogebra calculations and presentation20 pts Overall Presentation, Neatness, and Demonstration of Following Directions 10 pts Fair Division of Work – Paper Assessment & Oral Presentation

10 pts Promptness - Meeting all due dates (10 pts = on time or 10 pts off for each day late) Nov 5th

100 pts Total Possible Points

AP Calculus AB Disk, Washer, & Cross Sectional Method RubricThe goal of this project is to creatively demonstrate how

area and volume of solids use calculus concepts.1. 10 Groups using both classes. 2. Each group will need three different items to demonstrate the disk, washer, and cross

sectional volume of the selected items. (Popular items goblets, pottery, butterfly wings, Hershey kisses, and fresh/plastic food items.)

1. Each group will need to accurately trace the graph of each selected item on graph paper. Use data analysis to find a function to model the curve. (Calculator regression, Excel, etc) Use calculus techniques to calculate the volume of the selected item. Be careful of units. Include units with your groups’ graph.

2. Part 1 – Disk Method Item : – Collect data, obtain data list, create data function, graph selected item, provide calculus procedures including a sketch, & calculated area and volume.

3. Part 2 – Washer and/or Shell Method Item: – Collect data, obtain data list, create data function, graph, provide calculus procedures including a sketch of the solid, calculated volume, & use specific units.

4. Part 3 – Cross Section Method Item: – Collect data, obtain data list, create data function, graph, provide calculus procedures including a sketch of the solid, calculated volume, & 3 – D graph demonstrating the cross sections. The cross section can be any shape except a rectangle. The materials can be no thicker than 0.25 inches. Your model must be at least 6 inches long. There must be at least 24 laminations. The base can be any non-linear function. Combinations of functions are allowed.

(For examples, do a Google search for “calculus cross section project/images”.)5. Prior Approval must be obtained from the teacher; no two groups may have the same

object as part of his/her project. 6. Part 4 – Lab Write - up : – Each student will need to write a "Clean & Legal" 3-

paragraph Written Summative paper using the Right-hand corner margin technique from previous projects. The 1st paragraph of the paper needs to contain an explanation of what each person in the group contributed to the project activity, the 2nd paragraph needs to explain what was gained from this activity (other than weight – ha ha ha), and the 3rd paragraph needs to connect this activity with a career and tell the benefits of using calculus that involves the area of shapes and the volume of solids.

7. Within the calculations of the area & volume state the solution method used: vertical, horizontal slices, disks, washers, or shells etc. Several calculus approaches should be accurately demonstrated within this project.

8. Grading Scale:10 pts

Group PortionPart 1 – Disk Method results

20 pts Part 2 – Washer and/or Shell Method results 20 pts Part 3 – Cross Section Method results10 pts Part 4 – Lab Write – up of Written Summative (TYPED!)

20 pts Accuracy of Mathematics & Level of Difficulty

10 pts Neatness, Creativity, Visual Appearance, and Presentation of Project Material

10 pts Promptness (10 pts = on time or 10 pts off for each day late) Dec 17th

100pts Total Possible Points9. Good Luck & Have Fun!

CALCULUS PROJECT Volumes of Solid with Known Cross-Section and Area using

left, right and trapezoidal Riemann SumMake a physical model of a solid with a known cross section on a base with a standard function. You may work with a partner on this project. You will also be computing the area of the base. The following guidelines apply:

1) The cross section can be any shape except a rectangle. 2) The materials can be no thicker than 0.25 inch. Your model must be at

least 6 inches long. There must be at least 24 laminations.3) The base can be any non-linear function. Combinations of functions are

allowed.

Your project must have the following information:1) Clearly defined functions (formula of the functions) (4

points)

2) Explanation in your own words of what the cross section looks like (4 points)

3) Curve of functions sketched accurately (different from the base of your model) (4 points)

4) In excel, calculate the area of each rectangle and trapezoid to compute estimated area (6 points)with approximations using left, right, trapezoidal Riemann sum (in excel).

5) The total area of the rectangles and trapezoids in your model.(9 points)

Using an excel spreadsheet; please also calculate the total area of the base using:

24 rectangles and trapezoids 50 rectangles and trapezoids 100 rectangles and trapezoids 200 rectangles and trapezoids

6) The actual area of the base using a definite integral.If your problem is not integrable, you may use the Numerical Integration feature of your calculator. (Please ask me if you’re not

sure!) (4 points)

7) The computed volume for each slice (in excel)(6 points)

8) The total volume of the slices in your model (9 points)Using an excel spreadsheet; please also calculate the volume of:

24 slices 50 slices 100 slices 200 slices

For each estimation, calculate the percentage of error versus theoretical volume. On your poster board, you will display the estimated volume using

24 slices only and the percentage error of each estimation. Send the excel spreadsheet electronically to me at jaharris@hbuhsd.edu

9) The actual volume as defined by a definite integral. If your problem is not integrable, you may use the Numerical Integration feature of your calculator. (4

points)

AP Calculus AB Treasure Hunt Rubric

The goal of this project is to creatively review the mathematical material covered in AP Calculus AB.

1. Each team of 3 students will create a “Clean and Legal” Mathematical Treasure Map using information from class on various types of treasure maps and or http://www.dltk-kids.com/crafts/pirates/mmap.html for map structural hints:

a. Navigator - Map Readerb. Accountant – Paper configurations / Writerc. Calculator Expert

2. Each “Treasure Map” and/or Scavenger Hunt of Clues will contain “Clean and Legal” AP Calculus AB – Chapters 1 – 7 review problems that when solved lead to the number of “paces” a person should take or a specific room location using the 2nd floor of the school as the “Island” for the “Treasure”.

a. 12 math review problems (approximately 2 per chapter)b. Mathematical Limerick as a clue to the “treasure’s secret” (example: “The Teacher”)c. Pictures to create the “Island’s” terrain.

The TeacherThere is a teacher from Leeds

Who swallowed a packet of seedsAnd in less than an hourHer nose was a flower

And her hair was a bundle of weeds.(From Stephen Gallagher –

www.funnypoets.com)

Limerick Pattern: a a b b a

3. Students will hide a “Clean and Legal” small treasure: candy bar, small gift, etc. for their classmate treasure hunters to find using the Treasure Chests provided by the Instructor.

4. Students will devise a way to be recognized as “Treasure Hunters” abiding by the School Dress Code Policy i.e. T-shirts, badges, costumes, etc.

5. Grading Scale:30 pts Treasure Map30 pts Mathematical problems (Solutions & problems given to instructor)20 pts Team Work & Treasure Hunting Identification10 pts Small Treasure10 pts Promptness & Neatness (10 pts = on time or 10 pts off for each day late) Jan 6th 100 pts Total Possible Points

6. This Assignment will count as Unit Test 8 and Lab 4 (Grades = 100, 95, 90- tentative, 80, 70, 60, 50, 20, or 0) if all requirements are met.

7. Good Luck “Mates” & Have Fun!

Scenario 1 – Ford Mustang Cobra

The accompanying table shows time-to-speed data for a 1994 Ford Mustang Cobra accelerating from rest to 130 mph. How far had the Mustang traveled by the time it reached this speed? (Use trapezoids to estimate the area under the velocity curve, but be careful: the time intervals vary in length.) Find the same statistics for a popular racing type car on the market today and repeat the process. Were the good old days better days for muscle cars?

Scenario 2 – Filling a Swamp

Scenario 3 – Marathon

Scenario 4 –Coal Gas

Scenario 5 – Swimming Pool (Trapezoid Rule)