JMisciaUbD

Janine Miscia Curr530 UbD

4.2 C 4.3 A,B,C,D 4.5 A,B,C,D,E,F (All Grade 12 Standards) see: http://www.state.nj.us/education/cccs/2004/s4_math.pdf || Geometric and algebraic methods of solving real-life problems are supported by calculus concepts |||| 1. What connections exist between a function and its first and second derivatives?
 * ** Title of Lesson ** || Applications of Derivatives Unit || ** Grade Level ** |||| 11-12 ||
 * ** Curriculum Area ** || Calculus (AP and Honors) || ** Time Frame ** |||| 2 – 3 weeks ||
 * ** Developed By ** |||||||| Janine Miscia ||
 * ** Identify Desired Results (Stage 1) ** ||
 * ** Content Standards ** ||
 * 4.1 A,B,C
 * ** Understandings ** |||||| ** Essential Question(s) ** ||
 * ** Overarching Understanding ** |||| ** Overarching ** || ** Topical ** ||
 * Derivatives and associated computation can be applied to several other disciplines

2. How are derivatives used to maximize or minimize a function that represents a real-life application? || 1. What steps are taken to find absolute and local extrema when given a function?

2. What conditions could exist in a given graph in which there are NO extrema?

3. How are the Mean Value Theorem and the Intermediate Value Theorem applied to justify extrema?

4. What role does the Chain Rule play when solving Related Rate problems?

|| Incorrect use of Chain, Product and Quotient Rules Arithmetic and Algebra errors in simplification || Students will… |||||| ** Skills ** Students will be able to… || 2. Use the derivative to determine the Maximum & Minimum, Concavity, Points of Inflection and Anti-derivative 3. Use Sign charts to justify local and absolute extrema 4. Apply differentiation to real world problems concerning optimization and related rates |||||| 1. Establish connections between a function and its first and second derivative and be able to describe these relationships using the Rule of 4 2. Define critical values as points in which a derivative either equals zero or does not exist 3. Use the derivative to identify the interval over which a function is increasing, decreasing, or equal to zero 4. Describe a local maximum as the point in which the derivative changes from positive to negative 5. Describe a local minimum as the point in which the derivative changes from negative to positive 6. Create equations to minimize or maximize a value or number of objects, and interpret real-world examples in economics, manufacturing, shipping, land surveying and marketing 7. Identify “common” real-world problems in which related rates are present: i.e. ladder sliding down a wall, baseball player running between bases, hot air balloon rising, vehicles commuting to and from various location || Online Tutorials (on HwkNow) Written Assessments: Pop-Quizzes, Formal Quizzes,Chapter 4 Test, Review Dittos, AP Barrons Prep, Old AP Exams Extra Activities: Write your own Calculus Song! Match Game!!!: Pair the original graph and derivative graph || Review concepts from prior calculus unit. Revisit all derivative rules || Performance task as introduction to unit || Utilize activities esp. derivative match game. Online animations and tutorials will assist in learning process. || Interpret results of real-life problems and applications. Use solution rubrics from AP Exams || Constantly use the Rule of 4: students should understand material graphically, algebraically, intuitively and analytically/algebraically. || Unit Packets that outline example problems, theorems, steps, concepts and technology infused exercises Text Assignments: __ 4.3: __ ** #1-6, 8, 11, 13, 18*, 23*, 24*, 25, 29, 31, 32, 33, 37, 41, 45 ** __ 4.4: __  ** #1-18 + #27 and #32 ** __ 4.6: __ 1-6, 9, 10, 12, 13, 16, 20, 22, 23, 30
 * ** Related Misconceptions ** ||
 * Incorrectly identification of given graph as original, first or second derivative.
 * ** Knowledge **
 * 1. Describe the Extreme Value Theorem, the Mean Value Theorem and its Corollary, Rolle’s Theorem in conjunction with identifying absolute and relative extrema on a given function
 * ** Assessment Evidence (Stage 2) ** ||
 * ** Performance Task Description ** ||
 * ** Goal ** |||||||| Optimization in Marketing ||
 * ** Role ** |||||||| Packaging Manufacturer ||
 * ** Audience ** |||||||| Company of Packaged Product (Soup) ||
 * ** Situation ** |||||||| A soup company is hiring your design team to think of new packaging for a soup line. ||
 * ** Product/Performance ** |||||||| Using calculus and methods of optimization, design a soup box that will maximize volume while minimizing cost. Identify all assumptions, variables, costs, materials, etc. Include all geometric and algebraic formulas used. ||
 * ** Standards ** |||||||| All aforementioned CCS ||
 * ** Other Evidence ** ||
 * Homework: Students will work cooperatively to review and discuss problems as well as self-check work with an answer key
 * ** Learning Plan (Stage 3) ** ||
 * ** Where **** are your students headed? Where have they been? How will you make sure the students know where they are going? **
 * ** How will you hook students at the beginning of the unit? **
 * ** What events will help students experience and explore the big idea and questions in the unit? How will you equip them with needed skills and knowledge? **
 * ** How will you cause students to reflect and rethink? How will you guide them in rehearsing, revising, and refining their work? **
 * ** How will you help students to exhibit and self-evaluate their growing skills, knowledge, and understanding throughout the unit? ** Group collaboration and student opportunity to teach the class. ||
 * ** How will you tailor and otherwise personalize the learning plan to optimize the engagement and effectiveness of ALL students, without compromising the goals of the unit? **
 * ** How will you organize and sequence the learning activities to optimize the engagement and achievement of ALL students? **
 * Also: **
 * __ 4.1: __**** 1, 3, 6, 7, 8, 11, 12, 18, 19, 2, 25, 40, 45-48 +32 and 36 by graphing **
 * __ 4.2: __**** 2, 7, 8, 13, 16, 21, 27, 28, 30, 31, 34, 39, 42, 43, 48 **

Extras: 4.1 worksheet, Barron’s Workbook: __ Part IV: Max/Min __ Test I: 2, 45 Test II: 36, 5 Test IV: 11, 20, 35 Test V: 6, 20, 33, 43 Test VI: 38

__ Part V: 2nd Deriv __ Test I: 13, 15, 21, 39 Test II: 8, 25 Test III: 21, 35 Test IV: 28 Test V: 10, 23 Test VI: 38

__ Part VI: Related Rates __ Test I: 3 Test II: 23 Test III: 4 Test IV: 16, 36 Test V: 27, 35 Test VI: 35 || From: Wiggins, Grant and J. Mc Tighe. (1998). //__Understanding by Design__//, Association for Supervision and Curriculum Development ISBN # 0-87120-313-8 (ppk)

I had a rough draft of this unit on file that I used last year, however, it was not nearly as detailed. Having taught this class for 6 years, I have so many plans in my head that, like many other teachers I’m sure, I could teach the class without a single lesson plan. When I was first introduced to UbD, however, it really forced me to critique my teaching methods and alternate assessments I was using for the class. Some lessons and specific subject areas lend itself to this format more than others especially in mathematics which typically is more concrete and has less subjective matter. I found it difficult to fit various skill sets into such tight categories. I felt that the information I wrote in the knowledge and skill “boxes” could be revised to fit either category. Also, going back and identifying the individual content standards and strands was tedious, especially in a math class on which so many previous math skills are built.

I regularly use unit packets which technically reflect the main ideas of UbD: reverse planning. My units take HOURS for me to prepare because they include sample problems, activities, dittos, SAT/HSPA problems, practice tests, answer keys and the homework sets for the ENTIRE chapter. Students do not take notes in their notebooks…they take notes directly into these color-coded packets. (I will bring in a sample if anyone is interested) My unit packets allow my students and me, of course, to stay very organized and on top of assignments. They are no surprises! But, once I put together a packet, I can breathe easily for at least 2 weeks….sometimes 3 or 4.