city of burlington public school district curriculum · a texas instruments 84 plus graphing...

CITY OF BURLINGTON PUBLIC SCHOOL DISTRICT CURRICULUM Revision Date: 08/20/2012 Submitted by: Jill Winegar Antoinette Margaretta

Calculus Honors

Advance Mathematics beyond Common Core Curriculum Standards

“Calculus need not be made easy, it is easy.” Jaime Escalante

Table of Contents

Topic Page

Overview 3

Standards Overview 6

Notes on Curriculum 8

Scope and Sequence 9

Domain and Standard Key 19

Common Core State Standards for Calculus 25

COURSE OVERVIEW

Introduction Calculus is the study of limits, derivatives and integrals through the Fundamental Theorem of Calculus and some applications of the definite integral. This study corresponds to slightly more than the first semester course taught at many colleges and universities. The Honors Calculus course is designed as a full year, challenging course on limits, derivatives and integrals which will focus on real world methodology and application and lead the successful student to a working knowledge of single variable Calculus. Students who elect to may receive college credit through the CAP program at BCC. This course requires that students will have successfully completed Precalculus (B or better) and are capable of self motivation and direction. All students will have and use a graphing calculator on a daily basis. While most students have their own, the school issues graphing calculators for those students who do not. Given the successful completion of this course, students will be prepared to undertake a college level Calculus course beginning with the 2nd level. By successfully completing this course, the student will be able to: • Work with functions represented in a variety of ways and understand the connections among these representations. • Understand the meaning of the derivative in terms of a rate of change and local linear approximation, and use derivatives to solve a variety of problems. • Understand the relationship between the derivative and the definite integral. • Communicate mathematics both orally and in well-written sentences to explain solutions to problems. • Model a written description of a physical situation with a function, a differential equation, or an integral. • Use technology to help solve problems, experiment, interpret results, and verify conclusions. • Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement. • Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

A Balanced Approach

Current mathematical education emphasizes a “Rule of Four.” There are a variety of ways to approach and solve problems. The four branches of the problem-solving tree of mathematics are: • Numerical analysis (where data points are known, but not an equation) • Graphical analysis (where a graph is known, but again, not an equation) • Analytic/algebraic analysis (traditional equation and variable manipulation) • Verbal/written methods of representing problems (classic story problems as well as written justification of one’s thinking in solving a problem—such as on our state assessment)

Technology Requirement A Texas Instruments 84 Plus graphing calculator in class regularly. The calculator in a variety of ways including: • Conduct explorations. • Graph functions within arbitrary windows. • Solve equations numerically. • Analyze and interpret results. • Justify and explain results of graphs and equations. Please note, students selecting this Calculus course should have already met the college and career ready standards plus standards marked with a +. Calculus is not covered by the Common Core Standards as it exceeds the requirements expected in the high school curriculum. This course is certified by Burlington County Community College as meeting the requirements for CAP credit and is a dual enrollment course. The Standards for Mathematical Practice remain an integral part of student academic development as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years.

of 35

Primary Resources: Textbook Title: Calculus (Single Variable, Early Transcendentals) Authors: Finney, Demana, Waits, Kennedy Publisher: Pearson Education, Inc. Copyright: 2004 Title: Calculus Author: Rogawski, Cannon Publisher: BFW Copyright: 2012 Supplementary Resources: In order to develop mastery of stated objectives, the following may be used: - Extra practice problems - Reinforcement exercises - Transparencies - On-line applets for motion and development problems - Maintenance of a notebook and organizational material - Computer assisted instruction - On-line quizzes and exercises - Group explorations and chapter projects - Self assessments - Graphing calculator explorations Internet Resources: The website associated with our text (provides dynamic resources including TI graphing calculator downloads, online quizzing, study tips, Explorations and end-of-chapter projects, and InterAct Math tutorials. In addition, students are encouraged to use www.khanacademy.com , a free math video tutoring site. During class time, there is a daily PowerPoint, demonstration videos (www.education.ti.com) and use of Discovery Learning videos for real world applications. All class resources as well as class work, homework and guided notes are available via the electronic gradebook and the class web site (www.burlington-nj.net link Winegar). There are also several college on line learning resources that my students have permission to use to view classroom lectures and/or work sessions for additional insight and learning. These include MIT on line which is an excellent, high level calculus course for entering freshman (ocw.mit.edu).

http://www.khanacademy.com/

http://www.education.ti.com/

http://www.burlington-nj.net/

of 35

Mathematics Standards for High School Overview

1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to de-contextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 4. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables,

of 35

graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6. Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7. Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 8. Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

of 35

Notes on the Curriculum A unified approach to problem solving will be used in all math courses. The UMAR model adopted by the District Math Committee in 2012 includes four key steps:

By consistently applying these steps, students will forge critical math practices that transcend the complexities and intricacies of the many concepts that will be studied. The Essential Questions noted in the scope and sequence will be summations of understanding and applications of the unit’s key concept. These questions, in addition to other math journal prompts, will be assigned for students to reflect upon and respond in report or essay form. They will use the appropriate math vernacular and reasoning and may utilize any resources available to construct these essays. The Suggestive Student Output column in the Standards section details specific knowledge and skills pertaining to the standard and objective. Evidence of demonstrated understanding may take the forms of illustrated or visual vocabulary, diagrams, graphic organizers, and other devices that can be incorporated into the student notebook.

of 35

SCOPE AND SEQUENCE

Unit Description

Common Core

Standard(s) Domain & Standard

Suggested Timeline

Pacing

Benchmarking Suggested Interdisciplinary Activities

Example for Each Subject Area

Unit P: Introductory Unit Course Overview and Expectations

General information about Calculus and the scope of the course

Student Resources- text, in class and online Course Outline Useful Acronyms

Sept. 5 days phil4 AP Calculus AB Entrance Exam

Unit 1: Limits and Continuity Unit Expectations: By the conclusion of unit one, the learner will demonstrate an understanding of the behavior of functions. Specific demonstrations of ability include:

Demonstrate an understanding of limits both local and global.

o Calculate limits, including one-sided, using algebra.

o Estimate limits from graphs or tables of data.

Recognize and describe the nature of aberrant behavior caused by asymptotes and unboundedness.

o Understand asymptotes in terms of graphical behavior.

o Describe asymptotic behavior in terms of limits involving infinity. Compare relative magnitudes of functions and their rates of change.

Identify and demonstrate an understanding of continuity of functions.

o Develop an intuitive understanding of continuity. (Close values of the domain lead to close values of the range.)

F.TF.3 N.RN.1 N.RN.2

Sept. –Oct. 30 days Lesson Quizzes Chapter

Checkpoints Chapter Tests

Benchmark 1

(week 6) 20 multiple choice

questions and 4 free response questions

covering PreCalculus basics, limits and

continuity.

Evaluation Criteria: Students are evaluated by results of homework and classwork, labs, a minimum of 2 formal quizzes and a formal

Career Education: Students will develop an intuitive understanding of the nature of limits and explore the connection between the graphical, numerical and analytical approaches to determining limits. Students will complete this activity by investigating behavior of several functions numerically using and t-chart and calculator tables and then graphically using calculators. This lab is completed at the beginning of the Limits Unit. After students have covered the material relating to analytical methods of finding limits, students again revisit exploring the connection between evaluation of limits graphically, numerically and analytically by rechecking their results using Calculus.

Health/PE – While exercising, a person’s recommended target heart rate Is a function of age. The recommended number of beats per minute, y, is given by the function y = f(x) = -0.85x + 187, where x represents a person’s age in years. Determine the number of recommended heart beats per minute for the

of 35

o Understand continuity in terms of limits. o Develop a geometric understanding of graphs of

continuous functions. (Intermediate Value Theorem and Extreme Value Theorem).

Unit Essential Questions

How are the value of a function and the value of a limit the same? Different?

What is an indeterminate form and what does it mean algebraically and graphically?

How can the limit be used to create a tangent line from a given secant line?

assessment at the conclusion of the unit.

Authentic Assessment:

Students will develop an intuitive understanding of the nature of limits and explore the connection between the graphical, numerical and analytical approaches to determining limits. Students will complete this activity by investigating behavior of several functions numerically using and t-chart and calculator tables and then graphically using calculators. This lab is completed at the beginning of the Limits Unit. After students have covered the material relating to analytical methods of finding limits, students again revisit exploring the connection between evaluation of limits graphically, numerically and analytically by rechecking their results using Calculus.

following ages and explain the results. What is the limit of the human heart?

English Language Arts & Literacy – Congressional Academic Funding amounts are published annually. Let x = 0 represent 1980. Perform regression analysis to build a prediction model for this funding. What does the model predict about future funding? Document and present the results. Science – Population models in federal wildlife reserves are based on current count, growth rate and sustainable populations. Choose an animal to research how their model influences hunting policy. History/Social Studies – Using the results of the animal population models, determine what laws have been enacted to either increase or decrease an animal population.

Technical Subjects- Students will use Microsoft Excel to calculate the area of regular polygons that are both inscribed and circumscribed about a circle of radius one. Students should realize that as the number of sides increase, the area of the polygons should approach the value of pi from the left and the right.

World Languages – When analytic geometry was developed in the seventeenth century, European scientists still wrote about their work and ideas in Latin, as the universal language. How has that influenced the vocabulary of calculus?

Unit 2: The Derivative and its Applications Derivative Techniques

F.TF.3

Oct.-Dec.

25 days

Lesson Quizzes Chapter Checkpoints

Career Education- The members of the Blue Boar society always divide the pavilion rental

of 35

Unit Expectations: By the conclusion of unit two, the learner will demonstrate an ability to recognize, construct and solve derivatives in all 12 families of functions. Specific demonstrations of ability include:

Explore and interpret the concept of the derivative graphically, numerically, analytically and verbally.

o Interpret derivative as an instantaneous rate of change.

o Define derivative as the limit of the difference quotient.

o Identify the relationship between differentiability and continuity.

Apply the concept of the derivative at a point. o Find the slope of a curve at a point. Examples are

emphasized, including points at which there are vertical tangents and points at which there are no tangents.

o Find the tangent line to a curve at a point and local linear approximation.

o Find the instantaneous rate of change as the limit of average rate of change.

o Approximate a rate of change from graphs and tables of values.

Interpret the derivative as a function. o Identify corresponding characteristics of graphs of

ƒ and ƒ'. o Identify relationship between the increasing and

decreasing behavior of ƒ and the sign of ƒ'. o Investigate the Mean Value Theorem and its

geometric consequences. o Translate between verbal and algebraic

descriptions of equations involving derivatives.

Demonstrate fluency and accuracy in the computation of derivatives.

o Find the derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.

o Use the basic rules for the derivative of sums,

Chapter Tests

Evaluation Criteria: Students are evaluated by results of homework and classwork, labs, a minimum of 2 formal quizzes and a formal assessment at the conclusion of the unit.

Benchmark 2 (Take Home – Thanksgiving Break): Cumulative assessment comprised of 20 multiple choice questions and 4 free response questions covering limits, continuity and rules of differentiation. (Jon Ragowski).

Benchmark 3 (Christmas Break Take Home): Cumulative assessment comprised of 20 multiple choice questions and 4 free response questions covering limits, continuity, rules of differentiation and applications of differentiation. (Jon Ragowski).

fee for their picnics equally among the members. Currently, there are 65 members and the pavilion rents for $250. The pavilion cost is increasing at a rate of $10 per year while the membership is increasing at a rate of 6 members per year. What is the current and predicted rate of change in each members’ share of the pavilion rental fee? This model is the basis for condominium and other membership fee structures. Health/PE – While acetaminophen relieves pain with few side effects, it is toxic in large doses. One study found that only 30 % of parents who gave acetaminophen to their children could accurately calculate and measure the correct dose. What is the best model for dosing based on age, weight or height? How does the rate of change of the dosage relate to the changing growth rate of children? English Language Arts & Literacy – Each student will write a verse as a memory aid for derivative rules. The class together will combine them into an ode, a song, or a rap. The class may opt to make this into a video. Science – If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to volume V by a formula composed of variables and constants. What is the differential and how many of these gas/pressure formulas are there? History/Social Studies – Bear Creek, GA, is a favorite river of kayaking enthusiasts dropping more than 770 feet over one stretch of 3.24 miles. By reading a contour map, one can estimate the elevations (y) at various distances

of 35

products, and quotients of functions. o Use the chain rule and implicit differentiation.

Derivative Applications Unit Expectations: By the conclusion of unit three, the learner will demonstrate an ability to recognize, construct and solve derivatives for a variety of applications. Specific demonstrations of ability include:

Interpret the second derivative.

Identify the corresponding characteristics of the graphs of ƒ, ƒ', and ƒ".

o Identify the relationship between the concavity of ƒ and the sign of ƒ".

o Identify points of inflection as places where concavity changes.

Apply the derivative in graphing and modeling contexts. o Analyze curves, with attention to monotonicity and

concavity. o Optimize with both absolute (global) and relative

(local) extrema.

Model rates of change, including related rates problems.

Use implicit differentiation to find the derivative of an inverse function.

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

Interpret differential equations geometrically via slope fields and the relationship between slope fields and solution curves for differential equations.


How can we discuss a function’s behavior using Calculus?

How can modeling a situation mathematically lead to optimizing the quantities involved?

A.CED.3 F.IF.4 F.IF.5

G.MG.3

Dec. – Jan.

25 days

(x) downriver from the start of the kayaking route. Using calculus techniques how would you identity the most dangerous section of the river? Technical Subjects- Students will determine the rule for taking derivatives of composite functions (chain rule). Students will be given composite functions and asked to guess the derivative and then determine the derivative using a TI-84 calculator. Students will complete the activity by explaining in correct sentences how to take the derivative of any composite function and how to confirm graphically. World Languages – Gottfried Wilhelm Leibniz, a German mathematician, writing on differentials was a foundation block for Calculus. To Leibniz the key idea was that the differential was an infinitely small quantity that was almost like zero but could be used in the denominator. Research his work on the differential (in English of course) and discover why many current mathematicians still read, research and publish in German.

Unit 3: The Definite Integral and its Applications Techniques of Integration

F.TF.3 Feb 20 days Lesson Quizzes Chapter Checkpoints

Career Education- - Students will work as the owner of a canning company whose goal is to

of 35

Unit Expectations: By the conclusion of unit four, the learner will demonstrate an ability to recognize, construct and solve definite and indefinite integrals using a variety of methods. Students will thoroughly explore the Fundamental Theorem. Specific demonstrations of ability include:

Explore and interpret the concept of the definite integral. o Compute Riemann sums using left, right, and

midpoint evaluation points. o Find the definite integral as a limit of Riemann

sums over equal subdivisions. o Find the definite integral of the rate of change of a

quantity over an interval interpreted as the change of the quantity over the interval:

o Identify basic properties of definite integrals.

Apply standard techniques of anti-differentiation. o Find anti-derivatives following directly from

derivatives of basic functions. o Find anti-derivatives by substitution of variables.

(including change of limits for definite integrals).

Apply and interpret the Fundamental Theorem of Calculus.

o Use the Fundamental Theorem to evaluate definite integrals.

o Use the Fundamental Theorem to represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined.

Applications of Definite Integrals Unit Expectations: By the conclusion of unit five, the learner will demonstrate an ability to recognize, construct and appropriate integrals to solve area, volume and accumulated change applications. Specific demonstrations of ability include:

Define and use appropriate integrals in a variety of

F.IF.6

Mar

20 days

Chapter Tests


Benchmark 5 (February): Cumulative assessment comprised of 20 multiple choice questions and 4 free response questions covering limits, continuity, rules of differentiation, applications of differentiation and the integral. (Jon Ragowski).

minimize the surface área of a can given a certain volume. They will compare their results obtained mathematically with the results of actual cans on the market today. They can discuss the reasons for discrepancies in the projects solution or in a journal. Health/PE – The number of liters of blood your heart pumps in a fixed time interval is called your cardiac output. How can a physician measure a patient’s cardiac output without interrupting the flow of blood? Download the statistics from actual tests and perform the calculations.

English Language Arts & Literacy – Students are given a table of values showing time elapsed and the download speed of a file from the internet in kb/sec. Using this data of rate of change versus time, the students will compute midpoint Riemann sums to approximate the size of the downloaded file. Then they will calculate acceleration at a specific time, approximate the size of the downloaded file by using a trapezoidal sum, and the average download speed. They will then document and present their results.

Science – Archimedes Parabola. Students will prove Archimedes formula regarding the area under a parabola. They will start with a few concrete examples and then move on to the general case.

History/Social Studies – The notion of a function’s average value is used in economics to study things like average daily inventory. An inventory function I(x) gives the number of radios, shoes, tires or whatever product a firm

of 35

applications. o Interpret the integral of a rate of change to give

accumulated change. o Find the area of a region. o Find the volume of a solid with known cross

sections. o Find the average value of a function. o Find the distance traveled by a particle along a

line.


How does the antiderivative relate to the derivative of a function?

How can simple shapes be used to approximate the area under a curve?

How are the concepts of integration and differentiation related?

How do logarithmic and exponential functions play a role in Calculus?

How can a rate of change of a variable quantity be used to find a total amount?

has on hand. The average value of I over a time period is the firm’s average daily inventory for the period. How did this form the basis for Just In Time Manufacturing theory first used by the military and then by industry world wide?

Technical Subjects- Use the SMART Board/calculators to show the volumes of the regions when applying discs, washers, shells, and cross sections.

World Languages- German Mathematician Georg Riemann (1826 – 1866_ is considered the founder of mathematical methods of integrals. Write a paper about his life and his contributions to mathematics, in particular his contributions to what is now called Riemann Sums.

Unit 3: Differential Equations and Mathematical Modeling Unit Expectations: By the conclusion of unit five, the learner will demonstrate an ability to recognize, construct and solve differential equations using a variety of methods. Students will thoroughly explore math modeling in growth and decay. Specific demonstrations of ability include:

Define and use appropriate integrals in a variety of applications.

o Find specific anti-derivatives using initial conditions.

o Solve separable differential equations and use them in modeling. In particular, study the equation y' = ky and exponential growth

o Use differential equations within slope fields to solve for general antiderivatives.

F.IF.8 F.LQE.5 F.TF.6 F.TF.7

Apr 20 days Lesson Quizzes Chapter

Checkpoints Chapter Tests


Benchmark 5 (End of

Career Education- McDonalds and Dunkin Donuts have strict rules about the shelf life of products based on Newton’s Law of Cooling. What are these rules and what are their bases?

Health/PE: Students will gain an understanding of differential equations by using TI-84 software to plot slope fields. Given data concerning the amount of land necessary to provide food for one person and also given the population of the world at different time, students will calculate the rate of growth of the world’s population using growth and decay formulas to calculate when the maximum sustainable population of the earth will be reached. Student will research their methods based on the methods used to calculate herd

of 35

o Apply initial conditions to find specific solution equations.


How can models be built to describe rates of change in real-world applications?

March):

Cumulative assessment comprised of 20 multiple choice questions and 4 free response questions covering limits, continuity, rules of differentiation, applications of differentiation; the integral and differential equations (Jon Ragowski).

size via the Department of Agriculture.

English Language Arts & Literacy – Carbon-14 dating uses radioactive elements to date events from the earth’s past. Research the formulas used for 4 of the radioactive elements and how and hen they can be used. Use one of the formulas to prove or disprove the date of either the Shroud of Turin or the Haddonfield dinosaur. Present your results.

Science – Students will research the rotary engine which is used by some car manufacturers. They will understand what the engine looks like and use a cross-section of it to calculate its cross-sectional área. They must discuss different ways to find the área and actually use one way to come up with an approximation. Then they must discuss what they can do to improve their method so that their approximation will become closer to the actual área.

History/Social Studies – By modifying the exponential growth model we can créate a model for the current world population. Students will compare and contrast the exponential model with the population model and discuss problems with the current rate of population growth in the world.

Technical Subjects- Students will use Microsoft Excel to develop a spreadsheet to compare the cost of car purchase given a variety of interest rates and terms of contract.,

World Languages – The estimate of 15,500 years for the age of the cave paintings at Lascaux France is based on carbon-14 dating. Research the caves. Is the dating accurate

of 35

according to modern or historical methods? John Napier (1550-1617) was the Scottish Laird who invented logarithms. He Ws the first person to answer the question’ What happens if you invest an amount of money at 100 % yearly, compounded continuously?” What did he mean? What is the answer?

Unit 4: Parametric and Polar Systems

Define parametric equation o Graph curves parametrically using a graphing

calculator o Solve application problems using parametric

equations

Define polar equations o Convert points and equations from polar to

rectangular coordinates and vice versa o Graph polar equations o Determine the maximum r-value and symmetry of

a graph


How can other coordinate systems be used to better describe real-life relationships?

N.CN.4 N.CN.5

May 10 days Career Education- Some baseball stadiums are known as home run parks. What makes a home run park? Explain parametrically. Health/PE – Parametric motion is basis of many sports especially field, shooting and ball sports. How does an athlete calculate the trajectory of say a shot put in order to maximize horizontal distance? Explore golf course design as a product of parametric analysis. Present the findings. English Language Arts & Literacy – A human cannonball is fired. The circus performer hopes to land in a special cushion without hitting the roof of the tent. Explain how the performed can be fired to the cushion without striking the ceiling? If they can, calculate the cannon’s angle of elevation. Science – Explain how drag force is calculated and how it acts in a direction opposite to the velocity of the projectile. How are calculations adapted in ballistics to account for the drag force? History/Social Studies – To open the 1992 Summer Olympics in Barcelona, bronze medalist archer Antonio Rebollo lit the Olympic torch with a flaming arrow. What calculations had to be made in order for this impressive shot to be successful? Was it?

of 35

Technical Subjects- Students will use Microsoft Excel to establish a distance formula that can be used to calculate the distance between two points in polar coordinates. World Languages - The graph of an equation of the form r = aӨ where a is any non zero constant, is called an Archimedes spiral. Is there anything special about the widths between the successive turns of such a spiral? How is Archimedes and how did he die?

Unit 5: Exam Preparation

See all above

June 10 days Final Exam Preparation

In addition to formal review and individual preparations, students will have their choice of the following: Culminating Project 1 (ongoing through unit 7): Each student in the class will pick a topic we have covered in the previous units and teach a review lesson about it to the class. Student presentations will include a written lesson plan (to turn in), an oral presentation, a mini-quiz for the end of class, and an answer key to that quiz with detailed explanations, in complete sentences, of each problem solved.

of 35

Culminating Project 2- The Sudoku Challenge: Students work in small groups to develop their own system of notation for describing, solving, and writing theorems about the Sudoku puzzle. Students get the opportunity to see how difficult it can be to come up with a concise, clear, intuitive, and suggestive way of symbolizing a problem. Students also get to do their own “mathematical work”, which is presented to the teacher as a written project for a grade and to their classmates as oral presentations. Culminating Project 3- Iron Chef: Using their favorite family recipe, students will rewrite the instructions in calculus. They must have at least one original problem from each of the units. They will provide a copy of the “calculus” recipe and may choose to prepare the dish as part of their class

of 35

presentation. All projects will become part of a multi-year class notebook.

Common Core Standard Domain Key (Bold Apply to Fourth Year Courses)

The Complex Number System N-CN Perform arithmetic operations with complex numbers. 1. Know there is a complex number i such that i2 = −1, and every complex number has the form a + bi with a and b real. 2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Represent complex numbers and their operations on the complex plane. 4. (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. 5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°. 6. (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. Use complex numbers in polynomial identities and equations. 7. Solve quadratic equations with real coefficients that have complex solutions. 8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). 9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Vector and Matrix Quantities N-VM Represent and model with vector quantities. 1. (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v). 2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. 3. (+) Solve problems involving velocity and other quantities that can be represented by vectors. Perform operations on vectors.

of 35

4. (+) Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. 5. (+) Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy). b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). Perform operations on matrices and use matrices in applications. 6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. 7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. 8. (+) Add, subtract, and multiply matrices of appropriate dimensions. 9. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive proper-ties. 10. (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. 11. (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. 12. (+) Work with 2 Reasoning with Equations and Inequalities A-REI Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Solve systems of equations.

of 35

5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 7. Solve a simple system consisting of a linear equation and a quadratic equation in two varia-bles algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. 8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. 9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (us-ing technology for matrices of dimension 3 Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation in two variables is the set of all its solutions plot-ted in the coordinate plane, often forming a curve (which could be a line). 11.Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear,

polynomial, rational, absolute value, exponential, and logarithmic functions.★

12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Interpreting Functions F-IF Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 2. Use function notation, evaluate functions for inputs in their domains, and interpret state-ments that use function notation in terms of a context. 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative

maximums and minimums; symmetries; end behavior; and periodicity.★

5. Relate the domain of a function to its graph and, where applicable, to the quantitative rela-tionship it describes. For example, if the function h(n) gives the number of person-

hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★

6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

Analyze functions using different representations.

7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

of 35

e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Building Functions F-BF Build a function that models a relationship between two quantities.

1. Write a function that describes a relationship between two quantities.★

a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★

Build new functions from existing functions. 3.Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 4. Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1. b .(+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain. 5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Linear, quadratic, and exponential F-LE

Construct and compare linear, quadratic, and exponential models and solve problems.

of 35

1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

2.Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). 3.Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Interpret expressions for functions in terms of the situation they model.

5. Interpret the parameters in a linear or exponential function in terms of a context.

Trigonometric Functions F-TF Extend the domain of trigonometric functions using the unit circle. 1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. 2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counter-clockwise around the unit circle. 3.(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π /3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number. 4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Model periodic phenomena with trigonometric functions.

5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★

6.(+) Understand that restricting a trigonometric function to a domain on which it is always in-creasing or always decreasing allows its inverse to be constructed. 7.(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms

of the context.★

Prove and apply trigonometric identities. 8.Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant. 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Expressing Geometric Properties with Equations G-GPE

of 35

Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 2. Derive the equation of a parabola given a focus and directrix. 3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Use coordinates to prove simple geometric theorems algebraically. 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Geometric Measurement and Dimension G-GMD Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

Visualize relationships between two-dimensional and three-dimensional objects. 4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Using Probability to Make Decisions S-MD Calculate expected values and use them to solve problems. 1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. 2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probabil-ity distribution. 3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. 4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Use probability to evaluate outcomes of decisions.

of 35

5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. 6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Common Core State Standards for Calculus

Domain &Standard

HS Calculus Cluster

Standard &Student Learning Objectives (SLO)

References/ Resources

Suggested Instructional

Activities

Suggested Student Output

Assessments: Portfolios,

Evaluations, & Rubrics

Multimedia Integration Accommodation of Special Needs

Students (SE, ELL, 504, G&T)

F.TF.3

F.TF.6

F.TF.7

Extend the domain of trigonometric functions using the unit circle.

Model periodic phenomena with trigonometric functions.

F.TF.3: (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π /3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.

F.TF.6: (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. F.TF.7: (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in

terms of the context.★

Calculus Textbook 1.6, 3.5, 3.8

Open-ended questions. Non-routine problems Teacher will periodically provide essential background information including key formulas, methods, and properties via notes and PowerPoint presentations. Review of common errors associated with topic Guided instruction Lesson check- Do you understand?

Calculate a limit using a table of values

Calculate a limit graphically

Use and understand the precise definition of a limit.

Calculate a limit algebraically using the limit laws.

Calculate a limit that is in an indeterminate form

Discuss the continuity of a function at a point

Discuss the continuity of a function on a domain

Guided Notes

Homework

Calculator Lab

Reports/essays

Alternative or project-based assessments will be evaluated using a teacher-selected or created rubric or other instrument

Utilize various Websites, including but not limited to:

PowerAlgebra.com

Study island

Classtools.net, phschool.com,

measinc.com (NJ assessments website),

usu.edu

National library of virtual manipulative

kutasoftware.com, edhelper.com

Definedstem.com Online problems. Audio /Themed Music You/Teacher Tube Smart Board Email I-pads (as available) Videos

-Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. - Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or

of 35

Common Core State Standards for Calculus

Domain &Standard

HS Calculus Cluster




Activities






Understand and apply the Squeeze Theorem

Calculate limits that approach infinite

Calculate limits at infinity

Understand how limits relate to vertical and horizontal asymptotes

English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers.

Domain &Standard

HS Calculus Cluster




Activities






N.RN.1

N.RN.2

Extend the properties of exponents to rational exponents.

N.RN.1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

N.RN.2: Rewrite expressions

Calculus Textbook 1.2, 2.1, 2.3, 3.6, 3.7, 6.5, 6.6, 8.1

Open-ended questions. Non-routine problems Teacher will periodically provide essential background information including key formulas, methods, and properties via

Apply the power rule, constant multiple rule, and sum and difference rules to calculate a derivative

Find the equation of a tangent line

Apply the product and quotient rules to

Guided Notes

Homework

Calculator Lab

Reports/essays

Alternative or project-based assessments will be evaluated using a


PowerAlgebra.com

Study island



usu.edu


kutasoftware.com,

- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the

of 35

Domain &Standard

HS Calculus Cluster




Activities






involving radicals and rational exponents using the properties of exponents.

notes and PowerPoint presentations. Review of common errors associated with topic Guided instruction Lesson check- Do you understand?

calculate a derivative

Apply the chain rule to derivatives

Calculate higher derivatives

Apply derivatives to Physics: position, velocity, acceleration

Calculate a derivative implicitly

Understand the concept of a tangent line approximation

Compute the value of a differential and compare it with the actual change

Estimate a propagated error using a differential

Find the differential of a function using the differentiation formulas

teacher-selected or created rubric or other instrument

edhelper.com


number of students that need further support and scaffolding, versus those that need more challenge and enrichment. - Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers.

of 35

Domain &Standard

HS Calculus Cluster




Activities






F.IF.4

F.IF.7

Interpret functions that arise in applications in terms of the context.

Analyze functions using different representations.

F.IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated

cases.★

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F.IF.5: Relate the domain of a function to its graph and, where

Calculus Textbook 1.1 – 1.6, 2.1, 2.1 – 2.4, 4.3, 4.3, 4.5, 5.1, 5.5, 6.1, 6.5, 6.6, 7.1, 7.4,


Understand the definition of extrema of a function on a closed interval

Understand the definition of relative extrema of a function on an open interval

Find extrema on a closed interval

Determine intervals on which a function is increasing or decreasing

Determine interval(s) on which a function is concave up and concave down

Find any points of inflection of the graph of a function

Determine the horizontal and vertical asymptotes of a function

Analyze and sketch the graph of a

Guided Notes

Homework

Calculator Lab Reports

Quiz- Relative and Absolute Extrema, the First Derivative Test, Rolle’s and Mean Value Theorem

Quiz-Concavity, Points of inflection, Second Derivative Test, asymptotes

Test-Analysis of a Curve

Quiz-Graphing polynomial and rational functions

Quiz-Graphing Radical and Trigonometric functions

Test-Graphing functions

Quiz-optimization and Newton’s Method

Quiz-Differentials and Business Applications

Test-Optimization, Newton’s Method, and differentials

Alternative or


PowerAlgebra.com

Study island



usu.edu




- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. - Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing

of 35

Domain &Standard

HS Calculus Cluster




Activities






applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the

function.★

F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)

t, y =

(0.97)t, y = (1.01)

12t, y =

(1.2)t/10

, and classify them as representing exponential growth or decay.

F.IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

function

Solve applied minimum and maximum problems

Find the domain of a function given a real-world scenario

Approximate a zero of a function using Newton’s Method




Find the differential of a function using the differentiation formulas

Understand terms in business and their relationship to

Calculus.

project-based assessments will be evaluated using a teacher-selected or created rubric or other instrument


strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers.

of 35

Domain &Standard

HS Calculus Cluster




Activities






A.CED.3

Create equations that describe numbers or relationships.

A.CED.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Calculus Textbook 4.4, 6.5, 6.6,




Guided Notes

Homework

Calculator Lab

Reports/essays



PowerAlgebra.com

Study island



usu.edu




- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. - Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study

of 35

Domain &Standard

HS Calculus Cluster




Activities






guides, and/or suggestions from special education or ELL teachers.

Domain &Standard

HS Calculus Cluster




Activities






G.MG.3

Apply geometric concepts in modeling situations.

G.MG.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Calculus Textbook 5.1, 4.4, 4.6, 7.1 – 7.4


Find the domain of a function given a real-world scenario

Approximate a zero of a function using Newton’s Method


Guided Notes

Homework

Calculator Lab

Reports/essays



PowerAlgebra.com

Study island



usu.edu



Definedstem.com Online problems. Audio /Themed Music You/Teacher Tube Smart Board Email I-pads (as available)

- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. - Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or

of 35

Domain &Standard

HS Calculus Cluster




Activities






Videos English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers.

Domain &Standard

HS Calculus Cluster




Activities






F.BF.1

F.BF.4

Build a function that models a relationship between two quantities.

F.BF.1: Write a function that describes a relationship between

two quantities.★

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon

Calculus Textbook 1.2, 3.6, 3.7, 3.8, 3.9, 5.4, 6.2, 6.3,

Open-ended questions. Non-routine problems Teacher will periodically provide essential background information including key formulas, methods, and properties via notes and PowerPoint presentations. Review of common errors associated with topic

Write the general solution of a function

Use indefinite integral notation for antiderivatives

Find a particular solution of a function

Use sigma notation to write and evaluate a sum

Use the concept of area

Guided Notes

Homework

Calculator Lab

Reports/essays

Quiz-Antiderivatives

Quiz-Calculating area under the curve using Riemann Sums

Test-Antiderivatives, calculating area under a curve by Riemann Sums, Trapezoidal Rule, and Simpson’s Rule


PowerAlgebra.com

Study island



usu.edu



Definedstem.com Online problems. Audio /Themed Music

- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. - Modifications may

of 35

Domain &Standard

HS Calculus Cluster




Activities






as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

F.BF.4: Find inverse functions.

a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x

3 or

f(x) = (x+1)/(x-1) for x ≠ 1.

b. (+) Verify by composition that one function is the inverse of another.

c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

Guided instruction Lesson check- Do you understand?

Approximate the area of a plane region

Find the area of a plane region using limits

Analyze the approximate error in the Trapezoidal and Simpson’s Rule

Project (To be completed before calculating the área under a curve) Students will research the rotary engine which is used by some car manufacturers. They will understand what the engine looks like and use a cross-section of it to calculate its cross-sectional área. They must discuss different ways to find the área and actually use one way to come up with an approximation. Then they must discuss what they can do to improve their method so that their approximation will become closer to the actual área.


You/Teacher Tube Smart Board Email I-pads (as available) Videos

also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers.

of 35

Domain &Standard

HS Calculus Cluster




Activities






F.LE.4

F.LE.5

Construct and compare linear, quadratic, and exponential models and solve problems.

Interpret expressions for functions in terms of the situation they model.

F.LE.4: For exponential models, express as a logarithm the solution to a b

ct = d where a, c,

and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. F.LE.5: Interpret the parameters in a linear, quadratic, or exponential function in terms of a context.

Calculus Textbook 6.4 – 6.6


Use exponential functions to model growth and decay in applied problems

Use initial conditions to find particular solutions of differential equations

Recognize and solve differential equations that can be solved by separation of variables

Use a differential equation to model and solve an applied problem

Guided Notes

Homework

Calculator Lab

Reports/essays



PowerAlgebra.com

Study island



usu.edu




- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. - Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred

of 35

Domain &Standard

HS Calculus Cluster




Activities






seating, study guides, and/or suggestions from special education or ELL teachers.

city of burlington public school district curriculum · a texas instruments 84 plus graphing...

Documents