High School Curriculum Map Calculus

Transcription

1 High School Curriculum Map Calculus Marking Period 1 Topic Chapters Number of Blocks Dates Functions /5 9/ /27 10/12 PRE-TEST 2 Limits and Continuity /16 11/ MP 1 ASSESSMENT 2 11/4 & 11/6 Marking Period 2 Topic Chapters Number of Blocks Dates Derivatives /13 12/5 12/7 1/7 1/9-1/22 MP 2 ASSESSMENT 2 1/24 & 1/28 Marking Period 3 Topic Chapters Number of Blocks Dates Applications of Derivatives /31 3/4 Definite Integral /7 4/5 MP 3 ASSESSMENT 2 4/2 & 4/4 Marking Period 4 Topic Chapters Number of Blocks Dates Definite Integral /18 5/8 Differential Equations and Mathematical Modeling /10 5/20 MP 4 ASSESSMENT 5/22 6/13 2 6/9 6/11

2 Content Area: Calculus Unit Title: Prerequisites for Calculus Target Course/Grade Level: Duration: 2 Weeks Unit Overview Description: This unit introduces the use of a graphing utility as a tool to investigate mathematical ideas, to support analytic work, and to solve problems with numerical and graphical methods. The emphasis is on functions and graphs, the man building blocks of calculus. Functions and parametric equations are the major tools for describing the real-world in mathematical terms, from temperature variations to planetary motions, from brain waves to business cycle, and from heartbeat patterns to population growth. Many functions have particular importance because of the behavior they describe. Trigonometric functions describe cycles, repetitive activity; exponential, logarithmic, and logistic functions describe growth and decay; and polynomial functions can approximate these and most other function. Students identify and write the relationships between parallel lines, perpendicular lines, and slopes. And also, they recognize even functions and odd functions using equations and graphs. In addition, they also write and evaluate compositions of two functions. More- over, they know how to identify a one-to-one function, how to use parametric equations to graph inverse functions, how to apply the properties of logarithms, how to use logarithmic regression equations to solve problems. Finally, students know how to convert between radians and degrees, and find arc length, how to identify the periodicity and evenodd properties of the trigonometric functions, how to generate the graphs of the trigonometric functions and explore various transformations upon these graphs and how to use the inverse trigonometric functions to solve problems. The student will get evaluated by quizzes and by tests after completion of sections and the chapter. Concepts Increments Slope of Line Parallel and Perpendicular Lines - Equations of Lines Functions Domains and Ranges Viewing and Interpreting graphs Even functions and Odd Functions Symmetry Functions Defined in Pieces The Absolute Value Function Composite functions Exponential functions Parametric Equations Logarithmic functions - Trigonometric Functions CPI Codes AAPR.HS.05 A-CED.HS.01 A-CED.HS.03 A-SSE.HS.01 A-SSE.HS.03 Concepts & Understandings Learning Targets Understandings Identify and write the relationships between parallel lines, perpendicular lines, and slopes. -recognize even functions and odd functions using equations and graphs. Write and evaluate compositions of two functions. Identify a one-to-one function. Apply the properties of logarithms. Use logarithmic regression equations to solve problems. Convert between radians and degrees, and find arc length. Identify the periodicity and even-odd properties of the trigonometric functions. Generate the graphs of the trigonometric functions and explore various transformations upon these graphs. Use the inverse trigonometric functions to solve problems.

3 F-BF.HS.01 F-BF.HS.05 F-IF.HS.02 F-IF.HS.04 F-IF.HS.06 F-LE.HS.02 F-LE.HS.04 F-TF.HS.01 F-TF.HS.02 F-TF.HS.03 F-TF.HS.04 F-TF.HS.06 F-TF.HS.07 G-CO.HS.02 Math Practices: See Addendum Garfield High School See Addendum 21 st Century Themes and Skills Guiding Questions How to find the linear regression? How to identify the domain and range of a function? How to recognize even and odd functions? How to graph the function? How to analyzing exponential growth function? How to analyzing exponential decay function? How to determine whether the functions are one-to-one? How to use properties of logarithms to solve the equation? How to analyzing graphs of trigonometric functions? How to analyzing graphs of the inverse trigonometric functions? Unit Results Students will... Graph functions and determine their fundamental characteristics. Determine the characteristics of exponential functions and to apply the growth and decay models to real-life problems. Calculator to graph parametrically. Determine if one-to-one functions have inverses and find them as well as using the definitions and properties of logarithmic functions to examine logarithmic functions. Examine the characteristics of the unit circle as it applies to the various trigonometric functions. determine periodicity of functions as well as how various values in an equation affect the shifting of a trigonometric graph Suggested Activities The following activities can be incorporated into the daily lessons: Use the calculator to graph functions and determine domain, range, and symmetries. Use growth/decay models to evaluate real-life situations. Use the calculator view the graphs of functions in order to determine their properties. Graph piecewise functions. Write equations for piecewise functions form graphs. Form and evaluate compositions of functions. Explore the functions of the calculator to graph and view the characteristics of parametric equations. Use the horizontal line test.

4 Find time (t) in exponential growth problems. Determine angle, arc length, or radius in radian or degree context. Determine appropriate period length for various trig. Equations. Content Area: Calculus Unit Title: Limits and Continuity Target Course/Grade Level: Duration: 2 Weeks Unit Overview Description: The concept of limit is one of the ideas that distinguish calculus from algebra and trigonometry. In this unit, the student knows how to define and calculate limits of function values. The calculation rules are straightforward and most of the limits we need can be found by substitution, graphical investigation, numerical approximation, algebra, or some combination of these. Students calculate average and instantaneous speeds, define and calculate limits indirectly, and find and verify end behavior models for various functions. In addition, they also apply directly the definition of the slope of a curve in order to calculate slopes, find the equations of the tangent line and normal line to a curve at a given point, and find the average rate of change of a function. Moreover, they apply to models of natural behavior. They also have special mathematical properties, not otherwise guaranteed. The students get evaluated by quizzes and by tests after completely finishing sessions and the chapter. Concepts CPI Codes Average and Instantaneous speed Definition of Limit Properties of Limits Sandwich Theorem End behavior Models limits as x approaches to positive or negative infinity vertical and horizontal asymptotes Continuous functions composites IVT for continuous Functions Average Rates of change Tangent to a Curve Slope of a Curve Speed revisited ASSE.HS.01 F-LE.HS.01 G-MG.HS.01 Math Practices: See Addendum See Addendum What is a limit? What is the meaning of infinity? What does it mean if something is continuous? How did we calculate average speed? Concepts & Understandings Understandings Learning Targets 21 st Century Themes and Skills Guiding Questions Find the rate of change and limits. Calculate average and instantaneous speeds. Define and calculate limits indirectly. Find and verify end behavior models for various functions. Apply directly the definition of the slope of a curve in order to calculate slopes. Find the equations of the tangent line and normal line to a curve at a given point. Find the average rate of change of a function. Apply to models of natural behavior.

5 Unit Results Students will... Determine average and instantaneous speeds of falling objects. Determine limits involving infinity, to look for horizontal and vertical asymptotes, and to find end behavior models for functions. Evaluate the continuity of functions and to determine types of discontinuities and where they occur. Calculate average rates of change, the equations of tangent lines, slopes of a curve, and the equations for normal lines to a curve. Suggested Activities The following activities can be incorporated into the daily lessons: Determine limits about a point by substitution. Determine average and instantaneous speed. Determine limits about a point from a given graph. Determine one-sided limits. Find limits as x approaches infinity, neg. infinity of a function as well as its asymptotes. Find points of discontinuity for functions, piecewise functions and graphs of functions. Write extended functions to eliminate removable discontinuities. Use the limit definition to find the slope and equation of the tangent line. Use the limit definition to find the equation of the normal to the curve. Find the instantaneous speed. Determine the average rate of change for a function. Content Area: Calculus Unit Title: Derivatives Target Course/Grade Level: Duration: 7 Weeks Unit Overview Description: In this unit, students are to understand the limit of the slopes of secant lines. The study of rates of change of functions is called differential calculus, and the formula - 1/a^2 was our first look at a derivative. The derivative was the 17th-century breakthrough that enabled mathematicians to unlock the secrets of planetary motion and gravitational attraction - of objects changing position over time. Students will concentrate in this chapter on understanding what derivatives are and how they work. They can calculate average speeds and instantaneous speeds, calculate limits for function values and apply the properties of limits. Also, they use the Sandwich Theorem to find certain limits indirectly. And also, students learn how to graph original functions from the graph of first derivative functions, graph the first derivative of a function from the graph of an original function, and graph the derivative of a function given numerically with data. In additional, they will learn how to use the rules of differentiation to calculate derivatives, including second and higher order derivatives. Students will know how to use the rules for differentiating the six basic trigonometric functions. Moreover, they will use derivatives to analyze straight line motion and solve other problems involving rates of change. The student will get evaluated by quizzes and by tests after completion of sections and the chapter. Concepts Definition of Derivative Differentiability Rules for differentiation Velocity and Other rates of change Derivatives of Trigonometric Functions Chain Rule Implicit Differentiation Derivatives of Inverse Trigonometric Functions Concepts & Understandings Understandings The derivative of the function f with respect to the variable x is the function f' whose value at x is the derivative of f(x) is equal to limit h approaches to zero [f(x + h) - f (x)] / h provide the limit exists. To apply the rules of differentiation to all types of functions in order to explore and understand how they may be used to examine their physical properties and applications.

6 CPI Codes Garfield High School Derivatives of Exponential and Logarithmic Functions. limit notation finding the slope of a curve at a point finding the slopes of lines relationship between the slope of a line and its perpendicular negative exponent properties logarithms Learning Targets AREI.HS.01 A-REI.HS.02 A-REI.HS.11 A-REI.HS.12 F-BF.HS.01 F-BF.HS.03 F-BF.HS.04 F-IF.HS.01 F-IF.HS.02 F-IF.HS.04 - F-IF.HS.05 F-IF.HS.06 F-IF.HS.07 F-IF.HS.08 F-IF.HS.09 F-LE.HS.02 F-LE.HS.05 F-TF.HS.09 G-MG.HS.03 Math Practices See Addendum 21 st Century Themes and Skills See Addendum Guiding Questions What does finding the derivative mean with regard to a function? Why is finding the limit sometimes difficult? What problems existed with finding limits of functions? What problems apply to derivatives? What was the meaning of finding the limit of a function? What is the rate of change? What does instantaneous mean? Are trigonometric functions differentiable? Why? If two functions are differentiable, should their composition be differentiable? _Do some functions have more than one variable? How can we differentiate these functions? What are the inverse trig functions? Should exponential and logarithmic functions be differentiable? Why? Unit Results Students will... Use the definition to find the derivative with respect to variable functions and to determine the graph of the derivative as

7 well. Graph the Derivative using NDER. Determine failures in differentiability and establish continuity. Apply the rules for differentiation to first and higher order derivatives. Determine instantaneous rates of change, velocity, acceleration, and marginal cost / revenue. Determine derivatives involving the six basic trigonometric functions, simple harmonic motion, and jerks. find the derivative of composite functions by applying the chain rule Find derivatives of implicitly defined functions as well as derivatives of higher order and rational powers. find derivatives of functions involving inverse trigonometric functions Find derivatives of functions involving exponential and logarithmic functions. Suggested Activities The following activities can be incorporated into the daily lessons: Apply the definition of derivative. Apply the alternate definition of derivative? Define relationship between the graph and its derivative? Establish the use of one-sided limits. Evaluate the causes for failure in differentiability; corner, cusp, vertical tangent, discontinuity. Finding where failures occur. Study continuity through differentiability, theorem 1. Find horizontal tangents. Determine higher order derivatives Apply definition of instantaneous velocity. Model motion. Apply definition of instantaneous rate of change. Use derivatives to describe simple harmonic motion. Use the rules of differentiation on functions containing sine and cosine. Use derivative rules of remaining trig functions to find tangents and normal lines. Determine the "jerk". Apply the power chain rule to functions. Work the chain rule from "outside-in". Apply the formulas of derivatives of inverse trig functions Differentiate a function implicitly. Use the power rule for rational powers to find derivatives. Determine derivatives of exponential functions. Find the derivative of functions involving the common log and the natural log. Content Area: Calculus Unit Title: Applications of Derivatives Target Course/Grade Level: Duration: Unit Overview Description: In this unit, the student shows how to draw conclusions from derivatives about the extreme values of a function and about the general shape of a function's graph. Students learn how a tangent line captures the shape of a curve near the point of tangency, how to deduce rates of change we cannot measure from rates of change we already know, and how to find a function when we know only its first derivative and its value at a single point. And also, they will know how to find linearization and use Newton's method to approximate the zeros of a function. In additional they can solve application problems involving finding minimum or maximum values of functions. Moreover, the student estimates the change in a function using differentials. The student will get evaluated by quizzes and by tests after completion of sections and the chapter.

8 Concepts Extreme Values of Functions Mean Value Theorem Connecting f' and f" with the Graph of Modeling and Optimization Linearization and Newton's Method Related Rates CPI Codes AREI.HS.01 A-REI.HS.02 A-REI.HS.11 A-REI.HS.12 F-BF.HS.01 F-BF.HS.03 F-BF.HS.04 F-IF.HS.01 F-IF.HS.02 F-IF.HS.04 F-IF.HS.05 F-IF.HS.06 F-IF.HS.07 F-IF.HS.08 F-IF.HS.09 - F-LE.HS.02 F-LE.HS.05 F-TF.HS.09 G-MG.HS.03 G-SRT.HS.01 G-SRT.HS.02 G-SRT.HS.04 G-SRT.HS.06 G-SRT.HS.08 Math Practices Concepts & Understandings Understandings Learning Targets Finding maximum and minimum values of functions, called optimization, is an important issue in real-world problems. The Mean Value Theorem is an important theoretical tool to connect the average and instantaneous rates of change. Differential calculus is a powerful problem-solving tool precisely because of its usefulness for analyzing functions. Historically, optimization problems were among the earliest applications of what we now call differential calculus. Engineering and science depend on approximations in most practical applications; it is important to understand how approximation techniques work. Related rate problems are at the heart of Newtonian mechanics; it was essentially to solve such problems that calculus was invented

9 See Addendum 21 st Century Themes and Skills See Addendum Guiding Questions Explain the Extreme Value Theorem and how to finding the Absolute Extreme? Explain Mean Value Theorem and how this theorem applies in real-life? How do you find the interval (decreasing and increasing)? How do you apply first derivative test for local extrema? How do you apply second derivative test for point of inflection? How do you find the minimum or maximum values of functions using by first derivative test and second derivative (mathematics)? How do you find minimum or maximum values of functions using by first derivative test and second derivative (In business and industry)? How do you apply linear approximation on real-life problems? How do you find the differential dy and evaluate dy for the given values of x and dx? How do you solve the related rate problems in real-life problems? Unit Results Students will... Find the absolute, relative extrema and critical points of a function. Apply the mean value theorem as it applies to increasing and decreasing functions. The antiderivative will also be introduced. Use the first and second derivative tests for local extrema and to determine concavity as well as points of inflection of functions. Maximize of minimize some aspect of a real-life situation. Suggested Activities The following activities can be incorporated into the daily lessons: Apply the Mean Value Theorem. Determine if a function is increasing or decreasing. Use the antiderivative to study motion. Apply the first derivative test. Determine the concavity of a function. Use f ' and f " to graph f. Model examples from industry. Model mathematical problems. Model real life economic problems. Content Area: Calculus Unit Title: The Definite Integral Target Course/Grade Level: Duration: 5 Weeks Unit Overview Description: We have seen how the need to calculate instantaneous rates of change led the discoverers of calculus to an investigation of the slopes of tangent lines and, ultimately, to the derivative - to what we call differential calculus. In addition to a calculation method (a ""calculus"") to describe how functions were changing at a given instant, to describe how those instantaneous changes could accumulate over an interval to produce the function. That is why Newton and Leibniz were investigating areas under curves, an investigation that ultimately led to the second main branch of calculus, called integral calculus. Once they had the

10 calculus for finding slopes of tangent lines and the calculus for finding areas under curves-two geometric operations that would seem to have nothing at all to do with each other-the challenge for Newton and Leibniz was to prove the connection that they knew intuitively had to be there. The discovery of this connection (called the Fundamental Theorem of Calculus) brought differential and integral calculus together to become the single most powerful insight mathematicians had ever acquired for understanding how the universe worked. In this unit, students interpret the area under a graph as a net accumulation of a rate of change and to approximate the area under the graph of a non-negative continuous function by using rectangle approximation methods. They apply the Fundamental Theorem of Calculus to compute the Volume of Sphere, Definite integrals and Antiderivatives. And also, students will understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus. Moreover, students can approximate the definite integral by using the Trapezoidal Rule and by using Simpson's Rule. They will get evaluated by quizzes and by tests after completion of sections and the chapter." Concepts Estimating with finite Sums Definite Integrals Definite Integrals and Antiderivatives Fundamental Theorem of Calculus Trapezoidal Rule CPI Codes A-REI.HS.01 A-REI.HS.02 A-REI.HS.11 A-REI.HS.12 F-BF.HS.01 F-BF.HS.03 F-BF.HS.04 F-IF.HS.01 F-IF.HS.02 F-IF.HS.04 F-IF.HS.05 F-IF.HS.06 F-IF.HS.07 F-IF.HS.08 F-IF.HS.09 F-LE.HS.02 F-LE.HS.05 F-TF.HS.09 G-MG.HS.03 Math Practices: Concepts & Understandings Understandings Learning Targets Learning about estimating with finite sums sets the foundation for understanding integral calculus. The definite integral is the basis of integral calculus, just as the derivative is the basis of differential calculus. Working with the properties of definite integrals helps us to understand better the definite integral. Connecting derivatives and definite integrals sets the stage for the Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus is a triumph of mathematical discovery and the key to solving many problems. Some definite integrals are best found by numerical approximations, and rectangles are not always the most efficient figures to use.

11 See Addendum 21 st Century Themes and Skills See Addendum Guiding Questions How to use Rectangular Approximation Method (RAM)? How to find volume of Sphere and Cardiac output using the RAM? Explaining Riemann Sums? How to use Riemann Sums to Definite Integral and Area? How to use FINT to find Integrals? What are the Rules for Definite Integrals? How to find the Average (Mean) Value? What is the definition of the Fundamental Theorem of Calculus (part 1)? What is the definition of part 2? What is difference in the Fundamental Theorem of Calculus (part 1) and the Fundamental Theorem of Calculus (part 2)? What is the Trapezoidal Rule and how does it apply? Unit Results Students will... Approximate the area under the graph of nonnegative continuous function by using rectangle approximation methods and interpret the area under a graph as a net accumulation of a rate of change. Express the area under a curve as a definite integral and as a limit of Riemann sums. And also, they are able to compute the area under a curve suing numerical integration procedure. Moreover, students will be able to use calculator to find NINT Apply rules for definite integrals and dined the average value of a function over a closed interval. Fundamental Theorem of Calculus and understand the relationship between the derivative and definite integral as expressed in both parts of the Fundamental Theorem. Approximate the definite integral by using the Trapezoidal Rule and by using Simpson's Rule, as well as estimating the error in using the Trapezoidal and Simpson's Rules. Suggested Activities The following activities can be incorporated into the daily lessons: Smart board lesson Content Area: Calculus Unit Title: Target Course/Grade Level: Duration: Unit Overview Differential Equations and Mathematical Modeling 5 Weeks Description: One of the early accomplishments of calculus was predicting the future position of a planet from its present position and velocity. Today this is just one of a number of occasions on which we deduce everything we need to know about a function from one of its known values and its rate of change. From this kind of information, students can tell how long a sample of radioactive polonium will last; whether, given current trends, a population will grow or become extinct; and how large major league baseball salaries are likely to be in the year In this unit, students examine the analytic, graphical, and numerical techniques on which such predictions are based using by Integration by Substitution or using by Integration by Parts. Also, they can use Newton's Law of Cooling to find the rate at which an object's temperature is changing at any given time is roughly proportional to the difference between its temperature and the temperature of the surrounding medium. They can construct antiderivatives using the Fundamental Theorem of Calculus. And also, they can construct slope fields using technology and interpret slope fields as visualizations of differential equations. Moreover, students will be able to solve problems involving exponential growth and decay in a variety of applications. Finally, they will get evaluated by quizzes and by tests after completion of sections and the chapter. Concepts & Understandings

12 Concepts Differential Equations Slope Fields The First Fundamental of Calculus Theorems The Second Fundamental of Calculus Theorems Substitution in Indefinite Integrals Substitution in Definite Integrals Product Rule in Integral form Tabular Integration Inverse Trigonometric and Logarithmic functions Separable differential Equations Law of Exponential change Continuously Compounded Interest Radioactivity Modeling growth with other Bases Newton s Law of Cooling How Population growth Partial Fractions The Logistic differential Equation Logistic Growth Models CPI Codes Understandings Learning Targets A-REI.HS.01 A-REI.HS.02 A-REI.HS.11 A-REI.HS.12 F-BF.HS.01 F-BF.HS.03 F-BF.HS.04 F-IF.HS.01 F-IF.HS.02 F-IF.HS.04 F-IF.HS.05 F-IF.HS.06 F-IF.HS.07 F-IF.HS.08 F-IF.HS.09 F-LE.HS.02 F-LE.HS.05 F-TF.HS.09 G-MG.HS.03 Math Practices: See Addendum 21 st Century Themes and Skills See Addendum Guiding Questions What is a differential equation? What is a slope field for the differential equation? How do you use substitution to evaluate indefinite integrals? Apply the fundamental theorem of calculus to solve the differential equation. Construct slope field and check using by the differential equation. understanding concepts of rates and temperature change

13 How do you use substitution to evaluate definite integrals? How do you evaluate antidifferentiation by parts? How do you evaluate for the unknown Integral? How are differential equations separable? How can you use the Law of exponential change to find continuously compounded interest? How do you find partial fraction decomposition? How does it get used with an antidifferentian equation? Unit Results Students will... Use the fundamental theorem of calculus to solve the differential equation and construct a slope field for the differential equation. Use substitution to solve antidifferentiation and evaluate the integral. Evaluate Antidifferentiation by Parts. Understand that the differential equation dy/dx = ky gives us new insight into exponential growth and decay. find antidifferentiating with partial fractions, to find the logistic differential equation using the general logistic formula. Suggested Activities The following activities can be incorporated into the daily lessons: Smart board lesson. Content Area: Calculus Unit Title: Application of Definite Integrals Target Course/Grade Level: Duration: 5 Weeks Unit Overview Description: Finding the limits of Riemann sums is a natural way to calculate mathematical or physical quantities that appear to be irregular when viewed as a whole, but which can be fragmented into regular pieces. With calculus it became possible to get exact answers for these problems with almost no effort, because in the limit these sums became definite integrals and definite integrals could be evaluated with antiderivatives. With calculus, the challenge became one of fitting an integral function to the situation at hand (the ""modeling"" step) and then finding antiderivatives for it. In this unit, students calculate values for the regular pieces using known formulas, and then sum them to find a value for the irregular whole. Students can find position from displacement and modeling the effects of acceleration. And also, they know how to find the area of a region between a curve and the x-axis, the area of a region that is bounded above by one curve, y= f(x), and below by another, y = g (x). Moreover, students find the volume of a solid by integration. They apply the volume formula to a solid with square cross section, with circular cross sections, and also with cylindrical shells. Students will be able to adapt their knowledge of integral calculus to model problems involving rates of change in a variety of applications, possibly in unfamiliar contexts. The student will get evaluated by quizzes and by tests after completion of sections and the chapter." Concepts Linear Motion Revisited General Strategy Consumption over time Net Charge from Data Area Between Curves Area Enclosed by Intersecting Curves Boundaries with Changing Functions Integrating with Respect to y Saving Time with Geometry Formulas Volume as an Integral Concepts & Understandings Understandings Calculate values for the regular pieces using known formulas, and then sum them to find a value for the irregular whole. Find position from displacement and modeling the effects of acceleration. Find the area of a region between a curve and the x- axis, the area of a region that is bounded above by one curve, y= f(x), and below by another, y = g (x). Find the volume of a solid by integration. Apply the volume formula to a solid with square cross

14 CPI Codes Garfield High School Square cross Sections Circular Cross Sections Cylindrical shells Other Cross Sections Fluid force and fluid Pressure Learning Targets section, with circular cross sections, and also with cylindrical shells. Adapt their knowledge of integral calculus to model problems involving rates of change in a variety of applications, possibly in unfamiliar contexts. A-REI.HS.01 A-REI.HS.02 A-REI.HS.11 A-REI.HS.12 F-BF.HS.01 F-BF.HS.03 F-BF.HS.04 F-IF.HS.01 F-IF.HS.02 F-IF.HS.04 F-IF.HS.05 F-IF.HS.06 F-IF.HS.07 F-IF.HS.08 F-IF.HS.09 F-LE.HS.02 F-LE.HS.05 F-TF.HS.09 GMG.HS.03 Math Practices: See Addendum 21 st Century Themes and Skills See Addendum Guiding Questions How do you find position from Displacement? How do you find total distance traveled by the particle? How do you find the area of a region enclosed by between curves? How do you apply the volume formula to a solid with different cross sections? How do you finding the Work Done by a Force? Unit Results Students will... solve problems in which a rate is integrated to find the net change over time in a variety of applications use integration to calculate areas of regions in a plane. use integration to calculate volumes of solids and surface areas of solids of revolution. use their knowledge of integral calculus to model problems involving rates of change. Suggested Activities The following activities can be incorporated into the daily lessons: smart board lesson

15 1. CCSS.Math.Practice.MP1 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. CCSS.Math.Practice.MP2 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 decontextualize 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. CCSS.Math.Practice.MP3 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

16 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. CCSS.Math.Practice.MP4 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, 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. CCSS.Math.Practice.MP5 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

17 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. CCSS.Math.Practice.MP6 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. CCSS.Math.Practice.MP7 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 , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as 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. CCSS.Math.Practice.MP8 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 2 + x + 1), and (x 1)(x 3 + 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.

18 Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word understand are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards which set an expectation of understanding are potential points of intersection between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.