Mathematics – AP/IB AB

Mathematics Curriculum Map

AP/IB Calculus AB

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Mathematics – AP/IB Calculus AB AP/IB Calculus AB – At a Glance NOTE: Mathematical standards are interwoven and should be addressed throughout the year in as many different units and tasks as possible in order to stress the natural connections that exist among mathematical topics.

AP/IB Calculus AB Curriculum Map

Semester 1 Flex Semester 2 Unit 1 Unit 2 Unit 3 Flex Unit 4 Unit 5 Unit 5 Limits and Continuity Applications of the Integration Application of Differential Derivative Equations 1.1A Notation 2.1A Definition of the 2.3A Interpreting the 3.1A 3.4A Applications of 2.3D Solving Problems Derivative Meaning of the the Definite Involving Rates of 1.1B Estimating Limits Derivative 3.2A Riemann Sums Change 2.1B Estimating 3.4B Average Value of 2.3B Linear 3.2B Approximating 2.3E Verifying Solutions 1.1C Evaluating Limits a Function Approximation Definite Integrals to Differential Equations 1.1D Function Behavior 2.1C Calculating 2.3C Related Rates, 3.2C Properties of 3.4C Integrals and 2.3F Estimating and Limits Derivatives Optimization, Definite Integrals Rectilinear Motion Solutions to Differential Rectilinear motion Equations 1.2A Continuity 2.1D Higher Order 3.3A The Fundamental 3.4D Area and Solids Derivatives 2.4A The Mean Value Theorem of Calculus 3.5A General and 1.2B Applications of of Integration Theorem Specific Solutions of Continuity 2.2A Functions and 3.3B Calculating Differential Equations Derivatives Indefinite and Definite Integrals 3.5B Differential 2.2 B Differentiability Equations from Context

Mathematical Practices for AP Calculus (MPACs) All units will include the M athematical Practices MPAC 1: Reasoning with definitions and theorems MPAC 4: Connecting multiple representations MPAC 2: Connecting concepts MPAC 5: Building notational fluency MPAC 3: Implementing algebraic/computational processes MPAC 6: Communicating

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Mathematics – AP/IB Calculus AB AP/IB Calculus AB Overview Big Idea 1: Limits Big Idea 3: Integrals and the Fundamental Theorem of  The concept of a limit can be used to understand the behavior Calculus of functions.  Antidifferentiation is the inverse process of differentiation.  Continuity is a key property of functions that is defined using  The definite integral of a function over an interval is the limit of limits. a Riemann sum over that interval and can be calculated using a variety of strategies. Big Idea 2: Derivatives  The Fundamental Theorem of Calculus, which has two distinct formulations, connects differentiation and integration.  The derivative of a function is defined as the limit of difference quotient and can be determined using a variety of strategies.  The definite integral of a function over an interval is a mathematical tool with many interpretations and application  A function’s derivative, which is itself a function, can be used involving accumulation. to understand the behavior of a function.  Antidifferentiation is an underlying concept involved in solving  The derivative has multiple interpretations and application separable differential equations. Solving separable differential including those that involve instantaneous rates of change. equations involves determining a function or relation given its  The Mean Value Theorem connects the behavior of a rate of change. differentiable function over an interval to the behavior of the derivatives of that function at a particular point in the interval.

Mathematical Practices for AP Calculus (MPACs)

MPAC 1: Reasoning with definitions and theorems

MPAC 2: Connecting concepts MPAC 3: Implementing algebraic/computational processes MPAC 4: Connecting multiple representations MPAC 5: Building notation fluency MPAC 6: Communicating

Source: AP Calculus AB/BC Course and Exam Description

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Mathematics – AP/IB Calculus AB Semester 1 Unit 1 – Limits and Continuity Essential Question(s): How can the concept of limits be used to understand the behavior of functions? How can you define continuity as a key property of functions using limits? Topic College Board Standard Mathematical Resources Practices Express limits 1.1A1: Given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be MPAC 1 Finney, Demanna, Waits, symbolically made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit Kennedy, and Bressoud using correct exists and is a real number then the common notation is lim 푓(푥) = 푅. MPAC 2 2.1, 2.2 notation 푥→푐

1.1A2: The concept of a limit can be extended to include one-sided limits, limits at infinity, and Larson Edwards 2.1, 2.2, infinite limits. 2.4, 2.5

1.1A3: A limit might not exist for some function at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this Smith & Minton 1.1, 1.2 value, or if the limit from the left does not equal the limit from the right. Estimating 1.1B1: Numerical and graphical information can be used to estimate limits. MPAC 3 Finney et al. 2.1, 2.2 Limits of MPAC 4 Functions Larson 2.2

Smith & Minton 1.1, 1.2 Determining 1.1C1: Limits of sums, differences, products, quotients, and composite functions can be found MPAC 1 Finney et al. 2.1, 2.2, limits of using the basic theorems of limits and algebraic rules. Functions MPAC 2 Larson 2.3 1.1C2: The may be found by using algebraic manipulation, alternate forms of trigonometric functions or the squeeze theorem Smith & Minton 1.1, 1.2 MPAC 3

MPAC 4

Function 1.1D1: Asymptotic and unbounded behavior of functions can be explained and described MPAC 2 Finney et al. 2.1, 2.2 Behavior and using limits. Limits Larson 2.5, 3.2, 3.4, 4.5 1.1D2: Relative magnitudes of functions and their rates of change can be compared using limits. Smith & Minton 1.4 2016 The College Board Chandler Unified School District #80 Page 4 of 17 Revised: September 2017

Mathematics – AP/IB Calculus AB Continuity 1.2A1: A function f is continuous at x = c provided that f(x) exists, lim f (x) exists, and MPAC 1 Finney et al. 2.3 xc MPAC 2 lim f (x)  f (c) . Larson 2.1, 2.2, 2.3, 2.4, xc MPAC 3 2.5 MPAC 4 1.2A2: Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are Smith & Minton 1.3 continuous at all points in their domains.

1.2A3: Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical .

Applications of 1.2B1: Continuity is an essential condition for theorems such as the Intermediate Value MPAC 1 Finney et al. 2.3 Theorem, the Extreme Value Theorem, and the Mean Value Theorem. Continuity MPAC 2 Larson 2.4, 4.1 MPAC 3 MPAC 4 Smith & Minton 1.3

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Mathematics – AP/IB Calculus AB Semester 1 Unit 2 - Derivatives  How do we use derivatives to describe the rate of change of one variable with respect to another variable, and how do we use multiple representations to understand change in a variety of contexts? Topic College Board Standard Mathematical Resources Practices Definition of the 2.1A2: The instantaneous rate of change of a function at a point can be expressed by MPAC 1 Finney et al. 3.1, 3.2 푓(푎+ℎ)−푓(푎) 푓(푥)−푓(푎) MPAC 2 Derivative lim or lim , provided the limit exists. These are common forms of the 푥→ℎ ℎ 푥→푥 푥−푎 MPAC 3 Larson 3.1 definition of the derivative and are denoted f’(a). MPAC 4 MPAC 5 푓(푎+ℎ)−푓(푎) MPAC 6 2.1A3: The derivative of f is the function whose value at x is lim provided this limit Smith & Minton 2.1, 2.2 푥→0 ℎ exists.

2.1A4: For y = f(x), notations for the derivative include 푑푦, f’(x), and y’. 푑푥

2.1A5: The derivative can be represented graphically, numerically, analytically, and verbally. Estimating 2.1B1: The derivative at a point can be estimated from information given in tables or graphs. MPAC 1 Finney et al. 3.1 Derivatives MPAC 2 MPAC 3 Larson 3.1, 3.2, .3.3 MPAC 4 Smith & Minton 2.1, 2.2 MPAC 5 MPAC 6 Continuity and 2.2B1: A may fail to be differentiable at a point in its domain. MPAC 1 Finney et al. 3.2 Differentiability MPAC 2 MPAC 3 2.2B2: If a function is differentiable at a point, then it is continuous at that point. Larson 3.1 MPAC 4 MPAC 5

MPAC 6 Smith & Minton 2.2 Derivative 2.1C1: Direction application of the derivative of the definition of the derivative can be used to MPAC 1 Finney et al. 3.3, 3.5, 4.3, 4.4 Rules find the derivative for selected functions, including polynomial, power, sine, cosine, MPAC 2 Larson 3.2, 3.3, 3.6 exponential, and logarithmic functions. MPAC 3 MPAC 4 Smith & Minton 2.3. 2.4, 2.5, 2.1C2: Specific Rules can be used to calculate derivatives for classes of functions, including MPAC 5 2.6 polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse MPAC 6 trigonometric.

2.1C3: Sums, differences, products and quotients of functions can be differentiated using derivative rules.

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Mathematics – AP/IB Calculus AB The Chain Rule 2.1 C4: The chain rule provides a way to differentiate composite functions. MPAC 1 Finney et al. 4.1, 4.2 MPAC 2 Larson 3.4 2.1C5: The chain rule is the basis for implicit differentiation. MPAC 3 MPAC 4 Smith & Minton 2.7 MPAC 5 MPAC 6 Implicit 2.1 C4: The chain rule provides a way to differentiate composite functions. MPAC 1 Finney et al. 4.1, 4.2 Differentiation MPAC 2 Larson 3.4 2.1C5: The chain rule is the basis for implicit differentiation. MPAC 3 MPAC 4 Smith & Minton 2.8 MPAC 5 MPAC 6 Higher Order 2.1D1: Differentiating f’ produces the second derivative f”, provided the derivative of f’ exists; MPAC 1 Finney et al. 3.1, 3.3, 4.2 Derivatives repeating this process provides higher order derivatives of f. MPAC 2 Larson 3.3 MPAC 3 2.1D2: Higher order derivatives are represented with a variety of notations. For y = f(x), MPAC 4 Smith & Minton 2.3 2 MPAC 5 notations for the second derivative include d y , f”(x) and Higher order derivatives can be dx 2 MPAC 6 d ny denoted or f(n)(x). dx n

Curve 1.2 B1: Continuity is an essential condition for theorems such as the Intermediate Value MPAC 1 Finney et al. 5.1, 5.3 Sketching Theorem, the Extreme Value Theorem, and the Mean Value Theorem. MPAC 2 MPAC 3 Larson 3.1, 4.3 4.4 2.2A1: First and second derivatives of a function can provide information about the function MPAC 4 and its graph including intervals of increase or decrease, local (relative) and global (absolute) MPAC 5 Smith & Minton 3.3, 3.4, 3.5, extrema, intervals of upward or downward concavity, and points of inflection. MPAC 6 3.6

2.2A2: Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.

2.2A3: Key features of the graphs or f, f’, and f” are related to one another.

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Mathematics – AP/IB Calculus AB Semester 1 Unit 3 – Applications of Derivatives Essential Questions: In real-world application, how do we show understanding of a function through the use of the slope of the tangent line? How do we use graphs and charts in mathematical applications to further examine one’s understanding of derivative? Topic College Board Standard Mathematical Resources Practices Rectilinear 2.3C1: The derivative can be used to solve rectilinear motion problems involving position, MPAC 2 Finney et al. 3.4 Motion speed, velocity, and acceleration. MPAC 3 Larson 3.2

MPAC 4 Smith & Minton 2.1 MPAC 6 Related Rates 2.3C2: The derivative can be used to solve related rate problems, that is, finding a rate at MPAC 2 Finney et al. 5.6 which one quantity is changing by relating it to other quantities whose rates of change are MPAC 3 Larson 3.7 known. MPAC 4 Smith & Minton 3.8 2.3C3: The derivative can be used to solve optimization problems that is, find a maximum or MPAC 6 minimum value of a function over a given interval.

2.3D1: The derivative can be used to express information about rates of change in applied contexts. Optimization 2.3C3: The derivative can be used to solve optimization problems that is, find a maximum or MPAC 2 Finney et al, 5.5 minimum value of a function over a given interval. MPAC 3 Larson 4.7

2.3D1: The derivative can be used to express information about rates of change in applied MPAC 4 Smith & Minton 3.7 contexts. MPAC 6 Local 2.3B1: The derivative at a point is the slope of the line tangent to a graph at that point on the MPAC 1 Finney et al. 2.4, 5.5 Linearization graph. MPAC 2 and Tangent MPAC 3 Larson 4.8 Lines 2.3B2: The tangent line is the graph of a locally linear approximation of the function near the MPAC 4 point of tangency. MPAC 5 Smith & Minton 3.1, 3.2 MPAC 6

The Mean 1.2B1: Continuity is an essential condition for theorems such as the Intermediate Value MPAC 1 Finney et al. 5.2 Value Theorem Theorem, the Extreme Value Theorem, and the Mean Value Theorem. MPAC 2 MPAC 3 Larson 4.2 MPAC 4 2.4A1: Apply the mean value theorem to describe the behavior of a function over an interval. MPAC 5 Smith & Minton 2.9 MPAC 6 2016 The College Board Chandler Unified School District #80 Page 8 of 17 Revised: September 2017

Mathematics – AP/IB Calculus AB Indeterminate 1.1C3: Limits of the indeterminate forms 0 and ∞ may be evaluated using L’Hospital’s Rule. MPAC 1 Finney et al. 9.2 Forms and 0 0 MPAC 2

L’Hospital’s MPAC 3 Larson 8.7 Rule MPAC 4 MPAC 5 Smith & Minton 7.6 MPAC 6

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Mathematics – AP/IB Calculus AB Flex Unit Unit 4 - Integrals Essential Question(s): How are integrals used through practical and theoretical application?  Topic College Board Standard Mathematical Resources Practices Antiderivatives 3.1A1: An of a function f is a function g whose derivative is f. MPAC 1 Finney et al. 6.3, 7.2 MPAC 2 3.1A2: Differentiation rules provide the foundation for finding antideriviatives. Larson 5.1 MPAC 3 Smith & Minton 4.1, 4.6 x MPAC 4 3.3B1: The function defined by F(x)   f (t)dt is an antiderivative of f. MPAC 5 a MPAC 6

3.3B3: The notation means that F’(x) = f(x) and is called an indefinite integral of the function f.

3.3B4: Many functions do not have closed form antiderivatives.

3.3B5: Techniques for finding antiderivatives include algebraic manipulation such as long division and completing the square, substitution of variables.

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Mathematics – AP/IB Calculus AB Riemann 3.2A1: A Riemann sum, which requires a partition of an interval I, is the sum of the products, MPAC 1 Finney et al. 6.1, 6.2, 6.5 Sums and each of which is the value of the function at a point in a subinterval multiplied by the length of MPAC 2 Approximating that subinterval. Larson 5.2, 5.3, 5.6 MPAC 3 Area Under Curves MPAC 4 Smith & Minton 4.2, 4.3 3.2A2: The definite integral of a continuous function f over the interval [a, b], denoted by MPAC 5 b MPAC 6 f (x)dx , is the limit of Riemann sums as the widths of the subintervals approach 0. That is,  a b n * * th f (x)dx  lim f (x )x where x is a value in the i subinterval, xi is the width of   i i i max xi 0 a i1 th the i subinterval, n is the number of subintervals, and max xi is the width of the largest b n * b  a subinterval. Another form of the definition is f (x)dx  lim f (xi )x i where x i   n  a i1 n * th and xi is a value in the i subinterval.

3.2A3: The information in a definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral.

3.2B1: Definite integrals can be approximated for functions that are represented graphically, numerically, algebraically, and verbally.

3.2B2: Definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or non-uniform partitions.

3.4A3: The limit of an approximating Riemann sum can be interpreted as a definite integral.

Properties of 3.2C1: In some cases, a definite integral can be evaluated by using geometry and the MPAC 1 Finney et al. 6.2, 6.3 Definite connection between the definite integral and area. MPAC 2 Integrals Larson 5.3 MPAC 3 3.2C2: Properties of definite integrals include the integral of constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a MPAC 4 Smith & Minton 4.4 function over adjacent intervals. MPAC 5 MPAC 6

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Mathematics – AP/IB Calculus AB x The 2 MPAC 1 Finney et al. 6.2, 6.3, 6.4 3.3A1: The definite integral can be used to define new functions; for example f (x)  e t dt . Fundamental  MPAC 2 Theorem of 0 Larson 5.4, 5.5, 5.7, 5.8 MPAC 3 Calculus MPAC 4 Smith & Minton 4.5 d  x  MPAC 5 3.3A2: If f is a continuous function on the interval [a, b], then  f (t)dt   f (x) , where x is dx    MPAC 6  a  between a and b.

3.3A3: Graphical, numerical, analytical, and verbal representations of a function f provide x information about the function g defined as g(x)   f (t)dt . a

3.3B2: If f is continuous on the interval [a, b] and F is an antiderivative of f then b  f (x)dx  F(a)  F(b) . a

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Mathematics – AP/IB Calculus AB

Semester 2 Unit 5 – Applications of Integration Essential Question(s):  How can you use the definite integral to solve applications involving accumulation?  How do you interpret the definite integral in context? Topic College Board Standard Mathematical Resources Practices Applications of 3.4A1: A function defined as an integral represents an accumulation of a rate of change. Finney et al. 6.1, 6.2, 8.1, 8.5 the Definite MPAC 2 Integral 3.4A2: The definite integral of the rate of change of a quantity over an interval gives the net MPAC 3 Larson 5.1, 5.3, 5.4 change of that quantity over that interval MPAC 4 Smith & Minton 4.1, 4.2, 4.3 MPAC 6 Master Math Mentor Pp: 171- 184 Average b Finney et al. 6.3 1 MPAC 2 Value of a 3.4B1: The average value of a function f over an interval [a, b] is f (x)dx .  Larson 5.4 Function b  a a MPAC 3

MPAC 4 Smith & Minton 4.4

MPAC 6 Master Math Mentor Pp: 199- 212 Definite 3.4C1: For a particle in rectilinear motion over an interval of time, the definite integral of Finney et al. 6.1, 8.1 Integrals and velocity represents the particle’s displacement over the interval of time, and the definite MPAC 2 Larson 5.2, 5.3. 5.4, 5.5, 5.6 Rectilinear integral of speed represents the particle’s total distance traveled over the interval of time. MPAC 3 Motion MPAC 4 Smith & Minton 4.1, 4.4, 4.5

MPAC 6 Master Math Mentor Pp: 185- 198

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Mathematics – AP/IB Calculus AB Area and 3.4D1: Areas of certain regions in the plane can be calculated with definite integrals. Finney et al. 8.2, 8.3 Integration MPAC 2 Larson 5.2, 5.3, 5.4, 7.1 MPAC 3

MPAC 4 Smith & Minton 4.1, 4.4, 4.5, 5.1 MPAC 6 Master Math Mentor Pp: 171- 184; 199-212 Solids of 3.4D2: Volumes of solids with known cross sections, including discs and washers, can be Finney et al. 8.2, 8.3 Revolution calculated with definite integrals. MPAC 2 Larson 7.2 MPAC 3

MPAC 4 Smith & Minton 5.2

MPAC 6 Master Math Mentor Pp 199- 212

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Mathematics – AP/IB Calculus AB Semester 2 Unit 6 – Differential Equations Essential Question(s): How do slope fields relate to differential equations? How can separable differential equations be applied to Exponential Growth and Decay?

Topic AP College Board Standard Mathematical Resources Practices Separable 3.5A1: Anti-differentiation can be used to find specific solutions to differential equations with MPAC 1 Finney et al 7.1, 7.4 Differential given initial conditions, including applications to motion along a line, exponential growth and MPAC 2 Equations decay. Larson 6.1, 6.2, 6.3 MPAC 3 Smith & Minton 6.5 3.5A2: Some differential equations can be solved by separation of variables. MPAC 4 MPAC 6 3.5A3: Solutions to differential equations may be subject to domain restrictions.

Differential 2.3D1: The derivative can be used to express information about rates of change in applied MPAC 1 Finney et al 7.1, 7.4 Equations from contexts. MPAC 2 Larson 6.1, 6.2, 6.3 Context x MPAC 6 3.5A4: The function F defined by F(x)  c  f (t)dt is a general solution to the differential  Smith & Minton 6.4, 6.5 a dy x  f (x) F(x)  y  f (t)dt equation , and 0  is a particular solution to the differential dt a

equation satisfying F(a) = y0.

3.5B1: The model for exponential growth and decay that arises from the statement “The rate of dy change of a quantity is proportional to the size of the quantity” is  ky . dt

Slope Fields 2.3E1: Solutions to differential equations are function or families of functions. MPAC 1 Finney et al 7.1, 7.4 MPAC 3 2.3E2: Derivatives can be used to verify that a function is a solution to a given differential Larson 6.1, 6.2, 6.3 equation. MPAC 5 MPAC 6 Smith & Minton 6.6

2.3F1: Slope fields can provide visual cues to the behavior of solution to first order differential equations. 2016 The College Board Chandler Unified School District #80 Page 15 of 17 Revised: September 2017

Mathematics – AP/IB Calculus AB Mathematical Practices for AP Calculus (MPACs) The Mathematical Practices for AP Calculus (MPACs) capture important aspects of the work that mathematicians engage in, at the level of competence expected of AP Calculus students. They are drawn from the rich work in the National Council of Teachers of Mathematics (NCTM) Process Standards and the Association of American Colleges and Universities (AAC&U) Quantitative Literacy VALUE Rubric. Embedding these practices in the study of calculus enables students to establish mathematical lines of reasoning and use them to apply mathematical concepts and tools to solve problems. The Mathematical Practices for AP Calculus are not intended to be viewed as discrete items that can be checked off a list; rather, they are highly interrelated tools that should be utilized frequently and in diverse contexts. The sample items included with this curriculum framework demonstrate various ways in which the learning objectives can be linked with the Mathematical Practices for AP Calculus.

The Mathematical Practices for AP Calculus are given below,

MPAC 1: Reasoning with definitions and theorems MPAC 2: Connecting concepts  Use definitions and theorems to build arguments, to justify  Relate the concept of a limit to all aspects of calculus. conclusions or answers and to prove results.  Use the connection between concepts (e.g., rate of change and  Confirm the hypotheses have been satisfied in order to apply the accumulation) or processes (e.g., differentiation and its inverse conclusion of the theorem.  Apply definitions and theorems in the process of solving a problem. process, antidifferentiation) to solve problems.  Interpret quantifiers in definitions and theorems (e.g., “for all,” “there  Connect concepts to their visual representations with and without exists”) technology.  Develop conjectures based on exploration with technology  Identify a common underlying structure in problems involving  Provide examples and counterexamples to clarify understanding of definitions, to investigate whether converses of theorems are true or different contextual situations. false, or to test conjectures

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Mathematics – AP/IB Calculus AB MPAC 3: Implementing algebraic/computational processes MPAC 5: Building notation fluency 푑푦  Know an use a variety of notations (e.g., 푓′(푥), 푦′, )  Select appropriate mathematical strategies. 푑푥  Sequence algebraic/computational procedures logically.  Connect notation to definitions (e.g./ relating the notation for the  Complete algebraic/computational processes correctly. definite integral to that of the limit of Riemann sum)  Apply technology strategically to solve problems.  Connect notation to different representations (graphical, numerical,  Attend to precision graphically, numerically, analytically, and verbally analytical, and verbal) and specify units of measure.  Assign meaning to notation, accurately interpreting the notation in a  Connect the results of algebraic/computational processes to the given problem and across different contexts question asked. MPAC 6: Communicating MPAC 4: Connecting multiple representations  Clearly present methods, reasoning, justifications, and conclusions.  Associate tables, graphs, and symbolic representations of functions.  Use accurate and precise language and notation.  Develop concepts using graphical, symbolical, verbal, or numerical  Explain the meaning of expressions, notation, and results in terms of representations with and without technology. a context (including units).  Identify how mathematical characteristics of functions are related in  Explain the connections among concepts different representations.  Critically interpret and accurately report information provided by  Extract and interpret mathematical content from any presentation of technology. a function (e.g., utilize information from a table of values).  Analyze, evaluate, and compare the reasoning of others

 Construct one representational form from another (e.g., a table from a graph or a graph from given information).  Consider multiple representations (graphical, numerical, analytical,

and verbal) of a function to select or construct a useful representation for solving a problem.

Source: AP Calculus AB/BC Course and Exam Description

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