Mathematical Practices & Big Ideas (College Board AP® Calculus AB): Revision Note

Practice 1

Implementing mathematical processes

Determine expressions and values using mathematical procedures and rules

Practice 2

Connecting representations

Translate mathematical information from a single representation or across multiple representations

Practice 3

Implementing mathematical processes

Justify reasoning and solutions

Practice 4

Communication and notation

Use correct notation, language, and mathematical conventions to communicate results or solutions

Practice 1

Implementing Mathematical Processes

1.A

Identify the question to be answered or problem to be solved (not assessed).

1.B

Identify key and relevant information to answer a question or solve a problem (not assessed).

1.C

Identify an appropriate mathematical rule or procedure based on the classification of a given expression (e.g., Use the chain rule to find the derivative of a composite function).

1.D

Identify an appropriate mathematical rule or procedure based on the relationship between concepts (e.g., rate of change and accumulation) or processes (e.g., differentiation and its inverse process, anti-differentiation) to solve problems.

1.E

Apply appropriate mathematical rules or procedures, with and without technology.

1.F

Explain how an approximated value relates to the actual value.

Practice 2

Connecting Representations

2.A

Identify common underlying structures in problems involving different contextual situations.

2.B

Identify mathematical information from graphical, numerical, analytical, and/or verbal representations.

2.C

Identify a re-expression of mathematical information presented in a given representation.

2.D

Identify how mathematical characteristics or properties of functions are related in different representations.

2.E

Describe the relationships among different representations of functions and their derivatives.

Practice 3

Justification

3.A

Apply technology to develop claims and conjectures (not assessed).

3.B

Identify an appropriate mathematical definition, theorem, or test to apply.

3.C

Confirm whether hypotheses or conditions of a selected definition, theorem, or test have been satisfied.

3.D

Apply an appropriate mathematical definition, theorem, or test.

3.E

Provide reasons or rationales for solutions and conclusions.

3.F

Explain the meaning of mathematical solutions in context.

3.G

Confirm that solutions are accurate and appropriate.

Practice 4

Communication and Notation

4.A

Use precise mathematical language.

4.B

Use appropriate units of measure.

4.C

Use appropriate mathematical symbols and notation (e.g., Represent a derivative using , , ).

4.D

Use appropriate graphing techniques.

4.E

Apply appropriate rounding procedures.

Big idea 1
Change (CHA)

Using derivatives to describe rates of change of one variable with respect to another or using definite integrals to describe the net change in one variable over an interval of another allows students to understand change in a variety of contexts. It is critical that students grasp the relationship between integration and differentiation as expressed in the Fundamental Theorem of Calculus—a central idea in AP Calculus.

Big idea 2
Limits (LIM)

Beginning with a discrete model and then considering the consequences of a limiting case allows us to model real-world behavior and to discover and understand important ideas, definitions, formulas, and theorems in calculus: for example, continuity, differentiation, and integration.

Big idea 3
Analysis of functions (FUN)

Calculus allows us to analyze the behaviors of functions by relating limits to differentiation, integration, and infinite series and relating each of these concepts to the others.