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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 MPACs 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 exam questions included in the AP Calculus AB and AP Calculus BC Course and Exam Description (.pdf/5.92MB) demonstrate various ways the learning objectives can be linked with the MPACs.

MPAC 1: Reasoning with definitions and theorems

Students can:

  • Use definitions and theorems to build arguments, to justify conclusions or answers, and to prove results.
  • Confirm that hypotheses have been satisfied in order to apply the conclusion of a theorem.
  • Apply definitions and theorems in the process of solving a problem.
  • Interpret quantifiers in definitions and theorems (e.g., "for all," "there exists").
  • Develop conjectures based on exploration with technology.
  • Produce examples and counterexamples to clarify understanding of definitions, to investigate whether converses of theorems are true or false, or to test conjectures.

MPAC 2: Connecting concepts

Students can:

  • Relate the concept of a limit to all aspects of calculus.
  • Use the connection between concepts (e.g., rate of change and accumulation) or processes (e.g., differentiation and its inverse process, antidifferentiation) to solve problems.
  • Connect concepts to their visual representations with and without technology.
  • Identify a common underlying structure in problems involving different contextual situations.

MPAC 3: Implementing algebraic/computational processes

Students can:

  • Select appropriate mathematical strategies.
  • Sequence algebraic/computational procedures logically.
  • Complete algebraic/computational processes correctly.
  • Apply technology strategically to solve problems.
  • Attend to precision graphically, numerically, analytically, and verbally and specify units of measure.
  • Connect the results of algebraic/computational processes to the question asked.

MPAC 4: Connecting multiple representations

Students can:

  • Associate tables, graphs, and symbolic representations of functions.
  • Develop concepts using graphical, symbolical, verbal, or numerical representations with and without technology.
  • Identify how mathematical characteristics of functions are related in different representations.
  • Extract and interpret mathematical content from any presentation of a function (e.g., utilize information from a table of values).
  • 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.

MPAC 5: Building notational fluency

Students can:

  • Know and use a variety of notations.
  • Connect notation to definitions (e.g., relating the notation for the definite integral to that of the limit of a Riemann sum).
  • Connect notation to different representations (graphical, numerical, analytical, and verbal).
  • Assign meaning to notation, accurately interpreting the notation in a given problem and across different contexts.

MPAC 6: Communicating

Students can:

  • Clearly present methods, reasoning, justifications, and conclusions.
  • Use accurate and precise language and notation.
  • Explain the meaning of expressions, notation, and results in terms of a context (including units).
  • Explain the connections among concepts.
  • Critically interpret and accurately report information provided by technology.
  • Analyze, evaluate, and compare the reasoning of others.