Subtract the Hole from the Whole.

Sometimes I think textbooks are too rigorous. Behind every Riemann sum is a definite integral. So, authors routinely show how to solve an application of integration problem by developing the method starting from the Riemann sum and proceeding to an integral that give the result that is summarized in a “formula.” There is nothing wrong with…

MPAC 6 Communicating

Saving the best, or perhaps the most important, until last, MPAC 6 is the verbal part of the Rule of Four. Problems and real life situations are “translated” from ideas or words into the symbols, equations, graphs, and tables where they are examined and manipulated to find solutions. Once the solutions are found, they must…

MPAC 5 Notational Fluency

MPAC 5: Building notational fluency Students can: a. know and use a variety of notations (e.g., ); b. connect notation to definitions (e.g., relating the notation for the definite integral to that of the limit of a Riemann sum); c. connect notation to different representations (graphical, numerical, analytical, and verbal); and d. assign meaning to notation, accurately…

MPAC 4: Multiple-representations

We used to call it the “Rule of Four.” Maybe that’s why its MPAC 4. MPAC 4: Connecting multiple representations Students can: a. associate tables, graphs, and symbolic representations of functions; b. develop concepts using graphical, symbolical, verbal, or numerical representations with and without technology; c. identify how mathematical characteristics of functions are related in…

MPAC 3 Computing

Continuing our look at the Mathematical Practices today we consider computations. We require students to do computations so that they will learn how to do computations; the answer and the check are just the last steps.  MPAC 3: Implementing algebraic/computational processes Students can: a. select appropriate mathematical strategies; b. sequence algebraic/computational procedures logically; c. complete…

MPAC 2 – Connections

-Steve Jobs Continuing the series on the Mathematical Practices for AP Calculus (MPACs) today we look at MPAC 2. MPAC 2: Connecting concepts Students can: a. relate the concept of a limit to all aspects of calculus; b. use the connection between concepts (e.g., rate of change and accumulation) or processes (e.g., differentiation and its inverse process,…