We used to call it the “Rule of Four.” Maybe that’s why its MPAC 4.

MPAC 4: Connecting multiple representationsStudents 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 different representations;

d. extract and interpret mathematical content from any presentation of a function (e.g., utilize information from a table of values);

e. construct one representational form from another (e.g., a table from a graph or a graph from given information); and

f. consider multiple representations (graphical, numerical, analytical, and verbal) .of a function to select or construct a useful representation for solving a problem.

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AP, The College Board, New York © 2016. Full text is here.^{®}Calculus AB and AP^{®}Calculus BC Course and Exam Description Effective Fall 2016

This is another concept not just of use in the calculus. Students should be using symbols, geometric representations (not just graphs), and numerical ideas along with reading and writing about mathematics from their first days in school.

The symbolic (analytic) aspect of the Rule of Four is perhaps a bit more important in doing mathematics. Things have to be proved analytically. The proper use of symbols in mathematics is the subject of MPAC 5.

The verbal part of the Rule of Four also includes writing and explaining mathematics. This is the subject of MPAC 6.

Technology makes using graphs and table of values very easy. Back in ancient times (that is, in BC – before calculators) when I was in high school getting a graph or a table of values required a lot of work. Now these things are easy and quick when using a calculator; now we can spend our time on what the graphs and numbers mean and what they tell us about the situation we’re investigating.

**How/where can you make sure students use these ideas in your classes. **

The Rule of Four is definitely not restricted to calculus. Using and relating the parts of the Rule of Four should start way back to the students’ earliest work in mathematics long before Algebra 1. “Graphically” should be expanded to “geometrically;” students should be using drawings and pictures and the like before they learn graphing; and continue to use non-graph representations where appropriate after they learn graphing.

While symbolic or analytic work (working with equations, matrices, etc.) is still where you go when you want to be sure something is true (i.e. to prove things), the others have their place in investigations, in helping to form conjectures, and helping to understanding meaning. By the time they get to the calculus, students should be familiar with looking at functions and other mathematical objects from all four perspectives.

Many problems lend themselves to working with only one or two of the Four. This is natural. While you do not have to force all four aspects into every problem, always consider the others. It is not unusual that one of the other might make things clearer. Students who are required to explain verbally or in writing what they are doing (MPAC 6) will benefit even if that is not strictly required.** **

When AP exam questions are written the writers reference them to the LOs, EKs and MPACs. The released 2016 Practice Exam given out at summer institutes this summer is in the new format and contains very detailed solutions for both the multiple-choice and free-response questions that include these references. (This version is not available online as far as I know.) About 1/4 of the multiple-choice and about ½ of the free-response questions on both AB and BC exam reference MPAC 4.

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