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# Precalc vs. AP Precalc: What's the Difference?

The College Board is launching their newest AP course, AP Precalculus, this fall. You may be wondering what the difference is between the Precalculus course your school has offered for decades and this new AP course (besides the fact that students can earn college credit if they pass the AP Exam). Are the topics covered the same? Which of your teaching resources can you re-use?

Current Precalc classes across the country vary widely in what topics they include. While an in-depth study of various function types is fairly common, treatment of additional topics like polar coordinates, conic sections, sequences and series, and vectors varies from school to school. Perhaps the biggest shift, then, in a course like AP Precalculus is the standardization of topics covered and the emphasis and development of particular themes and throughlines.

The main themes of AP Precalculus are:

• Covariation (being able to describe how two quantities change with respect to each other for a variety of function types)

• Equivalence (producing algebraically equivalent forms of equations and expressions through symbolic manipulation and describing the same function through multiple representations)

• Modeling (constructing new functions that satisfy a mathematical or contextual scenario or model a bivariate data set)

So how else does AP Precalc differ from other Precalc courses? We’re going to answer this question in four broad categories:

• Topics OMITTED in AP Precalc

• Topics ADDED to AP Precalc

• Topics EMPHASIZED in AP Precalc

• Notable pedagogical shifts in AP Precalc

## Topics OMITTED in AP Precalc

There are some topics you just won’t find in AP Precalculus. These are topics that may or may not have been included in your regular Precalc course, depending on where you teach. They are:

• Degrees! (Yup, this is a big one. The whole course is done exclusively in radians. This means there are no conversions between degrees and radians, and the input of a trigonometric function and the solutions of a trigonometric equation are always angles measured in radians.)

• Series and sigma notation

• Non-arithmetic/geometric sequences

• Systems of equations (except for its applications to linear programming and matrices in the optional final unit)

• Right triangle trigonometry (by which we mean solving for sides and angles in a right triangle in isolation, the idea of right triangles IS used to understand the unit circle)

• Factoring polynomials of degree 3 or greater by hand (unless there is a greatest common factor that can be factored out, leaving a quadratic expression)

• Rational Root Theorem/Descartes’ Rule of Signs (these are considered Algebra 2 topics)

• Partial fraction decomposition

• Synthetic division (good riddance, in our opinion; we much prefer using an area model!)

• Half angle identities

• Tangent sum and difference identities

• Law of Sines and Law of Cosines (these are included in the optional final unit but only as a means of finding angles and magnitudes of vectors involved in vector addition)

## Topics ADDED to AP Precalc

• Concavity (determining graphically and analytically how the rate of change of a function is changing)

• Relative and absolute extrema (not just identifying their values but being able to justify them using function behavior near the extrema, e.g. f changes from increasing to decreasing)

• Regression for linear, quadratic, cubic, quartic, exponential, and logarithmic functions

• Instantaneous rate of change (estimating)

• Residual plots

• Semi-log plots

• Limit notation for end behavior and vertical asymptotes

• Polar functions and their rate of change (note that students will NOT need to know the names of various polar graphs but should be able to reason about the behavior of a polar function from a table of values and point-by-point graphing)

• Selecting and validating function models, articulating assumptions and limitations of a model

## Topics EMPHASIZED in AP Precalculus

• Determining the type of function that best models a data set, table of values, or contextual scenario by distinguishing the rate of change of linear, quadratic, cubic, quartic, exponential, and logarithmic functions

• Describing the covariation of a function (e.g. “As inputs grow additively, outputs grow multiplicatively”)

• Exponential and logarithmic functions (the Calc Medic AP Precalculus curriculum has a whole unit on each!)

• Connection between arithmetic and geometric sequences and linear and exponential functions

## Notable Pedagogical Shifts in AP Precalculus

#### 1. Function concepts spread throughout the course

There are a few key differences in the sequence of topics that will affect how the course is structured and taught. While most Precalculus courses have an early unit on function concepts (domain, range, transformations, combinations, compositions, and inverses), in AP Precalculus these topics are introduced throughout the course, with a new function concept being introduced when it makes sense with a particular function type. For example, compositions and inverses are saved until the study of exponential and logarithmic functions, as exponential and logarithmic functions should be studied in the context of their inverse relationship. Transformations are introduced in the context of constructing new functions, rate of change is introduced in the context of linear and quadratic functions (which are both being defined by the rate of change of their rates of change: zero, and a constant value, respectively). Periodicity is introduced in the unit on trigonometric functions, though the idea of periodic behavior is introduced at the very beginning of that unit BEFORE students explore sinusoidal graphs.

#### 2. Conceptual understanding first, procedural skill later

Algebraic manipulation of function types is often saved until the end of the unit, rather than when the function is first introduced and explored. For example, students study the zeros and factors of polynomial functions well before they use polynomial division to fully factor the polynomial expression. We believe this aligns well with the Experience First, Formalize Later model, as students’ understanding increases in formality and rigor not just within a lesson, but also throughout a unit.

#### 3. Greater use of graphing calculators

The use of graphing calculators is also perhaps more pronounced than in a regular Precalculus course. Students in this course should feel very comfortable using their graphing calculator to graph a function and find its intercepts, extrema, and points of intersection with other functions (specifically a constant function, such as when solving an equation). Students should be able to choose an appropriate window to view a graph and use the table or trace feature to evaluate the function at various values. Students also need to know how to use their graphing calculator to create a regression model for a data set.

#### 4. Focus on multiple representations and communication

Finally, AP Precalculus lays a very heavy emphasis on multiple function representations and the ability to communicate one’s reasoning precisely. A traditional Precalculus course tends to focus far more on analytical representations of a function whereas AP Precalculus expects students to transition flexibly between a verbal, graphical, numerical/tabular, and analytical representation of a function. Additionally, a correct numerical solution to a problem is often insufficient in this course, as students will be expected to justify their answer, interpret their answer, and reason about the constraints of their answer or model. Both of these topics of representation and communication will have to be repeatedly addressed, practiced, and reinforced throughout the course in order for students to be successful on the AP Precalculus Exam.