CAMT 2016

Getting More "AP" into your PreAP Classes

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Catherine Baker - Nederland High School

16 Years Teaching Experience

2000-2002 Algebra I & Geometry


2002-2011 AP Calculus & Geometry


2011-present AP Calculus & PreAP PreCalculus

AP Calculus Exam Format - 2017

Multiple Choice

30 Questions - No Calculator - 60 minutes


15 Questions - Calculator - 45 minutes

Free Response

2 Questions - Calculator - 30 minutes


4 Questions - No Calculator - 60 minutes


More than 60% of the exam is WITHOUT a calculator. Students should be proficient at skills both with and without a calculator.

PreCal Skills - No Calculator

Transformations of functions (quadratic, absolute value, reciprocal, square root, exp, log, trigonometric)

Polar graphing with given equation and table

Graphing piecewise functions

Graphing parametric functions

Complex number operations

Sketching graphs of higher polynomial functions

Solving non-linear equalities

Graphing rational functions

Trigonometric values at special angles

Graphing conic sections

PreCal Skills - With Calculator

Polar graphing: Generate table with calculator and graph.

Using “Table” and Intermediate Value Theorem (IVT) to find integer intervals for zeros.

Finding/Using non-special angles in trigonometry.

Finding zeros of functions.

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Graphing Calculator Capabilities for the Exams

The committee develops exams based on the assumption that all students have access to four basic calculator capabilities used extensively in calculus. A graphing calculator appropriate for use on the exams is expected to have the built-in capability to:
  • plot the graph of a function within an arbitrary viewing window
  • find the zeros of functions (solve equations numerically)
  • numerically calculate the derivative of a function
  • numerically calculate the value of a definite integral

Examples for extending graphing calculator skills

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Teach students how to store numbers and use Y1 or f1(x).
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The Rule of Four

The Rule of Four: AP Calculus

Students are expected to be able to solve problems in four different ways.


1. Analytically


2. Graphically


3. Numerically


4. Verbally

Even and Odd Functions Using the Rule of Four

Analytically
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Graphically
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Numerically
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Verbally
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Algebra Skills

Rational and Negative Exponents

Algebra I or II
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Calculus
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Add problems like this to Algebra II or PreCal.
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Composite functions: Algebra II or PreCal

Easy
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More Difficult
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Even More Difficult
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Difference Quotient

Alg I, Alg II or PreCal


Simplify until original denominator is gone

Easy
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Easy
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More Difficult
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More Difficult
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Logs and Exponentials

Most solving problems in Algebra II and PreCal are like this:
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Here is a separable differential equation question from AP Calculus. Only the first two steps need Calculus skills. The rest can be done in Alg II or PreCal.
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Logarithmic and Exponential Graphs

Graphs can be classified in four ways based on if they are increasing or decreasing, and if they are doing so at an increasing or decreasing RATE.

Geometry

Proofs

Students need a solid understanding of how to prove a concept. They should generate proofs on their own throughout the year, not just in one chapter. As an instructor, you should provide proofs for as many formulas and theorems as possible.
Pythagorean Theorem
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Area of a Trapezoid
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Surface Area of a Cylinder
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Area by Integration

Introduce the integral as a new mathematical symbol meaning to find the area between a function and the x-axis. Area under the x-axis is negative.

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Area without Integration

You could also just introduce area under the x-axis as negative area without formally talking about integration.



Find the area under the curve f(x) from -5 to 0.

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Shapes by Rotation

Graph the line segment that gives a cone of radius 4 and height 3 when rotated about the y-axis.

Trigonometry

Laws of Sines and Cosines are not part of the Geometry TEKS, but it would be beneficial to include the basic questions in Geometry. Then, PreCal can build upon those skills.


Combine Special Right Triangles with Trigonometry to get students familiar with those values. The more times the students can see those values, the easier it becomes to remember them.

Additional Topics

Composite and Inverse Functions

Composite: Using a table
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Inverse: Using a table
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Inverse: f(a) = b and g(b) = a when f and g are inverses.
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Concavity

Quadratic Functions: Say "The parabola is concave up" instead of "The parabola opens up."


Cubic Functions: The parent function y = x^3 is concave down for x<0 and concave up for x>0.

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Transformations of Functions

Prepare students to find volumes of solids that rotate around vertical and horizontal lines that are not axes.
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Writing Equations as Functions of a Specified Variable

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Intermediate Value Theorem

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Slope

Talk about slope as a rate of change

Algebra II or PreCal – Find the slope of the secant line for functions. Talk about the slope being the average rate of change of the function over that interval.

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Physics Applications

Algebra I, Algebra II or PreCal

Introduce physics notation and possible units of measurements.

Position: x(t) feet

Velocity: v(t) feet/sec

Acceleration: a(t) feet/sec/sec

Emphasize units when finding average velocity or average acceleration.

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Graphing Rational Functions

Students find x-int, y-int, vertical asymptotes and end behavior asymptotes.

Give students a few points on the graph (two or three).

Give students intervals for inc, dec and/or concavity.

Have them graph the function without a calculator

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Maximums and Minimums

When talking about max/min (either with parabolas or finding with calculator) ask these questions:

Where does the function reach a maximum value? (answer is x-coordinate)

What is the maximum value? (answer is y-coordinate)

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