Getting More "AP" into your PreAP Classes
Catherine Baker - Nederland High School
16 Years Teaching Experience
2002-2011 AP Calculus & Geometry
2011-present AP Calculus & PreAP PreCalculus
AP Calculus Exam Format - 2017
15 Questions - Calculator - 45 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.
Graphing Calculator Capabilities for the Exams
- 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
The Rule of Four
The Rule of Four: AP Calculus
Even and Odd Functions Using the Rule of Four
Rational and Negative Exponents
Composite functions: Algebra II or PreCal
Simplify until original denominator is gone
Logs and Exponentials
Logarithmic and Exponential Graphs
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.
Area without Integration
Find the area under the curve f(x) from -5 to 0.
Shapes by Rotation
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.
Composite and Inverse Functions
Cubic Functions: The parent function y = x^3 is concave down for x<0 and concave up for x>0.
Transformations of Functions
Writing Equations as Functions of a Specified Variable
Intermediate Value Theorem
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.
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.
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
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)