# CAMT 2016

### 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

## Multiple Choice

15 Questions - Calculator - 45 minutes

## Free Response

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

1. Analytically

2. Graphically

3. Numerically

4. Verbally

## Even and Odd Functions Using the Rule of Four

**Analytically**

**Graphically**

**Numerically**

**Verbally**

## Algebra Skills

## Rational and Negative Exponents

**Algebra I or II**

**Calculus**

**Add problems like this to Algebra II or PreCal.**

## Composite functions: Algebra II or PreCal

**Easy**

**More Difficult**

**Even More Difficult**

## Difference Quotient

**Alg I, Alg II or PreCal**

Simplify until original denominator is gone

**Easy**

**Easy**

**More Difficult**

**More Difficult**

## Logs and Exponentials

**Most solving problems in Algebra II and PreCal are like this:**

**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.**

## Logarithmic and Exponential Graphs

## Geometry

## Proofs

**Pythagorean Theorem**

**Area of a Trapezoid**

**Surface Area of a Cylinder**

## 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

**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.

## Shapes by Rotation

## Trigonometry

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

**Inverse:**Using a table

**Inverse:**f(a) = b and g(b) = a when f and g are inverses.

## 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.

## Transformations of Functions

**Prepare students to find volumes of solids that rotate around vertical and horizontal lines that are not axes.**

## Writing Equations as Functions of a Specified Variable

## Intermediate Value Theorem

## 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.

## 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.

## 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)