PreCalculus: Lesson Plan * 11/1/2006

Quickstart
Writing About Math
p.265

Project
Predicting Basketball Accuracy
Project Goal: Do an experiment to find how your ability to make a basket changes with distance.

The three-point rule was instituted by the National Basketball Association in 1979. Why do you think the rules were changed?

Part I
Investigating your Basketball ability

Part II
Making a Model

Part III
Writing a Report

Algebra I: Lesson Plan * 10/31/2006

Quickstart
Strand E Opener #5

Lesson
Chapter 5 Project Fast Talker Objective: Gather data and use tables, graphs, and function rules to create models of this real-life data. Students will practice creating and using scatter plots as they investigate and display relationships among sets of data.

Some websites to further explore tongue twisters
Tongue-twister - Wikipedia
1st International Collection of Tongue Twisters

PreCalculus: Lesson Plan * 10/31/2006

Quickstart
1. Solve for x: 7^x = 3; x ≈ 0.5646
2. Solve for x: log(x + 4) + log(x + 1) = 1; x = 1 (x = -6 is not in the domain.)

Lesson
3.5 Exponential and Logarithmic Models
Objective: Students will know how to fit exponential and logarithmic models to sets of data, and how to use exponential growth models, exponential decay models, logistic growth models, and logarithmic models to solve applications.

I. Introduction
The five most common types of mathematical models involving exponential functions and logarithmic functions are as follows.
1. Exponential growth model
2. Exponential decay model
3. Gaussian model
4. Logistic growh model
5. Logarithmic models
See p.258 for their general forms and basic shapes

Check Understanding
p.266 #1 - 14

II. Exponential Growth and Decay
Example 1: Population Growth p.259

Map of countries by population. From wikipedia.com

Example 2: Modeling Population Growth p.260
Example 3: Carbon Dating

IV. Logistic Growth Model p.263
Some populations initially have rapid growth, followed by declining rate of growth. One model for describing this type of growth pattern is the logistic curve, called a sigmoidal curve.

Example 5: Spread of a Virus p.263

V. Logarithmic Models p.264
Example 6: Magnitudes of Earthquakes

VI. Fitting Models to Data pp.264 - 265
Example 7: Fitting a Logarithmic Model
Example 8: Fitting an Exponential Model

Algebra I: Lesson Plan * 10/30/2006

Quickstart
Checkpoint Quiz #2 p.267

Lesson
12-1: Inverse Variation Objective: To solve inverse variations using the correct operations and to use graphs and tables to compare inverse and direct variation.

Investigation: Inverse Variation p.636
Think-Pair-Share

Definition: Inverse Variation p.637
An equation in the form xy = k or y = k/x, where k ≠ 0, is an inverse variation.
The constant of variation is k.

Example 1: Writing an equation in inverse variation given a point
Check Understanding #1
Example 2: Find the missing coordinate

Key Concept: Summary - Direct and Inverse Variation
Direct Variation
y = kx
Graph is an increasing straight line.
y varies directly with x.
y is directly proportional to x.
the ration y/x is constant

Inverse Variation
y = k/x
Graph is a decreasing curve.
y varies inversely with x.
y is inversely proportional with x.
The product xy is constant.

Example 4: Determining direct or inverse variation
Example 5: Real-World Problem Solving

Math in the Media
p.641 #44

6-1 Rate of Change and Slope Objective: To solve problems by finding the rate of change and to find slope using a coordinate plane.

Investigation: Exploring Rate of Change p.282

Ski lifts are used to transport skiers up ski slopes.
Slope is rate of change,
the change in the dependent variable compared to the change in the independent variable.

Example 3: Finding Slope Using a Graph

Formula
Slope = rise / run = (y₂ - y₁) / (x₂ - x₁), where x₂ - x₁ ≠ 0

Homework
p.640 - 641 #3 - 45 x 3's

PreCalculus: Lesson Plan * 10/30/2006

Quickstart
p.254 #43 - 51

Assignment
Review #246
Solving exponential equations

Lesson
3.4 Solving Exponential and Logarithmic Equations
Objectives: Students will know how to solve exponential and logarithmic equations.

III. Solving Logarithmic Equations
There are two basic ways of solving logarihmic equations.
1. Isolate the logarithmic expression and then write the equation in equivalent exponential form.
2. Get a single logarithmic expression on each side of the equation, with the same base, then use the one-to-one property.

Example 2: Solve the following logarithmic equations and round your answer to three decimal places.
a) 2logx = 5
log x = 5/2
x = 10^5/2
x ≈ 316.228
b) ln √(x + 2) = ln x
e^{ln √(x + 2)} = e^{ln x}
√(x + 2) = x
x + 2 = x²
x² - x - 2 = 0
(x - 2)(x + 1) = 0
x - 2 = 0, x = 2
OR
x + 1 = 0, x = -1
-1 cannot be a solution because of the domain of the logarithmic function.
c) log x - log(x - 3) = 1
log [x/(x - 3)] = 1
x / (x - 3) = 10¹
x = 10(x - 3)
x = 10x - 30
-9x = -30
x = 10/3

Think-Pair-Share
p.255 #95, 97

IV. Approximating Solutions p.252
Some equations are beyond our algebraic skills and ony approximate solutions can be obtained.
See p.252 Example 11

Homework
p.254 - 256
#15 - 60 x 3's
#81 - 96 x 3's
#118

"Age affects how people experience time. The observations on this are well known, so it is only necessary to outline briefly what has been the experience of everyone I have ever talked to or read about: the years go faster as one gets older. At the age of four or six, a year seems interminable; at sixty, the years begin to blend and are frequently hard to separate from each other because they move so fast! There are, of course, a number of common-sense explanations for this sort of thing. If you have only lived five years, a year represents 20 percent of your life; if you have lived fifty years, that same year represents only 2 percent of your life, and since lives are lived as wholes, this logarithmic element would make it difficult to maintain the same perspective on the experience of a year’s passage throughout a lifetime." -Edward T. Hall, “Experiencing Time,” The Dance of Life: The Other Dimension of Time, Doubleday (1983)
From Quoteland.com