## Math

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## Mathematics: Polynomials & Parabolas

This is the fourth semester of a two-year Algebra sequence and a PBAT class. The PBAT will reflect an in depth studies of parabolas, the main topic of this course. Parabolas occur frequently in the real world and have many applications in art, architecture, engineering, physics and technology. For example, because of their U-shape and reflective properties, parabolic solar panels are constructed to harness solar energy: as solar rays hit the panels they reflect to converge towards a central focus, where heat concentrates, and from which it is transferred to a power generator.

Media Used:
http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/emat6690/Insunit/parabola/
http://tutorial.math.lamar.edu/Classes/Alg/Parabolas.aspx#Graph_Para_Ex1_a
http://www.cut-the-knot.org/ctk/Parabola.shtml
http://www.nrel.gov/csp/troughnet/solar_field.html
http://www.ucsusa.org/clean_energy/technology_and_impacts/energy_technologies/how-solar-energy-works.html
http://www.eere.energy.gov/de/csp.html
Significant Assignments:

Tests and quizzes. Work must be explained.

Sample PBATs:
“How the properties of parabolas are employed in producing solar energy”.
“What equations can be used for parabolic solar panels?"

## Algebra and Beyond

Essential Question: How do we use math to model real life situations?

Unit 1: Exploring Functions
• Review conceptual understanding of topics related to linear relationships.
• Review the skills required to graph a line, to calculate the intersections of two lines, and to calculate x and y intercepts.
• Become familiar with function notation and investigate functions with a graphing calculator.
• Become familiar with domain and range of a function.
• Become familiar with multiple representation of a linear relationship to model real life situations.

College Preparatory Mathematics: Volume One and Two
Interactive Mathematics Program: The Pit and The Pendulum
Media Used:
Graphing Calculators TI84
Calculator-Based Rangers (motion detectors)
Excel
Significant Assignments:

A test at the end of every unit

Significant Activities or Projects:

Unit 1 Pencil Sharpening Lab

Unit 2 The Bouncing Ball Exhibition: Part I and II

Unit 3 Fast Cars Struggle Problem and The Student Loan Problem

Unit 4 Parabolic Path Exhibition (take video of yourself throwing a ball and find the equation to model the path of the ball's flight)

Unit 5 Artistic design using the Sketchpad and parent graphs

The Mathematics of Corporate Responsibility Exhibition (use inequalities represent a corporation's profits and discuss equity issues connected to the math)

The Pendulum Exhibition (determine which variables, when changed, affect the period of a pendulum)

Sample PBATs:
The Bouncing Ball Exhibition
Changing the Period of a Pendulum
Additional PBATs grow out of the problem-solving activities detailed above

## An Introduction to Calculus Thinking

All seniors take an introduction to calculus thinking at ICE. This one-semester course will provide a culmination for students’ mathematical studies at ICE, and a bridge to possible future studies in college.

"What is Calculus About?" by W.W. Sawyer
"The Story of Zero" by Charles Seife
Media Used:
TI-84 Plus, Program Ranger, Geometer's Sketchpad, Microsoft Excel, PowerPoint
Significant Activities or Projects:

Students investigate the idea of limits and infinite sums through a cake-eating problem.

Students investigate and analyze speed by measuring time and distance of an actual moving vehicle and graph their results.

Students investigate the instantaneous speed of a quadratic function, such as a penny dropped from the George Washington Bridge.

Students investigate the instantaneous speeds of various functions by calculating the slope of a tangent line using Geometer's Sketchpad. Students also find patterns between the distance function and its corresponding velocity function (the derivative) and derive the power rule based on their conclusions.

Students prove the validity of the power rule for the derivative by creating and solving for an algebraic proof.

Students investigate the second derivative by dropping a basketball and measuring its velocity and acceleration, and analyzing the slopes of these functions.

Students derive the distance function for a given velocity function by using the anti-derivative or integral, and investigate the relationship between the area under a curve and the integral.

Sample PBATs:
Students present to a Math Roundtable in which one of the investigations above is presented by incorporating technology. Students create and solve their own problem based on their investigations.

## Math 12

In this course students will explore applications and expansions on the essential concepts behind Algebra 2 while prompting students to connect this knowledge to some of the geometeric concepts that will form the foundation of Precalculus.

The Car Dealership Problem
Data Analaysis Project
Sample PBATs:
Data Interpretation PBAT

## Calculus

This course is designed to teach students the fundamentals of calculus.  Students will begin the course by reviewing the concept of a function, then moving on to the ideas of limits and continuity, and finally will formally learn differentiation (they already learned about the general power rule last semester in their race car project) and how to apply it.  This will set students up for next semester, when students will learn integration and how to apply it to area and volume problems.  We will constantly be reviewing topics from last year’s pre-calculus course as needed to strengthen problem-

Calculus, James Stewart, 4th Edition , Barron’s AP Calculus 2008, 9th Edition
Calculus with Applications, Lial, Greenwell, & Ritchey, 8th Edition, Princeton Review’s Cracking the AP Calculus AB & BC Exams, 2008 Edition
Significant Assignments:

Problem sets include examples require critical thinking and diligence.  For example, curve sketching examples require an incredible amount of information gathering and application using graphing skills. Related rates problems entail the creation of diagrams and models, etc.

Significant Activities or Projects:

Midterm

1&2. Reduce each fraction completely (3 pts ea):

1. EMBED Equation.3 2. EMBED Equation.3

3&4. Find the inverse functions (3 pts ea):

3. f(x) = EMBED Equation.3 4. g(x) = EMBED Equation.3

5-7. Find the domain of each of the following (2 pts for each):

5. a(x) = EMBED Equation.3 6. b(x) = EMBED Equation.3
Domain: Domain:

7. c(x) = EMBED Equation.3
Domain:

8&9. Find the zeroes of each of the functions (2 pts ea):

8. e(x) = EMBED Equation.3 9. f(x) = EMBED Equation.3

10-15. Based on the diagram below, find each of the following limits (1 pt ea):

Research:

N/A

Sample PBATs:
The PBAT is embedded within exams that student take.

## PreCalculus

Precalculus concepts become clear using multiple representations. Instruction is therefore aimed to show the subject from many angles; illustrations include algebraic, graph-based, and real-world examples and feature the use of the graphing calculator as a calculation and visualization tool on advanced forms of algebraic math.

Review of Algebra: Review slope, lines, distance-time graphs, closed intervals, and inequalities with an explicit goal of creating a foundation for precalculus and calculus.
Functions and Graphing Functions: Use of graphing calculator; Find domain, range intercepts, zeros, asymptotes; predict shape of graphs; describe transformations; describe relationships; represent functional relationships in formulas, tables, and graphs.
Rational and Polynomial Functions: Given an equation, determine the number and nature of roots, find roots, graph them; describe, generalize, and use basic types of functions; work with properties and mechanics of functions.
Exponential and Logarithmic Functions: Use the laws of exponents and logarithms; define, find, and check inverse functions of logarithmic and exponential functions.
Sequences and Series: Understand and use summation notation; apply the sum formulas for arithmetic series and for finite and infinite geometric series.
Data Analysis: Find linear models using linear regression; decide which model gives a better fit; find a non-linear function to model a data set and explain the parameter of the model.
Media Used:
Incorporating new technology (HP 39/40 mg and TI-Nspire)which would allow for a wider variety of approaches (numeric, graphic, symbolic, and verbal) that appeals to all learners.
Sample Websites:
Jefferson Math Project website: jmap.org
Kuta Software: http://www.kutasoftware.com/free.html
Brunnermath: an immense directory of worksheets, tutorials and activities for precalculus
National Council of Teachers of Mathematics (NCTM) Illuminations: http://illuminations.nctm.org
M&M's web site: http://www.mms.com/
Significant Assignments:

Mathematical Lens: Using photos of amusement park rides, water fountains, landscape, house, etc., students model the functions that the shapes of the objects represent

Families of Graphs: Students identify the domain, range, intercepts, zeros, asymptotes, and points of discontinuity of functions. (Use paper and pencil methods and graphing calculators.)

Solve word problems involving applications of logarithmic and exponential functions.

Significant Activities or Projects:

M&M Activity: Using M&Ms, students will generate data to explore exponential growth and decay.

Helicopter Rotors Activity: Using cut-out paper models of short and long rotors, students will compare drop times and model functions.

“Shockwave” activities that illustrate real-time correlations between equations and graphs designed to help students visualize and experiment with the graphs of functions, systems, and complex numbers.

Writing: Explain the use the properties of the number systems and order of operations to justify steps of simplifying functions and solving equations.

Research:

Examples of exponential growth and decay occur often in the real world. In the world of finance, for example, there are savings accounts, mortgages, automobile loans. Population growth and the half-lives of radioactive material also provide numerous real world examples of exponential growth and decay. Students will research on selected real-life situation that models exponential growth or exponential decay. Students will produce data through sampling and experiment, organize the data obtained, and explain what information is revealed by their data (i.e., look for patterns and deviations from patterns.

Sample PBATs:
Modeling Movie Gross Income: Is it Exponential Decay? (Use data on top 10 movies (gross incomes), selected years)
Population Growth Model: U.S. Population from 1790-2000. (Students compare with others to see whose prediction model was most accurate.)
Growth of Investments in Stock Market