## Math

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## Precalculus/Calculus

In this course, students complete a unique 12-part year-long project ranging from algebra through calculus. This original project begins with students choosing three locations on the Earth and involves using linear equations, creating piece-wise functions to match data, using limits to test for continuity, finding and using first and second derivatives, using integrals to solve problems, and other topics such as average value, arc length, trapezoidal approximations, and even partial derivatives.

"A Short History of Nearly Everything"
"Functions in the Real World"
Media Used:
TI-83 Graphing Calculator
Significant Assignments:

Create your own homework assignments: Throughout the year, students create their own homework problems to complete and then exchange those problems with other students in the class. Students keep their problems in a safe place so that the final product is like a student-created math textbook they can bring to college.

Create your own exams: Students consistently create exams to assess their knowledge about the particular subject we are studying. Students exchange exams and work together to find and correct mistakes. On exam day, students each have their own unique exam.

Significant Activities or Projects:

The Map Project is a year-long 12-part project ranging from algebra through calculus. This is the main performance-based assessment students complete.

The Native Country Project, wherein students collect data from some aspect of their native countries (population, car accidents, disease, etc) and use the data to explore applications in math.

The Products Project, in which students try to save companies money by repackaging their products to use less surface area (keeping the volume constant) or hold more volume (keeping the surface area constant)

Sample PBATs:
The Map Project Part One
The Native Country Project

## Writing to Learn Math

Students review fundamental concepts from Algebra, Geometry, and Statistics and demonstrate their understanding via repeated short writing exercises which strengthen students' abilities to describe math operations and processes using words and statements. Students then complete indepth investigations on three topics: 1) Is the Coney Island Coin Toss Game Fair?; 2) How do manufacturers decide to package grocery store items?; and 3) How does the student government of BCS determine the ticket price for a dance?

Essays using math from trade magazines
Former students' Math papers
Significant Assignments:

Students write out every step involved in factoring a polynomial to extreme detail and compare with other students.

Students compare written analyses of a similar problem to see how different styles of writing help a writer to convey his/her point.

Significant Activities or Projects:

What do we know about carnival games? They may seem so easy to win, but in reality, we don’t win every time. Students investigate the original design of the Coney Island Coin Toss Game and decide if its design gives players a fair chance to win.

Walk into any supermarket and look at the shapes lining the shelves. Manufacturers consider dozens of factors before determining which shape will best suit the consumer and boost the company's profits. Students explore package design and uncover some of the reasons for the shapes that manufacturers have chosen. Students use their knowledge of geometry and algebra to investigate different ways to decide whether or not it is better for manufacturers to sell items in “bulk” or to sell individually packaged items and determine if manufacturers are packaging their items as efficiently as possible.

Suppose you are a student council member who is responsible for planning a BCS student dinner dance. Plans include hiring a DJ and buying and serving dinner. You want to keep the ticket price as low as possible to encourage student attendance. Identify and use systems of equations to analyze costs and make decisions. Then write a report detailing your choice of band, the cost of a catering service, and your ticket price recommendation. Present this report to Student Government.

Sample PBATs:
Are carnival games fair?
How can I plan a student event to maximize profit or minimize cost?

## Data Analysis

Knowledge of statistics is a valuable asset that will help students make sense of the vast amount of information we are presented with everyday. The goal of this course is to give the student a hands-on ability to apply the fundamentals of statistics to make informed decisions. This will be accomplished by examining the skills required to read data, interpret data and judge others' claims about data. In addition to manual calculations, we will look at the usefulness of technological tools such as the graphing calculator and Microsoft6 Excel in data analysis.

Significant Assignments:

Analyze the difference between various kinds of studies; apply knowledge of statistics to critique data presented and conclusions reached.

Determine factors which may affect the outcome of a survey; consider outcomes of various hypothetical situations on outcomes.

Significant Activities or Projects:

Calculate the measures of a central tendency with group frequency distributions

Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations

Sample PBATs:
Choose a subject of interest to you that is appropriate for a statistical analysis. Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data.

Algebra 2 is a continuing exploration of algebra for eleventh graders who are preparing for graduation-level Performance Based Assessment Tasks. This course is the second and third part of three-semester series. By the end of this year, all 11th graders should have written a graduation level report and presented her or his project for consideration of graduation PBAT.

All About Alice unit in IMP
Meadows or Malls unit in IMP
Small World, Isn't It? unit in IMP
High Dive unit in IMP
Media Used:
http://www.cssjbronx.org
Alice in Wonderland film
Significant Activities or Projects:

All About Alice project: exploring in-depth application of logarithms and scientific notation

Parks or Malls project: solving a system of multi-variable (at least 6) linear inequalities using matrices

Small World, Isn't It? project: exploring in-depth application of exponential functions and data analysis

High Dive project: in-depth application of trigonometric functions, polar coordinates, and parametric functions

Sample PBATs:
Multi-constraint linear programming
Population data analysis and prediction using exponential functions
Application of parametric and trigonometric functions in polar coordinates

Students will explore content focusing on: Reasoning, Numbers and Numeration, Operations, Modeling/Multiple Representation, Measurement, Uncertainty and Patterns/Functions.

Some of the major topics that are covered include: Geometric Proofs, Circles, Complex Numbers, Quadratics, Functions, Conic Sections, Transformations, Logarithms, Regressions, Trigonometry, Binomial Theory, Probability and Statistics.

Pearson:// Elementary Statistics
Media Used:
www.aw-bc.com/triola
Sample PBATs:
"Texas Tech -vs- Oklahoma" A comparative statistical analysis that exhibits how data can be manipulated to convey a variety of messages.
"Plinko" Students design their own 'Plinko Board' then use Pascal's Theory to compare their empirical data against expected outcome data.
"Trigonometry in Aviation" Students test various scenarios in relation to Visual Flight Rules.
"Calculating the Magnitude of an Earthquake" Students measure Magnitude and other variables using scale developed by Charles Richter.

## College Algebra, Trig, & Statistics (with LaGuardia Community College)

Math 115 is College Algebra- Trigonometry. The course will start with a review of basic algebra (factoring, solving linear equations and equalities) then proceed to study polynomials, exponential, logarithmic and trigonometric functions. Students will use this mathematical modeling to improve critical thinking skills in inquiry based activies.

Math 120- Statistics

America's Climate Choices- http://dels.nas/edu/basc/climate-change/
Redefining Progress: www.myfootprint.org
Medline Plus- US National Library of Medicine and National Institutes of Health- http://medlineplus.gov
Healthy People- http://www.healthypeople.gov
Center for Disease Control and Prevention- http://www.cdc.gov/datastatistics
Investopedia- http://investopedia.com
World Health Organization- Report- http://www.who.int/en
Media Used:
E-Portfolio
MAPLE
Significant Assignments:

The first part involves reading information related to CO2 emissions from transportation, followed by data on CO2 emissions by cars and transportation choices by country. Using the data, students will
analyze and make recommendations for environmental performance.-

Co2 Emmisions by Cars - By Sreedevi Ande- Department of Mathematics, Engineering and Computer Science at Laguardia Collmunity College

Modeling Unemployment Rate Data- By Denise Carter- Department of Mathematics, Engineering and Computer Science

This project is meant for e-portfolio.
Students use MAPLE to analyze the unemployment rates data for the months January through September

Sample PBATs:
Project Quantum Leap--tasks change each year in response to the research conducted by students

## Functions

This course is based on the following Essential Questions: How can we use functions to make predictions about the world? Should judgments be based on quantified relationships?

Students will learn about and analyze three types of functions: Linear, Quadratic, and Exponential

Significant Assignments:

In this mastery project, you will investigate which kind of function best models your data.

Formulate:
1.Give an in depth description your situation, including any background information and images that make your investigation as clear as possible.
3.Explain why it makes sense for your data set to be modeled by that function.

Represent:
4.Show and explain in as much detail as possible:
a.Your variables: what do “x” and “y” stand for?
b.Tables: data table and a table that models your data
c.Graph: clearly indicate your scale and units
d.Equation: identify each of the coefficients
5.In your opinion, which representation (table, graph, or equation) best reveals the nature of your function or gives you the most information about your function?

Analyze:
6.Show and explain interpolated and extrapolated predictions for your function.
7.Choose and explain a domain and a range for your function.
8.Explain the meanings of your coefficients: Pick any coefficient. If it were to change, how would your function be affected? Do the same for all coefficients.
9.How do your table, equation and graph help people understand the problem or situation your data describes?
10.Analyze the graph – what do all the intercepts, extreme points, or important points mean in the context of the situation?
11.Each student gets one extension question from their teacher.

Interpret:
12.During a class lesson planned by your teacher, you will make sense about at least one other person’s problem.

Significant Activities or Projects:

Identify patterns in a variety of forms (stories, geometric patterns, linear patterns).
Represent patterns (xy-relationshps) using words, tables, graphs and equations.
Describe growth that either increases or decreases by a constant amount.
Write an equation of a linear function in slope-intercept form.
Identify the slope and the y-intercept from the equation of linear function.
Use a variety of situations to collect and organize data (Barbie bungee, counting beats, balloon drop).
Use linear regression to model real-world data.
Use a linear regression to make predictions about linearly modeled situations.
Given a situation that can be modeled using a step function (postage rates, etc…), create a graph to model the situation.
Recognize non-integer values within an interval of values.
Use inequalities to describe intervals.
Come up with situations that can be modeled using step functions.

Identify more than one way of describing the “change” in the Step Change pattern.
Identify one way of describing the change as linear and another way as “non-linear.”
Describe the nature of the change of the non-linear.
In their own words, explain the idea that the quadratic growth means that the difference is linear.
Develop a strategy for the “chip-swap” game.
Using the standard form of the quadratic equation, identify how far a runner is after various amounts of time running.
Plug in x-values to find y-values in quadratic functions.
Find out how far the runner is after further amounts of time by following the pattern (similar to Step Change and Chip Swap).
Identify the standard form of a quadratic equation.
Make up quadratic equations in standard form.
Identify the a, b and c-coeffieients of quadratic equations in standard form.
Develop rules for the effect of the a- and b-coefficients on the graph of the equation.
Apply –b/2a as the x-value of the turning point on the graph.
Given an equation for a quadratic equation in standard form, shift the parabola up/down, right/left, make it skinnier/wider, and flip it.
Determine an object’s height off the ground after various seconds using an equation (with factors for speed, initial height off the ground, and gravitational pull back towards the earth).
Given an object’s initial speed, initial height off the ground, and gravitational pull back towards the earth, write an equation for the object’s position as a function of time.
Determine the amount of time it will take an object to rise and then land in a well below the surface of the earth.
Given data for the relationship between time and an object’s position off the ground, write a quadratic equation.
Create a reasonable pricing system for pizzas based on square units of area in the pizza and another based on diameter.
Discuss the differences in the relationships between “size” and “price.”
Given a specific amount of fencing, sketch various rectangles with perimeter equal to the amount of fence. (for various amounts of fencing).
Determine the dimensions that provide the largest area for each.
Create a generalization for the dimensions with largest area with perimeter P.
Create a rulebook for the rules of exponents.
Identify the growth pattern of exponential functions.

Sample PBATs:
Coffee Cooling: How Long is Too Long?
Additional PBAT questions grow out of the activities listed above

12th Grade Math Analysis Course Description

In this course, students study Pre-Calculus using the College Preparatory Mathematics program www.cpm.org This is an advanced, challenging course where we will explore mathematical ideas such as Problem Solving, Analysis of Models, Trigonometry, Advanced Functions, and Algebraic Fluency and Accuracy. Students will learn a variety of approaches to problem solve and use math in real life contexts.

CPM College Peparatory Mathematics
Significant Assignments:

Polynomial Functions: How does this connect to linear and quadratic functions

Trigonometric Functions: How can we model this relationship (ie height over time, etc.) using a trigonometirc funciton?
How does the unit circle connect to the graph ofthe sine/cosine curve/
How are the trigonometric functions related to each other

Law of Sines/Cosines: How do you know which formula you should use?
How does this connect to right triangle trigonometry and trigonometric functions?

Probability: How can you use probability to predict outcomes?
How can you use probability to make policy decisions and analyze the world around you? Is that a fair situation/game? Why or why not?

Combinatorics: How many choices are there?

Statistics: How can you use statistics to advocate for social justice? What do these statistics tell you about the situation, is there a bias? How does statustics help you seek significvance in the study of mathematics? How can you use statistics to provide evidence and reinforce your argument?

Significant Activities or Projects:

Using trigonometric functions analyze the height, distance and speed of amusement rides at Six Flags.

By analyzing Vanguard's data distribution determine if the data distribution is "normal"

Sample PBATs:
Vanguard Six Flags: Trigonometric Functions
Skywalker: Law of Sines and Cosines
Your Game: Polynomial Functions, Probability, Combinations and Permutations
Is Vanguard Normal--Statistics Analysis

## Statistics

This one semester course will provide experience in collecting data, analyzing data, and writing statistical reports. Students will study statistical concepts through activities and projects that involve the collecting and analyzing of data. Techniques for summarizing, analyzing and interpreting large sets of data will also be discussed. Emphasis will be on practical problem-solving with real data sets using appropriate computer software packages. This course culminates in a math portfolio presentation.

Prerequisite: Both Algebra & Geometry portfolios with at least a “Competent” rating

College Preparatory Mathematics (CPM)
Significant Assignments:

Portfolio/Panel Presentation: Social Justice Statistics Research and Analysis using single variable regression

Sample PBATs:
“Prison Populations and Race”
“Relationship Abuse and Gender”

## Foundations of Pre-calculus

A year-long course designed to strengthen and expand the concepts and skills learned in algebra and provide further development of the concept of a function. Students will expand their knowledge and understanding of linear, quadratic, and exponential functions and be introduced to logarithmic and trigonometric functions. Students will analyze and identify relationships among functions using multiple representations (verbal, numeric, graphic, and algebraic). Students will apply mathematical skills to make meaningful connections to the physical world and write about their conclusions.

Sample PBATs:
How can Matrices be used to solve multivariable mathematical situations?
How can linear and quadratic functions be used to maximize a company's profits?
How can exponential functions be used to predict the future value of investments?