Mathematics

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Advanced Algebra (Mathematical Modeling)

In this course, the primary focus is the creation of linear and exponential models to represent rates of growth involving a variety of situations, particularly human population growth.

The course addresses the following focus questions:
*How do rates of change relate to the steepness of a graph?
*How do we use graphs to interpret data?
*What is the definition of slope and how does it relate to real-life situations?
*How do we represent slope graphically and algebraically?
*What is the relationship between slope and a derivative?

Readings: 
"Small World, Isn't It?" - Interactive Mathematics Program
"Population: 7 Billion" - National Geographic Magazine
"Fatima's Story" - ZPG Population Education Program
Research: 

Students research population factors that affect the population growth in their selected countries from various sources considering both quantitative and qualitative data.

Media Used: 
"World in the Balance" - NOVA DVD
"Aftermath: Population Zero" - National Geographic
http://www.unicef.org/statistics/index_countrystats.html
http://www.xist.org/cntry/
http://www.prb.org/DataFinder.aspx
Interim Assessments: 

Students explore mathematical patterns in world population data and make forecasts to extend the patterns. Students organize their thinking in writing to present to the instructor and to their peers.

Students analyze graphical representations of linear and exponential sequences and make inferences based on the behavior of the graph. Additionally, students interpret statements about the graphs of different sequences, justifying their thinking in writing.

Students construct an addition (linear) and a multiplicative (exponential) model representing new computer sales. Students graph the data, calculate the discrepancies between their mathematical models and the actual data, and then interpret which model is better for forecasting future sales. Students then forecast sales into the future and explain some reasons why forecasting future sales would be important to the company.

Significant Assignments: 

Students learn and analyze how scaling can affect the visual representation of graphs through the creation of several graphs utilizing the same data set.

Students find “addition” and “multiplication” growth numbers to increase a given number to a specified target in a specific number of steps.

Utilizing a mathematical model, students utilize growth numbers to calculate intermediate values and then extend the model into the future or conversely into the past.

Students represent sequences algebraically working with the concepts of inverse operations, reciprocals, multiples, and powers.

Students graph addition (linear) and multiplicative (exponential) sequences and make comparisons including how the value of the growth number affects the shape of the graph.

Students construct secant lines and tangent lines on the graph of an exponential sequence and differentiate between them.

Significant Activities or Projects: 

Students research demographic data about their native countries and complete a population pyramid representing the age groups. Students interpret their population pyramids by making inferences about factors (social, religious, political, cultural) that could impact the shape of their pyramid.

Students construct and analyze growth spirals representative of addition (linear) and multiplicative (exponential) functions. Students display and present their work on chart paper, indicating visual characteristics of each type of spiral.

Students calculate the discrepancy between mathematical models representing the population growth of the United States and interpret which model fits actual data more closely, and then make resulting forecasts.

Sample PBATs: 
Native Country Population Project: Students tabulate the actual population data of their native country using the time period from 1950-2000. Students calculate the rate of growth under both a linear and exponential model and compare the resulting forecasts to assess which is closer to the actual population data both numerically and graphically. Students construct the secant lines and tangent line on their graphs and predict the rate of change at a specific year. Students then use the graphing calculator to find an exponential regression model to compare to their prediction. Students also use an exponential base of e, to forecast future population data and to predict when their country if continuing at its current rate of growth would be completely filled with people. Students also reflect on the social, political, and cultural factors that may affect their mathematical forecasts.

Trigonometry

Essex Street Academy

Consortium school

Trigonometry is a one semester class during which students investigate the relationships between the sides and angles of triangles. They use trigonometric functions and their inverses to describe those relationships, the relationships of angles and coordinates on the unit circle, and the motion of waves. Students explore when, why, and how trigonometry can be used to solve practical problems and model real world phenomena.

Media Used: 
Graphing calculators
Significant Assignments: 

(1) By drawing and measuring, students investigate ratios of triangle sides with given angles in order to understand why the sine, cosine, or tangent of a particular angle is a fixed ratio. They then solve and create problems which require finding missing parts of a triangle given minimal information.

(2) Students use a clinometer and triangle trigonometry to find the height of a building or lamp post outside the school. They compare results from this with their prior estimates and with other mathematical methods of finding unknown lengths. Their write up includes a scale model and a description of their procedure.

(3) Each student presents a different proof of the Pythagorean theorem to the class. They then use the Pythagorean theorem to find ratios of sides in "special angle triangles" in simplest radical form.

(4) Students find the coordinates, in exact form, of points along the unit circle by drawing right triangles with the radius as the hypotenuse. They also use the circle and its subdivisions to find the radian measure of each special angle and formulate a rule for converting between radians and degrees.

(5) Students sketch a graph of y=sin(a) based on their knowledge of triangle and circle trigonometry by measuring the height of various points at intervals along the unit circle. The height, or y-coordinate, is sin(a) where a is the measure of the central angle. They then translate the height to a rectangular coordinate grid in order to discover the periodic nature of the sine curve. Students use the graph to find the sine of quadrantal angles (the x-intercepts, the maximums and minimums); determine and define the period, amplitude, and range; determine in which quadrants the value is increasing or decreasing; use the graphing calculator to verify their graphs. After students experiment with graphing calculators changing a,b, and d in the equation y=a*sin(bx)+d, they will explain how each one changes the appearance of the graphs in terms of amplitude, frequency, period, and vertical shift.

Significant Activities or Projects: 

(1) Ferris Wheel Project: Student convert situational descriptions involving periodic motion (of riders on a Ferris wheel) into equations and graphs. They create sine (or cosine) equations and graphs to model the height of riders and discuss the contextual meaning of frequency, period, amplitude, and vertical shift as they appear in the equation y=a*sin(bx)+d. They discuss how these components must change to describe changes in a Ferris wheel's speed, size, height, rotational direction, or mounting place.

(2) Students model and graph tidal patterns using trigonometric functions (using a reasonable scale and "window"). Using estimation based on the graph, and then increasingly more precise algebraic methods, they calculate the time and position of high tide, low tide, and average tide, as well as how frequently the tide comes in. Finally, they calculate how much time a sand-castle maker would have to build and admire a castle (before the tide submerges it) at various points on the beach. Students extend the problem in ways of interest to them.

(3) Students collect data on sunrise and sunset times for various locations around the world. They model data from a particular place with a trigonometric function (using sine regression). They describe the characteristics of daylight hours in their location, including the dates of solstices and equinoxes, using the characteristics of the trigonometric function they develop. Each student compares data with two other students in the class and uses latitude and Earth's tilt to explain differences in amplitude and phase. Students have to defend their use of a sine regression and describe what each constant in their equation means in terms of daylight hours.They then assess how well their regression fits their data by comparing their calculations and graphs based on the regression with calculations and graphs based on the actual data. They use their work to predict daylight hours in the future and solve inequalities to make recommendations about if and when solar energy is viable in that location.

Sample PBATs: 
Students collect data on sunrise and sunset times for various locations around the world. They model data from a particular place with a trigonometric function (using sine regression). They describe the characteristics of daylight hours in their location, including the dates of solstices and equinoxes, using the characteristics of the trigonometric function they develop. Each student compares data with two other students in the class and uses latitude and Earth's tilt to explain differences in amplitude and phase. Students have to defend their use of a sine regression and describe what each constant in their equation means in terms of daylight hours.They then assess how well their regression fits their data by comparing their calculations and graphs based on the regression with calculations and graphs based on the actual data. They use their work to predict daylight hours in the future and solve inequalities to make recommendations about if and when solar energy is viable in that location.

Geometry 1B

Essex Street Academy

Consortium school

Geometry 1B is a one semester course that builds off of definition-based, informal understandings of geometric concepts learned the previous semester (in Geometry 1A) to develop direct proof-writing skills, mathematical communication skills (oral, English and symbolic), and patient, creative problem solving.

Readings: 
Geometry by Harold R. Jacobs
Amsco's Geometry by Ann Xavier Gantert
Interim Assessments: 

Project 1: Triangle Congruency
Students receive a personal collection of three triangle congruency statements to prove with accompanied diagrams and given information. In proving these three statements, students demonstrate mastery of the use of triangle congruency postulates (SSS, SAS and ASA), basic properties and postulates, angle definitions and theorems and adding and subtracting angles and line segments. Students are required to give written evidence of making initial observations based on the assigned givens and conjectures, brainstorming, planning a proof, writing it, and revising it. Students present the rough drafts of their proofs for critique from the class. Then they write a process paper detailing their strategies for proving their triangle congruency statements and how they arrived at these strategies, explaining observations, flawed strategies, revisions and lines of reasoning as in depth as possible.

Project 2: Using Triangle Congruency to Prove Other Conjectures
Students receive a personal collection of three statements to prove, as in the first project. Although the predominant skill to demonstrate mastery of in this project is use of the 'Corresponding Parts of Congruent Triangles are Congruent' theorem, students must devise a wide range of strategies with greater complexity and appeal to a larger selection of thematic theorems to prove their conjectures than in the first project. Students also complete a rough draft presentation with a peer review and then a process paper as in project 1.

Significant Assignments: 

Homework Presentation:
Each student is assigned a biweekly presentation of a challenging/ novel proof from their homework for peer review. Classmates give warm and cool feedback and offer ideas to improve the proof/ delineate issues with its validity. There is at least one student presentation almost every day of class. This is intended as a catalyst for class discussion around main topics, a way to clear up common errors, and to give importance to individual assignments completed at home.

Significant Activities or Projects: 

PBAT: Prove Your Own Conjecture
Students are given a personal diagram from which they make observations and formulate conjectures. Students endeavor to prove one or more conjectures that "show off" the full range of thematic theorems that they learned during the course and a level of proof complexity, sophistication, and creativity that is their very best work.

Sample PBATs: 
(see significant activities or projects)

Trigonometry

Grade Level (can be multiple): 11-12

Brief Overview of Course:
This one semester course is designed for students to understand practical applications of trigonometry. It also emphasizes estimation and logic. The following topics are covered:
-Definitions of Sine, Cosine and Tangent
-Inverse functions
-sine law and proof
-cosine law and proof
-area of a triangle and proof
-Cartesian to polar coordinate conversion
-practical world problems
-Trig equations including factoring
-solving triangles given a minimum of information
-drawing diagrams to scale

Readings: 
Math Puzzles and Games by Michael Holt
Algebra and Trigonometry by James stewart, Lothar Redlin and Salem Watson
Precalculus Mathematics in a Nutshell By George F. Simmonms
Media Used: 
Calculators
Interim Assessments: 

Students work outdoors, using a building they can walk around completely. The Flat Iron Building at 23rd Street and 5th Avenue is a good example. Students are asked to determine the volume of the building to the nearest cubic meter. They then determine which formula to use and must show all calculations, the tools used, and how they used them. They then explain if their answer makes sense, using logic and examples (other than trig) to show how they could estimate the answer. Finally, they make a scale drawing labeling everything.

Significant Assignments: 

To make trigonometry relevant, other locations in the city are used. For example, students may determine the height of the smoke stakes on Roosevelt Island from our location in Manhattan at 63 Street and the East River. They then complete the following 4 pages:
page 1) Explain in detail exactly what your procedure was. What tools did you use and how did you use them?
page 2) Show all of your formulas and calculations. Does your answer make sense? Why or why not? Explain using logic and examples (other than trig) of how you could estimate the answer. State this estimation.
page 3) Sketch to scale, as best you can, the picture of this problem. Be sure to label everything and state the scale. Label your sheet of graph paper.
page 4)
A) What difficulties did you have at any point in your procedure? How could you have avoided them?
B) What other way could you have solved this problem? Be specific in your explanation.

Significant Activities or Projects: 

We take the Staten Island Ferry from Manhattan to Staten Island and return. Before we leave, students research the height of the Statue of Liberty from the water's edge. From this information and data they collect from both docks by using simple tools, they determine the distance from Manhattan to Staten Island.
They then complete a 4-page paper:
page 1) Explain in detail exactly what your procedure was. What tools did you use and how did you use them?
page 2) Show all of your formulas and calculations. Does your answer make sense? Why or why not? Explain using logic and examples (other than trig) of how you could estimate the answer. State this estimation.
page 3) Sketch to scale, as best you can, the picture of this problem. Be sure to label everything and state the scale. Label your sheet of graph paper.
page 4)
A) What difficulties did you have at any point in your procedure? How could you have avoided them?
B) What other way could you have "done" this problem? Be specific in your explanation.

Sample PBATs: 
Students apply the skills they have acquired through the semester, using trig to calculate distances between two well-known NYC sites. For example, a student may be asked to find the distance from the Empire State building to the Williamsburg Bank building, using a vantage point on Governor's Island in New York Bay and based on their knowledge of the heights of both buildings. Students are told: You have with you: Trig tables, 100 feet of tape, balloons, a scientific calculator, an axe, binoculars, a hammer, a ball of twine, graph paper and colored pencils, a Polaroid camera, a sextant (angle measurer), a thermos of tea (two liters), a yard stick, peanut butter sandwiches, tent pegs, and a mirror You may not go in the water, nor may you throw anything in the water. Your method must be efficient (minimize time and energy) and must involve as much trig as possible to solve the distance problem. Draw diagrams and label everything. Show which equations you will use and how you will use them. When you have done this estimate the distance without using trig. What difficulties did you have at any point in your procedure? How could you have avoided them? What other way could you have "done" this problem? Be specific in your explanation.

Polynomials and Graphing

Grade Level (can be multiple): 11-12

Brief Overview of Course:
This one semester course focuses on developing students' ability to look for patterns in mathematics. Each class starts with a small logic puzzle. There are opportunities for interim assessments and final PBATs (performance-based assessment tasks) as well as quizzes, a midterm and a final examination. In addition, there are many practical applications. The following topics are covered:
- nth terms
- letter equations with restrictions
-fractional exponents
-negative exponents
-use and proof
-coordinate geometry
-slopes

Readings: 
Math Puzzles and Games by Michael Holt
Algebra and Calculus by Dennis Christy and Robert Rosenfeld
Precalculus:Functions and Graphs by Raymond A. Barnett, Michael R. Ziegler and Karl E. Byleen
Media Used: 
Calculators
Interim Assessments: 

Logic puzzle
1 6 Urban students ran a race. Each wore a different color. There were no ties.
2 Amanda lost to blue.
3 The 6-letter named runners came in consecutive order and one was green.
4 Alison lost to the runner in red but beat Iliriana.
5 Iliriana beat Mac and white but lost to Amanda.
6 The runner in white lost to Mac who lost to Amanda.
7 Heru beat Amanda by 2 places.
8 Courtney was not orange and lost to Iliriana.
9 The rider in blue lost to the rider in red but beat the rider yellow.

A Name the 6 runners in order of finish and each color each wore.
B You must prove that your solution is the only possible conclusion, in good order. Use the numbered sentences as evidence for each statement you make. If you use charts or partial charts, these must be fully explained.

Please read all of these instructions before starting.
It is known that water freezes at 32 degrees Fahrenheit (F) or 0 degrees Celsius (C) and boils at 212 degrees F or 100 degrees C. Show that the two temperature scales F and C are linear related by completing the following steps. Be sure to explain all of your procedures and show your calculations.

1) Find a linear equation that expresses F in terms of C. Neatly draw a graph of this equation. Label the axes, label the line with your equation and title your graph.
2) If a European family sets its house thermostat at 20 degrees C, what is the setting in degrees F? Find a linear equation that expresses C in terms of F. If the outside temperature in Milwaukee is 86 degrees F, what is the temperature in degrees C?
3) Explain what the slope in question 1) means in terms of converting Celsius to Fahrenheit.
4) Since we know water freezes at 32 degrees F and 0 degrees C and boils at 212 degrees F and 100 degrees C, explain why -40 C = -40 F.
Note: Show all work neatly on separate sheets.

Significant Assignments: 

The Pirates

Redbeard, Graybeard and Bluebeard were separated while being chased by the French Navy. Graybeard found himself at (-2, 13) (see the G). Bluebeard at (-12, 7), and Redbeard at (-3, -9). Redbeard took a course of Y = 2X -3 and Bluebeard took a course Y = 0X + 7 (or Y = 7) and Graybeard took a course of 7-3X = Y. When Bluebeard and Greybeard met, they continued on Graybeard's course till they met Redbeard. Then all three took course Y = X/2 till they came to Treasure Island.

1) Locate the (0, 0) center point. Put a C .
2) Locate Bluebeard's and Redbeard's starting points put a B (-12, 7) and an R (-3, -9)
3) Chart Bluebeard's course till he met Graybeard at ( , ). Use X's on the graph.
Chart Graybeard's course till he met with Redbeard at ( , ). Use O's on the graph.
Chart Redbeard's course till he met the other two at ( , ). Use +'s on the graph.
4) Chart the new course to Treasure Island and place a T at ( , ). Use *'s 0n the Graph.
Show all work on separate sheets.

Significant Activities or Projects: 

The Year Book

Here is another puzzle for you to work on. Try and use algebra this time – it is a short cut for trial and error. Use the reference sentence numbers in your setup.

1) 8 students collect $88 for Urban’s Year Book.
2) Anthony collected $2 more than Yan Mei.
3) Jesse collected twice as much as Yan Mei.
4) Levi collected the average (mean) amount.
5) Lily collected $2 more than Jesse.
6) Rachel collected the same amount as Jesse.
7) Sasha collected $1 less than Yan Mei
8) William collected as much as Sasha and Jesse together.

HOW MUCH DID EACH COLLECT?

Sample PBATs: 
Series of problem solving questions based on application of polynomials and graphing: Construction Using Circle Equation and Graphing For example: Town B is located 36 miles East and 15 miles north of town A. A local phone company wants to position a relay tower so the distance from the tower to town B is twice the distance from the tower to town A. 1) In order to find all the possible tower locations show that they must lay on a circle. Find the center and radius of this circle, and graph it. Be sure to title your graph and label both axes. Think about what your scale needs to be. 2) If the company decides to position the tower on this circle at a point directly East of Town A, how far from town A should they place the tower? Show how this location meets the criteria set out above. 3) Explain in your own words each step you did and why. Include in your explanation why you used certain formulas. 4) Write a brief analysis about why your answer makes sense. This should include a logical explanation.
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