creating an authentic precalculus curriculum: why we study mathematics dr. mark a. jones chatham...

Creating an Authentic Precalculus Curriculum: Why

We Study Mathematics

Dr. Mark A. JonesChatham High School

[email protected]

Dr. Mark A. Jones -- Chatham High School 2

Brief Bio

• High School Teacher (CA, 1 year)

• University Professor (NY, 4 years)

• AT&T Bell Labs Researcher (NJ, 19 years)

• High School Teacher (NJ, 7 years)


Context is Important

• Everything we learn in life is contextualized.Context facilitates learning, recall and performance.

• Math should be no exception:– Historical context.– Theoretical context.– Application context.


Context is Important

• Welcome and initiate “Why” questions. They are a rich source of context.

• Context arises in higher-order questions:– What portion of the domain for the trig functions would

you have chosen for the range of the inverse trig functions? Why?

– Vectors are objects having magnitude and direction. What representation might convey this information graphically?


Historical Context• Convey the Leibniz and Newton calculus controversy (story

telling) .

• Reconstruct the bridges of Konigsberg (kinetic learning).


Historical Context• Explore π (on π day!) – its history,

computation, memorization, Buffon’s Needle, digits of π, celebration in song, etc. (subject-based learning).


Historical Context

• Have students conduct and present their own research on the history of certain ideas, notation, topics, or influential people (research).– The concept of zero was fundamental to the development of place-

value representations and the modern algorithms for basic arithmetic.– Imaginary numbers arose in the 15th and 16th centuries in the study of

cubic equations.– Euler was a pioneering 18th century Swiss mathematician and physicist

with numerous important contributions including graph theory. He introduced much of modern math notation, including f(x). The symbol e honors Euler.


Theoretical Context

• Relations vs. functions– Nondeterminism / Determinism

• Transformations– Horizontal/vertical shifts (x-h, y-k)– Horizontal/vertical stretching/compression (ax, ay)– Horizontal/vertical reflection (-x, -y)

• Symmetry– Hand graphing– Even/odd– Polar symmetries


The Explanatory Power of Transformations• Translations [horizontal/vertical shifts from (0, 0) to (h, k)]:

– Slope-intercept (lines)– Point-slope (lines)– Vertex form (parabolas, absolute value)– General form for figures with centers (circles, ellipses, hyperbolas)

• Rates [horizontal compression/stretching]:

– Slope (lines)– Growth and decay rates (exponentials)

• Positive/Negative Growth [horizontal reflection]

– Negative slope is a horizontal reflection of positive slope (lines)– Decay is a horizontal reflection of growth (exponentials).


The Explanatory Power of Transformations

• Transformations in periodic functions

– Amplitude: vertical stretching/compression

– Angular frequency: horizontal stretching/compression

– Phase shift: horizontal shift

– Trig. Identities: sin(x) = cos(x-π/2)


Application Context• Applications convince students of the relevance of

mathematics.– When are we ever going to use that?– Work with real data when possible.

• Applications allow students to “feel” mathematics.– Kinesthetic applications:

• Using astrolabes to measure angles in solving trig problems.• Doing projectile motion experiments to illustrate parametric

equations, derivatives, etc.– Interactive software:

• Internet applets, Wolfram Demonstration Project demos• Geometer’s Sketchpad animation


When are we ever gonna have to use this?


Application Context

• Applications make math fun and interesting.– Connect math to music and art.– Tap into student creativity with exploratory learning (Polar Art

Festival).

• Create a new state of mind in your classroom.– Skill and drill is conditioning and weight training.

Word problems are playing the game.– Skill and drill is basic nutrition. Word problems are dessert.– Lead by example -- emphasize applications in your assessments.


Honors Precalculus Units

• Review of Relations and Functions• Exponential and Logarithmic Functions• Trigonometric Functions• Polar Coordinates• Vectors• Conics• Data Analysis• Preview of Calculus (Limits, Derivatives, Integrals)


Relations and Functions – Big Picture

• Why do we study mathematics?

• Relations vs. functions– (Non-)Determinism. Your calculator is a function

machine. Computer databases are relational. Why?

• What is an algebra?– Elementary (numeric) algebra, matrix algebra, vector

algebra, …


Relations and Functions – Representations

• Importance of representations: mappings, ordered pairs, equations, set builder notation, graphs


Relations and Functions – Story-telling• Story-telling

– What has your altitude been since you got up this morning?– What does a housing bubble look like? Is it OK to buy one now?


Relations and Functions -- Transformations

• Transformations– Transformations underlie most of the formulas students

encounter in algebra.– Students can transform new functions they have never

seen before: logistic functions, gaussian functions (bell curves), trig functions.

– Function Mania activity: Contest format. Equation-to-graph, graph-to-equation, transformations-to-equation, transformations-to-graph.


Exponentials and Logs• Find ways to dramatize the difference between

linear behavior and exponential/log behavior.– 42 paper foldings to reach the moon, 94 foldings to reach

the end of the visible universe!! (http://scienceblogs.com/startswithabang/2009/08/paper_folding_to_the_moon.php)

– The Million Dollar MissionYou have your choice of two payment options: (1) One cent on the first day, two cents on the second day, and double your salary every day thereafter for thirty days; or(2) Exactly $1,000,000.

http://scienceblogs.com/startswithabang/2009/08/paper_folding_to_the_moon.php

http://scienceblogs.com/startswithabang/2009/08/paper_folding_to_the_moon.php


Exponentials and Logs – DiscoveryExperimenting with the equation solver (discovery learning).When will you be a millionaire? Computing yields. Present value / future value. Pricing zero coupon bonds.


Exponentials and Logs – e

For advanced students, you can motivate how e arises:


Exponentials and Logs – eThen you can derive the continuous compounding formula from the discrete compounding formula:


Exponentials and Logs – Rule of 72• Rule of 72 in investing.

– Approximate the doubling time by dividing 72 by the interest rate as a percentage. Why does this work?

– It should be called the rule of 69. Why is 72 used instead?


Trigonometry

• Trigonometry has two faces.– As its name implies, it involves measuring triangles.– But it also is the first exposure to periodic functions.

• Measuring triangles activities.– Widescreen TV project.– Distance measuring (with astrolabes) project.

• Periodic functions activities.– Electromagnetic spectrum project.– Math and Music assembly.


Trigonometry – Widescreen TV Project

Let d be the distance to the TV in feet. Let w be the screen width in inches. Let x be the screen diagonal in inches. Let θ be the viewing angle. http://www.myhometheater.homestead.com/

1. Electrohome: Electrohome suggests a viewing distance of three (minimum) to six (maximum) screen widths for video. This corresponds to the point at which most people will begin having trouble picking out details and reading the screen. Probably too far away to be effective for home theater, OK for everyday TV viewing. Most people are comfortable watching TV between this distance and half this distance.

2. SMPTE: The Society of Motion Picture and Television Engineers (SMPTE) standard EG-18-1994 recommends a minimum viewing angle of 30 degrees for movie theaters. This seems to be becoming a de facto standard for front projection home theaters also. Viewing from this distance or closer will result in a more immersive experience, and also lessen eye strain caused by watching a smaller image in a dark room.

http://www.myhometheater.homestead.com/



Let d be the distance to the TV in feet. Let w be the screen width in inches. Let x be the screen diagonal in inches. Let θ be the viewing angle. http://www.myhometheater.homestead.com/

3. THX: THX publishes standards for movie theaters and home systems. THX certification requires that the back row of seats in a theater have at least a 26 degree viewing angle and recommends a 36 degree viewing angle.

4. Viewing Distances based on Visual Acuity: These distances are calculated based on the resolving power of the human eye or visual acuity. The human eye with 20/20 vision can detect or resolve details as small as 1 minute of a degree of arc. These distances represent the point beyond which some of the detail in the picture is no longer able to be resolved and "blends" with adjacent detail. At full resolution, an HDTV picture is 1920 pixels wide by 1080 pixels high.[Hint: The optimal visual acuity distance is thus the distance at which 1 minute of arc sweeps out a distance of inches.]

http://www.myhometheater.homestead.com/


Trigonometry – Widescreen TV ProjectPart 1: General Formulas

For each of the 4 criteria above, determine how big a TV should be. State the TV’s size in terms of the diagonal, x, rather than width and height. (Note that the width and height of a TV can both be expressed in terms of x.) Show all of your work. In each case, write the formula for the diagonal x in terms of the distance d: x = . . . some function of d . . .

For Electrohome: minimum x = ______________________maximum x = ______________________

. . .For SMPTE: minimum x = ______________________. . . For THX: minimum x =______________________

recommended x = _________________. . .For Visual Acuity: optimal x =_______________________



Part 2: Your Home HDTVMeasure the distance d in your home from the prime viewing location to the place where your existing or new HDTV will stand (or hang on the wall if it is a flat screen).

What is d? _________________________________Use the formulas from part 1 to compute the appropriate size HDTV for your home. Show all of your work.

Part 3: Your Math Teacher’s HDTVSuppose that your favorite math teacher has a 46” HDTV. The viewing distance, d, to his TV is 11 feet. Use the formulas from part 1 to compute the appropriate size HDTV for his distance of 11 feet.


Trigonometry – Measuring Distances

In this assignment, you’ll be using an astrolabe, a device capable of measuring vertical angles. It is best used by two students, one to do the sighting (using the straw) while the other student reads the angle from the side. Therefore, the first task is to select a partner.

To mark off a known distance to use as a measurement in your calculations, each group must acquire a piece of string of known length (e.g., 20 feet) or a measuring tape.



We will be using the CHS Auditorium as our measurement site.

Objective #1: Position yourself at, near or on the stage. Determine the height of the bottom of the curtain hanging above the stage.

Objective #2: Position yourself on the stage. Determine how tall the curtain is.

Objective #3: Position yourself on the stage. Determine how far above the stage the sound booth is (the bottom of the window at the back of the auditorium).

Objective #4: Position yourself up near the back of the auditorium at an elevation even with the bottom of the curtain. Determine the distance to the curtain.



After obtaining your measurements, complete the assignment by producing a typed report with your observations. Each team may submit a single team report. For each objective:

1. (1 pt per objective) Describe and diagram the observation setting,

2. (2 pts per objective) Provide the raw data that you collected and compute the required values,

3. (2 pts per objective) Discuss the possible errors from your observation and how they would influence your answers. For each quantity that you measure, estimate the amount that you could be off by (the possible minimum and maximum values, as well as the measured value). For each quantity that you compute, figure the computed value, but also the range of possible values given the measurement errors you estimated.



Possible Techniques:#1 Align the eyes of the sighter to be level with the base of the triangle. This may not be

practical if you are measuring something that stands at ground level or the base of the object is too high for you to reach eye-level. Use right triangle trig to find y, given x and θ.

(more sighting techniques are also given, including ones that utilize Law of Sines, etc.)


Trigonometry – Electomagnetic SpectrumThe EM Project is a short assignment which drills home the ubiquity of the electromagnetic radiation that surrounds them and the many practical uses of it. Here is the set up:

Trigonometry naturally models waves of all types. Electromagnetic waves are produced by the motion of electrically charged particles. These waves are also called "electromagnetic radiation" because they radiate from the electrically charged particles. The waves are carried by massless particles called photons. They travel through empty space as well as through air and other substances. The energy E (in Joules) of the photons directly varies as the frequency f (in cycles/sec) of the wave by the formula E = h * f, (sometimes the Greek letter ν (“nu”) is used rather than f for the frequency) where h is Planck’s constant, h = 6.626 × 10-34 J·s The frequency f (in cycles/sec) inversely varies as the wavelength λ (in cm) with the photons traveling at the speed of light, c: f = c / λ where c = 29,979,245,800 cm/sec


Trigonometry – Electomagnetic Spectrum. . . Fill out the table on the reverse of this sheet with as many different regions of the spectrum as you can find. (Use additional sheets if necessary.) Examples of man-made signals include AM radio, TV, FM radio, CB radio, cordless phones, radio controlled cars, garage door openers, cell phones, GPS, etc. Natural radiation includes microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays, etc. Order the table by increasing frequency ranges for each type of radiation. The ranges may overlap since different uses may coexist depending upon the power of the signals and their distribution. (Do not consider each band of light separately or each radio station separately.)


Trigonometry – Electomagnetic Spectrum


Trigonometry – Math and Music

• Math and Music is a two hour assembly, a subject-based presentation that includes perspectives on music from math, physics, psychology and music theory.

• The relationships to math include:– The mathematics of scales (reference frequencies, intervals,

etc.). The 12 tone equal temperament scale is a geometric progression.

– The mathematics of standing waves (wavelength, period, frequency, harmonic series, overtones, noise cancellation).

– The mathematics of pitch in strings and open pipes.– The mathematics of timbres (real sampled waveforms of student

voices and instruments).– The mathematics of digital music (mp3, synthesizers, etc.).



• The demonstrations include:– Guitar and synthesizer: Songs and phrases from songs

that illustrate math/music concepts (amplitude, tunings, scales, harmonics, distortion, human frequency response).

– Decibel readings (log scale for amplitude/volume).– Software sampling of student voices and instruments.– Animusic.– Synthesizer demos.


Just vs. Equal Temperament# Interval Note Just Freq 12-TET 440 Freq Diff

1 tonic A 440*1/1 440.00 440.00 440.00 0.00

2 minor 2nd A# 440*16/15 469.33 *2^(1/12) 466.16 3.17

3 major 2nd B 440*9/8 495.00 *2^(2/12) 493.88 1.12

4 minor 3rd C 440*6/5 528.00 *2^(3/12) 523.25 4.75

5 major 3rd C# 440*5/4 550.00 *2^(4/12) 554.37 -4.37

6 perf. 4th D 440*4/3 586.67 *2^(5/12) 587.33 -0.66

7 dim. 5th D# 440*7/5 616.00 *2^(6/12) 622.25 -6.25

8 perf. 5th E 440*3/2 660.00 *2^(7/12) 659.26 0.74

9 minor 6th F 440*8/5 704.00 *2^(8/12) 698.46 5.54

10 major 6th F# 440*5/3 733.33 *2^(9/12) 739.99 -6.66

11 minor 7th G 440*16/9 782.22 *2^(10/12) 783.99 -1.77

12 major 7th G# 440*15/8 825.00 *2^(11/12) 830.61 -5.61

13 octave A 440*2/1 880.00 *2^(12/12) 880.00 0.00


The Math of Pitch in Strings

For strings, the velocity depends on such factors as the tension in the string and the linear density of the string.

The frequency (or pitch) depends on the velocity and length of the string.


Frequency – Pitch• Frequency for musical sounds is usually expressed in terms of

cycles/sec or Hz. If f is the frequency in Hz, then: ω = f cycles/sec*2 radians/cycle = 2 f radians/sec so y(t) = A sin(ωt - φ) = A sin(2 ft - φ)

• For A 440: y(t) = A sin((2 440)t – φ) = A sin(880 t – φ) ~ A sin(2764.6t – φ)


Digital Music• Compact Discs

– 44.1 kHz, stereo, uncompressed(sample rate must more than double the frequency)44,100 samples/sec x 16 bits/sample x 2 [stereo]= 1,411,200 bits/sec ~ 1,400 Kbps

• MP3– 32-320 Kbps, stereo, compressed and lossy, uses mathematical and

psychoacoustic compression

• AAC (default Apple iTunes format)– 96-320 Kbps, stereo, compressed and lossy, uses mathematical and

psychoacoustic compression; better quality at lower bitrate than MP3 but less widely supported; optionally protected (encrypted)


Polar Coordinates – Polar Art Festival

• I sponsor an annual Polar Art Festival to motivate students to explore polar equations, their symmetries, transformations and shapes.

• Refreshments and low-volume music set the atmosphere.• I set up a video camera on a copy stand pointing down.

Students first place a paper with their entrant number, name and the title of their artwork. Then they place their calculator, display the equations, then display the window settings and then graph.

• Certificates are awarded for the Most Humorous, Most Bizarre, and Most Artistic.


Vectors – Information RetrievalOne of the fundamental approaches to information theory is the vector space model. It is adapted extensively in various forms for similarity matching throughout the Internet.


Conics

I have designed a number of activities that use the following examples:

• Parabolas– Suspension Bridges, Satellite Dishes, Telescopes, Flashlights, St. Louis

Arch, Arranging Students

• Ellipses– Whispering Rooms, Planets, Arranging Students (such as ‘0’, ‘6’, ‘8’, ‘9’

for the class yearbook picture)


Data Analysis – Using Excel

• One of the things that I experienced as a researcher in AI and natural language processing was how the availability of data and its impact had grown, particularly with the rise of the Internet.

• Data analysis and statistics have transformed every subfield of AI, linguistics and psychology (fields that I was directly involved in) and nearly every other academic field as well.

• To prepare our students, they need to be conversant with the language, tools and techniques of data analysis.

• I devote a full week or more to data analysis. Students learn to use Excel, including its data analysis facilities. They work with real data.



• Part 1 of the data analysis project is scripted. Students learn:– To obtain data that is reliable, relatively error-free, and appropriate to

the task. They obtain roller coaster data from http://www.rcdb.com/.

– To enter data, convert it to a desired form, and to clean the data of obvious errors. Skills include copy/paste, find/replace, cell types.

– To obtain summary statistics to gain an initial understanding of the data (central tendencies, histograms, etc.).

– To learn how to create and interpret scatterplots.

– To learn how to identify and model relationships which exist in the data using curve fitting, regression and an understanding of functions.

– To understand the concepts of correlation and goodness-of-fit.

http://www.rcdb.com/



• Part 2 of the data analysis project is open-ended. Students learn:– To frame a central question.

– To find data to answer it.

– To analyze that data.

– To write persuasive, quantitative prose with accompanying graphs and tables to answer the central question.

• Part 2 is far more challenging than part 1 for most students.



0 200 400 600 800 1000 12000

5

10

15

20

25

f(x) = − 2.07703165616062 ln(x) + 21.6755181038442R² = 0.721032831472869

Suicide Rate (# suicides / 100,000 people) v. Population Density (people / sq mile)

For year 2005

Population Density (people/ square mile)

Suici

de R

ate

(# su

icide

s / 1

00,0

00 p

eopl

e)



1880 1900 1920 1940 1960 1980 2000 20200.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

f(x) = − 0.00339816358456476 x³ + 19.8281598924107 x² − 38552.7983993705 x + 24978697.3198521R² = 0.989554974217085

Total Cigarettes (billions)

Time (years)

Ciga

rett

es (b

illio

ns)


Preview of Calculus / Parametric Equations

• The Projectile Motion Project is a group project that brings many topics together and has the following features:

– It is a good example of kinesthetic learning. Students directly interpret aspects of physical motion with a mathematical model.

– It illustrates the value of parametric equations and gives them practice in using them in their calculator.

– It features group dynamics and cooperation.

– It relates concepts from algebra (vertex of a parabola) to concepts of calculus (maxima).


Preview of Calculus – Projectile Motion

This is a team project which will give you some direct experience with projectile motion equations. Create a group with either four or five team members that will play the following roles. (Some roles may overlap in a four person team.) Fill in your names.

The thrower will throw the ball:The receiver will catch the ball:The timer will time the throw:The recorder will record the dataThe spotter will measure arm height, distances, etc.



PlanningOn the day of this activity, each participant must bring the appropriate equipment:

– The thrower is responsible for bringing a glove and ball.– The receiver is responsible for bringing a glove.– The timer is responsible for bringing a stopwatch or a watch with a

second hand.– The recorder is responsible for bringing paper and a pencil to record

the data.– The spotter is responsible for bringing a measuring tape.– The timer, recorder and/or spotter must bring a calculator so that

results can be checked.



Project Description

The group will submit a single report by completing and handing in this project description. Each member will separately complete and submit the final critique page.

You may use the equation solver in your calculator to check your answers, but you must show the algebra for how your computed values were obtained. Be sure to include the units in each of your answers. Each group will carry out the objectives below. You may need to conduct several trials for each objective before you get trustworthy data.



1. Vertical Toss: The thrower will throw the ball as close to straight up as possible and allow it to hit the ground. The timer will start timing when the ball is released and stop when the ball hits the ground. The recorder will record the data. The spotter will observe the vertical release point (thrower's hand height). The team should calculate the other parameters and analyze the situation.

2. Distance Toss: The thrower will throw the ball at least 100 feet to the receiver. The timer will start timing when the ball is released and stop when the ball is caught. The recorder will record the data. The spotter will observe the initial height (thrower's hand height), the final height (receiver's glove height) and the distance. The team should calculate the other parameters and analyze the situation.


Preview of Calculus – Projectile MotionFor each scenario:• Students sketch the experimental situation.

• Students work through a series of computations which lead them to a computation for the:

– initial horizontal/vertical/combined speed of the throw– angle of the throw– time to reach maximum height– height of the throw– instantaneous speed at the maximum height– instantaneous speed at the receiver

• Students complete a questionnaire on participation in their project, and how the project contributed to their understanding.


Summary

• Skill and drill is important, but for math to become ingrained, intuitive, memorable, and useful . . . context is everything!!

• Math begins to seep into their everyday thought processes.

– One student reported that the laser patterns at a rock concert reminded him of hyperbolas.

• Context, particularly application context, makes math come alive.

– A student commented after Math and Music, “Now I understand why we study mathematics.” Indeed.

creating an authentic precalculus curriculum: why we study mathematics dr. mark a. jones chatham...

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