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Creating an Authentic Precalculus Curriculum: Why We Study Mathematics

Creating an Authentic Precalculus Curriculum: Why We Study Mathematics. Dr. Mark A. Jones Chatham High School mark.jones@chatham-nj.org. Brief Bio. High School Teacher (CA, 1 year) University Professor (NY, 4 years) AT&T Bell Labs Researcher (NJ, 19 years)

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Creating an Authentic Precalculus Curriculum: Why We Study Mathematics

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  1. Creating an Authentic Precalculus Curriculum: Why We Study Mathematics Dr. Mark A. Jones Chatham High School mark.jones@chatham-nj.org

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

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

  4. 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? Dr. Mark A. Jones -- Chatham High School

  5. Historical Context • Convey the Leibniz and Newton calculus controversy (story telling) . • Reconstruct the bridges of Konigsberg (kinetic learning). Dr. Mark A. Jones -- Chatham High School

  6. Historical Context • Explore π (on π day!) – its history, computation, memorization, Buffon’s Needle, digits of π,celebration in song, etc. (subject-based learning). Dr. Mark A. Jones -- Chatham High School

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

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

  9. 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). Dr. Mark A. Jones -- Chatham High School

  10. 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) Dr. Mark A. Jones -- Chatham High School

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

  12. When are we ever gonna have to use this? Dr. Mark A. Jones -- Chatham High School

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

  14. 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) Dr. Mark A. Jones -- Chatham High School

  15. 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, … Dr. Mark A. Jones -- Chatham High School

  16. Relations and Functions – Representations • Importance of representations: mappings, ordered pairs, equations, set builder notation, graphs Dr. Mark A. Jones -- Chatham High School

  17. 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? Dr. Mark A. Jones -- Chatham High School

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

  19. 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 Mission You 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. Dr. Mark A. Jones -- Chatham High School

  20. Exponentials and Logs – Discovery Experimenting with the equation solver (discovery learning).When will you be a millionaire? Computing yields. Present value / future value. Pricing zero coupon bonds. Dr. Mark A. Jones -- Chatham High School

  21. Exponentials and Logs – e For advanced students, you can motivate how e arises: Dr. Mark A. Jones -- Chatham High School

  22. Exponentials and Logs – e Then you can derive the continuous compounding formula from the discrete compounding formula: Dr. Mark A. Jones -- Chatham High School

  23. 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? Dr. Mark A. Jones -- Chatham High School

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

  25. 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/ • 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. • 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. Dr. Mark A. Jones -- Chatham High School

  26. 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/ • 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. • 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.] Dr. Mark A. Jones -- Chatham High School

  27. Trigonometry – Widescreen TV Project Part 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 =_______________________ Dr. Mark A. Jones -- Chatham High School

  28. Trigonometry – Widescreen TV Project Part 2: Your Home HDTV Measure 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 HDTV Suppose 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. Dr. Mark A. Jones -- Chatham High School

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

  30. Trigonometry – Measuring Distances 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. Dr. Mark A. Jones -- Chatham High School

  31. Trigonometry – Measuring Distances Dr. Mark A. Jones -- Chatham High School

  32. Trigonometry – Measuring Distances Dr. Mark A. Jones -- Chatham High School

  33. Trigonometry – Measuring Distances 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 pt per objective) Describe and diagram the observation setting, • (2 pts per objective) Provide the raw data that you collected and compute the required values, • (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. Dr. Mark A. Jones -- Chatham High School

  34. Trigonometry – Measuring Distances 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.) Dr. Mark A. Jones -- Chatham High School

  35. Trigonometry – Electomagnetic Spectrum The 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 Dr. Mark A. Jones -- Chatham High School

  36. 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.) Dr. Mark A. Jones -- Chatham High School

  37. Trigonometry – Electomagnetic Spectrum Dr. Mark A. Jones -- Chatham High School

  38. 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.). Dr. Mark A. Jones -- Chatham High School

  39. Trigonometry – Math and Music • 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. Dr. Mark A. Jones -- Chatham High School

  40. Trigonometry – Math and Music Dr. Mark A. Jones -- Chatham High School

  41. Trigonometry – Math and Music Dr. Mark A. Jones -- Chatham High School

  42. Just vs. Equal Temperament Dr. Mark A. Jones -- Chatham High School

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

  44. 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fradians/secsoy(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) Dr. Mark A. Jones -- Chatham High School

  45. 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) Dr. Mark A. Jones -- Chatham High School

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

  47. Polar Coordinates – Polar Art Festival Dr. Mark A. Jones -- Chatham High School

  48. Polar Coordinates – Polar Art Festival Dr. Mark A. Jones -- Chatham High School

  49. Polar Coordinates – Polar Art Festival Dr. Mark A. Jones -- Chatham High School

  50. Polar Coordinates – Polar Art Festival Dr. Mark A. Jones -- Chatham High School

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