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Seeking Depth in Algebra II

Seeking Depth in Algebra II. Naoko Akiyama nakiyama@urbanschool.org Scott Nelson snelson@urbanschool.org Henri Picciotto hpicciotto@urbanschool.org www.picciotto.org/math-ed. The Urban School of San Francisco 1563 Page Street San Francisco, CA 94117 (415) 626-2919

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Seeking Depth in Algebra II

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  1. Seeking Depth in Algebra II Naoko Akiyama nakiyama@urbanschool.org Scott Nelson snelson@urbanschool.org Henri Picciotto hpicciotto@urbanschool.org www.picciotto.org/math-ed The Urban School of San Francisco 1563 Page Street San Francisco, CA 94117 (415) 626-2919 www.urbanschool.org

  2. The ProblemTeaching Algebra II • Too much material • Too many topics • Superficial understanding • Poor retention • Loss of interest

  3. A Partial Solution Choose Depth over Breadth

  4. Our Hopes • Access for everyone • No ceiling for anyone • Authentic engagement • Real retention • Depth of understanding

  5. Math 3 Course Overview Themes: Functions Trigonometry “Real World” Applications

  6. Math 3A • Linear Programming • Variation Functions • Quadratics • Exponential Functions, Logarithms • Unit Circle Trigonometry

  7. Math 3B • Iterating Linear Functions • Sequences and Series • Functions: Composition and Inverses • Laws of Sines and Cosines • Polar Coordinates, Vectors • Complex Numbers

  8. The course evolves Collaboration makes it possible

  9. Our Colleagues • Richard Lautze • Liz Caffrey • Jee Park • Kim Seashore

  10. Workshop Outline • Iterating Linear Functions • Quadratics • Selected Labs • Complex Numbers

  11. Iterating Linear Functions Introduction to Sequences and Series

  12. The Problem: Opaque Formulas

  13. Select the number of the day of the month you were born Divide by 2 Add 4 Repeat! The Birthday Experiment

  14. Time Series Tablefor the Birthday Experiment

  15. Iterating a linear function input y = mx + b output

  16. Time Series Graph for the Birthday Experiment

  17. Modeling Medication:FluRidder • FluRidder is an imaginary medication • Your body eliminates 32% of the FluRidder in your system every hour • You take 100 units of FluRidder initially • You take an additional hourly dose of 40 units beginning one hour after you took the initial dose Make a time-series table and graph.

  18. FluRidder Problem What equation did we iterate to model this?

  19. Recursive Notation for the FluRidder Model for n ≥ 1

  20. Special Case: m = 1Iterating y = x + b Example: b = 4

  21. Time Series Graph for y = x + 4

  22. Special Case:b = 0Iterating y = mxfor 0 < m <1 Example: m = 0.5

  23. Time Series Graph for y = .5x

  24. Special Cases:b = 0Iterating y = mxfor m > 1 Example: m = 1.5

  25. Time Series Graph for y=1.5x

  26. Outcomes • Grounds work on sequences and series • Makes notation more meaningful • Enhances calculator fluency • Introduces convergence, divergence, limits • Makes arithmetic and geometric sequences look easier!

  27. Introducing Arithmetic and Geometric Series: Algorithms vs. Formulas

  28. Arithmetic Series 3+5+7+9 Staircase Sums 3 + 5 + 7 + 9 9 + 7 + 5 + 3 (12 +12 +12 +12)/2

  29. Geometric Series:multiply, subtract, solve S = 3 + .6 + .12 + .024 .2S = .6 + .12 + .024 + .0048  multiply .8S = 3 – .0048  subtract S = 2.9952/.8 = 3.744  solve a1 = 3, r = .2, n = 4

  30. Generalize: S = a1 + a2 + a3 + … + an r ·S = r (a1 + a2 + a3 +… + an) multiply = a2 + a3 + …+ an + an+1 (1-r)S = a1 – an+1 subtract  solve

  31. Outcomes A way to understand — the algorithms are more meaningful than the formulas for most students A way to remember — the formulas are easy to forget, the algorithms are easy to remember A foundation for proof of the formulas

  32. Quadratics Completing the Square

  33. The Problem What does this mean?

  34. We use a geometric interpretationto help students understand this.

  35. The Lab Gear

  36. Make a rectangleusing 2x2 and 4x

  37. x (2x + 4) = 2x2 + 4x

  38. 2x (x + 2) = 2x2 + 4x

  39. Lab Gear “The Box” Algebra

  40. Making Rectangles Make as many rectangles as you can with an x2, 8 x’s and any number of ones. Sketch them.

  41. Solving Quadratics: Equal Squares

  42. Making Equal Squares = =

  43. Completing the Square

  44. Outcomes • Concrete understanding of completing the square and the quadratic equation • Connecting algebraic and geometric multiplication and factoring • Connecting factors, zeroes and intercepts • Preview of moving parabolas around and transformations • Better understanding of “no solution”

  45. Selected Labs Inverse Variation Exponential Decay Logarithms

  46. Perspective • Collect data: apparent size of a classmate as a function of distance • Look for a numerical pattern • Notice the (nearly) constant product • Find a formula

  47. Review Similar Triangles  Constant product  Inverse variation

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