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Integrating CAS

Integrating CAS. casmusings.wordpress.com Chris Harrow Atlanta, GA casmusings@gmail.com Twitter: @chris_harrow. 1. Why CAS?. Levels the playing field/scaffolding; you focus on how to ask good questions and interpret the answers. Keeps the focus on student THINKING

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Integrating CAS

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  1. Integrating CAS • casmusings.wordpress.com • Chris Harrow • Atlanta, GA • casmusings@gmail.com • Twitter: @chris_harrow 1

  2. Why CAS? • Levels the playing field/scaffolding; you focus on how to ask good questions and interpret the answers. • Keeps the focus on student THINKING • Has forced me to rethink assessment – what is my goal for each question? • Research: CAS Teaches Form EquivalenceOutput format is sometimes unpredictable. Just because an answer LOOKS different doesn’t mean it IS different. • Teacher isn’t the sole expert – instant/pressure-free confirmation & sometimes new math is motivated. 2

  3. In my classes • In our first year of a 1:1 laptop program • 11th & 12th graders have had CAS Nspires for almost a decade • Implication = Full-time CAS access & free computer software. • Encourage use of Nspire CAS, Wolfram Alpha, & Geogebra, especially now that we’re 1:1 • CAS is assumed, but isn’t the focus • It’s about asking good questions in 3 languages: English, Math, & CAS • Split assessments 3

  4. Warning: Math Blast Ahead • I’m uploading to my ‘blog tonight’s presentation in Keynote, PPT, & pdf formats along with all .tns and .ggb files. • casmusings.wordpress.com • My goal tonight is to show several examples of how CAS has dramatically enhanced my math classroom. 4

  5. Linear Regressions • What does a linear regression do? • This is 100% understandable by any algebra student. • In Foerster Instructor’s Resource Book (CAS Activities 3-2a & 3-2b) • TI Nspire CAS file: LinReg 5

  6. Quadratics • Unexpected formatting sometimes leads to additional math. Try • What does each coefficient of a standard form quadratic equation do to the corresponding graph? • KEY INSIGHT: • Deep understanding comes from parameters as sliders • In PreCalculus Transformed • ‘blog posts here and here • TI Nspire CAS file: QuadSurprises 6

  7. Quadratics II • What pattern emerges when you compute the areas of quadratic sectors of equal width? • Evolved from a student’s exploration that was published in the Mathematics Teacher • TI Nspire CAS file: QuadAreaSurprises 7

  8. Quadratics III • How many curves are uniquely defined by 3 given points? • KEY INSIGHTS: • Form is no longer a boundary. • Vertex, intercept, standard form are all equally accessible (plus others) • CAS Lesson: Different forms tell different stories about the underlying function/data. • CAS makes most forms equally accessible. • In Foerster Instructor’s Resource Book (CAS Activity 1-3a) plus extensions • TI Nspire CAS file: Quadratic_Forms • Geogebra for graphing relations & points: 3points.ggb 8

  9. Quadratics IV • But there really are infinitely many parabolas containing these points if you allow rotations • Serious CAS use • KEY INSIGHT: • This algebra is gross, but knowing what to do (& letting a CAS do it) keeps the problem in focus. • TI Nspire CAS file: 3points_rotated.tns • GeoGebra file: 3points_rotated.ggb 9

  10. Cubics • A cubic has inflection point at (1,3) and contains (0,-4). • Name one other point. • How many cubics contain those 3 points? Write an equation for each. • In PreCalculus Transformed • ‘blog post here • TI Nspire CAS file: Cubics 10

  11. Probability • Find the probability of exactly 3 heads in 10 tosses of a coin. • KEY INSIGHTS: • See the entire sample space at once • No longer restricted to binomial distributions. • ‘blog post here • TI Nspire CAS file: Probability 11

  12. Straightening Data • What do Power & other Regressions really do? • KEY INSIGHTS: • Connect Linear, Power, and Exponential Regressions • Equation manipulation via CAS levels the field. • In Foerster Instructor’s Resource Book (CAS Activities 3-4a) • TI Nspire CAS file: Straightening 12

  13. Graphing in Cartesian & Polar • New Idea submitted to Mathematics Teacher & planned for the next edition of PreCalculus Transformed. • Trig function centers, ceilings, & floors • Polar function centers, ceilings, & floors • TI Nspire CAS file: CeilingsFloors 13

  14. Transformations II • Trig identities via the SQ transformation 14

  15. Misc • Number of zeros in 200! ? • One term of (Ax+By)^n is 27869184x^5y^3 . Define A, B, & n. • Unexpected sum of squares of reciprocals result. 15

  16. Sources • PreCalculus Transformed • http://bit.ly/ypCgft • Precalculus with Trigonometry, 3rd ed, Foerster, (TE & Instructor’s Resources) • http://bit.ly/yjplJP • http://casmusings.wordpress.com 16

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