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Enhancing Math Education with CAS Technology

Discover the benefits of using CAS technology in math education such as leveling the playing field, focusing on student thinking, and rethinking assessment. Explore examples of how CAS enhances learning in linear regressions, quadratics, cubic equations, probability, and more.

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Enhancing Math Education with CAS Technology

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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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