1 Integrating CAS casmusings.wordpress.com Chris Harrow Atlanta, GA Twitter:

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 Equivalence Output 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.

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

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.‘blog 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: LinRegLinReg

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 herehere  TI Nspire CAS file: QuadSurprisesQuadSurprises

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: QuadAreaSurprisesQuadAreaSurprises

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_FormsQuadratic_Forms  Geogebra for graphing relations & points: 3points.ggb3points.ggb

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.tns3points_rotated.tns ◦ GeoGebra file: 3points_rotated.ggb3points_rotated.ggb

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 herehere  TI Nspire CAS file: CubicsCubics 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 herehere TI Nspire CAS file: ProbabilityProbability

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: StraighteningStraightening

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: CeilingsFloorsCeilingsFloors

14 Transformations II Trig identities via the SQ transformation

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

16 Sources PreCalculus Transformed Precalculus with Trigonometry, 3 rd ed, Foerster, (TE & Instructor’s Resources)