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Integrating CASIntegrating CAScasmusings.wordpress.com

Chris HarrowAtlanta, GAcasmusings@gmail.comTwitter: @chris_harrow

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Why CAS?Why CAS?Levels the playing field/scaffolding; you

focus on how to ask good questions and interpret the answers.

Keeps the focus on student THINKINGHas 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.

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In my classesIn my classesIn our first year of a 1:1 laptop program11th & 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 1:1CAS is assumed, but isn’t the focusIt’s about asking good questions in 3

languages: English, Math, & CASSplit assessments

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Warning: Math Blast Warning: Math Blast AheadAheadI’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.

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Linear RegressionsLinear RegressionsWhat 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

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QuadraticsQuadratics

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

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Quadratics IIQuadratics 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

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Quadratics IIIQuadratics IIIHow 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

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Quadratics IVQuadratics IVBut there really are infinitely

many parabolas containing these points if you allow rotations

Serious CAS useKEY 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

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CubicsCubicsA 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

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ProbabilityProbabilityFind 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

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Straightening DataStraightening DataWhat 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

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Graphing in Cartesian & Graphing in Cartesian & PolarPolar

New Idea submitted to Mathematics Teacher & planned for the next edition of PreCalculus Transformed.

Trig function centers, ceilings, & floorsPolar function centers, ceilings, &

floors

◦ TI Nspire CAS file: CeilingsFloors

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Transformations IITransformations IITrig identities via the SQ

transformation

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MiscMiscNumber 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.

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SourcesSourcesPreCalculus Transformedhttp://bit.ly/ypCgft

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

http://bit.ly/yjplJP

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