integrating cas
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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
http://casmusings.wordpress.com
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