CAS in the Maths Classroom: An Australian Experience

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CAS in the Maths ClassroomAn Australian
Experience

• Mitchell Howard B.Sc, B.Ed, M.Ed

Current email mitchellhoward_at_caulfieldgs.vic.e
du.au 2008 Lincoln High School, Christchurch
(NZ Pilot school for CAS)
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NZ pilot School
CAS
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Who is this Aussie anyway?

• Im not a salesman
• I currently teach at a large co-educational,
  independent, metropolitan school in Melbourne.
• My school currently use TI89s but are up-dating
  soon to . . .

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Who do we have here today?

• RED What is this CAS you speak of.
• AMBER I know what it is but I havent really
  used it in the classroom
• GREEN - I use CAS with my classes already

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What is CAS?

• Computer Algebra System
• A CAS has the ability to perform symbolic
  manipulations in much the same way as we might do
  ourselves with pen and paper.
• For example, expand and factorise algebraic
  expressions.
• CAS has also powerful numerical computational
  capabilities and the ability to represent and
  analyse mathematical problems graphically and in
  spreadsheets.
• But CAS can also be used as a learning tool

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Why do I think that CAS is good for 14 16 year
olds?

• For learning rules from pattern recognition.
• For scaffolding of Algebra.
• For multiple representations and making
  connections between them.
• Can use parameters to explore graphs or equations
  to find generalised solutions to big questions.

And all on the one portable piece of plastic
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1. For learning rules from pattern recognition

• How would you normally teach
• Solving quadratic equations
• The Null factor law

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1. For learning rules from pattern recognition

• E.g. Solving quadratic equations
• Use your calculator to solve the following
• x (x - 5) 0
• (x 3) (x - 2) 0
• Without your calculator, guess the answers to the
  following
• x (x 7) 0
• (x - 8) (x 4) 0
• Why is it so? Use your calculator to graph
• y x (x - 5)
• Then Null factor law

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1. For learning rules from pattern recognition

• Surds
• Type the following in your calculator
• v5 v5
• va va va va
• Make a conjecture about
• 2v5 3v5
• Technique has many applications
• Basic Algebra
• Indices

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2. For scaffolding of Algebra.

• How do you currently teach Equation solving?
• Balancing a see-saw
• Back tracking
• Doing a strip-tease

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2.For scaffolding of Algebra.

• E.g. solve
• TI 89 SMG Symbolic Math Guide
• Casio Classpad 300 ALGY

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2. For scaffolding of Algebra.

• Training wheels for solving equations
• Allows a safe environment for students to make
  mistakes and learn from the effects.
• Undo the mistake and try something else.
• Particularly good for weaker students.
• Demonstrates that there isnt just one algorithm
  to solve something.
• How many ways can you solve it?
• Can work backwards to generate own questions.

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3. Multiple representations and making
connections.

• How do you currently teach Simultaneous
  Equations?
• Substitution?
• Elimination?
• Graphically?

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3. Simultaneous Equations what we did with
year 10

• Tell me a story
• Worded problems
• Table of values
• Graphically
• Home screen
• Substitution and elimination by CAS
• Saved by hand techniques for extension and
  until year 11.

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3. Tell me a story

• A picture is worth a thousand words
• In small groups, students were given a theme and
  asked to decide what was happening in the graph
  and then present their story to the rest of the
  class.
• Themes included Cyclists, Cars, Planes,
  Bushwalkers, Mobile phones, Filling a beaker,
  Taxi fare, movies, goal scoring, Chinese
  characters

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3. Tell me a story

• Hints
• Decide what each of the axes represent
• What is happening
• At the start
• Before the lines cross
• When the lines cross
• After the lines cross

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• E.g. Two groups went to the movies. The first
  group included 5 adults and 5 kids and paid a
  total of 115. The second group included 2 adults
  and 7 kids and paid a total of 107. If ticket
  prices were the same for each group, find the
  cost of each type of ticket.

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Examine the following screen from a CAS
calculator, which has been used to find a
solution to the simultaneous equations x y
5 and 3x 2y 11.

• Explain how the CAS has been used to find a
  solution
• Use this method to check the solutions you
  obtained earlier

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3. Multiple representations and making
connections.
Nothing new
Sketch-pad add on
CAS

• Next generation CAS

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4. Can use parameters to explore graphs or
equations.

• What happens to the volume of a sphere if the
  radius is doubled?
• Beyond most kids algebra, but raises questions so
  we can then explore why.

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Belt around the Earth

• Consider a belt that is placed to fit around the
  equator of the earth. If 6m is then added to the
  belt circumference, can you
• A slip a piece of paper under it?
• B slide your hand under it?
• C crawl under it?
• D walk under it?

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CAS as a learning tool

• Represents a move away from algorithms, providing
  opportunities to develop thinking and a deeper
  understanding.
• Moving away from compartmentalised Mathematics
• Instead of skill, skill, skill, application
• Now start with a real-life problem as a hook
  and learn the skills because we need them
• Also represents a move to less contrived
  Mathematics
• Provides Motivation

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CAS the black box

• Great way to get answers
• Great way to Generate Questions

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But wont it mean my student will lose their
algebra skills?

• Yes CAS gives the Answer
• Want to know why? Must do by hand
• Still need algebra skills, in fact more of them
  to interpret CAS output as it doesn't always come
  up as expected. (e.g. transposing some formulae)

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Issues

• Cost
• Theft
• Class sets
• Syntax Can be a pain to start with
• i-pods and mobiles are here to stay
• Qualified staff - Some staff prefer the
  algorithmic approach
• Assessment

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Assessment

• Tech Free/Tech Active Model (European)
• VCE Examination (Year 13 External examination)
• 1/3 Tech Free
• 2/3 Tech Active (Includes Analysis)

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Assessment

• Dont ask traditional type questions
• Assessment of students understanding of
  mathematical concepts
• Not assessing how well students have memorised
  algorithms

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Benefits

• Top of the tree analogy (Tony McRae)
• We can see the destination, where we are headed.
• Allows you to quickly see where you are going.
• F1 race car analogy (Tony McRae)
• Safety car holds back all the cars.
• Ear piece, can give instruction but let your
  better students fly ahead at their own pace.
• First golf game analogy (Peter Fox)
• Driving range first to get skills? Or . . .
• Play the game first, then want to learn skills .

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Some resources

• ALGY
• www.stepsinlogic.com
• RITEMATHS - Melbourne University project
  http//extranet.edfac.unimelb.edu.au/DSME/RITEMATH
  S/
• CASCAT
• http//extranet.edfac.unimelb.edu.au/DSME/CAS-CAT
  /
• VCAA Victorian
• Curriculum (Junior CAS)
• http//vels.vcaa.vic.edu.au/support/domainsupport/
  maths/cas.html
• - Curriculum (Senior CAS)
• http//www.vcaa.vic.edu.au/vce/studies/mathematics
  /cas/casindex.html
• CAS exam papers
• http//www.vcaa.vic.edu.au/vce/studies/mathematics
  /cas/casexams.html