CAS in the Maths Classroom

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CAS in the Maths Classroom:An Australian Experience : 

CAS in the Maths Classroom:An Australian Experience Mitchell Howard B.Sc, B.Ed, M.Ed Current email: mitchellhoward@caulfieldgs.vic.edu.au 2008: Lincoln High School, Christchurch (NZ Pilot school for CAS)

NZ ‘pilot’ School : 

NZ ‘pilot’ School CAS

Who is this Aussie anyway? : 

Who is this Aussie anyway? I’m not a salesman I currently teach at a large co-educational, independent, metropolitan school in Melbourne. My school currently use TI89’s but are up-dating soon to . . .

Who do we have here today? : 

Who do we have here today? RED – What is this ‘CAS’ you speak of. AMBER – I know what it is but I haven’t really used it in the classroom GREEN - I use CAS with my classes already

What is CAS? : 

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

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

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

1. For learning rules from pattern recognition : 

1. For learning rules from pattern recognition How would you normally teach Solving quadratic equations The ‘Null factor law’

1. For learning rules from pattern recognition : 

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’

1. For learning rules from pattern recognition : 

1. For learning rules from pattern recognition Surds Type the following in your calculator √5 + √5 √a + √a + √a + √a Make a conjecture about 2√5 + 3√5 Technique has many applications Basic Algebra Indices

2. For scaffolding of Algebra. : 

2. For scaffolding of Algebra. How do you currently teach Equation solving? Balancing a see-saw Back tracking Doing a strip-tease

2.For scaffolding of Algebra. : 

2.For scaffolding of Algebra. E.g. solve TI 89 SMG – Symbolic Math Guide Casio Classpad 300 – ALGY

2. For scaffolding of Algebra. : 

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 isn’t just one algorithm to solve something. How many ways can you solve it? Can work backwards to generate own questions.

3. Multiple representations and making connections. : 

3. Multiple representations and making connections. How do you currently teach Simultaneous Equations? Substitution? Elimination? Graphically?

3. Simultaneous Equations – what we did with year 10 : 

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.

3. Tell me a story : 

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

3. Tell me a story : 

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

Slide 17: 

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.

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. : 

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

3. Multiple representations and making connections. : 

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

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. 4. Can use parameters to explore graphs or equations.

Belt around the Earth : 

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?

CAS as a learning tool : 

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

CAS the ‘black box’ : 

CAS the ‘black box’ Great way to get answers Great way to Generate Questions

But won’t it mean my student will lose their algebra skills? : 

But won’t 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)

Issues : 

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

Assessment : 

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

Assessment : 

Assessment Don’t ask traditional type questions Assessment of students understanding of mathematical concepts Not assessing how well students have memorised algorithms

Benefits : 

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 .

Some resources : 

Some resources ALGY – www.stepsinlogic.com RITEMATHS - Melbourne University project http://extranet.edfac.unimelb.edu.au/DSME/RITEMATHS/ 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