Showing posts with label CAS calculator. Show all posts
Showing posts with label CAS calculator. Show all posts

Sunday, July 14, 2013

Big discussions in education

There are some big discussions happening at the moment in education.  Australian Curriculum, standardised testing, year 7 transition, public private partnerships, emerging social issues, CAS calculator implementation.  Each are having a lasting impact on the way education is progressing.

Australian curriculum seems destined to repeat the primary mistakes of OBE in that it is being run to political timeframes, is being introduced without effective assessment policies and guidance is a bit haphazard about its implementation schedule.  The one size fits all model, being implemented across K-12 with missing blocks of understanding scattered throughout each year group indicates that the success will be limited to higher SES schools that already approached the norm expected by the curriculum.  The curriculum does not support our kids and the enforcement of A-E grading / inappropriate curriculum just ensures the feedback reinforces their position in society.  It will take schools to make a stand, change their approach and find innovative ways to smooth the learning curve to help these students succeed.

As schools struggle to reach the norms required they are trying publicly to show they are ready to maintain their competitive position in standardised testing.  Being based on averages, even if a low SES school catered well for its higher achieving kids, this result is hidden within the average.  To counter this effect small schools are putting vast amounts of effort "teaching to the test", something most teachers are vigorously opposed to.  I was hearing an anecdote last night from a friend talking about their kids playing schools and saying,
"And after maths we'll have NAPLAN"
Since when did NAPLAN become a formal class in year 3?  If we want this to stop, we have to stop publishing these figures.  By all means run the tests and direct funding to schools based on test results,  but schools are biasing the test so badly I question its relevancy as a standardisation tool.

Year 7 transition has become a non issue.  In many public schools there aren't any coming to high school.  The delay of the decision to move 7's means that many parents of higher ability students made the decision to send their kids (along with younger siblings) to private schools and get specialist teaching assistance.  The remaining kids in many cases lack support at home - many are the most at risk students.  Public school numbers that were quite stable at 500 are dropping sub 300 which makes smaller metro high schools unviable and there is no indication that this number will bounce in the next 5 years.  Smaller schools can't compete with private education and facilities, lacking a marketing budget or effective USP to drive students to the school.  The end result is that more public schools will close and our education system will become more and more dependent on private education, ultimately further disadvantaging and marginalising low SES students.

With smaller schools and reduced funding through lack of scale to minimise costs, our smaller schools will need to increasingly devote time to managing public/private funding agreements to maintain programmes.  This is a clear diversion from classroom first (as it diverts resources from the classroom), will bias schools towards areas required by industry or areas easy to support through volunteers.  This is an issue in itself as cyclical industries may leave highly at risk generations of kids in geographical areas without employment opportunities, potentially creating ongoing social issues for communities and creating situations for schools where difficult to staff specialist programmes or expensive subjects to teach will become unsupportable.

The marginalisation of the poor is already occurring with accumulations of cultural groups in low SES areas now not integrating with large sections of the community (as those children are in private education), something in the past restricted to exclusion from high SES students in a few independent schools.  Without any real hope of employment due to a lack of social support and poor levels of education, some low SES students are now focused on the quick wins available to them through crime and social loafing, others are facing low self esteem, poor job prospects and mental illness.  The lack of positive peer support is having a clear impact on our communities and schools.  The edges of this is starting to be reported in the media and has the potential to create another drug and alcohol effected generation that will again require large amounts of funding to address.

The last issue is a math issue and one we face right now, but is still related to the issues above.  Math itself is becoming marginalised with the cost of participation rising above the level of a growing number of students within the school.  CAS calculators at $200, revision guides, course costs and texts can account for 50% of year 11/12 fees.  Low participation rates are precluding students from higher study.  Able students are now choosing other subjects with lower costs as families cannot pay the cost (costs that may have been able to be found within the school when numbers of at risk students were lower, access to support bars were set lower and more discretionary funds were available).  A further question exists about whether we need these calculators as they are creating exams that test the corners of courses to create bell curves rather than teaching students solid mathematics.  Many teachers are still struggling with CAS calculator integration and I'm beginning to fall in line with the thought they are not  an effective teaching tool, tablet technology in the classroom (not in assessment) may be a better pathway for our high performing students.  I'm sure issues like this are apparent in other learning areas.

Public education is beginning to fail the students that it is most needed for, to ensure "the fair go" is still a national objective.  I hope we have the courage to address it early, rather than be forced into reactionary measures later.

Tuesday, April 3, 2012

CAS calculator and differentiation

There are a host of ways to find the first derivative on the CAS calculator and to solve calculus problems quickly.  Sometimes I think exam writers are well behind what these calculators can do and fail to understand how trivial some problems have become.

Let's start with simple substitution into an equation:

a) Find y at x=3 for y = 2x^2 + 2x + 2

The "|" is important between the equation y = 2x^2 + 2x + 2 and x=3

Let's now find x if we know y.  This is a little harder as we have to solve the equation.

b) Find x at y=14 for y = 2x^2 + 2x + 2

The easiest way to do this is to type 14 = 2x^2 + 2x + 2, highlight it using your stylus (this is important!) and then go

Note that it find both possible solutions (unlike using numsolve with the incorrect range specified)

Let's find the first derivative.  For this use the 2D template in the soft keyboard

Go keyboard -> 2D -> Calc ->

c) Find the 1st derivative of y=2x^2 + 2x + 2

Note that I removed the "y=" this time.  I differentiated the expression on purpose as it makes the next part easier.

Finding the 1st derivative/gradient/instantaneous rate of change at a point is also easy.

d) Find the 1st derivative of y = 2x^2 + 2x + 2 at x = 3

As you can see, it is a mix between c) and a)

Last but not least we can find a point for a particular gradient.

e) Find x at y' = 14 for y = 2x^2 + 2x + 2


To find y itself we could repeat a)

It's very much a case of thinking what you need and then finding it.  As you can see it can all be done with one line on a CAS calculator, things that would take multiple steps on paper.  TanLine is also a useful function that can be investigated and used to quickly find tangents.


Click here for more CAS calculator tutorials

Saturday, May 22, 2010

Functions and the Casio Calculator fog(x)

When finding fog(x) the CAS calculator does a good job of simplifying algebraic steps, often to the point of making traditional questions trivial in the calculator section.

I'll start by defining a function f(x) =2x+1 and g(x)=x^2 and attempting to find fof(x), fog(x) and the inverse of (fog(x)).

To define a function f(x) go to the Main window and select


A window will appear

Enter the function name (eg. f)
Variable (eg. x)
Expression (eg. 2x+1)

The main window will reply:
Define f(x)=2(x)+1


Enter the function name (eg. g)
Variable (eg. x)
Expression (eg. x^2)

The main window will reply:
Define g(x)=x^2

To find fof(x)

In the main window
f(f(x)) (using the soft keyboard)
press exe

The main window should say

To find fog(x)

In the main window
f(g(x)) (using the soft keyboard)
press exe

The main window should say

To find inverse of fog(x)

In the main window
y=f(g(x)),x))) (using the soft keyboard= don't forget the "y=")
press exe

The main window should say

Here is a link to other CAS calculator posts.

Inverse Functions and the Casio classpad 330

The 3CD course requires knowledge of domain, range, co-domain and inverse functions. The classpad can handle these in a number of ways.

The most obvious way is in graph mode (Menu-> graph tab). Set up a graph (say y=(x+1)/(x+2), and visually examine it to find the domain and range. To find the inverse, click on the graph and select Inverse (Analysis->Sketch->Inverse). The equation of the inverse can be found by selecting the inverse graph and examining the equation bar at the base of the screen. Be careful with this method, as the resultant inverse graph is not written in "y=" form.

The less obvious way to do it is in the main window. I have not found an "inverse" function yet, but the following is a workaround to find the inverse.
Action->Transformation->Simplify (to simplify the resultant equation)
Action->Assistant->Invert (to swap the x & y variables around)
Action->Advanced->Solve (makes x the subject of the equation)
Enter the function
Press ",x)))" using the soft keyboard (to make x the subject of the new equation)

It would look something like this when finished:


In a way I prefer the main window method as it mirrors the algebraic method. I think students need to really understand the parts of an equation to effectively find the domain and range. I explicitly draw students attention to critical information such as the inability of the function to equal zero or where the function is undefined

a) Look for possible values of x where y=1/0 will occur (eg. {x≠-1} for y=1/(x+1) ).
b) If it is not possible for the numerator to be zero (eg. {y≠0} for y=2/(x+1))

If the range is not obvious it is often easier to examine the domain of the inverse of the function.

Here is a link to other CAS calculator posts.

Purplemath has a good explanation of inverse functions.

Sunday, January 10, 2010

Combinations & Permutations on the CAS Classpad330

The factorial(!) button is found in the soft keyboard under the abc tab (hit the up arrow next to 'z' and the '1' turns into a '!' (like with a computer keyboard). It's probably worth looking for (or making) an eActivity to speed it up a little.

Going through the first two 3C exercises in Saddler, the easiest way I've found to use the calculator is to open the softkeyboard, press cat tab, scroll to N (using the alphabet at the base of the softkeyboard), highlight nCr, press INPUT and seperate the n & r values with a comma from the hard keyboard.

(Update 1/2/2009): One of my students found another way via softkeyboard->mth tab-> calc->nCr which is a lot simpler and nPr is also there!

When you need to use multiple functions just rehit the INPUT button with nCr still highlighted. Completed the whole of 3C Ex 1A & B with a combination of no calculator and this function.


I don't have my calculator books here, so this will have to do until I get back to school.

Here's a link to an index of other CAS calculator posts.

Sunday, May 17, 2009

3A MAT Ex. 8B Annuities and Amortisation

Ok.. AP's and GP's are now a thing of the past (what?? huh?? when did that happen - in 8A of course!).. we're now onto applications of growth and decay.. nope.. (we did that in 8A too.. huh?? what??)..

MAT 8B We're now onto Annuities and Amortisation - growth and decay with payments.

The calculator handles this under the Financial, Sequences or Spreadsheet.

Starting with Financial:
Once in Financial, select Compound interest.

n - represents the number of installment periods
I% - is the interest p.a.
PV - is the present value (the initial investment)
PMT - is the payment per period
FV - is the future value (the investment at period N)
P/Y - is the number of installment periods per year (how often a payment is made)
C/Y - is the number of times interest is compounded

Let's look at a simple problem say 8B q.3 in 3A MAT. Kelvin invests $620,000 into an account giving 5.8% pa. interest compounded annually from which her withdraws $50,000 at the end of every year.

a) How much is left after 10 withdrawals (N=10, FV=?).

Leave the cursor on FV and press solve (at the bottom left hand corner of the window)

b) For how many years will Kelvin be able to withdraw 50000 per year

Find when the account is exhausted of funds (eg. N=? when FV=0)

Leave the cursor on N and press solve (at the bottom left hand corner of the window)

N=22.52 therefore for 22 years.

If anyone can explain why PV is negative I would be very appreciative. I know from last year's course that it is but have no idea why.

Now Sequence:
This could also have been done through the Sequence tool using recursion
a) Tn+1=Tn*1.058-50000; T0=620000. Find T10
b) Tn+1=Tn*1.058-50000; T0=620000. Find n Where Tn=0

I'll leave the spreadsheet method for another day.

Here is a link to other CAS calculator posts.

3A MAT recursive formula, AP's & GP's

Those of you attempting Exercise 8A without students with a thorough grounding in AP's & GP's in year 10 will be scratching your head at this chapter.

Q1: creating recursive formula from word descriptions
Q2 -5: creating sequences (tables of values) from recursive formula in the form Tn=...
Q6: identifying AP's, GP's or neither from sequences
Q7-9: creating sequences (tables of values) from recursive formula in the form Tn+1=...
Q10: creating sequences (tables of values) from recursive formula in the form Tn-1=...
Q11-12,15,16,17: creating sequences from multiple previous terms
Q13: recursive formula using the term counter(n) in the formula
Q14: finding unknowns in recursive formula
Q18-22: Growth and decay problems

My recollection of when we did this in Discrete was that these topics were covered over multiple chapters. When attempting 8A students faced difficulties in that the calculator has some limitations. I can usually maintain 1 chapter per lesson but in this case I let it run over three lessons and found some extra resources to supplement the topic as it left many students scratching their heads. This was hard as it chewed into the revision time I had left for exams.

Things to remember for next year:
1. Present multiple examples of recursive formula for the same sequence for Tn, Tn+1 and Tn-1.
2. Introduce the limitation that the calculator (in sequence mode) can only use up to two previous terms in its definition (eg. Tn+2=Tn+1+ Tn not Tn+3=Tn+2+Tn+1+Tn). We wasted a lot of time on this.
3. You cannot move freely between Tn-1, Tn, Tn+1 and Tn+2 representations if n itself is used in the formula. eg. Tn+1=Tn +3 is equivalent to Tn=Tn-1 +3 but Tn+1=Tn + n is not equivalent to Tn = Tn-1 + n
4. Be careful with the position when dealing with growth and decay. It is usually much easier to define T0 (Tzero) and Tn+1=.. as the initial value and formula. Thus when solving for n, n is the answer rather than n-1 (which caused no end of confusion amongst students).
5. Make students do the examples without a calculator unless it states otherwise. A lot of time can be wasted trying to make Sequence mode do things it is not intended to do.

Sequence mode and Calculator usage (What not to do).

Most sequences can be done in Sequence mode. Some cannot. Here's how to get into Sequence mode.

Let's put in Tn = 2Tn-1 -5 where T1=3 (q.11 from 3a MAT). We are looking for T1 to T5. Press Type in the menu.
Figure 1.

You will notice that the notation to the text is different in that "an" is used instead of Tn. Ignoring that, you will also notice that there is no option for "an", only for "an+1" or "an+2". (Blogger can't do subscripts so just put them in where needed!).

In this case it is not such a problem, we can just transpose our equation to Tn+1 = 2Tn + 5 as n itself is not used in the formula; The given value T1=3 now becomes T2=3, remembering that we are looking for T2 to T6 now (which is really T1-T5 of the original formula).

So now we put in T2.. Easy no? NO! If you look at Figure 1 we have options for Tn+1 where we are given a0 (the zero term) or a1(the first term). No option for a2.

The Saddler text does a pretty good job of making the calculator look clumsy and painful to use compared to paper and pen.

Here is a link to other CAS calculator posts.

Sunday, March 29, 2009

Variables on the classpad

There has been some confusion about how to define variables on the classpad in my class. Here is what we have discovered.

If we use a variable found under the mth tab -> var on the soft keyboard (the variables that are italicised) it is treated as a normal pronumeral in algebraic equations (multiplication is assumed with adjacent pronumerals). The x,y,z on the keypad is also treated this way.

eg x = 10, y=20; therefore xy=200

The multiplication sign is automatically added.

If we name a variable using the abc tab in the soft keyboard(the variables that are not italicised) then we are naming a variable that has multiple letters.

eg xy =10; x & y are undefined.
m = rise ÷ run

Potential Gotcha!

We have to be careful not to confuse functions defined under the mth tab (eg. trig ratios) and variables that we have created when using NumSolve. One of my students entered this on their calculator.


It would return the fractional value adj÷hyp rather than the value for theta. This is because the student had defined a variable "Cosθ" by typing Cos via the soft keyboard rather than entering the function Cos via mth->trig->Cos.

Superscripts and Subscripts

Later on students will want to use subscripted characters when creating variable names. One example is the gradient formula.


The subscripts are found in the soft keyboard under abc->math at the bottom of the screen. Superscripts are on the line above it. Only numbers at this stage (more will be possible as more fonts are released) can be superscripted or subscripted as far as I can see.

Here is a link to other CAS calculator posts.

Saturday, March 21, 2009

Trigonometric equations and the CAS calculator

There are lots of ways of solving trigonometric equations on the Classpad but I have avoided using Trisolve as it takes away the thinking aspect of trigonometric equations. Instead I focussed on setting up equations in eActivities with the intent to complement them with the Geometry section later.

eActivities are a great place to store frequently used equations. In this instance, I wanted to keep all of the trigonometric and circle equations in one place ie sine, cosine & tan ratios, sine rule, cosine rule, sector, segment equations, circumference, area.

To do this I opened an eActivity from the main menu.

Then I started a new eActivity by going File -> New. Then I saved it by going File->Save. I called it Trig Formulae.

So then I inserted a Numsolve strip to hold my equation.

Once the strip was added I used the soft keyboard to name it the Sine ratio. Then I pressed solve to put the equation in.

Using the mth tab in the soft keyboard and then selecting the Trig option at the base of the soft keyboard I entered 'sin('. Directly below the mth tab, the theta button can be found and then closed the bracket. Don't type the word 's' 'i' 'n' using the soft keyboard as it won't work - it will treat it as s x i x n.

Then using the 2D tab, I created a fraction and using the var option entered o ÷ h. I hit exe, then closed the equation using the x at the top left hand corner of the window.

I then tried it out using the example opp=7, hyp=14, theta = ? I left theta blank, made sure the angle was selected (with the dot next to it coloured in) and pressed solve in the toolbar. Viola, theta = 30°. If you get some weird answers check that the calculator is set in degrees mode. If the answer is still weird, reset the calculator and it seems to work.

Update (25/3/09): After using this with the class for a few days (especially with radians) I noticed a few strange results where the calculator would return unexpected answers (eg for the above example -330°). To fix this, set the Lower bound to 0 and the upper bound to 180 (for degrees) or pi/3.14159 (for radians) and the results will appear as expected.
I then set about putting in the cosine ratio.

It's a great tool for things like the cosine rule where students find it hard to transpose equations and forget negative signs or for circle, segment and sector equations that are commonly forgotten.

Here is a link to other CAS calculator posts.

Thursday, February 5, 2009

Absolute value and the 3A MAT course

Ok. Absolute value - easy enough, to take the absolute value of a number, make it positive if it wasn't already. easy peasy...
... until you start to look at IyI=IxI and ask students to graph it..
... then ask them to find the intersection of Ix-3I=I2x+4I algebraically
... then ask them to find Ix-3I<=I2x+4I

Students really bogged down when they reached inequalities. The approach I used was similar to that by Sadler in his 3A book for MAS. The problem was that I really wasn't sure they understood what they were doing.. they could follow the algorithm but understanding was eluding them.
I started by looking at absolute numbers and explained models for solving using number lines, graphing and algebraically. Then I used a composite approach to assist students visualise what it was they were doing with problems like:

I started by displaying the graph on the board using the overhead gadget for the Classpad.
I entered Graph&Tab from the menu workpane (using the menu icon at the base of the workpane).

I selected Graph&Tab and entered Ix-3I for y1 and I2x+4I for y2. For some reason the graph workpane doesn't allow you to use the 2d tab absolute value option - so use the abs() function under the cat tab in the soft keyboard. When you hit enter it will restore the absolute value notation.

Make sure both y1 and y2 are ticked (if they are not place the cursor on the line using your stylus and hit exe). Hit the graph button in the toolbar (the first icon with the top formula pane selected). The following graph should appear:

We then looked at the original inequality again and I asked what did it really mean?
One way of thinking about it was, "when is the graph y=Ix-3I less than or equal to the graph of y=I2x+4I?"
We looked at the graph and found that the part marked red on the line y=Ix-3I satisfied the inequality.

Using the intersect function under analysis in the menu bar we know that the two lines intersect at -1/3 and -7 therefore the interval is x<=-7, x>=-1/3.
We had discussed that we could also do this algebraically by using the property if IxI=IyI then x=y or x=-y to find the points of intersection.


x-3 = 2x+4
-x = 7
x-3 = -(2x+4)

I then asked students to draw a number line with the intervals marked and substitute the values back into the original inequality. We numbered the three intervals. The first interval represented x<=-7, the second -7<=x<=-1/3 and the last x>=-1/3.

We then selected a value within each of the intervals and substituted them into the inequality. If they were true then this indicated values of x that satisfied the inequality.
Interval 1 (x=-8)
11<=-12 (true)
Therefore x<=-7 is a valid interval.
Interval 2 (x=-5)
8<=-6 (false)
Therefore -7<=x<=-1/3 is not a valid interval.
Interval 3 (x=0)
3<=6 (true)
Therefore x>=-1/3 is a valid interval.
The inequality Ix-3I<=I2x+4I is valid over x<=-7, x>=-1/3
Drawing students attention from the graph and back to the algebraic representation released the tension in the room, the screwed up faces and suddenly lights went back on.
Thank goodness!

Here is a link to other CAS calculator posts.

Wednesday, February 4, 2009

Eureka.. one problem solved!

Teaching students to solve equations with the balancing method can be difficult as many different skills are required. Collecting like terms, fractions, multiplying pronumerals, dealing with coefficients and the like. When adding all the complexities together students can really struggle.

Surfing around yesterday I found the following link using a classpad calculator to assist students check their understanding of how to use the balancing method. It has worked fabulously well and yr10 students that typically hate algebra (and maths) are all smiles...

Here's the sequence of lessons up to this point (first week of term one)..

  1. review of algebraic terminology
  2. review of collecting like terms
  3. review of multiplying algebraic terms
  4. solving simple equations
Students were shown x + 5 =7 and asked what was a possible value for x. They responded 2 and we discussed how the observation method is often a good method for solving equations. We discussed how this was good for simple cases but with more complicated examples it became too difficult.

I then introduced the balancing method saying we could get to the same result by making x the subject of the equation by examining the LHS and thinking what operation could we do to isolate the x value.

A student suggested that we subtract 5 and I said great.

Then I said to students that the crux of the balancing method was that anything we did on the LHS of the equation has to be done to the RHS. I wrote on the board

x + 5 = 7
x + 5 -5 = 7 - 5
x = 2

and asked how did that compare with our original answer. We then did the following example:

5x + 5 = 20

A student offered the following step:
5x + 5 - 5 = 20 -5
5x = 15

Typically students get stuck at this stage as 5÷5 =1 is not an intuitive step. For once I told them that I would divide by five and showed them how it works.

5x ÷ 5 = 15 ÷ 5 (please excuse the division symbol, I actually used fractions but it is too hard in html)

x = 3

And here's the real magic.. I then took out CAS calculators borrowed from the senior school and they did a number of examples with them. For the following example:

2x - 2 = 15

Their brains started making connections and they actually were using the calculators to check that their logic was correct rather than to give them just answers.

You can see from the example that each step in the calculator mimics the steps to answer the problem on paper. It is easy to see how after each operation (+2, ÷2) x becomes the subject of the equation and ultimately becomes solved with x=8.5

Common errors become obvious earlier. Students decide what operation needs to be done and see what that operation would do. Take this common case:

The student has multiplied by 2 before they have subtracted. They can instantly see their mistake (the LHS of the equation looks more complex rather than simpler so the student starts again. The second attempt subtracting 5 gets them closer to making x the subject of the equation.

For many of us, this is how we learnt to transpose equations - a little trial and error. Lots of practice. Lots of heartache. Lots of looking at the back of the book.

The students found using the calculator fun... and the calculator only gave them guidance - not just solving the answer. It was a mix between the old inverse operations method (change the sign/change the sign that causes all sorts of difficulties when fractional terms/multiple terms are introduced) and the balancing method. To be honest, I've never found the 'scales' explanation that typically accompanies the balancing method useful - but the CAS introduction way I think may have real promise.

The other great thing is that they were recording their answers really well on paper.

For the above example I would see (with equals signs aligned):


To see mid tier students lay out work like this rather than
1) 8.5
was fantastic.

Here is a link to other CAS calculator posts.

Thursday, January 22, 2009

Trigonometry and CAS calculator I

During exams last year we noticed that those who had CAS calculators were not using them for Trigonometry problems. During summer school we sought to rectify this. To start with we looked at properties of triangles.

Students stated that the calculator would give no solution when solutions existed. This sounded doubtful but I had a good idea where they were going wrong.

I started with the following. Take three lines, 4cm, 2cm, 1cm. Now obviously this can't make a triangle. Right?

No matter how we change the angles at A & B they cannot form a triangle. This was to inform students that a no solution result in their calculator had meaning.

So now we had a look at a problem that they were having difficulty with.

In their work pads they had written A=x, a=5cm, B=72°, b=72 and labelled the triangle correctly. So far so good.

They could tell it was a sine rule problem but had difficulty entering it into the calculator. Where did the sin, cos, tan buttons go from the calculator?

The first thing to do is find the sine function. Open the soft keyboard, select the mth tab and press the Trig button at the bottom of the keyboard.

Next enter the sine equation with the substituted values. They needed the fraction template inside the soft keyboard under 2D

Hitting execute at this point gives no solution. Huh?

Well.. we still have to solve for x. So highlight the equation and go to the interactive menu (in the menu bar at the top of the work pane), select Advanced and then solve. The Equation should be there and the variable listed should be x.

After a rather long wait a huge expression appeared with a strange looking answer.

I'll rekey it here as the whole answer does not appear on the work pane on the calculator.

{x=360.00.constn(1)+137.209, x=360.00.constn(2)+42.791}

The answers are the two bits in red (at this point we had a bit of a chat about the ambiguous case with the sine rule). To get to the answer you have to navigate with the style and the left/right arrows at the edges of the equation.

So the answer is x=137° or x=43° (0 d.p.)

We then drew these triangles to give students a better understanding of the ambiguous case.

In the next example we'll look at the perils of rounding and go back to the case above with the impossible triangle.

Here is a link to other CAS calculator posts.

Tuesday, January 20, 2009

Solving Simultaneous Equations using the Classpad 330

There are a number of ways of finding where two lines intersect. Let's solve this example.

"Where do the equations y=x and y=-2x+3 intersect?"
One way to find the solution is to solve the two equations algebraically using simultaneous equations.

First open and clear the main work pane.

Press the blue Keyboard button and bring up the soft keyboard. Select the 2D tab.

You should be able to see a button with a bracket and two small boxes (circled below in red). Press it.

You should see the simultaneous equation template in the main pane.
(Update 1/6/2010: Press it twice to add a third equation line!)

Click on the first box and type y=x
Click on the box below it and type y=-2x+3

In the third box to the right of the vertical line type x,y (the variables we wish to solve).

Hit the exe button.

The answer (x=1, y=1) should appear.

Here is a link to other CAS calculator posts.

Saturday, October 25, 2008

Casio Classpad 330, Finding the mean and missing values

I posed the following question to my year 10's in order to continue our learning of the new calculator. It is an example of solving a problem where the mean is known but a value in the sample is not.

"Q: A class had 5 students. Student results in the last test was {50,56,64,72,81}. Isabella joined the class and the new mean became 68. Did Isabella score higher than the old mean and what was her score?"
H: If the mean of {50,56,64,72,81} is less than 68 then Isabella has scored higher as a higher score by Isabella will raise the mean. Since we know the new mean (68) we can work out Isabella's score by working backwards.
Set up a working pane with a main application and a list editor. Title a column 'list1'. Add the 5 student results to the list editor.

Click in the main application and type mean(list1) using the soft keyboard. Hit the blue exe button. The answer is 64.6 .
A: The old mean 64.6 is less than 68 therefore Isabella has scored higher.

To find Isabella's score click in the list editor and tap the next empty cell in list1. Press the x button. Click in the main application pane on the line that says mean(list1). Press the blue exe button.

This will return a sum to work out the mean of the list for values of and value of x i.e. (x+323)/6.

As we know the new mean alter the first line to read mean(list1)=68. Highlight the solution sum and tap Edit in the menu bar and then Copy. Paste the sum on the next line in the main application pane. Highlight the sum, tap Interactive on the menu bar, then tap Advanced on the sub menu and then tap solve. Tap ok at the base of the dialog box. The answer is x=85.
A: Isabella's test score was 85.

Click here for other CAS calculator articles

Wednesday, October 22, 2008

Casio Classpad 330, Creating a Histogram

Today in class we looked at how to produce a Histogram using the list editor. A Histogram is used when data is continuous (there is no gap between intervals).

Class interval (Frequency)
0 <= x <>=80 (1)

Tap in the list editor. Tap Edit in the menu bar. Tap Clear All. and tap Ok in the dialog box. If a graph is open tap the StatGraph pane to select it. Tap the cross in the top corner of the window to remove the graph.

Name a column in the list editorclassmid’using the soft keyboard. Put the midpoint of each class into the classmid column. eg. {5,15, 25, .., 85} (make sure you name the column before putting the data in!).

Name a column in the list editor ‘freq’ using the soft keyboard. Add each corresponding frequency into the freq column. eg. {3,10,16,..,1}.

Tap SetGraph in the menu bar. Tap Setting. Select Histogram in the Type dropdown, select classmid in the XList dropdown and freq in the Freq dropdown. Make sure the Draw option is on. Tap Set at the base of the dialog box.

Tap the StatGraph icon in the icon bar to display the graph. Make HStart 5 (midpoint of first interval) and HStep 10(size of intervals).

A Histogram will appear. Tap the StatGraph pane and then tap Analysis in the menu bar. Tap Trace in the menu.

A flashing crosshair should appear above the first column of the graph. Use the blue cursor key to navigate column values in the graph. You can use these values to create your histogram on graph paper. The xc at the base of the graph are horizontal axis values and the Fc are your vertical axis values.


Other educationWA articles on CAS calculators
How to navigate through menus (what's a menu bar?) Click here
How to create a list (what's a list editor??) Click here

Here's a link to an index of other CAS calculator posts.

Monday, October 20, 2008

Casio Classpad, day 1 with students

As I play with the calculator things become a little more obvious. It was good fun with my year 10's showing them how to find the mean of


with the CAS calculator during p5 on a 35°C day and then set Maths for WA3 10C with 50 items in the sample. I was upfront in saying to my students that learning all the new content next year and learning how to use the calculator was a bad idea (lights went on... ahh, that's why I need to get one this year!!). For those students still unsure, I made them find the mean of a 50 item sample with their scientific calculators. They promised to buy a CAS calculator tomorrow.

Anyhow.. this is one way of finding the mean with the CAS calculator. There are many better ways but the idea was to learn how the calculator works (the picture is the end result).

Open a main application in the work pane.
  1. The last icon in the tool bar should be a graph. Click the dropdown to the right of the graph. Tap the icon that looks like three columns in the sub menu. The list editor will open in the bottom pane below the main application.
  2. We need to give our list a name. Tap the top of the first column. “list =” should appear at the base of the list editor.
  3. Press the blue Keyboard button. The list editor will temporarily move to the top pane. The soft keyboard will appear in the bottom work pane.
  4. There are four tabs in the soft keyboard. Tap the abc tab with the stylus. A qwerty keyboard should appear. Name the first column in the list editor ‘list1’ if it is not already. You may need to click again in the list editor list= textbox first.
  5. Press blue Keyboard to get rid of the soft keyboard. The main application should reappear in the top pane and the list editor in the bottom pane
  6. Use the stylus, tap the first cell in list1.
  7. Using the number keys press 10 then exe (bottom right hand corner of the keypad). This should put the first number in the list. Not that the cursor has dropped to the next item in the list without having to use the stylus. Now enter 12 then exe. Your list should now have two entries. Add the remaining entries.
  8. Click in the main application. Raise the soft keyboard with the blue Keyboard button. Open the abc tab and type list1 and press exe. {10,12,13,14,15} should appear.
  9. Click Action in the menu bar and tap List-Calculation. Tap mean from the options provided. 'mean(' should appear in the main application.
  10. Complete the action by typing ‘list1’ using the soft keyboard and the button ‘)’. You should now have ‘mean(list1)’ displayed. Press exe. The answer 64/5 will appear. To get a decimal representation, highlight ‘64/5’ with the stylus and click the first icon in the icon bar.
viola. You should be able to finish the tutorial by finding the median yourself. (An alternate way is to type list1, highlight it, tap the Interactive item in the menu bar, tap list calculation in the sub menu and then median and then select ok at the base of the dialog box.) You could also use statistics mode (tap Main on the icon bar, then tap Statistics.) The Statistics application is very similar in structure to the stats mode on the fx graphics calculator).

Here's a link to my last article on learning how to use a CAS calculator.
Here's a link to an index of other CAS calculator posts.

Tuesday, October 14, 2008

Blog entries on CAS calculators.

Other educationWA articles on CAS calculators:

My first use of the CAS calculator Click here
Naming conventions Click here
How to navigate through menus (what's a menu bar?) Click here
Naming variables Click here

How to create and use a list of data (what's a list editor??) Click here
How to create a graph? (What's a StatGraph?) Click HereHow to find the mean and missing values of a data set? (how do you solve an equation?) Click here
How to find probabilities with Normal Distributions? Click Here
Finding simple moving averages Click Here
Combinations and Permutations Click Here

Balancing equations. Click Here
Solving simultaneous equations. Click Here
Absolute Value and Inequalities. Click Here
Absolute Value and Inequalities 2. Click Here
Functions (Inverse) Click HereFunctions (fog(x)) Click Here

How to find an unknown angle from a triangle using the sine rule. Click here
Storing formulae on the CAS calculator. Click Here

Annuities, Reducible Interest and Amortisation (Finance). Click Here
AP's & GP's. Click Here

Finding and solving problems involving the 1st derivative. Click Here

The articles should be completed in order as they build upon previous entries. They use the Casio Classpad 330.