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
interactive->advanced->solve->ok

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.

Viola!

Click here for more CAS calculator tutorials

1-1 laptops and utilisation in mathematics classes

Last year I was struggling with using laptops in the mathematics classroom with students of significant disadvantage.  I had thousands of dollars of equipment and yet felt unable to utilise them in a way that raised performance in my classroom.  Use of technology outside of CAS calculators was near nil.  We called in our local laptop reseller to help.  It was a big surprise to hear from the reseller that utilisation of laptops in 1-1 laptop mathematics classrooms was around 5%.   Schools hadn’t done much more than load textbooks, do the occasional assignment and play with math games.

This was a defining moment in the use of laptops at our school and freed the math department to think outside the box.  These machines were unlikely to assist us in solving problems in a mathematics classroom.  It wasn’t going to revolutionise our teaching, but with such a high level of available resourcing we realised it may have the potential to revolutionise learning.

The first thing we did was look at problems of the low socioeconomic student, resourcing, organisation, attendance, engagement, completion of work (in and outside the classroom), skill and knowledge deviation/development.  This is where lights turned on and we started towards real reform of learning.

Our reseller jump started our thinking by showing us an application that filled a need for a learning management system at the school.  Interactive whiteboards were creating masses of information that students wanted and digital resources were becoming more prevalent.  We needed a platform to distribute electronic information to students in real time.  Something cloud based was our preference as we did not have the resources to maintain something onsite.  It needed to be something reliable and already tested, we didn’t want to be doing and paying for cutting edge research.  Edmodo, the software suggested by the reseller, filled a need.

We needed to seek ways to increase our influence outside of the classroom to better utilise classroom time.  School grades were posted online allowing students access to their progress.  Practice tests and screencasts of key lessons were posted online for students to use as revision for tests.  Students started to post question online for answering by the teacher or peers – more importantly students started responding to posts before I had a chance to.

Last Friday I sat with a disengaged student learning BIMDAS.  I noticed that although he could identify what to do, by the time the student reached the part of the sum to work with, he had forgotten what he had to do.  A simple idea was to underline the next operation to be performed.  This changed his demeanour and he engaged with me as his teacher, working through a page of sums.  A new (although I imagine obvious to many) teaching method though was not the main outcome from this incident.

By far the most unexpected outcome of our IT journey is watching the doors between classrooms open, with math teachers in the school sharing ideas.  Ideas such as the underlining during BIMDAS are starting to pollinate between teachers – teachers that found it hard to meet due to timetabling were now able to share and comment on resources.  Ideas are being recorded and are retained for use in following years.  Although the laptops were not engaging students, the idea of using them more effectively was solving some of the problems that I had expected that laptops would resolve directly.

This idea of technology helping engage students goes beyond direct use of laptops.  The online environment itself was proving a motivating factor with the year eights.  They could see that we were making an effort to extend into their world, where physical appearance, socioeconomic status, age, gender, religion were insignificant factors to interaction.  Where rapport is so important with low socioeconomic students, any connection we can make between their world  and ours is beneficial.  It helped that edmodo had adopted an interface very much like that of Facebook, a generally banned interface in the government school system.  

Now that we had a system that can store boardwork and assist absentee students work whilst outside school (such as those doing VET courses) we had the potential to ameliorate some of difficulties of absentee students and students with academic difficulties.  It removed some of the excuses and put more of the onus of responsibility back onto the student to perform.

It is only a beginning but one that is exciting.  We know IT has been traditionally an underperformer, but in this instance I am excited by the response both by students and teachers.  We will need to continue examining how the system is working going forward and drive towards more tailored resources for students.

Dominance games and adapting to a new position.

Being of ethnic background, some things are just not nice.  Four senior staff have called me boy in conversation.  Given that I actively seek to remain connected to the current generation, I can understand where they are coming from, but I must admit to finding it extremely irritating especially coming from a white person.  I have more life experience than most, if not all the staff (I packed a lot into my twenties) and I'm two years off 40.  I'm hardly boy anymore.

Culturally it is really inappropriate (my grandfather was darker than most of our boys of African descent) and although I don't think a racist element is present, it probably is an unconscious dominance tactic at worst, a really poor idea for an affectation at best.

The only thing I know is that it is annoying.  Coming from a professional background outside of teaching, politeness becomes an artform, professionalism a requirement.  In teaching the rules are blurred and seemingly become moreso the higher you reach.

I have managed to stay out of school politics to some degree by being being predominantly in a classroom isolated from other teachers.  I've had to move into the building and it makes me more accessible and requires more contact with elements of the organisation than I am used to.  It will take time to adjust, knowing what people need to know and what can be just dealt with.

Even jurisdiction is a problem, knowing who, what and where is your responsibility and what should be done by others.  What do you do when things that need to be done are neglected by those that should be doing them?  Do you ignore it, act to fill a need or report it to get fixed (and to who?).

Given that no time is given to resolve anything (as TiC), I know there is little within the role I can do, other than ensure administrative tasks are done.   This of course means that staff training, monitoring of progress, curriculum development and other roles traditionally in the hands of a HOLA will be neglected at times of the year as my teaching load demands attention.

At least things have calmed down this week - my classes are planned, notes are done and assignments are ready.  From a teaching point of view it should all be good.  Let's hope there's not too much else going on for a little while.

Transition

I'm finding transition from one position to another is irksome.  We just completed stage two of our IWB rollout, with eight more boards installed today.  Testing each board and ensuring computers are set up correctly for teachers is becoming a troublesome task.  Each installation had it's own quirk.  One had a failed USB cable, another a bad monitor cable, another three had driver issues on their computer.  Rooms had to be rearranged, teachers needed basic instruction.  It's been uncontrolled chaos.

My own classes are a mix of classes transitioning from the old teacher in charge to my own.  I'm finding our styles are vastly different, in that I prefer to teach each topic explicitly rather than rely on the text.  It will be interesting watching how each group performs.  Small groups have not been set, so it will be at least two weeks until programmes are finalised.

The eights were a mess as the room that the monitor cable failed was mine, leaving me without a working whiteboard.  I believe it's fixed so I will need to deliver a ripper class to get them back on track.

I could have done without having to move from a class I had been in for 5 years.  That in itself has drawn a lot of time that could have been used for other things.  The students don't like it and I'm not a fan of the new room.  I think the logic that I need to be in the main building is flawed.  I need to be settled and organised - not subject to the constant disruption caused by being in the main building.

Picking up and dropping off my daughter to and from daycare takes time away before and after school that I have always spent on planning.  You make the best of any situation, so I have to settle and get my classes running.  It's time to just get on with it.

TAFE roles and stage 3 subjects

Classes where students are taken out of school for a day and then remain in stage three courses in year 11, in the majority of cases are setting students up to fail.  A student missing for a whole day each week, in a challenging course, lacking any evident independent learning skills is not destined to pass a stage three course.

The argument that they will make it up in directed study, to my mind, holds no water.  These students fail because they have to catch up multiple classes in these study periods, lack specialist assistance at these times (teachers have to keep teaching)..  and are generally borderline students in the first place.

I'm fully supportive of students participating in TAFE courses to complement their mainstream courses, I just believe that their mainstream courses should be kept at stage two levels to ensure that they can cope and succeed when year 11 reality and expectations kick in.  Stage two courses such as 2C and 2A MAT are better suited as these can be learned independently more easily.

If we think about it, it becomes obvious why they fail.  Firstly they lack a connection to the class, as better students tend to pair up with students that are regularly available.  Notes on the board do not replace the peer and teacher assistance that they lose from not being in class.  During directed study they cannot get immediate assistance from the teacher, particularly where the text is lacking in a particular area.  Students tend to shy away from them as the questions they ask are perpetually behind the level of the class and take considerable time both from the teacher and from other students trying to catch up on a regular basis.  This catch up process is exacerbated each week in multiple subjects.  It also causes issues with homework and assignments as that day is often lost due to travel commitments.

I have said this before and I will say it again.  We have to be careful with our subject selections and review them when students are out of class on a regular basis.  We should not underestimate the impact that this has on learning for the student.

Teacher in charge

Nothing like a promotion to bring in the new year.  One of the stranger structures in the public education system is the Teacher in Charge(TiC).  It's a position between teacher and Head of department(HoD).  The pay rise is minimal, the teaching load the same as a classroom teacher and the responsibility is similar to HoD(albeit not officially).

The main issue with the TiC role is that in many implementations the actual responsibility for traditionally HoD roles lies with team and year leaders, yet little is done by them in this regard as team and year leaders are swamped with administrative and behavioural concerns.  This has caused friction between the TiC and team/year leaders in prior years, as responsibilty is passed around like a hot potato depending on the amount of effort required. In particular, student guidance for non-performance is an area that is often neglected - particularly with students that are passively resisting work (work avoidant behaviours).

Secondly, early in the year, with full load, senior school teachers(the breeding ground for HoDs) are busy bedding down NCOS and small groups, leaving little time for assisting new teachers, examining curriculum, setting up IT structures(such as mathsonline/mymathsonline/matheletics), doing research on best practices (eg. IWBs, Tablets, NC texts, new pedagogy), entering competitions and addressing minor and niggling issues across the department.

TiC is a strange position as it is a promotional pathway to Head of department, which is a large pay rise and includes a reduction in teaching load.  Small schools use TiCs where level 3 FTE is not available for HoDs.  Unfortunately what this causes is a drain of talent from small schools to larger ones as TiC becomes a pathway to HoD - something prevalent in these times, as many maths HoDs are reaching retirement age.  Our school is on it's third TiC in three years.

So, here I am entering the fray of promotional positions, doing the TiC role this year.  It's exciting in that I have increased access to management thinking and can better understand the direction of the school.   The negative is that I am used to senior management positions where robust discussion is a part of the job - my experience is that robust discussion is not always welcome in a school.

Here comes the new year!

Summer school

Today was the fifth and last day of summer school. It's an event we run for students destined for stage 3 maths in year 12. Students benefit from being ready from the start of term, are more organized and are given guidance on material to be presented during the new year.  Teachers and tutors get experience working in a university environment.

Two of the presenters cancelled just before the first day, I had been sick for the majority of the holidays and I was left with the decision of cancelling the event or trying to do the event with a relief teacher, four pre-service teachers and some student graduates from 2011 that offered to help. I had three streams of students from year 10-12, 42 students and I couldn't do it on my own.

We went ahead anyway and everyone stepped up. The students were positive from the start all the way to the last day. Some had cancelled holidays overseas to be there. Brian, our ever ready math relief teacher, stepped in when I would have been overwhelmed otherwise. The pre-service teachers responded to each of their strengths - leading through preparation, personality, warmth and enthusiasm. They completed the tasks assigned to them with the year 10's and then stepped into the gaps left by the missing presenters for the 11's and 12's. It was a real team effort and awesome to watch. 30 of the 32 students that attended on the last day said that they would be back next year, with the 2012 year 12's showing real enthusiasm to come again and give the benefit of their experience.

All of the year 12 student helpers from 2011 were fantastic, but one really shone, as he showed the new year 12's how to navigate a wide range of problems. He stayed for the entire week and demonstrated what hard work could really achieve. He had nothing to prove by coming, he was our school dux in 2011, and I hope he saw what a difference he can make in the world.

With such a large number of people coming together and actively achieving something, (especially when I was at 50%) it quickly became another of the inspirational events of my career. To watch four summer schools, and then watch the benefit for attending students during the year and the growth in the pre-service teachers, it tells me that it is more than just another school event. Hopefully those involved understand the wonderful thing they have done. There's no doubt it could and will be done better, but that as they say, is a problem for another day.

Equation editor

The equation editor in activeInspire broke with the tickover of the year and a javascript error. This has been fixed in the latest update.

Russ.