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=?).
N=10
I%=5.8
PV=-620000
PMT=50000
P/Y=1
C/Y=1

Leave the cursor on FV and press solve (at the bottom left hand corner of the window)
FV=436670

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)
I%=5.8
PV=-620000
I%=5.8
PMT=50000
FV=0
P/Y=1
C/Y=1

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.

Me Tarzan.. You Jane...

Yep sure.. that's me big alpha male...

ha!

Went to get a flu shot today.. No man flu this year I thought.. No sniffling and whining about how sick I feel.

Anyhow, two minutes after the needle was put in (which took about a second and didn't hurt at all(really!)) I woke up on the ground. The nice nurse (who was a great grandmother) had to catch me and guide me to the ground as I was out cold.

What is it about men and needles? The nurse assured me that it was not at all an uncommon occurrence. I can stand in front of a bunch of hostile adolescents and win them over.. but can't overcome the fear of a one inch bit of metal stuck in my arm.

Next time I'll just take the flu.

Thinking back though, I didn't feel weak until I watched Kendra get her needle. I think the thought of a loved one being hurt is also a factor.. of course I could just be rationalising what is a pretty pathetic incident.

I still feel horrible.

And now for my maths joke so you can all feel terrible too..

If a five sided figure is a pentagon and a six sided figure is a hexagon, what is it when your grandmother passes out?

A Nanagon of course! (Not to be confused with a Nonnagon (an italian grandmother) or a pollygone (a missing parrot)).

Must be the residual chemicals in the system!

Turning around struggling schools

I read this article with interest about schools being turned around.

"The basic theory is that middle-class kids enter adolescence with certain working models in their heads: what I can achieve; how to control impulses; how to work hard. Many kids from poorer, disorganized homes don’t have these internalized models. The schools create a disciplined, orderly and demanding counterculture to inculcate middle-class values."

Duh!

To not provide strong behaviour models is to invite failure. The ethic that work = results = success is not inbred in these kids by their immediate environment. Developmental programmes for these kids are inappropriate as they do not have a drive to develop.. we must first create this drive.

"Basically, the no excuses schools pay meticulous attention to behavior and attitudes."

I like that.. "no excuses".. and it is so applicable to our public schools. In our society, everyone gets to have their say, anything can be rationalised as true and the time wasted unpacking excuses is.. well.. inexcusable.

"They teach students how to look at the person who is talking, how to shake hands. These schools are academically rigorous and college-focused."

Under the guise of political correctness and multiculturalism we accept a range of behaviours that interpreted under this model is inappropriate. Whether this is right or not is not a question I can easily answer but as a nation we need to decide what is acceptable behaviour and then teach it. Teaching a class of thirty under this model would require some bending of cultural mores in order to encourage a class environment of like behaved students rather than a group of individuals. To drive these kids towards developing the rigour for higher education would be fantastic.

"Promise Academy students who are performing below grade level spent twice as much time in school as other students in New York City. Students who are performing at grade level spend 50 percent more time in school."

I love the idea that those behind have to work extra hours.. how obvious.. if you're behind you need to do extra work to catch up, unlike our current policy of teach what you can in the time allowed which results in students that need more time to learn each topic falling further behind each and every day.

The sad fact is that creating a no excuse environment would take much commitment/courage and would create much heartache within the school community. I don't know if it could be done within the Perth environment with our inclusiveness of multiple cultures. It would be a move back towards creating "Australians" rather than a nation of multiple cultures. I don't think as a nation we have been driven to this yet.

If we could decide on minimum standards (minimum attendance requirements, obedience to teachers, zero aggression, completion of homework, minimum expectations of results before progression) perhaps it would be a small step to replicating the results found in Harlem.