Exponential Growth

Good morning from Jack, your tutor from Campbell River.  It’s brilliantly sunny here, with a high of 27 (or 81 F) expected.  This kind of weather can be so hard for kids just returning to school.

The people entering math 12 will probably encounter exponential growth very soon.  It’s not a term you hear very often, but compound interest is an example of it.  Most natural things grow the same way – while they have the resources.

Exponential growth means that the growth is a percentage of how much is already there.  So if your growth rate is 10%, then you go from 10 to 11 in one year.  If you start with 100, though, you go to 110 in that year.  Interest is the obvious example:  everyone knows that if you have \$1000 on deposit, you’ll get more interest than if you’ve only got \$100.

Step two is the one that surprises some people.  Let’s imagine you start with 1000 individuals at 10% growth.  At the end of year one, you’ve got 1100 – true enough.  The mistake many people make is that they assume that the following year, the population increases by another hundred, making 1200.  That’s not true.  If you start the second year with 1100, still at 10% growth, the population will increase that year by 110 – which, of course, is 10% of 1100.  So by the end of year two, you’ll have 1210:

1000 + 100=1100  (100 is 10% of 1000).

1100 + 110=1210  (110 is 10% of 1100).

The difference is only 10 at the end of year two, but that difference keeps getting larger because it contributes to the growth of the population.  That’s why, believe it or not, the population will be over 2000 in less than 8 years.  In 32 years it’ll be over 21 000.

The rule of 72 for compound interest says that

(interest rate)x(doubling time)=72

(Of course, that law is an approximation, but actually a very good one.)

Since compound interest is just an example of exponential growth, that law works for anything that grows exponentially.

Enjoy this beautiful day.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Algebra: basic

When you tutor math, you see various approaches to algebra.  Here’s my interpretation:

1)  Decide which side you want “x” (or whatever variable you have).

2)  Get all the x terms to that side, and all the numbers to the other side.

To do this, you perform a series of operations:  always the same to both sides.

3)  Divide out the coefficient of x.

Here’s an example:

7x – 1 = 5x + 9

Let’s get the x’s on the left.  To get rid of the 5x on the right, we subtract 5x from both sides:

7x – 1 = 5x + 9

-5x         -5x

On the left, 7x – 5x = 2x.  On the right, 5x – 5x = 0.  We are left with:

2x – 1 =         9

Now, let’s get the numbers to the other side.  To get rid of a -1, we add 1 (as always, to both sides):

2x  – 1 =        9

+1          +1

2x       =        10

Now we divide both sides by the coefficient of x, which is 2, to cancel it. Then we get the result:

x         =         5

I’ll be talking more about algebra, of course, but for the first day back, this is probably good.

Jack of Oracle Tutoring By Jack and Diane, Campbell River, BC.

Have a great day.  All the luck this school year, and come visit me often for more tips!

School Supplies: A tutor’s point of view

Last year, listening to French radio, I heard a surprising report:  school supply shopping is the second most stressful occasion for a great many parents, second only to Christmas shopping.

If that’s true, it doesn’t need to be.  What’s more, I can explain why:

1) With Christmas shopping, you’re not told what to get – whereas with school supply shopping, you usually are.

2) While, with Christmas shopping, there is a deadline, there really isn’t one in the same way when it comes to school supply shopping.

I’ve heard that a lot of people fear the expense of back-to-school shopping.  I can’t comment on the other dimensions of it (clothes, for instance), but I can tell you this:  school supplies don’t have to be (that) expensive.

In front of me I’ve got two big-box store flyers from Friday’s paper.  I’ll admit that if you have to shop today, you might pay around \$15 for a zipper binder (of course, you can pay a lot more if you want), then another \$15 or more for a school bag.  (This is all before tax.)

Coloured pencils, if you need those, might run around \$2.50 to \$6.00.  Buy a good name – not cheap ones.  When you’re a tutor, you see a lot of school supplies.  I’ve noticed that many kinds of coloured pencils – especially cheap ones – don’t hold a sharp point.  You don’t want your kid to be stuck with coloured pencils whose leads keep breaking.  Ask what the good names are.  I use Staedtler, but there are other good ones around.  I’d guess a name like Hilroy would be pretty trustworthy – although I’ve never tried their coloured pencils.

For just normal pencils (rather than colour), I prefer mechanical rather than wood.  Get 0.7 leads – they last longer and break way less often (as opposed to 0.5).  A twelve pack of Bic 0.7 plastic pencils will cost maybe \$4.  If they don’t get lost, your kid probably wouldn’t use more than half of them the whole year.  Pens are even cheaper than pencils – unless you want to pay more for something special.

One note, though, about mechanical pencils:  you can’t get them for most kids until they’re in grade 5 or later.  The reason is that the kids just play with them.  If your kid is earlier than grade 5, you probably want wooden pencils.

Erasers:  get white ones.  Staedtler is one kind I use, but most white erasers are pretty good.  At one place, they’re on for less than a dollar apiece right now.  If it’s not lost, one could last you for years.

Paper – both loose leaf and graph – might be the most variably priced item.  You might have to pay a lot on a given day at a given place.  However, my wife says she’s seen 150 sheets of loose leaf for less than a dollar recently. If you’re paying more than that today, you should probably look elsewhere.  Graph paper is usually more – you might have to pay 3 or 4 dollars for around 100 sheets – but you can sometimes get it for a lot less.  Kids don’t usually use much of it, anyway.

Markers, if you need them, might be around what coloured pencils cost.  A ruler you can get for less than a dollar.  A scientific calculator shouldn’t cost more than \$15.  I know I could get a good one for less.  The less fancy, the better.  As long as it has sin, cos, and tan on it, as well as square root, you’re probably pretty safe.  Of course, you shouldn’t need to buy a new calculator every year.

There are other odds and ends, but let’s make a rough total of the items I’ve mentioned.  You might be looking at around \$60 to \$75 before tax.  You could do better, depending on where you live and how much hunting you’re willing to do.

Where I live, some stores have school supply lists right at the front entrance.  You can find your kid’s school and grade, then pick up a list of supplies.

If your kid goes to the first day of school with just a few pencils and pens, some paper, a binder, a calculator, and the coloured pencils, they’ll probably be all right.  When they get home, they can tell you what they’re missing, and you can get the rest that night.

One final point:  Look for school supplies again in a month.  Watch the prices go up and down.  Eventually you’ll probably be able to get almost everything cheaper than you can today.  Stock up when it’s cheap.  Ask your kid what works and what doesn’t – and why.  If you familiarize yourself with school supplies, you’ll be on top.  Like most things, they’re best to buy before you need them.

Good luck!

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Reasoning: Inductive vs Deductive

A couple of years ago, a new topic came up during math tutoring:  deductive reasoning vs inductive reasoning.  What’s the difference?

There are many ways to explain how the two are different.  I’ll explain it this way:

With inductive reasoning, you notice a few similar situations that lead to the same outcome.  You ask, “Will that outcome always happen for situations like these?”  Here’s an example:  imagine you move to a new town in a foreign country.  You don’t know the local customs.  However, the first Sunday you are there, a festival happens in the streets.  The following Sunday, the festival repeats.  You ask yourself, “I wonder if every Sunday, there’s a festival in the streets?”  That’s inductive reasoning.

You use deductive reasoning to arrive at new facts from ones you already know.  For example, if the ticket price on an item is \$100 and the sales tax is 12%, you can deduce that you’ll be paying \$112 at the register.  In complex cases, such as a sudoku puzzle, the original few given facts lead the puzzler to new conclusions that lead, ultimately, to the solution.

Note that with inductive reasoning, you develop a hunch that may still be wrong.  With deductive reasoning, you are often funneled to one right conclusion.

Scientists who do experiments often begin by using inductive reasoning.  Deductive reasoning is the one a detective might more likely use.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Story Elements: Setting vs Character

During English tutoring, elements of story creation are sometimes explored.  Among them are plot, setting, character, symbol, and irony.

When I was in English 12, I was taught that great writers don’t develop a story by plot, but rather, by character.  The real story is how the main character adapts in the face of a conflict for which he or she is unprepared.

In fact, there is another angle of story development that gets less attention:  the setting.  A writer who focuses on the setting develops it so carefully that the events of the story are inevitable.

In every day life we see that people believe in setting.  For example: suppose, at a family function, there are two people you know don’t get along.  You try to keep them separate, so they won’t fight.  Therefore, you are trying to control the setting in order to control the outcome.

In studies of fiction, setting gets less attention than character.  However, short story writers tend to use setting more than novel writers do, since it’s hard for a character to adapt very much in the duration of a short story.

One time I picked up a book that just contained short stories about voodoo.  The setting was always some isolated place in tropical America, with lots of machetes lying around to cut cane with.  In the stifling afternoon heat, time would pass by slow drip.  Throw in a witch doctor and someone with a score to settle, and you have the inevitability of a gripping voodoo tale.  It was a great book.

I’m a setting writer.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Can an Endless String of Numbers have a Sum?

If you tutor math 12, you’ll have come across the premise

1 + 1/2 + 1/4 + 1/8 +….

Before we go any further, a reminder on how to multiply fractions:

(2/3) x (4/5) = 8/15.  You just multiply the two top numbers and put that number on top.  Then you multiply the two bottom ones and put that number on the bottom.

Now, back to 1 + 1/2 + 1/4 + 1/8 + 1/16+….(the terms continue forever)

To generate the next term, you just multiply the previous one by, in this case, 1/2.

(1/4) x (1/2) = 1/8

1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 +….

In spite of the infinite number of terms, the answer is 2.

1 + 1/2 + 1/4 + 1/8 + 1/16 +1/32 + 1/64 + (the terms go on forever)….= 2.

How can there be a sum of an unending number of terms?  Well, there is a proof for it – but that doesn’t mean you have to believe it.  Perhaps the best answer is that from a mathematical point of view, you can add up an infinite number of terms.  However, in this context, two things have to be true:

1) You must always multiply the current term by the same factor to get the next one.

2) That multiplying factor must be less than 1.

So there isn’t a sum to 1+1/2+1/3+1/4 + (the terms continue forever)….because the next number is not a constant multiple of the one before.

There isn’t a sum to 1 + 1 + 1 + (the terms continue forever)….because we’re multiplying the current term by exactly 1 to get the next term.  We must multiply by the same number each time, but it must also be less than one.

There is a sum to 1 + 2/3 + 4/9 + 8/27 +….(infinite terms).  Note that we are always multiplying by 2/3 to get the next term.  The sum is 3.

The formula for the sum is

S=a/(1-r), where

a is the first term

r is the multiplying factor to get the next term (called the common ratio)

If you use a calculator, be sure to put the brackets around the denominator as shown in the formula.

Thanks for stopping by.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Julius Caesar: Cassius, V,iii

Caesar, thou art avenged, even with the sword that kill’d thee.

Cassius, Act V, scene iii, Julius Caesar

In Act V, scene iii, of Julius Caesar, Cassius commits suicide.  Having been misled that his dear friend Titinius is slain, Cassius decides he must join him.

For almost two years, I wondered why Cassius had decided to die at that moment.  He may have loved Titinius, but Cassius was a soldier; he’d surely lost dear friends before.  I read Julius Caesar in grade ten and loved it.  By December of grade twelve, though, I was still asking myself why Cassius gave up when he did.

However, I had a resource in December, 1987, that I’d lacked when I’d first read Julius Caesar:  a waiter with whom I worked, named Tim.  I assisted him some nights at a pretty fancy place.  One night, Julius Caesar just happened to come up somehow, so I asked Tim why Cassius had thrown in the towel so early in the battle.

“Simple:  guilt,” Tim replied.  “Cassius knew he shouldn’t have led the conspiracy to kill Caesar.  He regretted it.  The guilt was too much for him.”

At once, the question was resolved.  It’s remained so since December, 1987.  Of all the answers I’ve received my whole life, that one from Tim remains possibly the most satisfying.

Thanks for stopping by.  Please come again.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

The Semicolon

Welcome back.

While tutoring English, punctuation is a constant concern.  Using a variety of punctuation leads to sentence variety – something that always impresses English teachers.

The semicolon can be used in many ways, but its simplest use is to join two short, but complete sentences.

a)  The weather turned to rain.  They canceled the game.

can become

b)  The weather turned to rain; they canceled the game.

The construction in b) brings the writing to a much higher level.  Note that even though the semicolon joins two complete sentences, the resulting sentence can still be pleasantly short and inviting to read.

Don’t overuse semicolons.  A few times a page is enough.

Happy writing. See you next time.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Divisibility by 3, Divisibility by 9

During math tutoring, no matter what the level, the front line is arithmetic.

In that spirit, here’s a handy trick to tell if a number is divisible by 3:

Add up the digits.  If the sum is divisible by 3, then so is the original number.

Example:  consider the number 222.  2+2+2=6.  Since 6 is divisible by 3, so is 222.  (Many people might not expect that.)  In fact, 222 ÷ 3 = 74.  (Check for yourself.)

A similar law works for 9:  if the digits add up to something divisible by 9, then the original number is also divisible by 9.

Example:  consider 414.  4+1+4 = 9, so 414 is divisible by 9.  So is 864, since 8+6+4=18, and 18 is of course divisible by 9.

Of all the math tricks I know, these are two of the most useful.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

French: passé composé, imparfait, and plus-que-parfait: when to use them.

Hello:

Well, today I thought we’d discuss some French:  specifically, the passé composé, the imparfait, and the plus-que-parfait.  In what situation do you use each?

The passé composé is the first way to express the past tense that I learned in high school.  It is the French equivalent to the English verb with -ed (eg., I walked).  The passé composé has two parts:  the auxiliary, followed by the past participle.

The imparfait expresses, as my French teachers always explained, “a state of things that had no particular beginning.  It may not yet be finished.”  For instance, “When I was young ….”  Additionally, “It was raining….”  Both use the imparfait tense.  In French, it’s among the easiest constructions, consisting of a stem with a subject-specific ending.

The plus-que-parfait expresses a completed action that happened before another completed action.  Consider  the sentence:  “I had finished the laundry when you called.”  “I had finished” is the plus-que-parfait tense, whereas “you called” is the passé composé.

Well, there you have it:  the passé composé, the imparfait, and the plus-que-parfait.  My wife (who is French and does our French tutoring) explained them to me.

Have a great day.  Come again.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.