Math: the “fraction button”

Tutoring math, you’ll become aware of the fraction button.  As a math tutor, I don’t recommend using it.  Nonetheless, people commonly do….

Looking at a scientific calculator, you’ll likely see a button that looks like this: abc. It’s the fraction button.

If you enter 5abc7 + 1abc3 =, you’ll probably get 1r 1r 21. This means 1121. Now press shift (or 2nd F or just 2nd, whichever your calculator has) abc; you’ll see 22r21, which means 2221.

The fraction button saves you from having to get a common denominator to add or subtract fractions. Therefore, using it can save a lot of effort – especially for someone who’s weak at times tables.

My advice: don’t use it. In high school math, you’ll encounter algebraic fractions, which the scientific calculator can’t handle. You’ll have to do them “the old way”, by hand. Therefore, this math tutor recommends keeping in touch with how to do so.

The fraction button didn’t exist when I was in school. I watched, fascinated, as a student showed it to me in the late 90s. Normally, I don’t use it; however, even some study guides recommend its use now. In certain contexts, I guess using it could make sense.

Hope this helps:)

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

English: two common sentence faults

Tutoring English, you deal with these issues constantly.  In this post, the English tutor sheds light on a couple of common sentence faults.

You burnt the cake she is furious.      fused sentence

You burnt the cake, she is furious.     comma splice

Neither the fused sentence nor the comma splice is correct.  Both are found commonly in everyday writing.  The comma splice is even found in published writing.

The problem with both the fused sentence and the comma splice is that each produces a run-on sentence. A run-on sentence is incorrect.  A simple way to identify one is that its subject changes or is restated without a semicolon or conjunction in between. There are other punctuation fixes besides the semicolon, but a comma by itself won’t work.

You burnt the cake she is furious.           Subject change from you to she.
I went to Bill’s party, I had a great time.    I is restated.

Facing a fused sentence or a comma splice, the same fixes can be used.  One solution is to add a conjunction before the second subject; another is to use a semicolon.

You burnt the cake, so she is furious. The conjunction so fixes the sentence.
You burnt the cake; she is furious. The semicolon correctly separates the two
complete thoughts.
I went to Bill’s party and I had a great time. The conjunction and, probably overused in writing, corrects the sentence in this case.

Fused sentences and comma splices are best gotten rid of. Changing habits always seems easier at the beginning of the school year:)

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

Source: Harbrace Handbook for Canadians, sixth edition. 2003: Nelson Education Ltd.

Math: prime and relatively prime

Tutoring math, these definitions don’t come up often enough. The math tutor offers this brief read on them.

A factor of a number divides into it with no remainder.  For example, 5 is a factor of 15; 7 is a factor of 56.

A prime number has exactly two factors:  1 and itself.  Notice that 2 is prime, having only 1 and 2 as factors.  7 is also prime.  9 is not prime, since it has three factors:  1, 3, and 9 all divide into it with no remainder.  10 is not prime either.  1 is not prime, since it only has one factor:  1.

Numbers that are relatively prime don’t have to be prime, but they share no common factor except for 1.  12 and 5 are relatively prime, for instance.  10 and 15 are not relatively prime, since they have 5 as a common factor.

An application of relatively prime numbers is fraction reduction.  If the numerator (the top number) and the denominator (the bottom number) are relatively prime, the fraction is reduced.  Reduced might also be referred to as in lowest terms.

By that rationale, 2418 is not in lowest terms because 24 and 18 are not relatively prime.   In particular, they share the common factor 6.  Dividing 6 out of both, we get 43. Since 4 and 3 are relatively prime, the fraction is now reduced.

Hope this helps:)

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

English: Active vs Passive

With another school year upon us, we resume tutoring math, sciences, English, etc.  Tonight:  a post about English.

Active voice and passive voice are both easy to understand.  Behold:

Active:  Bob presented the speech.

Passive:  The speech was presented by Bob.

Since the passive voice gives a gentler rendering of an event, most people prefer it. Consider the following:

Active:  I broke your ruler.

Passive:  Your ruler got broken.

Which telling would you more likely expect?

While the passive voice is preferred by everyday people, the active voice is preferred by people who mark writing assignments.  Specifically, the active voice gives a livelier, more honest telling of an event.  Such is the opinion of most English academics nowadays.

Hope this helps.  By the way:  welcome back to school!  All the best this academic year from Oracle Tutoring by Jack and Diane, Campbell River, BC.

Source: Harbrace Handbook for Canadians, sixth edition. 2003: Nelson Education Ltd.

Math: hexadecimal numbers

Tutoring math, this is an interesting concept.  Its common application is in computer science.

You might have seen numbers like c6 or e9.  Depending on the context, these may be hexadecimal (aka hex) numbers.  The hexadecimal system is base 16.  It uses the following notation:

decimal hex
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
10 a
11 b
12 c
13 d
14 e
15 f

Since base 16 requires numbers that would be two digits in base ten, the number system is simply expanded to 15 using letters.

Continuing the logic from my previous post, we see that c6 means

12×161+6×160=12×16 + 6×1=192 + 6=198.

So, c6 in hex is 198 in decimal.

Since 162=256, you can represent any number from zero to 255 with two digits in hex. ff=255, as follows:


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

Math: Numbers in other bases

Tutoring math, you rarely hear of this now.  For computer science students, the math tutor gives a five-minute read on it.

In everyday life, the number system we use is base ten.

The number 4062 means

4×1000 + 0x100 + 6×10 + 2×1

More specifically, it means


Recall, of course, that x0=1, x being any real number.

So, what is the number 57 in base 2?

Well, the powers of 2 are as as follows:








and so on.

57, written as a combination of powers of 2, can be thought of as

1×32 + 1×16 + 1×8 + 0x4 + 0x2 + 1×1

or, in other words,

1×25 + 1×24+1×23+0x22+0x21+1×20

Therefore, we have

57=111001 in base 2.

Exercise: Verify that 129, when converted into base 5, is 1004.

Solution: 129=1×125 + 0x25 + 0x5 + 4×1. Of course, 125=53, 25=52, and so on.

In computer science, the hexadecimal system – aka, base 16 – is often used. We’ll take a look at that in a future post:)

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

Area: size of a pizza

As a math tutor, you notice these little surprises about everyday things.

We all know that, with a circle,


Years ago, I worked in a pizza place. A small was 10″ (diameter), while a large was 14″. Let’s compare the sizes.

First, we’ll agree that from an eating point of view, the size of a pizza is really its area, rather than its diameter.

For the small, the diameter being 10″, its radius is of course 5″. Its area is

A=Πr2=Π(5)2=Π(25)≈78.5 square inches

For the large, the diameter being 14″, its radius is 7″, so its area is

A=Πr2=Π(49)≈154 square inches

The ratio of the areas:


I think we can agree that 1.96≈2.

Therefore, the 14 inch pizza is really twice the size of the 10 inch. You can check what kind of deal you’re getting on the large:)

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

Math: interest vs inflation

Tutoring math, interest and inflation do come up, though not often enough.  To all my financial readers:  hang on to your hats!

When I earlier talked about inflation and interest here, I mentioned the rule of thumb that, generally, interest hovers around 3% above inflation. A saver might believe they are protected by that three percent spread.

Assuming interest stays three percent above inflation, you are (more or less) protected while both inflation and interest are low. However, witness the outcome of the following scenario:

Example 1: You have $1000 in a savings account. Inflation is high: 16%. Interest, following the 3 percent rule, hangs at 19%. Calculate your balance at the end of the year, adjusted for inflation.

First, the interest:

Converting 19% to a decimal, you get 0.19, which you then multiply by the principal of $1000:


Adding the interest at the end of the year, your $1000 becomes $1190.

Now, consider inflation:

To adjust for 16% inflation, your balance of $1190 is worth 84% of its nominal value. (100 – 16=84)

Convert 84% to its decimal of 0.84, then multiply it by your year end balance:


Adjusted for inflation, your $1000 has become $999.60: you’ve lost value.

Where did the lost value go? After all, you didn’t withdraw it.

The answer is that the borrowers got it. In an environment of high inflation, value is gradually transferred from lenders to borrowers. The reason is that the borrower pays back the lender with money that is worth less and less.

Through the 70s, many people’s mortgage payments remained basically constant while prices ballooned. Specifically, for the people who didn’t need to refinance, the payment got smaller and smaller relative to the paycheque. For those lucky people, inflation picked up a lot of the tab.

Of course, such high interest and inflation rates are very hard to imagine today. People likely felt the same in 1965….

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

English: What is a homonym?

As an English tutor, I’ll share a definition I learned in elementary school.  I had no idea it was controversial.

I remember my spelling text from grade three – yes, it was a textbook.  Each week had a new list of words.  However, the book went further:  it covered a new theme each week as well.

One magic chapter talked about homonyms.  Homonyms, it explained, are words that sound the same but are spelled differently; dear and deer are homonyms, for example.

What was my surprise to discover, 35 years later, that the Yanks don’t necessarily agree with that definition!  Leafing through Websters yesterday, I read that homonyms are spelled the same, but have different meanings.  An example is screen, in the following two contexts:

1)  We’ll screen the applicants carefully.
2)  At the airport, you watch the screen for the arrivals.

That’s homonyms from a Yank point of view.

I wondered if maybe I remembered the definition wrong, so I checked the Oxford Canadian Dictionary.  Its definition comprises both the one I remember and the Websters one.

By the way:  I love the -nym ending:)

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

Math: Expected Value

Tutoring math, you might be asked about statistics, in which expected value is an early topic.

In many everyday situations, the expected value is equal to the mean, aka the average.  The difference is more in definition than in practice.  Expected value involves probability, whereas mean involves fact.

Formally, the expected value is defined as follows:

Expected Value=Σoutcome_value*probability

The Σ symbol means “sum”.  In other words, the expected value takes each possible outcome and multiplies it by its probability.  Then, it adds all those products together.

Example 1: Give the expected value of rolling a fair six-sided die.

Solution:  We know that the probability of getting each result is 1/6.

The expected value, E, of the die roll is


Notice that the expected value is not necessarily a possible value. Not surprising, really: if you take the average height of ten people, you’ll likely arrive at a height that none of them is.

Example 2: Find your expected payoff in the following situation: 1000 people pay $1 each for a ticket. Then a number is drawn. The holder of the winning ticket gets $500.

Solution: The probability of winning is 1/1000, in which case you get $500. Really though, the payoff is only $499, since you had to spend $1 to buy the ticket. The probability of not winning is 999/1000, in which case you get nothing. Once again, you still had to pay $1 to lose, so the payoff is -$1. Applying our definition of expected value we get

E=(999/1000)(-1) +(1/1000)(499)=-0.999+0.499=-0.50

So, your expected payoff is -$0.50. It had to be a loss, since not all the money paid in was awarded.

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