Math: What is a Radian?

When you tutor math, you explain radians every semester to your grade 12 students.

Most people begin measuring angles in degrees.  However, you can also measure an angle in radians.  1 rad≈57.3°.

While degrees come from (I am told) Babylonia, or one of the ancient civilizations of that area, radians are a “natural” way to measure angles.  Behold:


In the above picture, CA is a radius. The arc from A to B is the same length as CA. Therefore, angle ACB is 1 radian. 1 radian is the angle that you traverse by following an arc the length of the radius. Said another way, it’s the angle subtended by an arc one radius long.

Recall that the circumference of the circle is 2πr, where r is the radius.  Since 2πr is the exact circumference, 2π radians is exactly 360°.

Radians can be referred to as rads, but are usually stated without any unit. That’s how you can tell which way the angle is measured:  if it’s in degrees, it will have a degree sign.  If it’s in rads, it won’t have any units.  Therefore, an angle of 54° means, of course, 54 degrees.  However, an angle of 32 means 32 rads.

Please keep enjoying this fine summer!

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

Practical math: Some easy conversions from metric to imperial

As a math tutor, you realize that conversions from metric to imperial are part of the grade 10 curriculum.  Let’s talk about a few that don’t need a calculator.

Even though the (Canadian) high school student grows up in a metric environment, the trades use both systems.  Moreover, the tutor likely grew up in the 70s, so still thinks as much in imperial as metric.

With a calculator, of course, you can easily convert any measurement to any other. Nowadays, you can just key a measurement into your browser and it will return the conversion.  In summer, however, such questions seem to arrive more often in everyday life – possibly when you’re not at your computer.

So, both for those in summer school, as well as those who might find these tricks useful in every day life, here are some simple conversions you can do in your head. While not exact (I think the temp conversion is), they get you within 2% of the answer.

kg to pounds:  double it, then add 10% of the answer.

example:  77 kg to pounds

step 1:  double the mass in kg:  77 times 2 = 154.

step 2:  add 10% more.  15.4 + 154= 169.4

So, 77kg is 169.4lbs.

metres to yards:  just add 10%.

55m is 55 + 5.5 or 60.5 yards.

inches to cm:  multiply by 5, then divide by 2.

4 inches = 5(4)÷2 = 10cm.

Fahrenheit to Celsius:

This conversion comes up a lot, but there is no convenient way without a calculator. You subtract 32 from the Fahrenheit, then divide by 1.8.

Example:  Convert 80F to C

step 1:  80-32=48

step 2:  48÷1.8=27 (rounded to the nearest whole degree).

So, 80F is 27C.

Here’s an irony about summer measurements:  According to Wikipedia, the Canadian football field is 110 yards, whereas the American is 100.  However, the Canadian football field is 100 m (since going from metres to yards you just add 10%).  So the American and Canadian are both 100 long in their own units.

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

Math: More about logs

Understanding logs is critical to anyone who needs to pass math 12 precalculus.  Your tutor will give you a new point of view about them.

Imagine you have this equation:

(1)   \begin{equation*}3^x=46\end{equation*}

The solution is

(2)   \begin{equation*}x=\frac{log46}{log3}\end{equation*}

Now let’s evaluate the decimal using a calculator:

(3)   \begin{equation*}\frac{log46}{log3}=3.4850\end{equation*}

(We have rounded to four decimal places.)

Next, let’s check our answer by plugging it into the original equation. In order to raise 3 to the exponent of 3.4850, you’ll likely use the yx key or else the ^ key.

(4)   \begin{equation*}3^{3.4850}=46.0010\end{equation*}

Close enough:)  Usually, during calculations, four decimal places is sufficient. This is true not only for logarithms, but trig as well.

With the pressure of final exams behind us, we will continue to provide light reading throughout the summer.  We tutor all year:)

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

English: Punctuation with Quotation Marks

As term-end essays soon come due, your English tutor mentions a couple of finer details….

When quotation marks enclose speech, you put the comma or period inside:

“Call them back,” he requested.

“I’ll never make that mistake again.”

You also put the other punctuation inside, if it is part of what’s said:

“I love your car!”

“When will I get to drive it?”

What about if you have quotes around a title or saying?  For periods and commas, you do the same:

When my daughter told me my car was “sick,” I didn’t realize she was complimenting it.

Reading Amy Tan’s “Rules of the Game,” I developed a new appreciation for the importance of rules.

Tonight she will finish James Baldwin’s “Sonny’s Blues.”

Note, however, that other punctuation (that is not part of the saying or title) goes outside:

Tonight, will you finish “Sonny’s Blues”?

I just found out my car is “sick”!

Quotes are used around the titles above because they are short stories; if they were novels, they would be underlined or italicized.

We wish you the best of luck with all your term-end efforts:)

Source:  TRU Open Learning Writer’s Style Guide.  Open Learning Agency, 2003.

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

English: Using conjunctive adverbs

When you tutor English, conjunctive adverbs come up – especially around due dates for important English compositions.

Following a semicolon, a conjunctive adverb can be used.  It leads to an attractive construction that can elevate your essay.

Their best pitcher was benched; nonetheless, they won the game.

I prefer pasta to meat; however, I really enjoyed your tacos.

In the above sentences, “nonetheless” and “however” are conjunctive adverbs.  In each case, the conjunctive adverb is placed after the semicolon.  The idea that follows – which must be a complete thought on its own – is rather surprising, given the idea that precedes the semicolon.

Conjunctive adverbs don’t have to lead to surprise.  Consider the following:

Put the cake in a preheated 350 degree oven; next, start the icing.

In the above sentence, “next” is a conjunctive adverb.

A conjunctive adverb is a word that links two ideas (hence, conjunctive), while describing a connection between the actions of each.  Often, the connection is irony: with however or nonetheless, the second idea seems surprising relative to the first. Likewise, the connection can be sequential – as with “next” – or cause-and-effect, as with “thus”.

Conjunctive adverbs can, of course, be used to begin sentences as well as after a semicolon.  Your English tutor encourages using them here and there in order to spice up your writing:)

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

English: Using they or their with a singular pronoun?

She-or-he and her-or-his are clumsy constructions.  Can you escape them?  The English tutor has looked it up to be sure….

Most writers face the situation commonly:

Leaving the shelter of the train, everyone put on ______ hat.

“Everyone” is, of course, singular.  If you knew the people were all women, you would love to say

Leaving the shelter of the train, everyone put on her hat.

What if the group is mixed – as usually it would be?

Leaving the shelter of the train, everyone put on her or his hat.

In today’s times, using “her or his” is the proper way.  Grammarwise, it’s correct because her or his, being singular, agrees with everyone.  Politically it’s correct, being gender-inclusive.  However, it complicates the sentence.

A common solution to the dilemma:

Leaving the shelter of the train, everyone put on their hats.

Can you actually get away with using their – which is plural – to refer to everyone, which is singular?  The answer depends on your context:  formal writing won’t let you. However, informal writing permits it.

In a world that seems increasingly informal, formal writing still has some strongholds.  An English professor likely won’t let you get away with using their in the situation we are discussing.

Here are some possible fixes that make formal writing a little more graceful:

Leaving the shelter of the train, everyone put on her/his hat.

Everybody realizes she/he needs to retrain.

Everybody realizes s/he needs to retrain.

Ask your professor what s/he will accept.  Remember:  when in doubt, go formal:)

Source:  McGraw-Hill Handbook of English, Fourth Canadian Edition, 1986.

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

Math: Factoring: Difference of Squares

The math tutor recommends a little light factoring on this beautiful Sunday morning….

Last post I discussed common factor, which we will be using in concert with difference of squares. Difference of squares factors x2 – 36 into (x + 6)(x – 6). By the foil method you can confirm:

F: x*x=x2

O: x*-6=-6x

I: 6*x=6x

L: 6*-6=-36

Displaying the terms in a row we get

x2 – 6x + 6x – 36 = x2 – 36

Example: Factor x2 – 49

Solution: We notice that the square root of 49 is 7. Therefore we write

x2 – 49 = (x + 7)(x – 7)

Difference of squares is easy to spot and factor if it’s plain. However, it may be “hidden” by a common factor:

Example: Factor 2x3 – 50x

Solution: We notice that 50x isn’t square rootable. However, we also notice that 2x can be taken out front as a common factor:

2x3 – 50x = 2x(x2 – 25)

Now, we’re getting somewhere: we follow with

2x(x2 – 25) = 2x(x + 5)(x – 5)

Removing the common factor of 2x allowed us to apply the difference of squares technique.

Have a nice day:)

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

Math: Factoring: Common Factor

As a math tutor, you notice the importance of this technique.

Factoring means breaking a number or expression into a product.  For instance, we’ll factor 45:

45=9×5

In earlier posts I’ve mentioned prime factorization:

45=3x3x5

Now we’ll look at factorization of polynomials using common factor.
Example: factor -2x6 + 8x5-12x2

Solution: With the common factor method, we look for the expression that divides into all the terms, then write it out front. What remains in the brackets is each term divided by the common factor.

In this case we notice that 2x2 divides into all the terms. Therefore, we “take it out front”. Actually, we take out -2x2 because whenever the lead term is negative, you take out the negative with the common factor. Inside the brackets we write each term divided by -2x2:

-2x2(x4 -4x3+6)

Common factoring doesn’t have to be as complicated as the example above.  Consider the following:

3x – 15 factors to 3(x – 5)

Working with polynomials, factoring is constantly used.  There are at least five factoring techniques, of which common factor is the first.  I’ll discuss the other techniques in future posts:)

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

Math: Direct Proportionality

As a math tutor, you’ll likely introduce this concept.  It’s used even more in physics and chemistry.

So often in my university science courses I’d read “the mass is directly proportional to the volume” or “the distance is directly proportional to the time”, etc.  Science people love direct proportionality because predicting the result of a given input is so easy.

Direct proportionality means that if you double the input, the output will also double.  If y is directly proportional to x, it follows that

y=kx

In the above equation, is called the constant of proportionality. Once you know k, you can find the result of any input.

Example:  The distance travelled by a long haul train is directly proportional to the time traveled.  The train travels 600 km in 9 hours.

a)  Find the equation to model this situation.
b)  How far will the train travel in 12 hours?

Solution:  y is always the “output”, while x is the “input”.  Some people like to use different letters in order to reflect the actual wording of the question.  In that case:

(1)   \begin{equation*}d=kt\end{equation*}

where, of course, d stands for distance, t for time.

To find k, we use the idea that the train travels 600 km in 9 hours:

(2)   \begin{equation*}600=k(9)\end{equation*}

Dividing both sides by 9, we get

(3)   \begin{equation*}\frac{600}{9}\ =k\end{equation*}

(4)   \begin{equation*}66.7=k\end{equation*}

We know now that the equation to model the train’s travel is

(5)   \begin{equation*}d=66.7t\end{equation*}

To predict the train’s distance over 12 hours, we simply put 12 in for t:

(6)   \begin{equation*}d=66.7(12)\end{equation*}

Therefore,

(7)   \begin{equation*}d=800\end{equation*}

We conclude that over 12 hours, the train will travel 800 km.

More will be said about direct proportionality in future posts:)

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

Biology: Energy and ATP

Tutoring biology, you need to be aware of the connection between ATP and energy.

In a factory or a mill, there likely is a power plant where fuel is burnt en masse. The energy is captured there (often in the form of electricity), then channeled to the other locations as needed.

The human body, though, releases and consumes energy in a different way.  Each cell receives fuel (glucose), burns it internally, then captures the released energy by using it to synthesize a high energy chemical bond.  That bond can be broken at will when energy is needed.

ADP is adenosine diphosphate (adenosine bonded to two phosphate groups). When the cell burns glucose (in its mitochondria), the energy released is used to bond another phosphate to the ADP, so it becomes ATP (adenosine triphosphate).

When the cell needs energy, it breaks an ATP into an ADP and a phosphate.  The breaking of that bond releases energy the cell can use to power any life process.  Later, when the cell burns more glucose, it will use the energy to bond the ADP and the phosphate back into ATP.

The burning of one glucose molecule produces 36 or 38 ATP.

Hope this helps:)

Source:  Inquiry into Life, Eleventh Edition, Sylvia S. Mader.  McGraw-Hill:  2006.

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