Math: Trigonometry: The Tangent Ratio

As a math tutor, you explain the tangent ratio a few times a year.

Trigonometry involves finding unknown sides and angles of triangles.  At first, it only involves “right” triangles – that is, ones that contain a 90º angle.

At beginner’s level, there are three trigonometric functions: sin, cos, and tan.  (Of course, tan is short for tangent.)  Note their presence on any scientific calculator.  By the way:  in most cases, if a calculator has sin, cos, and tan keys, it’s probably got all you need for high school.

Understanding sin, cos, and tan means understanding how the sides of a triangle are named.

The hypotenuse is always the longest side.

The remaining two sides are called the legs.  The leg touching the angle of interest is called the adjacent side; the other leg is the opposite.

Note that the following diagram, like most diagrams in trig, is not to scale.

The capital letters refer to angles A, B, and C.  If A is the angle of interest, then the adjacent side is 11, and the opposite is 13.  If, on the other hand, B is the angle of interest, then the adjacent side is 13, while the opposite is 11.

The definition of tan is as follows:

(1)   \begin{equation*}tan=\frac{opposite}{adjacent}\end{equation*}

Therefore, in the diagram above,

(2)   \begin{equation*}tanA=\frac{13}{11}\end{equation*}

Here’s where we get practical:  if you know the angle of interest, then your calculator knows its tan ratio.  For instance, tan32º=0.625, rounded to three decimal places.  (Make sure your calculator is set to degrees.)

Let’s use the tangent ratio (known affectionately as tan) to solve a height question:

Problem:

When the sun is at 40º elevation, a tree casts a shadow 13m long.  How high is the tree?

Solution:

First, we draw a diagram:

Note that the box in the corner means 90º.

Looking at the diagram, we see that relative to the 40º angle, the height, h, is the opposite side.  13m is the adjacent side.  Remembering that

(3)   \begin{equation*}tan=\frac{opposite}{adjacent}\end{equation*}

it follows that, in our case,

Of course,

So then

Using the method of cross-multiplication described previously in this post, we proceed:

so that we have

h(1)=13(tan40º)

h=10.9m

Apparently the tree is 10.9m high.

Hope this gets you on the way to calculating those heights that seemed out of reach until now:)

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

Biology: Arteries vs Veins

When you tutor Biology 12, you cover the circulatory system.

Pretty much everyone knows arteries and veins are different.  However, we’ll focus on their similarities first:

1)  They both contain a reinforcing layer of smooth muscle.

2)  For both veins and arteries, the central opening which conducts the blood is called the lumen.

Now the differences between them:

1)  Arteries carry blood away from the heart, whereas veins conduct blood back to the heart.

2)  Arteries carry blood under pressure, whereas the blood pressure in veins is minimal.

3)  Arteries have thicker walls than veins.  The reason:  since the blood in arteries is under pressure, the reinforcing layer of muscle in an artery is much thicker than in a vein.

4) Veins have valves, whereas arteries don’t.  Veins need the valves to prevent backflow; remember, the blood in them is under minimal pressure.

5)  Arteries are, for the most part, buried deep in the body, while veins are commonly visible through the skin.  (The wrist, where you take your pulse, is an exception: at that location an artery is close to the surface.)

Although some of the facts above are familiar, there might be a few surprises:)

Sources:

Biology 12, Module 3:  Human Biology I.  2007:  Open School BC.

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

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

Biology: Your friend bile

Tutoring Biology 12, you cover human digestion – which mentions bile.

Bile is an oft-referred-to secretion:  in literature, it’s related to anger.  Specifically, if someone’s “bile is up”, they’re in a bad mood.  I don’t understand the association, but there it is.

Bile is a greenish fluid made by the liver but stored in the gall bladder.  It’s made, partly, from worn-out red blood cells.

In the digestive system, bile emulsifies fats – which means it separates large fat droplets into many more small ones.  Once the fat is spread out in tiny droplets, it can be digested by the enzyme lipase.

Via the bile duct, bile is released in the duodenum – the lead section of the small intestine – so as to mix with the food passing through.

Hope this helps:)

Sources:

Biology 12, Module 3:  Human Biology I.  2007:  Open School BC.

Inquiry Into Life, Eleventh Edition, by Sylvia S. Mader.  2006:  McGraw-Hill.

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

Biology: Carbohydrates

When you tutor Biology 12, which is needed for nursing, you need to define sugars and carbohydrates.

Put simply, a carbohydrate is a compound consisting only of carbon, hydrogen, and oxygen.  The ratio between the three is roughly 1:2:1.  “Hydrate” suggests water (H2O): note that in carbohydrates, the same 2:1 ratio exists between hydrogen and oxygen. Sugar, starch, and glycogen are all carbohydrates.

In biology 12, sugars are either monosaccharides or disaccharides.  A monosaccharide is a simple sugar. Technically, it can have three to seven carbon atoms.  However, in Bi-12, we mainly think of glucose (6 carbons), fructose (6 C), galactose (found in milk, 6 C as well), or ribose (5 C).  All are single-ring structures.

A disaccharide is two monosaccharides fused together; hence, it’s a two-ring structure. Sucrose is an example:  it comes from the union of glucose and fructose.  Bond two glucoses together and you get maltose. Lactose is glucose plus galactose.

If you bond many monosaccharides together, you get a polysaccharide.  Three instances of polysaccharides are starch, glycogen, and cellulose.  All are polymers of glucose molecules – meaning that they consist of large numbers of glucose molecules strung together.  (Glucose is the monomer, whereas starch, for example, is the polymer.)  Starch is the molecule that plants use to store glucose; glycogen is what animals use.  In cellulose, the glucose molecules are joined so as to be indigestible; cellulose gives plants their erect, rigid structure.

There’s the “skinny” on carbohydrates:)

Sources:

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

Biology 12, Module I: Cell Biology I.  Open School BC: 2007.

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

Punctuation: Everyday use of the Colon

When you tutor English, punctuation is a constant concern.  Appropriate use of the colon can add a nice touch to a writing assignment.

My wife has been questioning my use of colons for months now:  “Don’t you need to follow a colon with a capital letter?” she asks.

Well, according to the McGraw-Hill Handbook of English (1986), you don’t.  A sentence like the following is perfectly acceptable:

My one goal this year is simple:  to fit into my size-32 jeans.

The first writer I ever noticed using colons in the middle of sentences was Charles Dickens.  I don’t remember his following them with capital letters.

The colon’s purpose is to call the writer’s attention to what follows.  You needn’t ever use a colon in the middle of a sentence, of course.    Some writers don’t write in a style that suggests such use of colons.  However, it helps to “change things up” in a longer piece of writing.

Good luck, if you decide to experiment:)

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

Quadratic Functions: Finding the Vertex from Vertex Form

Tutoring math 11 – which is needed for nursing, among other careers – you’ll need to explain how to identify the vertex of a quadratic function.

Vertex form is designed to easily yield the vertex of a quadratic function.  A quadratic function of the form

y=a(x-p) ² + q

has vertex at (p,q).

Example 1:  Find the vertex of y=-3(x-4)² +9

Solution: the vertex is at (4,9).

Notice the (“opposite, same”) pattern: the x-coordinate is opposite to what you see in the brackets, whereas the y-coordinate is the same as what you see added (or subtracted) at the end.

Example 2: Find the vertex of y=2(x+5)² -3

Solution: the vertex is at (-5,-3).

Notice that the number multiplying in front of the brackets does not affect the vertex.

Example 3: Find the vertex of y=(x-5)²

Solution: Remembering the form y=a(x-p)² +q, we need to discern the values of p and q. Clearly, p=5. q=0, because

y=(x-5)²

can also be written as

y=(x-5)² + 0.

Therefore, the vertex is at (5,0).

Example 4: Find the vertex of y=3x² + 7

Solution: Going back to y=a(x-p) ² + q, we realize that although q=7, we seem to be missing p. However, we can rewrite our equation as y=3(x-0)² + 7. Now, we realize that p=0. The vertex is at (0,7).

Identifying the vertex can be tricky, but I hope this helps.

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

Equation Solving: the legendary “cross-multiplying” method

As a math tutor, you’re bound to teach this quick method for solving proportions.

When you encounter

what do you do? Well, the easiest approach is probably what many call “cross multiplying”. I learned it in grade 7.

Cross multiplying argues that when two fractions are equivalent, their diagonals multiply to the the same amount. For instance, consider

Notice that the diagonals both mutliply to 10:

We can use the handy principle that equal fractions have equal diagonals to solve virtually any rate or proportion question – including percents.
Consider, again, our first problem:

We now know that, since the left fraction equals the right fraction, the diagonals must multiply to be the same:

Next, we divide both sides by 12:

We arrive at

We’ll be looking at some applications of this very old – and extremely useful – method in future posts.

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

Literary Terms: What is a Foil?

Tutoring English at the college level, you might be asked to explain the term foil.

There are hundreds of literary terms.  Most of them you never hear in everyday conversation, while many others are seldom used.  In the hallways of universities, however, literary terms are important.

Although most people never discuss them, literary terms can be very interesting.  Today we’ll define and discuss the term foil.

To a jeweler, a foil suggests a thin sheet of metal used as a backing for a gem.  The reflective quality of the foil brightens the gem from behind, enhancing its brilliance.

In literature, a foil is a character who, viewed close to another, emphasizes certain traits of the other character.  The most compelling cases that come to my mind are ones of contrast:  by noticing how good or innocent the foil is, we can’t help but realize how bad or manipulative the more central character is. Of course, the pattern can work the opposite way as well.

In Cinderella, the stepsisters can be seen as foils.  Never working at all, they live in complete self-indulgence.  By contrast, Cinderella works tirelessly to take care of them, never receiving any consideration.  The stepsisters’ selfishness and laziness emphasize how selfless and hardworking Cinderella is.

Another example of a foil might be Banquo from Shakespeare’s MacBeth.  After MacBeth and Banquo encounter the witches, Banquo wants to discuss what they said, but MacBeth himself seems less interested. MacBeth suggests that giving credence to supernatural predictions is unwise.  He seeks to give Banquo the impression that really, what the witches said is not important to him.

In fact, MacBeth murders the King in order to realize one of the witches’ predictions. He goes on to plot the murder of Banquo and Fleance, Banquo’s son, in order to thwart another of the witches’ divinations.

Banquo’s innocent wonder and curiosity about the witches’ predictions, contrasted with MacBeth’s pretended dismissal of them, show us how profoundly the witches affect MacBeth from the very beginning.  In that way, Banquo serves as a foil for MacBeth.

Of course, foil is also a math term.  You can find out about that meaning – among other topics – by searching my math category on the right side of this page:)

Source:  Literary Terms, Coles Notes Study Guide.

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

Why January can be the hardest month – and how to handle it

Tutoring, you notice which months are tough on students.  There are also your own recollections….

This year, Christmas came a few days after the public school students left for holidays.  As a result, their days off stretched longer at the back end.  Between Christmas and January 7, you get used to a new way of life that doesn’t include going to school.  It can be a hard habit to break.

Remembering my university days, January was always the toughest month for me.  The weather was grey and dismal.  It was so hard to face the cold, grey campus after the festive time of Christmas.  The courses were new, so you weren’t yet engaged with them.  Bottom line:  too many changes at once, against a dismal backdrop.

I found that when February came, I usually felt much better.  The weather was much better by then (of course we’re talking about Victoria, but up here is not too different).  As well, I’d developed some attachment to my courses.  Indeed, February was a much easier month.  Momentum carried me through March – in spite of its avalanche of new material.

Life is about habits.  Good or bad, pleasant or unpleasant, you’ll likely continue a habit.

Here, then, are some of my hints about weathering the month of January:

  1. Don’t expect much from any given day – but go to class every day, anyway.
  2. Remember that 45 minutes of homework is better than nothing, even if it’s not enough.
  3. Try, if possible, to focus on the main idea of what the instructor is discussing.  If you need to throw something away, cut out details.
  4. The days you really don’t want to go to class – but you go anyway – are the most important ones.
  5. Remember:  Everyone else is in the same boat.

Years ago, I recall seeing a student handbook on the ground.  When I picked it up, it opened to a page that showed the principal and vice principal pointing out at the reader.  They were smiling.  The caption read “Remember:  every day counts.”

Those two people understood about habits – and that attending school needs to be one of them.

Good luck this January.  I’ve been there and I know it’s tough.

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

English: Your old friends who and whom

Tutoring English, you’re bound to be asked when to use who vs when to use whom.

“Whom do you love?”  asked George Thorogood.  Actually, I think he asked “Who do you love?”  However, he should have used whom.

Whom is for object, whereas who is for subject.  Therefore, the following are correct:

Who loves you?

Whom do you love?

With whomever you discuss who and whom, remember the distinction:  whom for object, who for subject.  However, in a complicated sentence, the object of one verb can be the subject of another.  If it’s a subject, use who – even if it’s also an object of another verb or preposition.

Give the copies to whomever you like.

Give the copies to whoever asks for them.

In the second sentence, the “who” asks for the copies, making that person the subject of “asks”.  Therefore, use “who” rather than “whom”.

How do you tell object and subject apart?  With verbs, the subject does the action, while the object receives the action.  With prepositions, the object follows the preposition.

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