Math: The Statement and the Contrapositive

For a few years, this topic fell from view.  As a math tutor, I’m glad to see it back.

To a person studying logic, the statement “p implies q” also means “if p, then q”.  It can also be written

p→q

Example of a statement:

If a minute has passed, then sixty seconds have passed.

p and q, by themselves, might be called assertions.  Therefore, in the above statement, “a minute has passed” is an assertion.  So is “sixty seconds have passed.”

To form the contrapositive of a statement, you reverse its order, then negate both parts of the statement:

If sixty seconds have not passed, then a minute has not passed.

In logic notation, you negate an assertion by writing a line above it:


It follows that the construction of the contrapositive is

I’m told that, in general terms, the contrapositive is the logical equivalent to the statement itself. From what I’ve seen myself, I’ve no cause to doubt that assertion:)

There are other logical derivatives of a statement: the converse and the inverse, to name a couple. I’ll discuss them in future posts:)

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

Math: the meaning of a negative exponent

As a math tutor, you’ll need to remind students about this exponent law.

Most exponent laws people find pretty straightforward.  I’ll likely cover them in a future post. However, this particular one deserves its own; most people just don’t like it.  Let’s discover it’s really not so bad.

Rule:

Example:

Notice the fraction version:

Example:

A consequence of this rule is that a negative power, if on the bottom, can be moved to the top and made positive:

Example:

The rule needs to be followed literally. Like most rules in math, it often doesn’t lead to the final answer. Rather, it normally occurs as a step on the way to the final answer. Apply it exactly, then proceed!

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

Math: Evens and Odds

What’s the difference between evens and odds?  When you’re a math tutor, you might need more than the obvious answer.

Everyone knows that 0, 2, 4, 6….are even, whereas 1, 3, 5, 7, 9….are odd.  Negative numbers can also be even or odd:  -8 is even, whereas -7 is odd.  Formally, the mathematical definition of “even” is as follows:

2p, p is any integer.   The integers are  {….-3,-2,-1,0,1,2,3….}.

The definition of odds:

2q+1, q is any integer.

Therefore, 2(-11) = -22 is even.  On the other hand, 2(-8) + 1 = -15 is odd.

An even can’t divide (without a remainder) into an odd:  every even number has 2 as a factor, and 2 won’t divide into an odd number (by definition).

On the other hand, an odd can divide into an even.  3, for instance, divides into 12.

Here’s a fun fact:  the square of an odd is odd.

Proof:  assume the odd is 2t + 1.  Then its square is (2t + 1)2.  Multiplying by the foil method (see my post on foil here):

(2t + 1)2=(2t + 1)(2t + 1)=4t2 + 4t + 1.

Notice:

4t2+ 4t + 1 = 2(2t2 + 2t) + 1.

By definition, 2(2t2+ 2t) + 1 is an odd number:  it is written in the form 2(integer) + 1.

The nuances of even and odd can reveal some surprising discoveries, as we’ll see in future posts:)

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

Math: Fun with Calories

Tutoring math, you’re often asked about real-world uses of it.  Here’s an application we might all find useful now and again.

Recently it occurred to me to look up the calorie density of proteins, fats, and carbohydrates.  My reasoning was that peoples’ fear of fat must be motivated by a high calorie content.  Then again, I reflected, it’s sugary foods – desserts, for example –  that are often implicated as adding pounds.  Some people, though, suggest that red meat puts the weight on them.

Is there a particular culprit, or do the foods work in concert to fatten us up?  Well, courtesy of Wikipedia I can report the following calorie densities:

fats:                                                9 cal/g

carbohydrates (flour, sugar, etc):      4 cal/g

protein                                         :  4 cal/g

I decided to become my own calorie counter.  Selecting three foods, I read each food’s calorie count, then its grams of fat, carbohydrates, and protein.  Using the densities above, I calculated the food’s “theoretical” number of calories.  In each case it was spot on.

Food 1:

calories:  160 (reported on label)

fat:                     5g
carbohydrate:   27g
protein:              1g

To calculate the theoretical number of calories, we proceed as follows:

from fat:                     9cal/g x  5g =      45 cal
from carbohydrates:   4cal/g x 27g =    108 cal
from protein:              4cal/g x   1g =        4 cal
total calories:                                       157 cal

What do you know?  The difference between 157 and 160 – which is less than 2 % – is probably due to rounding.  For practical purposes, it’s an exact match.

Food 2:  the package said 5g of fat, 42g of carbohydrates, and 5g of protein.  It gave a calorie count of 230.  Here are my numbers:

from fat:                      9cal/g x 5g =     45 cal
from carbohydrates:    4cal/g x 42g =  168 cal
from protein:               4 cal/g x 5g =     20 cal
total calories:                                      233 cal

Once again, the package’s count is spot on.

Food 3:  The label says 8g fat, 3g carbohydrates, 4g protein, and 100 calories.  My calculation:

from fat:                     9cal/g x 8g =      72 cal
from carbohydrates:    4cal/g x 3g =     12 cal
from protein:               4cal/g x 4g =     16 cal
total calories:                                      100 cal

Our formulaic calorie count exactly matches the label.

You can use this fun method for predicting the calorie content of foods you make at home:)

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

Calculator tips: Entering scientific notation

Entering scientific notation on calculators is an important consideration.  When you tutor high school sciences, you’ll want to mention it.

If you’re a new arrival, you might want to read my previous post on scientific notation.  Assuming you’re good with it, we’ll continue.

Scientific calculators have specific keys you’re meant to use to enter numbers in scientific notation.  For best results, you should enter scientific notation the way that is intended for your model of calculator.

In front of me I have a Sharp, a Casio, and a Texas Instruments.  All are fairly plain scientifics that run between $10 and $20 last I checked.  By far most of the calculators I see students using are similar to one of these three.  However, I do see other makes occasionally that use different keys for scientific notation.

Example 1: Enter 7.29×10-3 on a Sharp EL-520W.

The Sharp calculators I’ve seen, including this one, use the Exp key for entering scientific notation:

7.29Exp-3 does it.  You’ll know you’ve entered it correctly because on the right hand side you’ll see “x10-03“.

Example 2:  Enter 7.29×10-3 on a Casio fx-260 Solar.

As much as I’ve seen, Casio also uses the Exp key for entering scientific notation.  Use the same key sequence as in Example 1.  With this model of Casio, you’ll see “-03” as a superscript.

Example 3:  Enter 7.29×10-3 on a Texas Instruments TI-30XA.

The Texas Instruments calculators I’ve seen use the EE key for scientific notation.  You will enter 7.29EE-3.  It will also accept 7.29EE3-.

Hope this helps!

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

Scientific Notation

Tutoring physics or chemistry, you need to explain scientific notation.

Scientific notation is very easy to use; it was designed to be.  To start with, we need to realize the “everyday” way we write numbers is called “float” (aka “normal”).

Another point to bear in mind is that scientists commonly space numbers in groups of three.  Therefore, 0.03445 might also be written as 0.034 45.  Similarly, 3467 might be written 3 467.

Example:  write 34 200 in scientific notation.

Solution: 3.4200×104

So we see that 34 200 is 3.4200×104 in scientific notation.

Example:  write 0.024 132 in scientific notation.

Solution: 2.413 2×10-2

The point to realize is that in scientific, you always write the decimal after the leftmost digit, then write x10p. The value of p is the number of places you need to move the decimal to return to its “normal” place. If you need to move the decimal to the left, p is negative.

Going from scientific back to float is easy as well; an example or two may help solidify the whole idea.

Example: write 3.24×10-5 in float.

Solution: The exponent tells us to move the decimal five jumps to the left.   It turns out the number is 0.000 032 4 in float.

Example: write 7.59×106 in float.

Solution: The exponent tells us to move the decimal six jumps to the right. We arrive at 7 590 000 in float notation.

Good luck with this new way of seeing numbers.

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

Math: Simple Interest and Compound Interest

Financial math gets more coverage in high school now.  As a math tutor, you need to explain the difference between simple and compound interest.

To discuss either type of interest, we need to define some variables:

A=the end amount:  the total value at time t

t=the elapsed time in years

r=the interest rate as a decimal (not a percent)

P=the principal amount (the amount of money deposited at the beginning)

I=the value of the interest earned

The simple interest on an investment is calculated as follows:

I=Prt

Of course, the total value includes the principal as well as the interest:

A=P + Prt

You can factor out P and get the other form:

A=P(1+rt)

Example 1:  calculate the value of a $5000 investment kept in the bank for 6 years at 3.2% simple interest.

Solution:  First, we note the value of each variable given:

A=what we have to find

t=6 years

r=0.032 (to get the decimal, divide the percent by 100).

P=$5000, which is the amount invested.

Plugging into the formula gives us

A=5000(1+0.032(6))

We simplify to arrive at

A=5960

So, if we put $5000 in an account that pays simple interest of 3.2% and leave it in there for six years, the balance will be $5960.

To explain compound interest, we need to define compounding.  In financial math, compounding means taking the interest earned and adding it to the principal.  Once that interest is added to the principal, it can earn interest as well.

Hence the difference between simple interest and compound interest:  with simple interest, only the original deposit can earn interest.  With compound interest, the interest itself can earn interest.

With compound interest, people usually find the total value at the end, A, rather than the interest itself.  Of course,

I=A-P

To calculate the end amount, A, using compound interest, you need to know how many times per year the interest is compounded.  For today’s post, we’ll start with the easiest case:  annual compounding.  Then our formula for the end amount, A, after time t is

A=P(1+r)t

Example 2:  Find the value of a $5000 investment kept in the bank for 6 years at 3.2% compounded annually.

Solution:

A=5000(1+0.032)6

From the calculator, we get

A=6040.16

So, if we leave $5000 in an account paying 3.2% compounded annually for six years, the balance at the end will be $6040.16.

Comparing Example 2 with Example 1, you see that with all else equal, the account paying compound interest grows faster than the one paying simple interest.  As time goes on, the difference gets more pronounced.  We’ll have more to say about that in a future post.

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

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.