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.