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:

tan=opposite/adjacent

Therefore, in the diagram above,

tanA=13/11

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

tan=opposite/adjacent

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