During the process of learning more math, one progresses from something known to something sought. Apply real life experience to concept along the journey. Allow nouns and adjectives to color the world in more detail. Math can be of more use than to simply ount things in black and white. A quick literary usage off my shelf mentions that nouns are words that represent a person, place, or thing, including groups of words that may together function 'noun-like.' In other words, English is a good language to describe highly technical subjects because words can be taken outside of their usage from nouns to verbs to adjectives. Adverbs not so important in algebra. If talking poetry, romance, or diplomacy, several other non-English languages do a better work because adverbs become number one. The phrase "really feels like" doesn't go far in math.

An adjective can also be one or more adjective-like elements that modify the noun.

400 meters.
'meters' is the noun and '400' is the adjective

dozen donuts
'Donuts' is the noun

2 dozens
'Dozens' is the noun. Remember that nouns are made plural by adding an 's' or an 'es' and that adjectives don't change.

400 meter raceways
Adjectives don't change spelling much.
2 dozens of donuts
'two' not '2s' or 'twos'
Note that "of donuts" is a prepositional phrase, so "dozens of donuts" acts noun-like.

Try not to object to this way of thinking, as some amount of openness of thought will later allow you to deal in the abstract with concepts like:

3 volumes
n dimensions
2 limits
3 infinities
infinite dimensions
undefined limits
finite infinities

When dealing in abstract terms, keep the nouns and adjectives together on through to your solution. The solution will be as abstract as the equations, yet there may be a firm solution. The theorem may have a proof. Proving there does not exist any solution is also an answer.

Prepositional phrases make more sense of the matter. Lengthy problems may be a string of short problems. With knowledge of the context, a large problem can be screened quickly to eliminate extra information already committed to memory, already regarded as common knowledge.

A familiarity of basics and atomic weights simplifies a chemistry question. The following question would be viewed by the chemistry student as "2 hydrogen gas + 2 oxygen gas = water + 1 oxygen gas:"

If a mole of any gas at standard temperature and pressure has a volume of 22.4 liters, how many moles of oxygen gas are left over when 2 moles of hydrogen gas combines with 2 moles of oxygen? What is the volume in liters of the left over oxygen gas? If the molar mass of water is 18 grams per mole, and liquid water has a density of 1 gram per cubic centimeter ( 1 cubic centimeter is 1 milliliter), how many milliliters of liquid water are formed from the combination of the hydrogen gas with the oxygen gas?


How many grams sodium in two moles of table salt?
The chemistry student familiar with salt would read the question in its short form as:
x = 23 × 2
leaving out the units entirely out of familiarity with NaCl.

Nouns and adjectives stay together and are not separated by algebraic laws like the associative, symmetric, and distributive ones.

ab = ba
( 4 cats )( 3 dogs ) does not mean ( 3 cats )( 4 dogs )
( 4 cats )( 3 dogs ) does mean ( 3 dogs )( 4 cats )
a(b+c) = ab + ac
2 ( 5 kilometers ) does not mean 2 ( 5 ) + 2 kilometers, or 12 kilometers.

Take the distributive matter in reverse.

ba + ca = (b + c)a
2 meters + 4 meters does mean ( 2 + 4 ) meters

This is of particular help when dealing with fractions. Keep in mind that halves, thirds, quarters, and fifths are often the nouns, not the adjectives.

1 half + 3 quarters = x

or the more abstract:

1/x + 2/x³ = 1

One of the more helpful tools to demonstrate what a variable is can be found on most computers. It is the spreadsheet. Put the variable number 5 into the cell in row 1, column 1, and put the equation formula into the row 1, column 2 cell.

x, or '5' is the variable, and
= 2 × x

is the equation.

When you change the value in row 1, column 1, then the value in column 2 changes because the computer is receiving a new value from input for the variable x in the cell's formula.

What is the variable 'x'? It is the unknown starting point, a value that determines the solution. It could be the number of widgets (whatever a widget is). It could be initial temperature, starting time, beginning dollars, initial population, or anything measurable that is represented by a math variable. How about 'y'?

The letter y follows the letter x in the alphabet, so in math let the variable 'y' be dependent on something done to the letter 'x.'

y is dependent on x
y is a function of x
y is a f(x)

or for the advanced student:

z is a f(x,y)

or for the really advanced student:

y is a transformation dependent on how x is transformed
y' is f'(x)

now replace "is a" with "equals"

y = f(x)
z = f(x,y)
y' = f'(x)

Explaining the range of x takes only one line of space on the page of notes, noting that the range would be from 1 to 2 for example, or from -5 to 5, or from (- infinity, + infinity). The problem would say that x is always going to be in that range, so no need to worry or address cases dealing with an 'x' outside the specified range.

x is continuous over ( ½, 2 )
x exists for ( -5, 5 )

Then comes the concept of domain(s), which never seem as easy to visualize. The domain seems to be another range of x, except using that answer would be marked wrong. So domain falls into the category of "a range by any other name is a domain." Another way of describing a domain becomes, "if it looks like a duck, sounds like a duck, and walks like a duck, then it is a domain for all ranges that are ducks."

At least it can be understood that the domain depended on the range.

What is the domain of x when x is between 0 and 5?

Graphing domains becomes a chore, taking up half the page. Graphing 1/x takes more than half the page, so the graph gets chopped with a few arrows pointing out into space to show what the domain of 1/x is.

f(x) has a domain. Think of the number of ducks as variable x. 'x' could stand for 1 duck (it could stand for 0 ducks), a flock of ducks, or many flocks of ducks. A function of x, 'f(x)', is the math description of something happening to the ducks, something dependent on 'x.'

f(x) = food consumed by ducks
f(x) = ducks born from original ducks
f(x) = miles flown by ducks
f(x) = something else that happens to ducks

From math class to math class, problems deal with more than life in black and white, or 'x' and 'y.'

While the angle x is fine in geometry, it doesn't do in trigonometry.

Equations start using the range of angle theta. Letters in the English alphabet are not enough.

Functions of multiple variables need multiple letters of the alphabet. 't' could be a variable of time, which is fine, but problems dealing with time need a partner variable, so 's' is used.

Sometimes understanding the time problem means several starts and stops, all independent variables, so 't0,' 't1,' and 't2' were put into use.

While 'y' continues to represent the domain of some independent variable, 'z' could take the place of 'y.'

z = f(x,y)

But then the end of the alphabet is reached. Move backwards to use more letters as appropriate.

y = f( u,v,w,x )

Occasionally letters actually made sense and stood for some recognizable word (variable).

d = distance, density, diameter, change (delta)
r = rate, ratio, return, radius
p = price, principal, profit
v = velocity, value, volume
t = time
s = seconds
h = height, hours
w = width, work

Note how many variables can be words that are 'noun-like'

'x' is a useful variable when nothing more specific seems to fit, and not many words in English begin with 'x.'

It may be important to note that legibility is important, both with one's own work to be reviews by others and when one is reviewing other's work. Zeros look like letter 'o's. Number '1's look like letter 'l's, and v's look like u's. In an ocean of small handwriting, bad scanning, bad copying from neglected machines that are low on toner, add dots and lines from dirty glass, or general errors in execution, it is important to prepare to have enough to see problems in the assigned problems. Don't wait until the deadline to make the first read. Before copying notes or a handout, go to the library staff and ask for the glass cleaner to use before you start up the machine.

Depending on what field one falls into, whether into one that uses paper, or one that disposes of it, additional standards of symbols fall into use as variables stand for something,