There are lots of units. Recently the jelly donut became a unit of measure in nutritional calorie education. Understanding what a jelly donut is allows it to be used as an noun like any other unit of measure. One bakery's donut may differ from another, so it doen't belong to any exacting national or international standard, yet everyone knows what a jelly donut is.
5 jelly donuts per week = 5 × d ÷ week
( a variable )
5 jelly donut calories of rice per day = 5D × r ÷ day
( a unit of measure )
In choosing a birthday gift for a co-worker's 5 year old boy, the decision came down to a compass, a protractor, and a ruler in hopes that when the time came later in life the future leader of America would not be the one in class asking, "what's a centimeter?" The price was right.
Units of measurement are nouns. They are in the category of "a person, place or thing." A British Thermal Unit means something. It is the amount of heat needed to raise one pound of water 1° Fahrenheit. Physicists reading this might argue about this definition being real or not in comparison with units of force, mass, and temperature, but be assured that the BTU is most certainly a noun that is not going away because of its importance in several categories of use.
There are several units of measurement, or nouns, or groups of words that together act like nouns, that enter into math equations. A centimeter, an hour, a caret of diamond mean something. The units may not be spelled out, but omitted, implied as being in a formula because of an assumption that those solving the equation know which units of measure are paired with which adjectives, the numbers.
Constants are nouns. If the constant looks like an adjective, it's because the adjective in use is the number 1, which was omitted.
1 × pi
pi
This is why we capitalize constants in computer programs and start variable names with lower case letters.
$PI = 3.1415;
$d = <stdin>;
print "circumference = ";
print $d * $PI;
In algebra, variables are used to represent numbers. After pairing the numbers with units of measure the answers mean something. There are several kinds of numbers, or sets of numbers that are put to use when solving problems. Natural numbers, zero, negative numbers, rational numbers, real numbers, irrational numbers, imaginary numbers, primes, perfect squares, Pythagorean triples, arrays, matrices, continuous numbers, discreet numbers, sets, groups, samples, and so on.
Negative numbers are just as real as positive numbers are, as are the rational and irrational numbers in between the dots of dotted number lines of natural numbers. Imaginary numbers are off the line, but still exist. Several problems are worded from everyday experience to encourage some visualization of what the problem describes and hopes to solve. These examples include elevators, train stations, miles driven in a car or flown in an airplane, long walks back and forth between buildings, gambling chips, payroll, income taxes, slices of food, people waiting in lines, atomic particles, or just about anything that is measured in some sort of line going back and forward, up and down, or side to side. Applications are often chosen when they can be represented on a number line.
Take for instance the ordinary question of age: Joe was born sixteen and a half years ago. How many years old is Joe? 16 or 17 years old?
The correct answer depends on whether you are starting out your count with zero or not. If the rules say no zero, then you are born in your first year of life, then Joe is 17 years old. He is in his 17th year, no zero. If you start with zero, then Joe is 16 years old. He has 16 years. These are two different units of measure both called 'years.'
As for effort spent working through word problems, be aware of what is being dealt with. The solution should be as simple as possible within the confined world of numbers that you are working with. Also be careful that the tool you are using is compatible with what you need.
In my experience, I estimate pi as 3, a whole number. Don't do that when accuracy counts, like when being graded. I prefer whole numbers and use them in my estimates of of interest rates, return on equity, the price of oil, the price of wheat or whatever. I've had two instructors memorize the natural log base e to 20 or so decimal places. You could memorize pi to that accuracy as well. One of those two instructors mentioned in lecture that China had used 22 sevenths as a substitute for pi a very long time ago. 22/7 is a very accurate estimate for when a scientific calculator or calculating app is not available.
Tools may include rulers, calipers, calculators, computers, slides, grids, meters, magnets, springs, or any of a endless list of measuring and calculating devices. Learn to program math into a computer, both the easy way, a spreadsheet, and the hard way, a compiler. A spreadsheet will handle estimates into the thousands of trials, but when dealing in the millions of calculations, writing a short piece of code to iterate the algorithm is best, and unlike a spreadsheet, the computer program can work unattended.
Pay attention to how measurements crossover from one dimension to another. Learn how time and motion relate, how mass and energy relate, how production and labor relate. Money may relate to time, motion, mass, energy, labor or production. Notice that the hours of a clock display numbers 1 through 12. Think of the scale of hours having more to do with 12 inches in a foot than to do with 12 of something else. This is how time inches along.
Understanding units of measure and putting them to use in equations requires more diligence. Thinking through the problem without thinking about the units of the result may lead to a result in the wrong units. Label units.
Working through equations means making choices. Terms can be added, constants divided, new terms multiplied, and often the pieces are multiplied by the number 1 in its many forms. It is important to learn how to multiply by the number 1 by choosing the best numerator that equals the best denominator.
Al went twice and Bo went 315 meters for one block, an equation is created with units of measurement.
street block = 315 meters
dividing the terms left and right of the equal sign create a ratio equal to 1.
block/315 meters = 1
315 meters/block = 1
These ratios allow us the choice of answering the Al and Bo distance in blocks or in meters.
2 ( 1 block ) + ( 1 block ) = x blocks
Use the right ratio. There are two choices, and one choice doesn't make the equation any easier to solve
( 3 blocks )( 315 meters/blocks ) = 945 meters
and the wrong choice would be
( 3 blocks )( blocks/315 meters ) = ( 3 blocks² ) ÷ ( 315 meters)
Multiplying one term or the entire side equation by 1 does not have an effect on the equation. 5 + x = y remains equivalent to ( 5 + x × 1 ) × 1 = y.
Multiplying and dividing by 1 can take many forms. If two measurements are equal, then one measurement divided by the other equals 1. Their ratio equals 1. The top divided by the bottom can be multiplied into one side or the other of the equation.
Label the units when multiplying by 1.
3 feet ÷ 1 yard = 1 (labeled)
3 ÷ 1 = 3, not 1 (unlabeled)
When the two units of measure are the same measurement, then they can be used to form a ratio equal to 1 and be used to multiply into an equation to chase after an answer appropriate for the instructor or the person needing the answer to the problem.
A convenient way for the instructor to evaluate if the students understand the process is to assign questions that use common units of measure: Units of length used may by meters, yards, inches, kilometers, miles, etc. Time also has many units that can be put to use.
1 = 1 = 1 = 1 = 1
km/1000 m = yard/36 in = 16 oz/lb = 1000 ml/liter = gallon/8 pints
Because 1 equals any other 1, it is possible to mash up a string of equivalent measurements moving from length to area to force to mass to time to whatever. The simplification follows later when units in a numerator cancel out with like units in an adjacent denominator.
The details fall into place with the proper application of the number 1. The ratio of numerator to denominator equals 1, so choose the correct unit on top and the correct unit on the bottom of the ratio in order to simplify the equation.
How many ounces are there in a 5 gallon bucket? One would be hard pressed to know this trivia without some calculation.x = 5 gal × 4 qt/gal × 2 pt/qt × 8 oz/pt Notice that with proper placement of the top and bottom of each ratio that the "× gal" cancels out with the "÷ gal", the "× qt" cancels the "÷ qt", and the "× pt" cancels the "÷ pt", leaving:x = 5 × 4 × 2 × 8 oz
Word problems create labeling problems, so label the units.