The first thing to know about CRAM Math is that algebra is a language.
- It is important to read these sentences for uderstanding, even if you have to read them out loud. In fact reading them out loud forces your brain to be more efficient in processing the information because it takes more neurons to read and speak what is written than it does to just look at the words.
- Take your time. Don't rush through this, or you will end up re-reading this stuff later for what you didn't get the first time around.
If algebra weren't a language...
- ...then it would make no sense even attempting to solve a word problem.
Now that you know...
There are some quick and easy conclusions that come from knowing that algebra is a language.
- Algebra is a subset of spoken or written English (or most other human languages you might happen to speak, but since this is written in English, that is the most likely language this website will refer to.)
- Algebra is not as expressive or flexible as ordinary English when it comes to emotions, aesthetics, or other concepts that are difficult or impossible to communicate with numbers.
- Algebra is composed of algebraic statements called equations or inequalities. These algebraic statements have a truth value of being either true or false. Generally, algebraic statements are true in most homework problems, quizzes or examinations.
- The specific type of subset that algebra is in relation to ordinary English is that algebra is a formal language of the class phrase-structure languages that are exactly the Turing-acceptable languages.
For now, all this should mean to any student is that if you can read, write, and think in English, then you already have the intellectual skills to lean and master algebra as well.
- Notice the containment: Algebra is fully contained in ordinary English. Every algebraic term is derived from concepts used in ordinary educated speech.
- Reading, writing, and thinking only in terms of short, non-specific statements ("That's hot.") does not qualify as ordinary educated English.
- For the clueful, there was a survey done many years ago by the Harvard School of Business that showed vocabulary to be the number one indicator of executive capacity. Put simply, people who sling the biggest words around tend to have the greatest responsibility and the best paydays. If you can't use words having more than two syllables, it is likely that the only people who pay attention to you are your drinking buddies whenever you are buying.
- Containment becomes more important as the semester wears on. Containing your thoughts in any algebra problem to only the methods used to solve that class of problems prevents thrashing by looking for answers in mental places where they cannot be found. The solving of algebra problems is a discipline.
- Algebra is a formal language.
- Nothing in algebra is outside the scope of spoken or written English used by ordinarily educated people.
"She made ten dollars per hour last week working full time." So how much did she make before taxes?
- Let p be her pay for the week.
- We know the wage in dollars per hour is 10.
- Assume that a full work week in terms of hours is 40.
- Pay for the week is equal to the product of the hours worked and the employee's wage per hour. (This is a standard statement of this class of work-for-pay problems.)
- p = 40(10)
- p = 400
She made $400 last week before taxes and other deductions.
As a language, algebra has usefulness in describing situations in the real world that can be quantified or expressed as numbers (see Practice above).
- Virtually every computer language in existence is an implementation of some form of algebra. This means that if you can use algebra as a description of reality, then you can generally make a computer program out of your model to run as a simulation.
- This is a rendition of the Practice above in an old version of the Beginner's All-purpose Symbolic Instruction Code (BASIC computer language) and should produce the output "400" on any Apple II emulator:
- 100 LET H = 40
- 110 LET W = 10
- 120 LET P = H * W
- 130 PRINT P
- 999 END
If you learned the above here instead of going to the typical algebra tutoring website, you probably just saved yourself $35 for the last hour of your time. Celebrate your savings in an appropriate manner.