Math/Writing
Math
Beneath all this writing is tables, tips, hints, and ictures to help you figure out math!
Math Links: https://www.math.com/ https://mathforum.org/dr.math/ https://math.usask.ca/help_centre/index.html
1 Fundamental Suggestions
Here are some hints on how to do basic math calculations.
These suggestions are intended to make your life easier. Some of them may seem like extra work, but really they cause you less work in the long run.
This is a preliminary document, i.e. a work in progress.
- 1. Be tidy and systematic. Whenever you have data in tables, especially when place-value is important, be sure to keep things lined up in neat columns and neat rows. See section 4 for an example of how this works in practice.
- 2. If your columns are wobbly, get a pad of graph paper and see if that helps. If you don’t have a supply of store-bought graph paper, you can make graph paper on your computer printer. There are freeware programs that do a very nice job of this.
If that’s too much bother, start with a plain piece of paper and sketch in faint guidelines when necessary.
- 3. Paper is cheap. If you find that you are running out of room on this sheet of paper, get another sheet of paper, rather than squeezing the calculation into a smaller space. See section 4 for an example of using extra paper to achieve a better result.
- 4. The whole calculation should be structured as a succession of true statements. The first statement is true, the next statement is true, and the statement after that, et cetera. Each statement is a consequence of the previous statements (in conjunction with known theorems, and the “givens” of the problem). Finally, we get to the bottom line, and we know it is true. An example of this can be seen in reference 1.
- 5. Sometimes, especially for long and/or complex calculations, it helps to organize your calculation in two columns. An example of this can be seen in reference 1. In the left column, you write an equation. In the right column, make a note as to how you derived that equation. (Numbering all your equations makes this easier.) If there isn’t enough width to do this easily, turn the paper 90 degrees, so it becomes wider and less tall. (Writing paper isn’t suitable for this, so use graph paper ... or lacking that, plain white paper.)
- 6. Avoid writing down an un-named number like 17, or an un-named expression like a+17 ... because you might forget the meaning thereof. Instead, whenever possible, write down an equation such as b=a+17. That way you can point to every item on the page and say that’s true, that’s true, that’s true... in accordance with the strategy described in item 4.
If the meaning of variables such as a and b is not obvious, write down a legend somewhere (like the legend of a map), explaining in a sentence or two the meaning of each variable. That is, a name is not the same as an explanation. Do not expect the structure of a name or symbol to tell you everything you need to know. Most of what you need to know belongs in the legend. The name or symbol should allow you to look up the explanation in the legend.
- 7. It is important to be able to go back and check the correctness of the calculation you have done. See section 4 and/or reference 1 for examples of what this means in practice.
- 8. As a corollary of item 7: Don’t perform surgery on your equations. That is, once you write down a correct equation, don’t start crossing out terms (or, worse, erasing terms) and replacing them with other expressions. Such a substitution may be “mathematically” correct, if the replacement is equal to the thing being replaced ... but it is a bad strategy, because it makes it hard for you to check your work. Instead, write a new equation. Leave the old equation as it is. Paper is cheap. An example of this can be seen in reference 1.
- 9. Keep track of the units for each expression. For example, the statement “x=2.5 inches” means something rather different from “x=2.5 meters” ... and if you shorthand it as “x=2.5” you’re just asking for trouble.
Writing
Writing Tip #1: Put off editing
Each of us works at writing on two levels:a creative, unconscious leveland a critical, conscious level.
The unconscious produces creative and powerful words and images. It makes surprising and original connections. It shuts down if the critical "editor" part of your mind goes to work too soon.If your High School English teacher's voice runs through your mind as you write, if you worry about spelling, grammar, or how to sell your book while you write, you are writing with a dull pencil.
There are many books written on how to unlock your unconscious and let the writing flow. Here are just a few ideas
- Brainstorm words or images about your topic. Don't stop to evaluate their worth. Keep writing down ideas. When you can't think of another word, wait a while. Often the most powerful idea will surface after you have cleared all the less valuable ideas out of the way.
- Write a page or two with your eyes shut. It doesn't matter if you can't read what you've written. You are giving your mind permission to make "mistakes" and just get on with it.
- Write with music in the background. Experiment to find the style that you like. I prefer baroque or classical music. One of my writing teachers needed country and western.
- Give yourself permission to be emotional. If your writing begins to move you, experience the full emotion. Before your writing changes others it will change you.
Edit your work only when you have drawn deeply from the well of your unconscious.
Spelling counts. So does good grammar. They support vibrant writing. They do not create vibrant writing. There are a great many correctly written lifeless sentences.
The best writing comes to life, and then is refined just enough to make it crystal clear.
First, give it life.
Writing Tip #2: Write what you know
Given the chance, what do you talk about endlessly? What drives you to seek out information? What are your passions? When you write what you know, you write with authority. People listen to you because you are one who knows. You are interesting because you are interested. Your knowledge is a gift to share.
Writing Tip #3: Research
Deepen the well. No matter what you know about the subject, there is always more to learn. Make sure you have the latest information available on your subject.
If there are differences of opinion in the area you are writing about, acknowledge the other side. Your statements will come across more strongly if the reader knows you have addressed the arguments others would raise.
Once you write something, at least some of your readers are going to believe you. You owe them accuracy.
"Yes, but...
I'm writing my autobiography."
Or, "This is my family history. I know this story like no one else."
That's true, but others have a perspective not like yours. Memories, even yours, can be faulty.
"Yes, but...
I'm writing fiction."
O.K. The details of fiction need to be as accurate as the details of nonfiction. Margaret Atwood won The Booker Prize for her novel The Blind Assassin. Her work is powerful on many levels. She took no chances with the details. At the back of her book is a list of acknowledgements 2 1/2 pages long: libraries, archives, museums...
"Yes, but...
My story is a fantasy."
Even when you invent a universe, you invent it to be understood by earthlings. If you are going to have impossible things happening, you need to offer some explanation that will make sense.
Writing Tip #4: Use a structure
For some writers, having a structure in place first makes the writing easier. These writers prefer to think things out ahead of time and then build to a plan.
Other writers put down all their ideas in a glorious profusion of words. Papers may be spread all over the house, the car, the office desk, in fishing tackle boxes.... These writers like to see all the material and then build the structure.
Both approaches work well depending on the personality of the writer. Both kinds of writers need to end up with a structure that supports the reader's understanding.
There is no one right structure for a book any more than there is one right structure for a house. Some will be linear, and take the reader step by step directly through to a conclusion like a long hallway opening into an inner courtyard.
Others will feature a spiraling staircase that takes the reader around and around the topic, always climbing higher to the secret chamber at the top, or to the rooftop view where everything becomes clear.
The fair thing to do is to use a reasonable route to the destination. It's unfair to take your reader up the staircase to the fourth floor and then to push him out a window so he can enjoy the inner courtyard.
Writing Tip #5: Use strong verbs and nouns
The verbs are the action words. They put things in motion. Make yours as strong as possible.
The verb to be (am, is, are, was, were) puddles on the floor. Eliminate it wherever possible. I spent a year in Ukraine and experienced Russian, where the verb to be exists, but almost never appears. People simply leave it out and I found the effect powerful. In English we can't leave verbs out of our sentences, but we can make those we use work hard for us.
Nouns name the people, places, and things in our world. English has multiple words for almost everything. A male parent can be father, dad, pop, daddy, the old man, pater, progenitor, sire, begetter, conceiver, governor, abba, papa, pa, pap, pappy, pops, daddums, patriarch, paterfamilias, stepfather, foster father, and other family nicknames. Choose the noun that does the best work for you.
Short words are usually best. They have more punch. They hit the gut hard.
The paragraph above has only one word with more than one syllable.
Writing Tip #6: Be wary of adverbs and adjectives
If your verbs and nouns are strong, you can get rid of many adverbs or adjectives. Don't know what they are? They are the "describing words" your elementary school teachers told you to use to make your writing "more interesting."
The boy ran to the store.
The tall, tanned boy ran quickly to the store.
The teacher gives you a check mark.
The reader goes to sleep.
Wake up your reader with
The surfer raced to the store.
Be particularly wary of words ending with -ly.
Writing Tip #7: Use correct spelling, punctuation, and grammar
Yes, there is a time to turn on the proofreader.A book is like housework.
No one notices when it is done well, but they see your mistakes clearly.
The guest who comes for tea concentrates on conversation and a developing friendship--unless the windows are streaky or a cobweb hangs in the corner. She is polite so she says nothing, but her attention is divided.
Those pesky flaws in your book will make some readers turn away in disgust. Mistakes distract even the most sympathetic reader. The reader does not necessarily even know the rule you've broken, but he feels uneasy.
The best reference book with writing tips about troublesome grammar, punctuation, and word choice is small, simple, and inexpensive. Affectionately called "Strunk and White" by generations of writers, it is still a required text in many writing classes. You can purchase this here through amazon.com or if you are in Canada and prefer to stay north of 49, here through amazon.ca
Writing Tip #8: Work the details
Your ideas come through more clearly when they are supported by details. Sensory details bring a scene clearly to mind. Most of us rely on sight, so visual details are most common in writing. But use other senses, too. Psychologists tell us the most evocative sense is smell.
Give specific names for things.
The pine is better than the tree.
Give evidence for your point of view. Anecdotes, quotes from reputable sources, statistics, all add credibility.(See Writing Tip #12.)
Writing Tip #9: Cut, cut, cut
Writers often fall in love with their own words and phrases. Cutting them can feel like killing a person.
It only feels like that.
Cutting words from writing is like pruning in the garden. When we get rid of the dead, diseased, and ugly, we are left with a stronger, more beautiful, fruitful plant.
Be ruthless with your writing. Chop out every unnecessary word.
How do you know what can go?
Read what you've written leaving out parts you question. If the piece still makes sense, leave out the excess. Compressed writing packs a punch.
Writing Tip #10: Use active voice
Technically, active voice puts the active agent first, followed by the verb (the action), followed by the object of the action.
Passive voice reverses the order.
Active - The boy hit the ball.
Passive - The ball was hit by the boy.
If you take care of the verb to be (See Writing Tip #5) you will be using active voice more often. (Notice was in the example.)
Active voice is stronger and moves the action along. Passive voice sounds like someone is trying to hide something or to avoid responsibility. We find passive voice in many government documents.
Hm-m-m. Do you aspire to write like the government?
Writing Tip #11: Use parallel structure
Doing the same thing in the same way creates a pattern that helps a reader follow along.
On this page I've used a parallel structure for the tips. Each one is written as a command. I used the imperative mood (the command) because these tips are vital parts of writing. I used it in each case because that creates a pattern your brain picked up by the time you reached Writing Tip #3.
If I had changed Writing Tip #8 to "Details are important," your brain would have registered the shift in structure and for a moment would have flickered away from what I want you to do:
keep reading,
accept these tips,
use them,
become a stronger writer,
sell lots of books,
advance the general quality of written English in the world.
Human brains love pattern. Give your reader's brain a pattern and your ideas will come through like sunshine through a window. Your reader will
keep reading,
take you seriously,
recommend your book,
change the world...
Writing Tip #12: Show, don't tell
If it's a sermon your reader wants, there are churches to oblige.
What does it look like, sound like, feel like, taste like, smell like? When you describe a person or event, your reader is there with you. When you tell, the reader relaxes to the point of mental slumber.
Not sure of the difference?
Telling: John was sad after Susan broke up with him.
Reader: Yawn!
Showing: John shut his cell phone and leaned against the wall. He heaved a sigh and dropped his head into his hands.
Hear the reader's mind working:
"What's with John? Oh, I get it, he feels Susan let him down."
In nonfiction, details show, generalities or opinions tell.
Telling: Children are out of shape these days.
Reader: "I don't think that's true. My neighbor's kid plays Little League."
Showing: Forty percent of 5 to 8-year olds are obese.
The reader's mind kicks in:
"Wow! Children are out of shape these days!"
Writing Tip #13: Use humor when you can
Not everyone cracks jokes all day long. But a light touch from time to time lowers a reader's guard and opens her to your ideas. Be careful that your humor is kind and tasteful, unless of course you are writing for seven-year-olds, when bodily function humor is high on the list.
Writing Tip #14: Build to the end
In English we expect the most important item to be at the end. When you write a list, put the most important, unusual, or powerful item last.
The final sentence in a paragraph ties up your ideas in a neat package or hints at what is to come.
Your most powerful paragraph comes at the end of the chapter.
Poets labour over their final word. Let yours linger in the mind.
Writing Tip #15: Choose a beckoning title
A good title is catchy and says, "Read me." Depending on your topic, you may want to steer clear of a "cute" or "witty" title in favor of one that makes a clear promise of what is inside.
Writers often discover a title as they write. Sometimes a phrase or reference in the book comes to stand for the whole work.
Take your time to find a good title. You want one that calls to a reader, insisting on a purchase.
English Language Grammar Lessons
Introducing the Eight Parts of Speech
English grammar uses words based on eight parts of speech: the verb, the noun, the pronoun, the adjective, the adverb, the preposition, the conjunction,and the interjection. Each part of speech explains not what the word is, but how the word is used. The same word can be a noun in one sentence and a verb or adjective in the next.
The verb is perhaps the most important part of the sentence. A verb or compound verb states something about the subject of the sentence. The verb depicts actions, events, or states of being.
A noun is a word used to name a person, animal, place, thing, or abstract concepts. A noun can function in a sentence as a subject, a direct object, an indirect object, a subject complement, an object complement, an appositive, an adjective or an adverb.
Pronouns as a part of speech can replace a noun, another pronoun, noun phrases and perform most of the functions of a noun.
An adjective modifies a noun or a pronoun by describing, identifying, or quantifying words. Many consider articles: "the, a, an" to be adjectives.
An adverb can modify a verb, an adjective, another adverb, a phrase, or a complete clause by indicating manner, time, place, cause, or degree.
A preposition links nouns, pronouns and phrases to other words in a sentence.
Conjunctions link words, phrases, and clauses.
An interjection is a part of speech used to show or express emotion or illustrate an exclamation.
Additional Parts of Speech Forms and Functions
VERBS
A transitive or sometimes called an action verb passes action on to a direct object.
An intransitive verb does not indicate a transfer of action.
A linking verb joins a subject with a word that describes it.
A main verb indicates the primary or principal activity.
An auxiliary verb helps the main verb describe an action or state of being.
A modal verb indicates ability, obligation, permission, or possibility. Modal examples: can, may, must, should, could, might, ought, would.
A finite verb describes a definite and limited action or condition.
A non-finite verb shows an unfinished action or condition.
A ditransitive verb takes two complements, an indirect object and a direct object.
Monotransitive verbs take one complement, usually a direct object
An intransitive verb does not have any complements. Examples: Fred cried. Sally slept.
A prepositional verb is a multi-word verb consisting of a verb and preposition.
Phrasal-prepositional verbs are multi-word verbs consisting of a verb, adverb and preposition.
Verb Forms called Verbals
Infinitives are the word " to + verb" and they act as nouns, adjectives, or adverbs.
Participles in past or present tenses act as adjectives
Gerunds in the "present tense participle form" act as nouns.
NOUNS
Proper nouns are capitalized and include: name of a specific person, place, or thing, days of the week, months of the year, historical documents, institutions, organizations, religions, holy texts and religious followers.
A common noun is a noun referring in general to a person, place, or thing.
A concrete noun is a noun which names everything (or everyone) that you can perceive through the physical senses of touch, sight, taste, hearing, or smell.
An abstract noun is a noun that names anything that you can not perceive through your five physical senses.
A countable noun (or count noun) names anything (or anyone) that you can count and is a noun with both a singular and a plural form.
A non-countable noun (or mass noun) is a noun which does not have a plural form, and which refers to something that you could (or would) not usually count.
A collective noun is a noun naming a group of things, animals or persons.
A possessive noun indicates ownership or possession.
PRONOUNS
A personal pronoun refers to a specific person or thing and changes its form to indicate person, number, gender, and case.
A subjective personal pronoun indicates that the pronoun is acting as the subject of the sentence. The subjective personal pronouns: I, you, she, he, it, we, you, they".
An objective personal pronoun indicates that the pronoun is acting as an object of a verb, compound verb, preposition, or infinitive phrase. The objective personal pronouns: "me, you, her, him, it, us, you, them".
A possessive pronoun indicates that the pronoun is acting as an indication of possession and defines who owns a particular object. The possessive personal pronouns: "mine, yours, hers, his, its, ours, theirs".
A demonstrative pronoun points to and identifies a noun or a pronoun. The demonstrative pronouns: "this, that, these, and those''.
An interrogative pronoun is used to ask questions. The interrogative pronouns: "who, whom, which, what''.
Relative pronouns link one phrase or clause to another phrase or clause. The relative pronouns: "who, whom, that, which.''
An indefinite pronoun refers to an unspecified person or thing. An indefinite pronoun depicts the idea of all, any, none, or some. The most common indefinite pronouns: all, another, any, anybody, anyone, anything, each, everybody, everyone, everything, few, many, nobody, none, one, several, some, somebody, and someone.
The reflexive pronouns identify the "self" such as: "myself, yourself, herself, himself, itself, ourselves, yourselves, and themselves.''
An intensive pronoun is a pronoun used to emphasize or highlight an attribute.
ADJECTIVES
An adjective usually precedes the noun or the pronoun which it modifies. Gradable adjectives have a base, comparative and superlative form. The adjective happy is intensified in the following examples: "very happy, extremely happy, quite happy, happier, and happiest". Adjectives can have stative or dynamic and inherent or non-inherent properties.
An adjective can be modified by an adverb or by a phrase or clause functioning as an adverb. Some nouns, many pronouns, and many participle phrases can also act as adjectives.
A possessive adjective is similar to a possessive pronoun. The possessive adjective modifies a noun or a noun phrase.
The demonstrative adjectives ``this, these, that, those, what'' are identical to the demonstrative pronouns, but are used as adjectives to modify nouns or noun phrases.
An interrogative adjective such as ``which or what'' is like an interrogative pronoun. The interrogative adjective modifies a noun or noun phrase rather than standing on its own.
An indefinite adjective is similar to an indefinite pronoun. The indefinite adjective modifies a noun, pronoun, or noun phrase.
ADVERBS
Adverbs have a complex grammatical relationship within the sentence or clause as a whole. An adverb can be found in various places within the sentence. An adverb can modify a verb, an adjective, another adverb, a phrase, a clause or entire sentence. Adverbs are gradable with intensification and comparison.
A circumstantial adverb indicates manner, time or place.
A degree adverb specifies the degree or cause to which some property applies and answers questions such as: how, when and where.
The conjunctive adverb can join two clauses together. The most common conjunctive adverbs: "also, consequently, finally, furthermore, hence, however, incidentally, indeed, instead, likewise, meanwhile, nevertheless, next, nonetheless, otherwise, still, then, therefore and thus."
A disjunct adverb comments on the sentence as a whole. Example: Honestly, I couldn't believe my eyes.
An interrogative adverb is used to construct interrogative sentences and "wh-questions" example: Why did you do that?
PREPOSITIONS
A preposition usually indicates the temporal, spatial or logical relationship of its object to the rest of the sentence.
The most common prepositions: "about, above, across, after, against, along, among, around, at, before, behind, below, beneath, beside, between, beyond, but, by, despite, down, during, except, for, from, in, inside, into, like, near, of, off, on, onto, out, outside, over, past, since, through, throughout, till, to, toward, under, underneath, until, up, upon, with, within and without."
Complex prepositions consist of more than one word: along with, out of, up to.
CONJUNCTIONS
Conjunctions are a part of speech and are a closed word class which includes coordinating words such as "and, but, and or", and subordinating words such as "because, if, and when". Some conjunctions can also appear as prepositions or as adverbs.
Coordinating conjunctions ``and, but, or, nor, for, so, or yet'' are used to join individual words, phrases, and independent clauses. The conjunctions ``but'' and ``for'' can also function as prepositions.
A subordinating conjunction introduces a dependent clause and indicates the nature of the relationship between the independent clause(s) and the dependent clause(s). The most common subordinating conjunctions: "after, although, as, because, before, how, if, once, since, than, that, though, till, until, when, where, whether and while".
Correlative conjunctions always appear in pairs -- you use them to link equivalent sentence elements. The most common correlative conjunctions: "both... and, either...or, neither...nor, not only.., but also, so...as, and whether...or." Usually correlative conjunctions consist of a coordinating conjunction linked to an adjective or adverb.
INTERJECTIONS
Interjections are used in speech to indicate emotion or transition. Interjections such as "yuk, ouch, eh" are used as exclamations in conversation.