- You are writing a report so you will always want to use good grammar
and correct English. Your explanation should:
- be factually correct, or nearly so, with only minor flaws (for
example, a minor mistake in a calculation).
- address the specific question or problem that was posed. It is
focused, detailed, and precise. There are no irrelevant or distracting
points.
- be clear, convincing, and logical. A clear and convincing explanation
is characterized by the following:
- The explanation could be used to teach another (college) student,
possibly even one who is not in the class.
- The explanation could be used to convince a skeptic.
- The explanation does not require the reader to make a leap of faith.
- Key points are emphasized.
- Supporting pictures, diagrams, and/or equations are used appropriately
and as needed.
- The explanation is coherent.
- Clear, complete sentences are used.
- Your spelling is correct.
- The explanation begins with an introduction and end it with a
conclusion.
- Use words that say what you mean; a dictionary will be useful.
- Be careful not to use pronouns with ambiguous antecedents. Consider
the statement:
We make the substitution u = 1+x2 . Since it is positive ¼.
It is unclear whether the ``it'' in the second sentence refers to u or x. In order to make yourself clear, say instead:
We make the substitution u = 1+x2 . Since u is
positive ¼.
- Use mathematical words correctly. Below are some words people misuse
or confuse. Learn the differences between them and how to use them correctly.
- expression - almost any kind of mathematical term; may be an
equation, a function, a formula, or a piece of one of these
equation - a mathematical expression which always has an equal sign
- Ex:
- 3x2+5x = x2+4x+3
function - a special mathematical definition, which is not an
equation
- Ex:
- f( t) = 3sin2t
- solve - to find the solution of an equation
- Ex:
- If one solves the equation 3x+2 = x+4, one finds x = 1.
evaluate - compute an expression for specific values of the variable
involved
- Ex:
- When I evaluate f( x) = cospx for x = 1, I get f( 1) = cosp = -1.
substitute - replace one expression with another
- Ex:
- Substituting 1+u for x in the expression x2-2x, we get u2-1.
- problem - a generic word for something requiring a solution
equation - a specific kind of problem, one which involves an equal
sign
- Use correct and complete mathematical notation.
- Use equal signs when you want to show that expressions are equal.
Without the equal signs, I will not know you mean they are equal. Do not
use arrows (Þ ), which are used to show logical implications.
On the other hand, do not use equal signs for unequal things. For example,
when reducing a matrix, we often use replacement. The expression R3+2R1 = R3 is an equation that implies R1 is a row of zeros.
The notation R3+2R1® R3 is better, though the sentence
``Replace row 3 by the sum of row 3 and twice row 1'' more accurately and
unambiguously explains the process.
- As a rule of thumb, we write small mathematical expressions in the
text and we display large expressions or major points on their own lines.
Consider the statement:
We have discussed in class equations involving vectors
in which a1,¼,an and b are vectors and x1,¼,xn are scalars and have seen that
this system has a solution if b can be written as a linear
combination of a1,¼,an.
The collection of vectors ``a1,¼,an and b'' are included in the text, but the vector equation is displayed.
Displaying these expressions make them much easier to read.
Something you should never do when writing a mathematical report is to write
a column of mathematics on the left and some explanation on the right, like
this:
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I start with this matrix. |
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Then I make the replacement |
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to get the reduced echelon form. |
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It is better to write:
I start with the matrix
Replacing row 2 by the sum of -3 times row 1 plus row 2 yields
the reduced echelon form
- Use references and cite them. In particular, use your book. If you use
a formula from the book, give a reference for it (such as the page on which
it appears). It would be better to give too many references than too few.
- Use your book as a model for how to write mathematics. You may also
want to look at some other mathematics books to see how other authors
incorporate text in mathematical writing.