# Flow

Written mathematics must be readable. This may seem
trivial, but it is an important point. You

should be able to read your work aloud to a classmate and have them understand
your solution. If

you need to add any explanations, these should be included in your written work.
The most common

mistake is to write mathematics without using enough words. All writing, even
mathematics, should

consist of complete sentences. These should explain the problem by providing
both the method and

justification for each step of the solution.

**Why are sentences important in mathematics?**

Although sometimes it seems hard to read textbooks, it
would be much harder to understand if

they only had equations and no sentences. The situation is similar in lecture:
if the professor just

listed formulas on the chalkboard without talking about them, leaving you to
figure out what was

being done in each step, how much could you understand from the lecture? Neither
of these would

be a good way for most students to learn, since sentences are necessary to
explain the mathematics.

**Why should students use sentences in a Mathematics
class?**

In a Mathematics class, you should explain homework
solutions using complete sentences. That

means linking together thoughts with words and embedding equations into
sentences. Going through

the extra work to do this will benefit you in several ways:

• Writing down your thoughts and organizing them into
complete sentences will help you to

understand the method of solution better.

• When you look back on homework to study for a test, or
later on in another class, you will

understand what you were doing on each problem and the mathematics behind it.

• Other people (teacher, classmates, grader,...) will
understand what you are doing at each step,

and why you are doing it. This way, you won’t lose points for skipping steps or
solving the

problem in an unusual way.

• Communicating your work will be essential in whatever
field you choose. Even though the

fields are stereotypically weak on writing, engineers and scientists spend a
surprising amount

of time writing reports and giving oral presentations.

**Examples:**

**How to put an equation into a sentence:
Good:** The derivative of the curve

**Bad:**The derivative for the curve y which is

**Bad:**is the derivative of which is .

**Using sentences incorrectly:**

Find an equation for the tangent to the curve
at the point (1, 3).

**Bad:**

so the equation for the derivative is

so the slope at (x, y) can be found by y −3 = −1(x−1) so y = −x+4 is the point-slope equation.

**Good:**

We first check that (1, 3) is a point on the curve by plugging these values in:
3 = 1 + 2/1. The

derivative of the curve
is

(1)

A line tangent to the curve at the point (1, 3) will have slope

Using the point-slope formula with m = −1, x_{0} = 1, and
y_{0} = 3 gives the formula for the line

y − 3 = −1(x − 1). Solving for y and simplifying gives

y = −x + 4.

This is the equation of the line tangent to the curve at that point.

**Vague and uninformative sentences**

**Bad: **We use calculus to find that y = 3x^{2} + 1 has
a slope of 3 at x = 1/2.

**Good:** To find the slope of the curve y = 3x^{2}+1 at the point x = 1/2 we
find
evaluated at x = 1/2.

**Bad:**

is the derivative.

**Good: **To take the derivative of a product of 2
functions, we use the product rule, (fg)' = f'g + fg'.

In our case we have

**Referring to previous equations and figures.**

You may have noticed that one of the equations above has
been labeled equation number one by

putting “(1)” at the right hand margin. If you need to refer back to an equation
or figure, label it

and then refer to it by its label. Do not draw arrows.

**Bad:** Using the equation from before, the slope is
−1. **Which equation?**

**Good: **Using (1), the derivative at x = 1 is −1.