Guidelines for Writing PoW Answers

Writing a math answer for a Problem of the Week is very different from writing an essay in English class or a term paper in History class, so we would like to give you some guidelines. You write only one document, but we receive lots of answers to each problem to read and analyze; when your presentation style is at its best, much time can be saved, a more efficient service can be provided, and everybody will be happier.

Readability

The first thing that would speed up our evaluation process can be called readability. Sometimes a student sends us an answer in one long continuous paragraph, with equations embedded in it. Such paragraphs are very hard to read. (This is true in other subjects besides math!)

The solution is simple: just break the long paragraph up into several short ones, each one with its own concept, and leave a blank line between paragraphs.

Another matter regarding readability concerns polynomial expressions and equations. Notice the difference between these items:

Easy to Read                      Harder to Read

x^2 + 2x + 1                      x^2+2x+1

x^3 + 4x^2 - 6x + 10              x^3+4x^2—6x+10

(3a + 4b)(3a - 4b)                (3a+4b)(3a-4b)

See how a space on either side of a plus or a minus sign makes the reading easier? (This is what good textbooks do.)

Similarly, when you show the steps in solving equations, add spaces and align the equals signs, like this (when you align text, never use tabs!):

Easy to Read                      Harder to Read

2x + 48 = 58                      2x+48=58
                                  2x=10
     2x = 10                      x=5
     
      x =  5                      
      

Getting Off To a Good Start

After carefully reading a problem, it is essential to determine just what you need to find to answer the question posed. You should now select a letter, or letters, that will represent your unknown quantity, or quantities. This is the famous "Let" statement. Then, and only then, are you ready to begin forming your expressions or equations.

Be careful here, however. Many times "Let" statements aren't clear. Examples:

Good: Let x = the number of apples in the basket

Not so good: Let x = apples

Use of "Guess and Check"

In general, the method of "guess and check" is not allowed as your primary strategy to solve the Problem of the Week. This is not saying guess and check is not a good way to solve problems. In fact, it is often a good way to start to understand a problem, and therefore recommended for that. But for most of our problems, you must define variables or unknowns, then form equations to solve by logical steps.

One of our mentors advises students in the following way:

Guess and check is valid problem-solving approach. However, it is also one of the most difficult to explain. If you are going to use guess and check, you must list every guess, along with the reason that you know the answer is incorrect. You also must explain why you know your final answer is the only possible answer. All in all, it can be a pretty long process. You are better off to try and use algebra techniques.

So, unless otherwise indicated, please do not use guess-and-check as your principal solving procedure.

Writing "Math" in Plain Text

Sometimes we cannot write certain symbols (like exponents or square roots) in plain text as we do using paper and pencil. Here are some examples:

Exponents

It is standard now in plain text to use the ^ (caret sign) found above the 6 on the keyboard for exponents. If we wish to say 'four squared', we write 4^2. For higher powers we do the same: 'The volume of a cube is e-cubed' would be V = e^3.

Square roots

Some people use the notation popular in spreadsheet applications, e.g. sqrt(16), to mean 'the square root of 16'. This even applies in formulas; for the Pythagorean theorem, we can write: c = sqrt(a^2 + b^2).

Other students 'draw' a square root symbol this way:

               __                 _____
              V64       or      \/a + b

[A few people try decimal or fractional exponents: 64^0.5 or 64^(1/2), but this method can be difficult to read, so it is not recommended. However, there are occasions in which such exponents are better.]

Fractions

Writing fractions is more complicated. There are two basic styles: vertical (sometimes called 'stacked') fractions, and horizontal fractions. Vertical fractions are what we are used to writing with pencil and paper, and are what you see in books. We can make them in e-mail as well; it just takes more effort and more keystrokes. But they are more readable when we need to write algebraic fractions.

 1               15               3a  +  4b
---             -----            -----------     
 2               25               5c  -  6d

Horizontal fractions consisting only of numerals are easy to write, as these examples show:

Two-fourths  2/4     five-sixths  5/6     etc.

Even fractions that contain binomials, as shown above, can be written horizontally, if you employ parentheses. Observe:

(3a + 4b)/(5c - 6d)

The difficulty arises when you need to express something like 'two-thirds of x'. If you write this as 2/3x, it could be misinterpreted as 2 over 3x. Luckily we have ways of clarifying our meaning:

           2
Vertical: --- x      Horizontal: (2/3)x
           3

Now if your intention really was 2 over 3x, you still have two options:

           2
Vertical: ----       Horizontal: 2/(3x)
           3x
           

Subscripts

Unlike exponents, which go above the line (that's why they're sometimes called 'superscripts'), subscripts go below the line. Unfortunately, the standard keyboard doesn't have a true subscript key. Some people write a_1 for 'a-sub-one', but since many cases that need subscripts occur in sequences, we could write the following:

a1, a2, a3, a4, ...

to stand for a sequence of terms (a-sub-one, a-sub-two, ...). In this context there is no real confusion with multiplying 'a' by 4. We universally write that as 4a.

Notice how nice the slope formula can look using vertical fractions with this subscript style:

     y1 - y2
m = ---------   instead of  m = (y_1 - y_2)/(x_1 - x_2)
     x1 - x2
           

The vertical equation looks almost like a line from a textbook, but even a horizontal equation like this one would be preferable:

m = (y1 - y2)/(x1 - x2)
           

Determinants

When you are using determinants to solve a system of equation by Cramer's Rule, they may be nicely formed as shown here:

     |   3      5  |
D =  |             |    =  (3)(6)  -  (-1)(5)  =  18 - (-5)  =  18 + 5 = 23
     |  -1      6  |

For a 3-by-3 case, the same idea applies:

     | a    b    c |
     |             |
D =  | d    e    f |   =  aei  +  dhc  +  gbf  -  gec  -  dbi  - ahf 
     |             |
     | g    h    i |

The method you use will often depend on the needs of the specific problem you are working; these comments should be understood as suggestions and general guidelines only.