The 1st stage of writing a solution : Apprentice
Writing up a solution can be scary. You stare at a blank screen and wonder: What do I need to write? How much do I need to write? Where do I start? Who is reading this? Why am I writing this?
There is no "correct" answer to these questions. By writing up your solution, you are explaining to someone else the thought process which you used to solve the problem. Just as there are many ways of approaching a problem, there are also many ways of writing up a solution. As you practice writing solutions, the experience you gain helps you get a better sense of how to write a good solution. There isn't an "absolutely the #1 best solution", but there are numerous great solutions that are waiting to be written.
Here are some guidelines for an Apprentice:
1. Use words to explain. Use complete sentences, correct grammar, punctuation and phrasing.
Remember that you are trying to express your ideas to others, and it is very hard to do so through interpretative dance. Use complete sentences and proper phrasing to explain what you are doing.
2. Check for careless mistakes.
Always double-check your math, and reduce errors. Mistakes happen all the time, and are only a problem if you do not catch them. If arithmetic mistakes carry through your work, it could likely result in no one believing or understanding your solution.
3. Check that your mathematical statements display correctly.
In the preview box, look at your equations again and check that they appear as you want them to. If an equation is too long, you might want to split it up.
Often, using the sign for multiplication instead italicizes the surrounding text. For example, "" appears as "22 = 4". Instead, use a capital X to denote multiplication, as in . You can avoid this by placing your equations with using brackets, as in , which would appear as . This is known as LaTeX, which allows you to write beautiful equations.
4. State the numerical answer clearly.
Remember to include the final numerical answer, and not just the steps that you took. This makes it much easier for your audience to understand your proof as a whole, and indicates to them how you arrived at your solution.
5. Ensure that all necessary parts are included.
If your solution used multiple steps, remember to show all of your working. Skipping steps to 'save time' will result in others not understanding what you did, and waste all of their time.
Now that we have outlined these guidelines, let's look at a few examples. Consider the following problem:
The quadratic expression can be rewritten as . What is the value of ?
Let's look at the following solution:
Solution 1: .
How can we use the above guidelines to help us improve this solution?
Guideline 1: While we may (eventually) figure out the exact thought process of the equations, adding some words to explain what you are doing is immensely helpful. We are only able to read what you wrote, and not what you are thinking in your head.
Guideline 2: There is an arithmetic mistake made in the first line, though it was corrected at the end. Can you spot it?
At the last step, the wrong values of and are given. Remember to check that you wrote down what you were thinking.
Guideline 3: Saying can be confusing, since often refers to the decimal value of , while the author most likely meant .
Guideline 4: What is the final answer?
Guideline 5: You should explain how we arrived at the values of and (namely by comparison). It is also helpful to explain that the initial line was obtained by the technique of Completing The Square.
Now, let's compare this to the following solution (of the same problem):
Completing the square is the technique.
can be expressed as , or
(The came from , which is the desired term of the perfect square.)
Simplifying, is the desired result. and are and , so the answer is .
This is a great solution. It is easy to understand and follow, and provides explanations for each of the steps taken. There are no mathematical errors, the equations display nicely due to using LaTeX, and the answer is clearly stated.
You can view this solution (and the problem) by clicking on the hyperlink Solution 2. If you enjoyed this solution, remember to vote it up!
Aspire to be better. Proceed on and be a Journeyman.
Note: You can now view Latex codes by hovering over the equation. Read Seeing actual for more details!