4 Explaining Your Work
At its heart, mathematics is a social endeavor. Even if you work on problems all by yourself, you have not really solved the problem until you have explained your work to someone else, and they sign off on it. Professional mathematicians write journal articles, books, and grant proposals. Teachers explain mathematical ideas to their students both in writing and orally. Explaining your work is really an essential part of the problem-solving process, and probably should have been Pólya’s step 5.
Writing in mathematics is different from writing poetry or an English paper. The goal of mathematical writing is not florid description, but clarity. If your reader does not understand, then you have not done a good job. Here are some tips for good mathematical writing.
Do Not Turn in Scratch Work: When you are solving problems and not exercises, you are going to have a lot of false starts. You are going to try a lot of things that do not work. You are going to make a lot of mistakes. You are going to use scratch paper. At some point (hopefully!) you will scribble down an idea that actually solves the problem. Hooray! That paper is not what you want to turn in or share with the world. Take that idea, and write it up carefully, neatly, and clearly. (The rest of these tips apply to that write-up.)
(Re)state the Problem: Do not assume your reader knows what problem you are solving. (Even if it is the teacher who assigned the problem!) If the problem has a very long description, you can summarize it. You do not have rewrite it word-for-word or give all of the details, but make sure the question is clear.
Clearly Give the Answer: It is not a bad idea to state the answer right up front, then show the work to justify your answer. That way, the reader knows what you are trying to justify as they read. It makes their job much easier, and making the reader’s job easier should be one of your primary goals! In any case, the answer should be clearly stated somewhere in the writeup, and it should be easy to find.
Be Correct: Of course, everyone makes mistakes as they are working on a problem. But we are talking about after you have solved the problem, when you are writing up your solution to share with someone else. The best writing in the world cannot save a wrong approach and a wrong answer. Check your work carefully. Ask someone else to read your solution with a critical eye.
Justify Your Answer: You cannot simply give an answer and expect your reader to “take your word for it.” You have to explain how you know your answer is correct. This means “showing your work,” explaining your reasoning, and justifying what you say. You need to answer the question, “How do you know your answer is right?”
Be Concise: There is no bonus prize for writing a lot in math class. Think clearly and write clearly. If you find yourself going on and on, stop, think about what you really want to say, and start over.
Use Variables and Equations: An equation can be much easier to read and understand than a long paragraph of text describing a calculation. Mathematical writing often has a lot fewer words (and a lot more equations) than other kinds of writing.
Define your Variables: If you use variables in the solution of your problem, always say what a variable stands for before you use it. If you use an equation, say where it comes from and why it applies to this situation. Do not make your reader guess!
Use Pictures: If pictures helped you solve the problem, include nice versions of those pictures in your final solution. Even if you did not draw a picture to solve the problem, it still might help your reader understand the solution. And that is your goal!
Use Correct Spelling and Grammar: Proofread your work. A good test is to read your work aloud (this includes reading the equations and calculations aloud). There should be complete, natural-sounding sentences. Be especially careful with pronouns. Avoid using “it” and “they” for mathematical objects; use the names of the objects (or variables) instead.
Format Clearly: Do not write one long paragraph. Separate your thoughts. Put complicated equations on a single displayed line rather than in the middle of a paragraph. Do not write too small. Do not make your reader struggle to read and understand your work.
Acknowledge Collaborators: If you worked with someone else on solving the problem, give them credit!
Here is a problem you’ve already seen:
Problem 16
Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1’s are separated by one digit, the 2’s are separated by two digits, the 3’s by three digits, and the 4’s by four digits.
Think / Pair / Share
Below you will find several solutions that were turned in by students. Using the criteria above, how would you score these solutions on a scale of 1 to 5? Give reasons for your answers.
Solution (Solution 1).
41312432
This is the largest eight-digit b/c the #s 1, 2, 3, 4 & all separated by the given amount of spaces.
Solution (Solution 2).
41312432
You have to have the 4 in the highest place and work down from there. However unable to follow the rules the 2 and the 1 in the 10k and 100k place must switch.
Solution (Solution 3).
41312432
First, I had to start with the #4 because that is the largest digit I could start with to get the largest #. Then I had to place the next 4 five spaces away because I knew there had to be four digits separating the two 4’s. Next, I place 1 in the second digit spot because 2 or 3 would interfere with the rule of how many digits could separate them, which allowed me to also place where the next 1 should be. I then placed the 3 because opening spaces showed me that I could fit three digits in between the two 3’s. Lastly, I had to input the final 2’s, which worked out because there were two digits separating them.
Solution (Solution 4).
1×1
2xx2
3xxx3
4xxxx4
Answer: 41312432
Solution (Solution 5).
4 3 2 4 3 2
4 2 2 4
4 1 3 1 4 3
*4 1 3 1 2 4 3 2
4 needs to be the first # to make it the biggest. Then check going down from next largest to smallest. Ex:
4 3 __________
4 2 __________
4 1 __________
Solution (Solution 6).
41312432
I put 4 at the 10,000,000 place because the largest # should be placed at the highest value. Numbers 2 & 3 could not be placed in the 1,000,000 place because I wasn’t able to separate the digits properly. So I ended up placing the #1 there. In the 100,000 place I put the #3 because it was the second highest number.
Solution (Solution 7).
41312432
Since the problem asks you for the largest 8 digit #, I knew 4 had to be the first # since it’s the greatest # of the set. To solve the rest of the problem, I used the guess and test method. I tried many different combinations. First using the #3 as the second digit in the sequence, but came to no answer. Then the #2, but no combination I found correctly finished the sequence.I then finished with the #1 in the second digit in the sequence and was able to successfully fill out the entire #.
Solution (Solution 8).
4 _ _ _ _ 4 _ _
4 has to be the first digit, for the number to be the largest possible. That means the other 4 has to be the 6th digit in the number, because 4’s have to be separated by four digits.
4 _ 3 _ _ 4 3 _
3 must be the third digit, in order for the number to be largest possible. 3 cannot be the second digit because the other 3 would have to be the 6th digit in the number, but 4 is already there.
4 1 3 1 _ 4 3 _
1’s must be separated by one digit, so the 1’s can only be the 2nd and 4th digit in the number.
4 1 3 1 2 4 3 2
This leaves the 2s to be the 5th and 8th digits.
Solution (Solution 9).
With the active rules, I tried putting the highest numbers as far left as possible. Through trying different combinations, I figured out that no two consecutive numbers can be touching in the first two digits. So I instead tried starting with the 4 then 1 then 3, since I’m going for the highest # possible.
My answer: 41312432