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Paragraph Proofs

Reference > Mathematics > Introduction to Proofs

Two-column proofs are very structured, very rigid, and they force students to analyze every single step in the process of the proof. It might surprise you to know, though, that most mathematicians don't really do two-column proofs, and once you get out of your geometry class, you probably won't either.

One of the problems with two-column proofs is that they are not necessarily very readable. Every property, every definition, every theorem has a name, and if you don't remember the names of all those "reasons" then the right-hand column of the proof becomes indecipherable. And clarity is very important in mathematics.

In addition to having to remember a lot of names in order to make sense of a two-column proof, there's also the issue that different curricula may not use the same names for the same theorems. For example, I remember when I was taking Geometry in high school, I learned a theorem that says, "Vertical Angles are Congruent." Now, don't worry if you don't know what vertical angles are, or what congruent means; that's not my point. My point is this: in the textbook I learned from, that Theorem was titled "Theorem 4.8". Yes, that's right, it was named after the chapter and section of the book in which it was taught. And 25 years later, I still remember that. On the other hand, the book I'm teaching from now simply calls it "The Vertical Angle Theorem."

So two different people will write two different reasons in their two-column proof. One will write "Theorem 4.8," which is horribly cryptic, unless you've got that textbook memorized, and the other will write "Vertical Angle Theorem." And neither one will know what the other is talking about.

So in a paragraph proof, we throw away the two column format, and simply explain what we did, in complete sentences. It takes more words, but in a lot of ways it's a lot easier, both for the writer and the reader.

If I was writing a paragraph proof that used the Vertical Angle Theorem, I would make my statement, and then say, "because vertical angles are congruent."

Now no one needs to remember a theorem name; the concept of the theorem is embedded right in the proof!

Let's try writing a paragraph proof. Remember that we must have complete sentences, with proper grammar and punctuation. Also, we should try to keep this very impersonal. Avoid saying "I" or "me." Most of the time you can avoid any sort of personal pronouns in your proof, but if you just simply must use a personal pronoun, use "we," since it includes both the proof writer and the proof reader.

Prove: If x2 + 5x + 6 = 0, then x = -2 or x = -3.

Proof: Using the sum and product rule, x2 + 5x + 6 factors into (x + 2)(x + 3). Thus, (x + 2)(x + 3) = 0, and the zero rule for multiplication tells us that x + 2 = 0 or x + 3 = 0. These two equations lead to the solutions x = -2 or x = -3.

And that's it! Note that when we got to the very end, we didn't even provide an explanation for how we solved those two equations. That's one of the nice things about paragraph proofs; when we get it down to the really obvious steps, we can make the assumption that the person reading the proof will quickly recognize how we got our conclusion.


Prove: If x2 - y2 = 40 and x + y = 8 then x - y = 5
Prove: If x + y =10 and x - y = 4, then x = 7 and y = 3.
Prove: If two positive numbers are both greater than zero, then their sum is greater than either of the two numbers.
Prove: If x3 + 4x2 + 4x = 0 then x = 0 or x = - 2
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