What is a Paragraph Proof? - Definition & Examples

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  • 0:00 What Is a Proof?
  • 0:56 Outline of a Paragraph Proof
  • 2:45 Examples
  • 4:39 Lesson Summary
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Lesson Transcript
Instructor: Beverly Maitland-Frett

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

Writing proofs is an important aspect of mathematical inquiry and discovery. This lesson will discuss one method of writing proofs, the paragraph proof. We will explore some examples and provide some guiding steps you may use to write an effective paragraph proof.

What Is a Proof?

Lisa forgot to punch her time card at work. She needs to prove to her boss that she got to work on time. She has a train ticket to show the time she took the train to work, and she has a Dunkin' Donuts receipt. She writes her boss the following email:

Dear Boss,

I have attached a copy of the train receipt showing the time I got on the train this morning. I also included another receipt to show that I bought the donuts that are in the staff lounge, and to show the time I bought the donuts. Henry, who is always on time, admitted that the donuts were in the staff room when he arrived. Had I punched in, he would have punched in after me. Therefore, based on the time Henry punched in, you can verify that I got to work on time.



PS. I'm glad to know that you enjoyed your favorite coconut-sprinkled donut.

Lisa has just written a paragraph proof. A proof is a logical argument, presented with factual statements, in order to arrive at a conclusion.

Outline of a Paragraph Proof

There are three main types of proofs:

  1. A two column proof
  2. A paragraph proof
  3. A flow chart proof

In this lesson, we will focus only on the paragraph proof. The paragraph proof is a proof written in the form of a paragraph. In other words, it is a logical argument written as a paragraph, giving evidence and details to arrive at a conclusion. Writing a proof is like solving a puzzle or using Legos to create a model of something; everything needs to fit in an appropriate place.

In order to write proofs we need certain items:

Givens: A mathematical proof always begins with the givens. Usually, we are given some details to build our proof. In Lisa's case, she had two receipts. Those are her givens; they already exist.

Diagram: If a diagram is not provided, draw a diagram based on the givens. A visual representation is very helpful before you start a proof.

Prior knowledge to connect statements: We must have prior knowledge of theorems, postulates, angle relationships, definitions, and other pertinent information in order to build on our givens. Lisa knew that Henry is always on time, that he had eaten a donut, and that he had punched in. These facts had nothing to do with her train ticket, but the donuts and Henry were connected to the donuts receipt.

Reasoning skills and patience: Writing proofs demands patience and time. Using necessary reasoning skills, we have to work through the ideas to ensure that they make sense.

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