Illuminations: Joking with Proofs

# Joking with Proofs

 In the same way that supporting statements are used to reach a conclusion in a paragraph-style proof, students modify a cliché or common phrase to use as a punch line to a humorous story. This lesson works well in aiding in the transition from two-column proofs to paragraph-style proofs.

### Learning Objectives

 Students will: create a story "joke" analogous to a paragraph-style proof. Learn that a proof is a chain of statements (agreed-upon assumptions, definitions, or previously proven statements) used as a convincing argument leading to a concluding statement. Learn that a proof is a presentation tool and not an effort to find unknown values or a solution. Understand that the conclusion of a proof is known before the proof is created.

### Instructional Plan

On the day before the lesson, distribute the Find a Cliché activity sheet. Instruct each student as homework to find a cliché or familiar phrase that is familiar, appropriate, and at least 5 words. Preferably, each student's phrase should be different from every other student's phrase, so encourage students to search for a unique cliché. Do not share the purpose of this homework.

As the students enter class, collect the cliché homework exercise. Quickly sort through the clichés to make sure they are appropriate and familiar, and remove any duplicates. Because students will be working in pairs, you will only need half as many clichés as you have students. You also may wish to have a few extra clichés of your own to add to the collection to make sure that each pair gets a unique cliché.

Ask: "What is a proof?" Through discussion, students should recognize that a proof is a group sentences leading to a conclusion. Remind students that those sentences create a chain of logically valid deductions using agreed-upon assumptions, definitions, or previously proven statements. Typically, a proof is used to show that the concluding statement must be true. Clarify that a proof is a presentation tool, not a solution method for a problem. In problem solving, the goal is to carry out appropriate, logical steps to find unknown values or a solution. In a proof, the conclusion is already known. The goal is to assemble supporting statements to make a convincing argument that the conclusion is correct.

A joke can be similar to a proof. Jokes are often a chain of statements called the "setup" that are used to reach an unexpected conclusion called a "punch line." The punch line should be surprising but familiar. In this exercise, you will be writing a joke that has the same structure as a paragraph-style proof. Show the Jokes as Proofs overhead to reinforce this comparison.

Group the students into pairs, allowing them to create their joke collaboratively. Distribute the Jokes as Proofs activity sheet and one cliché to each group. Students should not get their own cliché so they can later find humor in how someone modified it. Have students record their cliché for Question 1.

Explain that just as the conclusion of a proof is usually known before the proof is created, they will begin by creating the punch line of the joke. Warn students not to modify the cliché so much that the resulting sentence is incomprehensible or does not sound like the original cliché. For example, "Don't cry over spilled milk" could become "Don't lie over milled silk." It sounds like the original sentence and has the same number of words and syllables, but is slightly altered to give it an entirely new meaning. The phrase "Phone cry clover filled bilk" should not be used because it doesn't make sense as a sentence. Also, "Don't cry over spoiled mints" wouldn't work either because it doesn't sound similar enough to the original sentence. They also need to be careful to modify the cliché enough to make it interesting. You also may want to remind them to keep their jokes appropriate.

Next have students write the setup to the joke, creating a story in which the modified cliché (the punch line) is the conclusion. Like a proof, in which every premise is necessary to reach the conclusion, every sentence in the setup should lead the reader directly to the punch line. Also like a proof, in which every part of the conclusion is explained, the story should justify every part of the punch line.

Explain that when they are done, students will have successfully written a joke in a form analogous to a paragraph-style proof. While it may be funny to extend the joke when telling it to make it longer and heighten the imagery and suspense, this joke should be succinct like a proof. If done correctly, every sentence in the setup will be required. That is, the omission of any sentence would cause the punch line to make less sense.

Ask groups to exchange and edit each other's jokes to make them more concise. The goal is to have the shortest collection of premises reach the same conclusion, while still justifying every element of the conclusion.

When they have finished, have the students read the jokes to the class. Depending on the size of your class or the amount of time you have, you may want to limit the joke reading to a few volunteer groups. Reiterate that a proof is different from problem solving in that a proof is a presentation tool and that the conclusion is known before the proof is written.

### Questions for Students

 How are these jokes analogous to mathematical proofs? [The punch line of the joke is like the conclusion of a proof. In the joke, the supporting statements are needed for the punch line to make sense. Similarly, the conclusion is justified by supporting statements in a proof.] In what ways is a proof different than solving a problem? [A proof is a presentation tool to make an argument to convince someone of something. Before creating a proof, you already know what is to be proven. When solving a problem, you do not know the answer ahead of time. Also, while it is useful to show your work when solving a problem (especially when it is to be graded), it is not required for the process of solving a problem to write in a way that someone else should be convinced that you were correct.] How is your joke like a proof? [The statements in the joke were necessary to justify the punch line in the same way that the statements in a proof are necessary to justify the conclusion. Also, when writing the joke, the punch line was determined before the story was written to justify it. Analogously, in a proof the conclusion is known before the justifying statements are identified.] Were you able to recognize the original cliché? Was every element in the conclusion justified? Could the joke have been more concise? What improvements can be made to the joke?[Answer to these questions will vary. Use these questions to start discussions after students read their jokes.]

### Assessment Options

 Give the students an example of a paragraph-style proof on a familiar mathematical topic with an extra statement added that is not needed to support the conclusion. Ask the students to identify the extra statement. You also could delete a statement and have students identify parts of the conclusion that were not justified. Give the students a joke and ask them to identify sentences which are superfluous and elements of the punch line that should have been justified but were not.

### Extensions

 Logical arguments are similar to proofs as well. Ask students to analyze some of the arguments that they encounter in their daily lives to evaluate whether the conclusions follow from the premises. These arguments can be found in political statements, advertisements, etc. Most jokes have an identifiable punch line and have a setup which justifies the punch line, but often they contain additional information that, while it arguably makes the joke more interesting by adding imagery, isn't necessary for the punch line to make sense. Find one of your favorite jokes, identify the punch line, and analyze it for succinctness. Here is an example using a joke by George Carlin: "I have six locks on my door, all in a row. When I go out, I lock every other one. I figure no matter how long somebody stands there picking the locks, they are always locking three of them." [The information that the locks were "all in a row" wasn't necessary for the punch line "They are always locking three of them."]

### Teacher Reflection

 Were students able to make the connection between jokes of this style and mathematical proofs? If not, what changes could be made to make the connection clearer? Did your lesson fit into your curriculum appropriately? Did students have sufficient understanding of proofs prior to this lesson and a better understanding of proofs as a result of it? If not, would there have been a better place in your curriculum to put this lesson? Was students' level of enthusiasm/involvement high or low? Explain why. Did you set clear expectations so that students knew what was expected of them? If not, how can you make them clearer? Was your lesson developmentally appropriate? If not, what was inappropriate? What would you do to change it? Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?
 This lesson was prepared by Martin Funk as part of the Illuminations Summer Institute.

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