Mutilated Checkerboard Problem

MASTA Proof and Reasoning 9-12 Module Handout - Page 1 of 11 Mathematical Proof Definition of Mathematical Proof

With your colleagues at your table, discuss the similarities and differences of “proving” or “proof” in each of the following subject areas.

 Biology

 Philosophy

 Mathematics

 Psychology

After the different tables share their ideas, finish the following definition:

A mathematical proof is…

Quote: “The unique feature that sets mathematics apart from other sciences, from philosophy, and indeed from all other forms of intellectual discourse, is the use of rigorous proof.” (Krantz, 2007) MASTA Proof and Reasoning 9-12 Module Handout - Page 2 of 11 Mathematical Proof

Mutilated Checkerboard Problem MASTA Proof and Reasoning 9-12 Module Handout - Page 3 of 11 Mathematical Proof

Compare and contrast Scientific Theory with Mathematical Theory.

Write down the mathematical proof that solves the mutilated checkerboard problem. MASTA Proof and Reasoning 9-12 Module Handout - Page 4 of 11 Mathematical Proof

Prime Number Generator

n f(n) Prime? MASTA Proof and Reasoning 9-12 Module Handout - Page 5 of 11 Mathematical Proof Carpenter’s Levels of Justification

Appeal to Authority

Justification by Example

Generalizable Argument

After the following activity, return to this sheet and explain how “justification by example” can be further subdivided. MASTA Proof and Reasoning 9-12 Module Handout - Page 6 of 11 Mathematical Proof Chord-Chord Power Theorem MASTA Proof and Reasoning 9-12 Module Handout - Page 7 of 11 Mathematical Proof Methods of Proof

Proof by intimidation: "Trivial."

Proof by vigorous handwaving: Works well in a classroom or seminar setting.

Proof by omission: "The reader may easily supply the details"

Proof by cumbersome notation: Best done with access to at least four alphabets.

Proof by obfuscation: A long plotless sequence of true and/or meaningless statements.

Proof by appeal to intuition: Cloud-shaped drawings frequently help here.

Proof by general agreement: "All in favor?..."

Proof by imagination: "Well, we'll pretend it's true..."

Proof by convenience: "It would be very nice if it were true, so..."

Proof by necessity: "It had better be true, or math wouldn’t work.”

Proof by plausibility: "It sounds good so it must be true."

Proof by accident: "Hey, what have we here?!"

Proof by profanity: (example omitted)

Proof by lost reference: "I know I saw it somewhere..."

Proof by calculus: "This proof requires calculus, so we'll skip it."

Proof by lack of interest: "Does anyone really want to see this?"

Proof by clever variable choice: "Let A be the number such that this proof works..."

Proof by stubbornness: "I don't care what you say-it is true!" MASTA Proof and Reasoning 9-12 Module Handout - Page 8 of 11 Mathematical Proof Obstacles to Proof

Discuss at your table: what do students find intimidating about mathematical proofs?

A Proof that Proves

Prove: The sum of the first n positive integers is n(n+1)/2.

For n=1 it is true since 1=1(1+1)/2.

Assume it is true for some arbitrary k : S(k)=k(k+1)/2. Then:

S(k +1) = S(k) + (k +1) = k(k +1)/2 + (k +1) = (k +1)(k + 2)/2

Hence the statement is true for k+1 if is true for k. By induction it is true for all n.

MASTA Proof and Reasoning 9-12 Module Handout - Page 9 of 11 Mathematical Proof A Proof that Proves and Explains MASTA Proof and Reasoning 9-12 Module Handout - Page 10 of 11 Mathematical Proof Language and Logic

Determine whether each of the following statements is true or false. Explain your answers. a. All the white objects are squares. b. All the square objects are white. c. No square objects are white. d. There is a white object that is larger than every gray object. e. Every gray object has a black object next to it. f. There is a black object that has all the gray objects next to it. g. All objects that are not small are not gray. MASTA Proof and Reasoning 9-12 Module Handout - Page 11 of 11 Mathematical Proof Further Example

Write down the sentence which exactly expresses what it means for the sentence “All mathematicians wear glasses” to be false.

For your future reference, write down the student data about responses to the above exercise.

Suggestions for Instruction

1.As time permits(!), continue to ask your students to show their reasoning, and give them feedback.

2.Logic Puzzles help students develop mathematical reasoning skills (see Day 2) and work with precise language.

3.Activities such as the one with the shapes on the previous page can help students learn the difference between “all,” “every,” etc.