MASTER PROOFS through GAMES
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1 MASTER PROOFS through GAMES NCTM Annual Conference 2018 Washington D.C. Presented by Peter Sell and Paul Winston Peter and Quinn Paul and KenKen inventor Tetsuya Miyamoto
2 DIGITS (Mastermind with Numbers) Presented by Peter Sell Math for America, Master Teacher HS for Health Professions and Human Services New York City Department of Education
3 DIGITS D = Digits: This means that a digit in the guess is in the number P = Place: This means one of the digits is in the correct place in the number. If the secret number was 290, this is how we would score it. GUESS D (Digits) P (Place)
4 Copy the table and score 815 Guesses D (Digits) P (Place)
5 Copy the table and score 815 Guesses D (Digits) P (Place)
6 From each individual guess, what can we conclude? Guess D P Guess D P Guess D P
7 What can we conclude? Guess D P
8 What can we conclude? Guess D P
9 What can we conclude? Guess D P
10 What can we conclude? Guess D P And now? Guess D P
11 Class Challenge The class will try to determine the number that I have selected. See how many guesses it would take. Line GUESS DIGITS PLACE
12 Sample Puzzle: Can you determine the number?
13 Sample Proof
14 Determine the numbers on your puzzle. Justify your answer to your partner.
15 Extensions Given a set of clues with a mistake. Can students locate the contradiction? Can students devise a set of guesses that can guarantee them the right answer on the 9 th guess? 8 th guess? 7 guesses? (for 3, 4 digit numbers) Have the students work out the number of possibilities remaining after a certain number of guesses. Is there an optimal strategy? Try it out. Combinatorics & Probability. Design a program (for CS students) for the game.
16 Digits as an introduction to Proofs
17
18 Ken Ken Proofs--background In New York State, we used to teach logic proofs in the first two weeks of Geometry in order to get students thinking logically, and understanding the form of a Euclidean two-column proof. There were only 6 laws of logic used, in addition to Givens. Logic proofs are no longer part of the curriculum. Ken Ken, is a square number puzzle. In an n x n Ken Ken, the numbers 1 through n appear once each in every column and row. Cages separate some cells from others, and require the mathematical operation indicated. When no operation is indicated, the cage value is given.
19 Ken Ken proofs--utility With a small number of valid reasons, Ken Ken proofs offer the student a chance to think logically, explain their process, and to facilitate much more challenging Euclidean proofs (i.e. low entry, high ceiling, easy to differentiate.)
20 An easy 3 x 3 with addition only; ideal for 1 st proof.
21 Solutions to our first puzzle. In this puzzle, what is the sum of the numbers in each row? In each column? How about in a 4 x 4? A 5 x 5? This is puzzle 3 from KenKenPuzzle.com Reprinted with permission.
22 A first proof Statement Reason 1) B3 =1 1) Given 2) C1 =1 2) Given 3) A2 + A3 = 4 3) Given 4) A2 and A3 = 1,3 or 3,1 4) No other way (to make 4) with our numbers (NOW WON.) 5) A3 = 3 5) 1 is already used in Column C 6) A2 = 1 6) Steps #4 and 5 (If A2 is not 3, it must be = 1.) 7) A1 = 2 7) Last Cell in Row (LCIR) 8) B1 = 3 8) Last Cell in Column (LCIC) 9) B2 =2 9) LCIR 10) C2 = 3 10) LCIC 11) C3 = 2 11) LCIC or LCIR
23 Alternate proof Statement Reason 1) B3 =1 1) Given 2) C1 =1 2) Given 3) A2 + A3 = 4 3) Given 4) A1+ A2 + A3 = 6 4) The sum of the numbers of any row of a 3 by 3 is 6. 5) A1 = 2 5) Subtracting (St #4 St #3) 6) B1 = 3 6) LCIC 7) B2 = 2 7) LCIR 8) C2 = 3 8) 1, 2 already used in column, row. 9) A2 = 1 9) LCIC 10) A3 = 3 10) LCIR 11) C3 = 2 11) LCIR or LCIC
24
25 Extensions Grid size: 3 x 3, 4 x 4, 5 x 5, etc Operations: addition only, addition/subtraction, 4 basic operations, no operations specified. Number Systems: Counting Numbers, Negative Numbers. Fill in the blank: Statements provided & reasons required, vice-versa Challenge your students to create a KenKen puzzle with a unique solution. Competitions: Class, School, or National
26
27 From KenKenPuzzle.com Reprinted with permission.
28 From KenKenPuzzle.com Reprinted with permission.
29
30 Thank you! Peter Sell Paul Winston
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