What is MAT 220? Dana C. Ernst Northern Arizona University Mathematics & Statistics Department http://danaernst.com Friday Afternoon Mathematics Undergraduate Seminar October 3, 2014 D.C. Ernst What is MAT 220? 1 / 17 

Claims 1. The world is changing faster and faster. If I were to teach you all known information, it wouldn’t last a week. D.C. Ernst What is MAT 220? 2 / 17 

Claims 1. The world is changing faster and faster. If I were to teach you all known information, it wouldn’t last a week. 2. An education must prepare a student to ask and explore questions in contexts that do not yet exist. That is, we need individuals capable of tackling problems they have never encountered and to ask questions no one has yet thought of. D.C. Ernst What is MAT 220? 2 / 17 

Claims 1. The world is changing faster and faster. If I were to teach you all known information, it wouldn’t last a week. 2. An education must prepare a student to ask and explore questions in contexts that do not yet exist. That is, we need individuals capable of tackling problems they have never encountered and to ask questions no one has yet thought of. Lofty Goal Transition students from consumers to producers! D.C. Ernst What is MAT 220? 2 / 17 

What is MAT 220? In response to assessment reports identifying weaknesses in communication and reasoning of junior and senior mathematics majors, we have developed a 3-credit semester-long course that is required for all first-year mathematics majors. The course is called MAT 220: Introduction to Mathematical Reasoning. D.C. Ernst What is MAT 220? 3 / 17 

What is MAT 220? In response to assessment reports identifying weaknesses in communication and reasoning of junior and senior mathematics majors, we have developed a 3-credit semester-long course that is required for all first-year mathematics majors. The course is called MAT 220: Introduction to Mathematical Reasoning. The focus of this course is on reasoning and communication through problem solving and written mathematical arguments in order to provide students with more experience and training early in their university studies. D.C. Ernst What is MAT 220? 3 / 17 

What is MAT 220? (continued) The goal is for the students to work on interesting yet challenging multi-step problems that require almost zero background knowledge. The hope is that students will develop (or at least move in the direction of) the habits of mind of a mathematician. The problem solving of the type in the course is a fundamental component of mathematics that receives little focused attention elsewhere in our program. There will be an explicit focus on students asking questions and developing conjectures. D.C. Ernst What is MAT 220? 4 / 17 

What is MAT 220? (continued) The goal is for the students to work on interesting yet challenging multi-step problems that require almost zero background knowledge. The hope is that students will develop (or at least move in the direction of) the habits of mind of a mathematician. The problem solving of the type in the course is a fundamental component of mathematics that receives little focused attention elsewhere in our program. There will be an explicit focus on students asking questions and developing conjectures. In addition to helping students develop procedural fluency and conceptual understanding, we must prepare them to ask and explore new questions after they leave our classrooms—a skill that we call mathematical inquiry. D.C. Ernst What is MAT 220? 4 / 17 

Let’s play with a few problems that students will encounter in MAT 220. D.C. Ernst What is MAT 220? 5 / 17 

Let’s play with a few problems that students will encounter in MAT 220. Warning! I’m not going to tell you what any of the answers are. In fact, there is one that I still haven’t figured out. D.C. Ernst What is MAT 220? 5 / 17 

Let’s play with a few problems that students will encounter in MAT 220. Warning! I’m not going to tell you what any of the answers are. In fact, there is one that I still haven’t figured out. Problem 1 Three strangers meet at a taxi stand and decide to share a cab to cut down the cost. Each has a different destination but all are heading in more-or-less the same direction. Bob is traveling 10 miles, Sally is traveling 20 miles, and Mike is traveling 30 miles. If the taxi costs $2 per mile, how much should each contribute to the total fare? D.C. Ernst What is MAT 220? 5 / 17 

Let’s play with a few problems that students will encounter in MAT 220. Warning! I’m not going to tell you what any of the answers are. In fact, there is one that I still haven’t figured out. Problem 1 Three strangers meet at a taxi stand and decide to share a cab to cut down the cost. Each has a different destination but all are heading in more-or-less the same direction. Bob is traveling 10 miles, Sally is traveling 20 miles, and Mike is traveling 30 miles. If the taxi costs $2 per mile, how much should each contribute to the total fare? Remark In preparation for their book, The Five Elements of Effective Thinking, Burger and Starbird encountered “I don’t know” as the most common response. D.C. Ernst What is MAT 220? 5 / 17 

Problem 2 Multiply together the numbers of fingers on each hand of all the human beings in the worldapproximately 7 billion in all. What is the approximate answer? D.C. Ernst What is MAT 220? 6 / 17 

Problem 2 Multiply together the numbers of fingers on each hand of all the human beings in the worldapproximately 7 billion in all. What is the approximate answer? Remark This problem appeared in a recent Math Horizons article. D.C. Ernst What is MAT 220? 6 / 17 

Problem 3 An ant is crawling along the edges of a unit cube. What is the maximum distance it can cover starting from a corner so that it does not cover any edge twice? D.C. Ernst What is MAT 220? 7 / 17 

Problem 4 A mouse eats his way through a 3 × 3 × 3 cube of cheese by tunneling through all of the 27 1 × 1 × 1 sub-cubes. If she starts at one corner and always moves to an uneaten sub cube, can she finish at the center of the cube? D.C. Ernst What is MAT 220? 8 / 17 

Problem 5 Imagine you have 25 pebbles, each occupying one square on a 5 × 5 chess board. Tackle each of the following variations of a puzzle. D.C. Ernst What is MAT 220? 9 / 17 

Problem 5 Imagine you have 25 pebbles, each occupying one square on a 5 × 5 chess board. Tackle each of the following variations of a puzzle. 1. Variation 1: Suppose that each pebble must move to an adjacent square by only moving up, down, left, or right. If this is possible, describe a solution. If this is impossible, explain why. D.C. Ernst What is MAT 220? 9 / 17 

Problem 5 Imagine you have 25 pebbles, each occupying one square on a 5 × 5 chess board. Tackle each of the following variations of a puzzle. 1. Variation 1: Suppose that each pebble must move to an adjacent square by only moving up, down, left, or right. If this is possible, describe a solution. If this is impossible, explain why. 2. Variation 2: Suppose that all but one pebble (your choice which one) must move to an adjacent square by only moving up, down, left, or right. If this is possible, describe a solution. If this is impossible, explain why. D.C. Ernst What is MAT 220? 9 / 17 

Problem 5 Imagine you have 25 pebbles, each occupying one square on a 5 × 5 chess board. Tackle each of the following variations of a puzzle. 1. Variation 1: Suppose that each pebble must move to an adjacent square by only moving up, down, left, or right. If this is possible, describe a solution. If this is impossible, explain why. 2. Variation 2: Suppose that all but one pebble (your choice which one) must move to an adjacent square by only moving up, down, left, or right. If this is possible, describe a solution. If this is impossible, explain why. 3. Variation 3: Consider Variation 1 again, but this time also allow diagonal moves to adjacent squares. If this is possible, describe a solution. If this is impossible, explain why. D.C. Ernst What is MAT 220? 9 / 17 

Problem 5 Imagine you have 25 pebbles, each occupying one square on a 5 × 5 chess board. Tackle each of the following variations of a puzzle. 1. Variation 1: Suppose that each pebble must move to an adjacent square by only moving up, down, left, or right. If this is possible, describe a solution. If this is impossible, explain why. 2. Variation 2: Suppose that all but one pebble (your choice which one) must move to an adjacent square by only moving up, down, left, or right. If this is possible, describe a solution. If this is impossible, explain why. 3. Variation 3: Consider Variation 1 again, but this time also allow diagonal moves to adjacent squares. If this is possible, describe a solution. If this is impossible, explain why. 4. Variation 4: What about other size boards? D.C. Ernst What is MAT 220? 9 / 17 

Problem 6 Suppose you have 12 coins, all identical in appearance and weight except for one that is either heavier or lighter than the other 11 coins. Devise a procedure to identify the counterfeit coin in only 3 weighings with two-pan scale. D.C. Ernst What is MAT 220? 10 / 17 

Problem 6 Suppose you have 12 coins, all identical in appearance and weight except for one that is either heavier or lighter than the other 11 coins. Devise a procedure to identify the counterfeit coin in only 3 weighings with two-pan scale. Consider the situation in the previous problem, but suppose you have n coins. For which n is it possible to devise a procedure for identifying the counterfeit coin in only 3 weighings with a two-pan scale? D.C. Ernst What is MAT 220? 10 / 17 

Problem 7 We have two strings of pyrotechnic fuse. The strings do not look homogeneous in thickness but both of them have a label saying 4 minutes. So we can assume that it takes 4 minutes to burn through either of these fuses. How can we measure a one minute interval? D.C. Ernst What is MAT 220? 11 / 17 

Problem 8 I have 10 sticks in my bag. The length of each stick is an integer. No matter which 3 sticks I try to use, I cannot make a triangle out of those sticks. What is the minimum length of the longest stick? D.C. Ernst What is MAT 220? 12 / 17 

Problem 9 There are n very intelligent lions on an inhabited island. They are very hungry because they ate everything they could. So whenever there is a new food source, the closest lion eats all the available food and falls asleep to digest. A sleeping lion becomes prey. What happens if a helicopter drops a dead gazelle onto the island? D.C. Ernst What is MAT 220? 13 / 17 

Problem 10 The first vote counts of the papal conclave resulted in 33 votes each for candidates A and B and 34 votes for candidate C. The cardinals then discussed the candidates in pairs. In the second round each pair of cardinals with differing first votes changed their votes to the third candidate they did not vote for in the first round. The new vote counts were 16, 17 and 67. They were about to start the smoke signal when Cardinal Ordinal shouted “wait”. What was his reason? D.C. Ernst What is MAT 220? 14 / 17 

Problem 11 In a PE class, everyone has 5 friends. Friendships are mutual. Two students in the class are appointed captains. The captains take turns selecting members for their teams, until everyone is selected. Prove that at the end of the selection process there are the same number of friendships within each team. D.C. Ernst What is MAT 220? 15 / 17 

Problem 12 Suppose you have 6 toothpicks that are exactly the same length. Can you arrange the toothpicks so that 4 identical triangles are formed? D.C. Ernst What is MAT 220? 16 / 17 

Question What do you reasonably expect you will remember from your courses in 20 years? D.C. Ernst What is MAT 220? 17 / 17