What is MAT 220?

In this FAMUS talk, I will be speaking about a new course that we are offering for the first time next semester called "Introduction to Mathematical Reasoning." However, what I'll really be doing is posing problems that require clever solutions and letting the audience play around with them. Come join us! If you like puzzles, it'll be a lot of fun. All the problems don't require any sophisticated background knowledge. If you enjoy the talk, you might consider taking the course, which I will be teaching in the spring.

This talk was given at the Northern Arizona University Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) on Friday, October 3, 2014.

Transcript

1. 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
   

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2. 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
   

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3. 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
   

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4. 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
   

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5. 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
   

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6. 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
   

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7. 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
   

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8. 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
   

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

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10. 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
    

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11. 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
    

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12. 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
    

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13. 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
    

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14. 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
    

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15. 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
    

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16. 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
    

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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
    

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18. 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
    

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19. 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
    

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20. 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
    

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21. 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
    

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22. 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
    

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23. 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
    

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24. 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
    

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25. 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
    

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26. 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
    

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27. 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
    

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28. 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
    

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29. 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
    

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

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