The Sum of our Knowledge

Many times, we use the same old icebreakers at the beginning of a class:  go around the room, introduce yourself, your major, a hobby or something interesting about yourself . What if instead, we could get our students working together and engaging in mathematical problem-solving from the get-go? Today’s post is the first in a four-part series where we'll present several ideas and strategies to accomplish this, most of which are appropriate for any level of mathematics course. Review Dominos We have found that this activity works great in many situations-- Our favorites uses: An icebreaker on the first day of class, while also reviewing material from a previous class. Low stakes group work to practice basic computation relating to newly learned material on an average class day A group warm-up, reviewing material from a previous day that is crucial to know before beginning class. A small-group project before an exam review. This activity requires a small amount of preparation up fro

Long ago, damsels in shining armor battled dragons and rescued knights in distress on a daily basis.  This ultimately lead to the complete extinction of dragons, a major catastrophe for the business of thrilling heroics! Although many of the surviving stories don't stress it, the rescues frequently involved logical thinking and problem-solving. Once upon a time, a knight was captured by an evil dragon and imprisoned in a castle that was surrounded by a square moat. The moat was infested with extremely  hungry and alligators, piranhas, and -- just for good measure -- a few sharks.  The moat was exactly 20 feet across and had no drawbridge or any other means of safely crossing. After a long time, a damsel appeared, carrying two wooden beams, each 19 feet 8 inches long and 8 inches wide. The beams were strong enough to walk across, but wouldn't reach across the moat.  The knight was despondent.  He'd already waited for weeks to be rescued.  Now he'd have to wait more, unti

Imagine a block of cheese has been cut into thirds along each face. A friendly neighborhood mouse comes along and starts eating one of the corners. The mouse will eat the entire block of cheese, always progressing to an adjacent cube using only up, down, or cardinal directions (North, South, East, West;  sorry, no diagonals!).  Challenge: Can you find a path following these rules so that our mouse friend finishes by eating the center cube of cheese last? Hint: A diagram showing the block cut into cubes -- and another showing the faces made up of cubes --- may help you visualize the problem. Update: 1/25/21: read below for the solution.

This problem, along with a couple of others, was inspired by the work of John H. Conway* and appeared in  the October 15, 2020 edition of Quanta Magazine . Let a, b, c, d, e, f, g, h, i, j be the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in some order. Each digit appears exactly once. (For those that want a more precise mathematical phrasing, there is a one-to-one correspondence between the set {a, b, c, d, e, f, g, h, i, j} and the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, or better yet, there is a bijection between the two sets.)  We form other integers by concatenating digits, so for example, 'ab' is a two-digit number where 'a' is the tens digit and 'b' is the ones digit. If abcdefghij is a 10-digit number with the following properties: a is divisible by 1 (OK, that much is obvious) ab is divisible by 2 (So, b must be even) abc is divisible by 3 (Do you know the test for divisibility by 3?) abcd is divisible by 4 abcde is divisible by 5 abcdef is divisible by 6 abcdefg

COME ON DOWWWWNNNN!!!!!!   You’ve been chosen to be a contestant on Let’s Make a Deal!   You’re now standing in front of three doors -- behind one of these doors is a car, and behind each of the other two is a goat. Monty, the host, says, “Pick a door! If it has the car behind it, you’ll drive it home today!”  You choose door #1, and feel really good about your choice.  The host then opens door #3,  which has a goat behind it, and asks “Do you want to switch doors?”  Question: What is the probability that you will win the car now that you know there was a goat behind door #3? Should you switch? Assumptions: Before the show, the car was equally likely to be placed behind any of the three doors. Once you make your selection of door #1, Monty will choose to reveal what is behind one of the other two doors, but will definitely not show you where the car is. Update: 11/19/20: read below for the solution.

I was honored earlier this month to be recognized by Stephen Wolfram as one of the recipients of the Wolfram Innovator Award at the annual Wolfram Technology Conference.   (Here is the official announcement of the Award Winners . ) This award is given annually to educational and scientific pioneers that use Wolfram’s Mathematica ™ software to bring the future to today.   The awarded project was the result of a large number of people at the University of Minnesota working to lower the cost of college attendance by replacing expensive textbooks with technology better suited to the needs of 21 st century students. In addition, this technological innovation has improved the way we deliver instruction at MathCEP . The project, now known as the Minnesota Online Learning System (MOLS), delivers assessments to students using the Mathematica platform. It replaced a publisher’s automated homework system initially, and then expanded to house the University of Minnesota’s math placement exam.