This year, as I have moved from a role as classroom teacher to a math coach, I find myself sharing some of favorite lessons and approaches with my colleagues. Taking a cue from Oprah, this is the first of a series of my “Favorite Things”, hopefully without the rampant screaming or random things hidden under your seat.
The Monty Hall Problem on NLVM
The famous “stick or switch” problem is one which will generate tremendous discuss in a classroom. The premise is simple: you are offered the choice of one of three doors, behind one of which there is a great prize. After the host reveals a non-winning door, you are offered the chance to switch your guess. Should you stick or switch? The problem rose to a next level of fame with fierce debate in the “Ask Marilyn” article by Marilyn Vos Savant in Parade Magazine. After having included the problem in my prob/stat course for any years, the discussions rose to a crest as the problem was featured in the movie “21”, with even students I did not know stopping me in the hallways to ask me about its logic (note, this video has a number of ads on it):
As an emerging stats teacher, I would play the game using cardboard doors and hand-drawn pictures of cars and goats. In later years, I used an applet at the National Library of Virtual Manipulatives to play the game with my class. Look under Data Analysis and Probability for “Stick or Switch”. What I really like about this applet is that it will play the game 100 times quickly and display the results.
There are many ways to explain this game to your students. One straight-forward approach is to have students attempt to write out the sample space (all possible outcomes) for this game. For example:
Car behind 1. Pick 1. Shown 2. Stay. WIN
Car behind 1. Pick 1. Shown 2. Switch. LOSE
Car behind 1. Pick 1. Shown 3. Stay. WIN
Car behind 1. Pick 1. Shown 3. Switch. LOSE
Patterns in Pascal’s Triangle
In one of my favorite lessons each year, the many, many patterns in Pascal’s Triangle were discussed. This would come after an investigation of combinations, and students would often recognize that each entry in the triangle is actually a combination, along with other patterns like the counting numbers or perhaps the triangular numbers. But how about the Fibonacci sequence and the “square” numbers? What’s the “hockey stick” theorem?
Over the years, I have used a number of sites to display the triangle and invite students to share their ideas. One nice resource which explains patterns in the triangle can be found here.
The cherry on the Pascal sundae would come when I would invite students to approach the board and circle all of the even numbers. Often, only 15 rows or so would be visible, so the chance to make a generalization is rich and there to discuss. How would this look if we colored 100 rows, 200 rows, 10,000 rows? And do we only get patterns for even numbers? How about multiples of 3, or 5, or 24? Check out this site from Jill Britton, which includes an applet that will color multiples in Pascal’s Triangle. Watch the expressions from your students when you reveal multiples of 5 in the first 128 rows of the triangle:
Try different multiples and observe the great symmetries. Prime numbers like 17 and 23 provide the most surprising results. Digging deeper, start a discussion of the modulus function with your class (it’s just the remainder, no big deal….). What happens when we not only color multiples, but also color the remainders similarly. From the Centre for Experimental and Constructive Mathematics, we get this great applet, which will color remainders based on specifications you provide:
While I have been using the Monty Hall problem and Pascal’s Triangle in my classrooms for many years, this last resource is fairly new to me. Electric Teaching by David Johns is an excellent site with an effective interface which is ideal for SMART Boards. The site challenges students to match-up equations with data tables and their graphs.
The site has problems for linear, quadratic and trigonometric functions, along with conics and even problems which include derivatives. What a great way to not only have students participate, but verbalize their thoughts as they identify the matches. The YouTube channel has additional resources from Dave, including a tutorial of the Electric Teaching site.