In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission again comes from my former student Zacquiri Rutledge. His topic, from probability: expressing a probability as a fraction and as a percentage.
Probability involves any kind of situation where the outcomes are known, but are not 100% certain. Examples of this could be things as simple as flipping a coin, to trying to predict the next card while playing blackjack. However, for a student who is just beginning to understand probability, I thought a word problem involving the rolling of a die would be just challenging enough. “You and your best friend have been playing Monopoly for hours. After several times around the board, you own a large amount of the properties and your friend is nearly bankrupt. In fact, your friend does not have enough money to survive landing on your Boardwalk property in the corner of the board. In order for your friend to land on this space, he/she would need to roll a 12. First, calculate the odds that one die will roll a 6 and express it both as a fraction and as a percentage. Then, calculate the odds that both dice will roll a 6 and express it both as a fraction and as a percentage.”
After learning the basics of probability as fractions and percentages, students can then begin to learn about how to change them into “odds”, or the probability of a series of actions. Since probability is simply the ratio between the desired number of outcomes and the total number of outcomes, only knowing how to write ratios will not help the student in calculating odds. By changing the probability ratio into a fraction, this will allow the student to easily apply the multiplication principle to a series of actions to find the larger probability ratio. From there the student will be able use previous experience of changing probability into a percentage to state how likely or unlikely a situation is.
Probability is found in many areas of culture and science. However, one of the most widely known forms of probability is in gambling. People all over the world gamble for money, for fun, and sometimes even for sport. A few of the common games people play in casinos are Roulette, Blackjack, and Texas Hold’em. Each one of these games has its own way in which it uses probability to make it more difficult for a player to make money.
To play a game of Roulette, all a player has to do is bet on which number, color, or set of numbers they think might win. On the table are the numbers 00, 0, and 1 through 36. 00 and 0 are both their own color, but 1 through 36 alternate between red and black. After the player bets, the rest of the game is controlled by “The House” or the casino. A ball is placed on a rotating circle that has all of the numbers listed one time on it with a slot in the middle for each one. As the ball rolls, it slows down and drops into one of these holes. This is where probability comes into play. Depending on the player’s bet, they have an x/38 chance of winning. If they select one number, it gives 1/38 or 2.6% chance; for 1-12 its 6/19 or 31.5%. By using a combination of bets, a player can increase their probability of winning by selecting more than one number. Due to Roulettes simplicity, it would make a good beginning topic for a student who is beginning to learn about probability.
Blackjack is game that uses cards to determine who wins or loses, instead of a ball and a wheel. The object of this game is to get as close to 21 as possible without going over, as well as attempting to beat the hand “The House” is holding. While there are a lot of calculations that must go in when calculating probability in a game of blackjack, it is possible to do it on a smaller scale. To do so, a player would have to look at what cards had come up in the past and then look to see what card it is that they need. Since there are only 4 of each card in the deck, assuming the player nor “The House” is holding the card he/she needs, the probability would be 4/(52 – y), where y is the number of cards that have already been shown and are not the card the player needs. Texas Hold’em uses this same kind of idea, but instead is used when playing against other people rather than against “The House”. This version of poker has become so well known, it is featured on an ESPN sports channel, where people play in a live tournament and compete for millions of dollars. What is significant about this channel is that they show what cards each of the players are holding as well as what cards are on the table. Then, once every player’s cards are seen, the channel shows on one side of the screen a player’s percent chance of winning. This percent is calculated by an analysis of what cards are in the player’s hand, what cards are in everyone else’s hands, and what cards are on the board. After analyzing the cards, it is then calculated what the probability is that the best possible cards the player needs are going to come up. Even though only basic probability is being used here, this is still on a much higher difficulty due to the amount of numbers that must be processed. However, given its complexity and how the probability can change by the turn of a card can make both Blackjack and Texas Hold’em an interesting topic for a student of probability.