Subject: Mathematics Teacher: Your name Grade Level: 11 Date: Topic: Time (min): 70 mins Learning Goals 1. Students will be able to represent probabilities as a decimal, fraction, and percent. 2. Students will be able to recognize the probability of an event occurring using manipulatives. 1. Ministry Expectations Strand: Data Management (Applying Probability) Specific Expectation(s): 2.1 identify examples of the use of probability in the media and various ways in which probability is represented (e.g., as a fraction, as a percent, as a decimal in the range 0 to 1) 2.2 determine the theoretical probability of an event (i.e., the ratio of the number of favourable outcomes to the total number of possible outcomes, where all outcomes are equally likely), and represent the probability in a variety of ways (e.g., as a fraction, as a percent, as a decimal in the range 0 to 1) 2.3 perform a probability experiment (e.g., tossing a coin several times), represent the results using a frequency distribution, and use the distribution to determine the experimental probability of an event 2. Pre-Assessment What will students have to know before they can do this lesson? It is important that you understand what they need to know before you teach them something new Before the lesson, students should know: 1. The relationship between fractions, decimals, and percents. 2. The definition of probability. 3. Know what a frequency distribution is and how to construct one. 3. Required Resources There will be four stations each with two sets of manipulatives. (Two decks of cards, two sets of spinners, two sets of coins, two sets of polyhedral dice). Split the students into groups of four (a good way would be to deal out cards and all aces, 2’s, 3’s, etc., are in a group). Then ask them to number themselves 1-4 (numbered heads). If your number corresponds to your station, you are the scribe this time. Each station will have a set of instructions (see handouts below). At the end of the activity, there is a “slide” that should be shown on a projector (if not, have enough handouts, one per group). 4. The Main Lesson a) Agenda 1. Bell-work problems. 2. Minds On / Introduction 3. Activity 4. Consolidation b) Bell Work Questions 1. Converting fractions to decimals and percents. 2. Written on the board: I owe [another teacher] half of the pizza I brought to lunch. However, he/she said that I don’t have to do it if could represent my debt in four different ways mathematically. What do I do? c) Introduction – Minds On Total Time: 5 min List a bunch of statistics. Your probability of dying in a car accident (18 585 : 1), hit by lightning (576 000 : 1), attacked by a shark (300 000 000 : 1), the probability to pass this math course (calculate as passing students / total number of students). How would you determine these probabilities? Students work individually. c) Teaching Plan - Action Total Time: 60 Split them up into groups by using a deck of cards. All aces are in one group, all two’s are in another group, etc. Separate them around the room and then number every person in a group from 1 through 4. You have ten minutes at each station to complete the work-sheet at each station. Use http://www.online-stopwatch.com/ to put a stopwatch on the screen When groups are down at their fourth station, stop them and tell them that they aren’t done yet and they need to pay attention for more instructions Put on a SMART-slide the new instructions as well as a stop-watch widget. d) Consolidation & Assessment Total Time: 5 Have students label their work and hand it in (assessment). Bring it back to the introduction question: how do you go about getting the probabilities for those kinds of statistics. 5. Optional Home Activity Tell the students that you’ve emailed to all of them an applet that uses probability to calculate how trees burn down in a forest fire. http://www.shodor.org/interactivate/activities/FireAssessment/ Or you can generate a probability all of your own or at least, the way you would do it. For example, what’s the probability the toilet seat is going to be up when you get home? Station Instructions Attached Below Polyhedral Dice 1. What’s the probability of rolling a one on any of the dice? What affects this? 2. What is the average (or expected) value for one of the dice? Note: Many people get this wrong! The expected value doesn’t need to be a number that appears on one of the faces! 3. What is the average (or expected) value for rolling any number dice? For example, two six-sided dice or a six-sided, a twenty-sided, and two twelve-sided dice. 4. With the twenty sided die, what is the probability to roll a 16 or higher? 5. Consider this game: It costs $5 to play. You get to roll up to 4 20-sided dice and add that number to make your score. Your score times 10 is how much money you make back in cents. You can choose not to roll any of your twenty-sided die and instead roll a 4-sided die. 1. If you do this, you can multiply your score by this new amount. 2. Doing this costs you an extra $4. Why or why not might this game be unfair? Spinner 1. Pick a spinner. What is the chance to get any of the values? 2. What is the relationship between the size of a section and the probability to get it? 3. What is the probability of you spinning one value on a spinner and then spinning that exact same value again? You may wish to use a probability tree diagram to help you. An example of a probability tree diagram for flipping a coin three times. 4. Consider the popular game in casinos, Roulette. In this game, players bet on where a ball on a spinning wheel will land (in essence, a spinner). There are 38 slots on the spinner that are evenly spaced (in Europe, there are 37). You can bet on a single number in which case the payout is 36 times your bet. You may bet on a “column” which pays out twice as much. You may also bet on a “dozen” which also pays twice as much. Betting on whether an even or odd number comes up pays twice as much. On a 0 or a 00, you always lose. If you were playing, how would you bet and why? Why wouldn’t you bet a certain way? Playing Cards 1. After each of the following card draws, replace the card in the deck and reshuffle. You may wish to use a probability tree diagram to help you. a. Pick a colour (Red, Black). What is the probability of drawing a card with that colour? What is the probability of drawing a second card with that colour? A third? b. Pick a suit (Diamonds, Clubs, Hearts, Spades). What is the probability of drawing a card with that suit? What is the probability of drawing a second card with that suit? A third? Note that the total number of cards in the deck decreases by one with every draw. c. Pick a rank (Ace, Three, Eight, King, etc.). What is the probability of drawing a card with that rank? What is the probability of drawing a second card with that rank? A third? Note that the total number of cards in the deck decreases by one with every draw. 2. Repeat each of the card draws above, but do not return the card to the deck. How does this affect the probabilities? Note that with each draw, the total number of cards changes! You may wish to use a probability tree diagram to help you. 3. A person draws a card from a standard deck, and it is the Queen of Hearts. The card is returned, and the deck is reshuffled. What is the probability of drawing the Queen of Spades on the next draw? 4. Two cards will be drawn from a standard deck without being returned. What is the probability of drawing the Nine of Clubs followed by a red card? 5. What is the probability of drawing a Five or a Diamond? Coins 1. Flip a coin (carefully!). What is the probability of flipping Heads? Tails? 2. Flip a coin twice in succession (carefully!) a. What is the probability of flipping 2 Heads? b. What is the probability of flipping 1 Head and 1 Tail? 3. Flip two coins at the same time (carefully!). a. What is the probability of flipping at least 1 Head? b. What is the probability of flipping 1 Head and 1 Tail? c. What is the probability of flipping 2 Tails? 4. Flip three coins in succession (carefully!). a. What is the probability of flipping 3 Heads? b. What is the probability of flipping 2 Heads and 1 Tail? 5. Flip three coins at the same time (carefully!). a. What is the probability of flipping at least 1 Head? b. What is the probability of flipping at least 2 Tails? c. What is the probability of flipping 2 Heads and 1 Tail? d. What is the probability of flipping 3 Tails? 6. A coin is flipped 99 times and it comes up Heads each time. What is the probability that for the next flip the coin will show Heads? Create a Histogram! 1. Choose a set of possible outcomes for the manipulative at your station (Ex. "Heads/Tails" for coins, "Hearts/Spades/Diamonds/Clubs" for cards, "Red/Blue/Green/Yellow" for spinner, "1/2/3/4/5/6" for a die). 2. Calculate the probability of each possible outcome occurring. 3. Flip, draw, spin, or roll the manipulative multiple times, as required. Keep track of both the number of times you do so, as well as the number of occurrences observed for each outcome. Obtain as large a sample size as possible. The larger the better! 4. Put your results into a histogram. 5. a. Calculate the percentages that each possible outcome was observed. b. Does this agree with the predicted probabilities? c. Why or why not?