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LESSON PLAN Name: Amber Blakley and Philip Kromer Title of lesson: Probability: Counting Outcomes Date of lesson: March 8th, 2004 Length of lesson: 50 minutes Description of the class Name of course: 7th Grade Math Grade level: 7th Grade Honors or regular: Regular Source of the lesson: Mathematics: Applications and Connections, Course 3, Chapter 12, 518-19. Practice Masters, 93. Study Guide Masters, 93. TEKS addressed: TEKS: 14A, 15A, 15B 7.14 (A) Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models. 7.15 (A) Make conjectures from patterns or sets of examples and non examples; 7.15 (B) Validate his/her conclusions using mathematical properties and relationships. I. Overview Students should understand how to count outcomes by making a tree diagram or by applying the counting principle (multiplication of outcomes rule). Students should be able to do examples on their own and feel comfortable that they did so correctly. On the small scale, counting outcomes is the first step in many problems in probability. Students have to be able to evaluate how many possible outcomes exist in order to find the probability of a specific event. More broadly, probability is fundamental to making informed decisions in voting, games, finance, and other real-life situations involving expectation, risk, and reward. II. Performance or learner outcomes Students will be able to: Construct a tree enumerating all possible outcomes of a given scenario. Invent such a tree to find the probability of a simple outcome or set of outcomes. Apply the counting principle (multiplication rule) to find the number of outcomes for a given scenario. Explain how the tree diagram and the counting principle yield the same number of outcomes and how the counting principle encapsulates the information in the tree diagram. III. Resources, materials and supplies needed Pencils, Paper Giant Dice Chart for MASH game* (done on chalkboard) IV. Supplementary materials, handouts. A handout with practice problems and an outline of ideas covered. Five-E Organization Teacher Does Probing Questions Student Does Engage: Play MASH. MASH is a How many places can you end Students should understand that children’s game to predict the up living? there are four outcomes for future: the player chooses where where to live. they will live (usually Mansion, What is the probability of living Depending, of course, on where Apartment, Shack, and House – where you want? the student wants to live, the hence MASH), what they will student should understand that drive, # or kids. (Students the probability is 1/4 (2/4, 3/4). nominate the possibilities for the The student realized that this last categories). A simple What is the probability of getting question requires a way to scheme eliminates choices until what you want in each case? enumerate the outcomes. the player’s future is divined. The teacher shows the class how Students see the basic idea of to use an outcome tree to solve making a tree. this problem: List all the possibilities for the first choice (M,A,S,H). For each of those, list the possibilities for the second choice (Ferrari, Honda, Bike), and so on. Explore: The teacher starts off with an How many possible outfits can Students may offer guesses or outfit and how many ways can we make? wonder how to solve the we put that outfit together. problem. (Outfit = shirt + shorts) Teacher gives appropriate information on board. List that we have Red or Blue shirts Black, Gray, or White shorts The solution: Teacher will show Students may be confused on that the students that no matter how the red shirt may use and of what they do they will always the shorts and the blue shirt can need to start with either a red or use all the shorts. Clarify that we blue shirt, and can put any shorts were not wearing these clothes; with any shirt. we just want to know how many different ways we can wear them. We again show the students how The students may not first to form a tree diagram. understand how to draw the tree. Show the students that for each How many outcomes are there 3 outcomes for blue shirt path you can follow, it will for blue shirt? make a different outcome. How many outcomes are there 1 outcome for the white shorts for the white shorts and red shirt? and red shirt. Leave the diagram up; step out How many different outfits are Students should be able to of the way so the students may there? answer this now by counting the see the board clearly. different paths and adding them together; by doing this they come up with six different outfits. Explain to the students that you Do you think this method is Will pause to think about the start your tree diagram with your slow? Can you think of a faster given questions. first choices and the tree way? Is there even a faster way? diagrams end with the different types of outcomes. Also remind them to be careful drawing their trees and to make sure not to repeat any outcomes. Moving on…first we’ll start off How many outcomes with one By seeing the example students with the probability throwing of die? should be able to see that dice a die and dice. Show an Can you explain to me how you resemble a cube and it has six example of die and dice. got that answer? sides along with six different Then roll the die for the first What’s the probability of getting numbers on each face. Students question. Roll the die a couple a six? should already understand that of times to help the students see with six faces there are six equal that only one value can appear at outcomes. So probability of one time. rolling each of the numbers is 1:6. Next move on to two die. Roll How many outcomes with two Some students may already the dice so the students may see dice? know, that’s great, others that that every time they roll the dice How many outcomes for the first don’t will get refreshed and they will get two numbers. Here die? understand how the other we want to use the tree diagram How many outcomes for the students got their answer. and then introduce the Counting second die? Principle. Do you think there is an easier Go through tree diagram with why to get the outcome? the children. Then show how Do you know what might work? they can multiply the number of outcomes for each specific case. Explain: Teacher does Probing Questions Students Reponses Explain that we can multiply to Is this a faster way to count Student may not understand how find 36 outcomes. This is called outcomes? you are able to do this; they may the “Counting Principle.” Can we see all of the possible want a better explanation. outcomes using this method? Next give the example of tossing If you flip three coins, how will three coins. the tree look? Would any one like to try to One student gets up to the board make the tree diagram of this and draws the diagram, other outcome? students decide if he/she is right. The diagram of the tree should How many outcomes are there? Students will be able to counts split up into the first coin being How many outcomes have just these and decide upon eight. head or tails. The next coin will heads or just tails? Look at the diagram and see just either come up head or tails and Why is it ok to have both HTH two have heads or tails. so will the last one. So we have and THH? They will have to think about the two separate tree diagrams and last question and decide if the there outcomes. order the coins come out matter. Go back to an outfit and use the How many outcomes are there? Will take the two shirts and three counting principle of math to Was this method faster? shorts and multiple 3 * 2 to get find answer. six the same number of outcomes as before. Another example: there is a test For this one is it better to draw a Students may try to answer ten, of five true or false questions, tree diagram or use the Counting clarify their confusion and make how many outcomes are there? Principle? sure they do 2*2*2*2*2 and not just 2*10. The teacher shows an example What are the advantages of using Students should spot the pattern of a tree diagram and the the tree diagram? between making the tree and Counting Principle. Example What are the advantages of using multiplying the outcomes You have Dr. Pepper, Coke, and the Counting Principle? Big Red. That you can pour on two ice creams (vanilla and chocolate). Explain which methods are Is one method better then the better to use in which questions. other? Shows that the Counting Do you really need to know Depending on the question Principle is a short cut and if which is better for different case? students should be able to decide you need to know the specific which method would be better. outcomes, then the tree diagram would be better. Extend / Elaborate: Recall the game that we played How many outcome were there? Calculate the number of at the beginning of the class. Which method did you use to outcomes, by themselves and Have the students find the total find the number of outcomes? have them give their answers. number of outcomes for the game. Evaluate: Could the Tree Method and the Multiplication Rule ever give different answers? Which is better, the tree method OK: Multiplying: it’s easier or the multiplication rule? OK: Tree: it’s simpler Better: Either, they both give the right answer. Best: Tree is fine for small problems; multiplication rule is best for large # of outcomes or levels. Ask class if there are any Any about the tree diagram or Ask their question or remain questions. the Counting principle? quite. Hand out worksheets and walk Start to work on the worksheets. around room as students fill them out. Go over the worksheet. Ask the whole class, what the Students give their answers. right answer to one? Pick a student. Maybe go up and down the rows and ask for their answers. Percent effort each team member contributed to this lesson plan: _50_%___ ____Name of group member_Amber Blakely_____ _50_%___ ____Name of group member_Philip Flip Kromer__