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Using Tree Diagrams to Calculate Probabilities Objective To use tree diagrams to help calculate probabilities. www.everydaymathonline.com ePresentations eToolkit Algorithms Practice EM Facts Workshop Game™ Teaching the Lesson Family Letters Assessment Management Common Core State Standards Ongoing Learning & Practice Key Concepts and Skills Writing and Graphing Inequalities • Use a unit percent to calculate the percent of a number. [Number and Numeration Goal 2] Math Journal 2, pp. 261A and 261B Students write and graph inequalities to represent descriptions of various objects and situations in space. • Add and subtract fractions with like and unlike denominators. [Operations and Computation Goal 3] • Use an algorithm to multiply a fraction by a fraction and a whole number by a fraction. [Operations and Computation Goal 4] • Determine the number of possible outcomes for situations using diagrams or counting strategies. [Data and Chance Goal 3] • Calculate the probabilities of outcomes using a tree diagram. [Data and Chance Goal 3] Math Boxes 7 5 Math Journal 2, p. 262 Students practice and maintain skills through Math Box problems. Curriculum Focal Points Interactive Teacher’s Lesson Guide Differentiation Options READINESS Adding Fractions Students review methods for adding fractions with unlike denominators. EXTRA PRACTICE Experimenting with Coin Flips Study Link 7 5 Math Masters, p. 232 per partnership: 1 coin (optional) Students use a tree diagram to predict outcomes in a coin-flipping experiment. Math Masters, p. 231 Students practice and maintain skills through Study Link activities. 5-Minute Math Key Activities Students use tree diagrams to find expected outcomes and calculate the probabilities of those outcomes. EXTRA PRACTICE 5-Minute Math™, pp. 130 and 209 Students apply the Multiplication Counting Principle to solve problems. Ongoing Assessment: Informing Instruction See page 648. Ongoing Assessment: Recognizing Student Achievement Use journal page 261. [Data and Chance Goal 3] Key Vocabulary Multiplication Counting Principle Materials Math Journal 2, pp. 260 and 261 Study Link 74 transparency of Math Masters, p. 226 (optional) Advance Preparation Teacher’s Reference Manual, Grades 4–6 pp. 172, 173 Lesson 7 5 645 Mathematical Practices SMP1, SMP2, SMP3, SMP4, SMP6 Content Standards Getting Started Mental Math and Reflexes Students calculate the number of possible outcomes. Suggestions: rolling a die and tossing a coin 6 ∗ 2, or 12 choosing a dinner from 5 main courses, 3 vegetables, 2 salads, and 4 beverages 5 ∗ 3 ∗ 2 ∗ 4, or 120 selecting one hour of the day and one minute of the hour 24 ∗ 60, or 1,440 6.EE.8 Math Message Tim is late to school about 1 school day out of 10. At this rate, about how many school days out of 30 would you expect Tim to be late? About how many school days out of 200? Study Link 7 4 Follow-Up Review the answers as necessary. Briefly discuss how the lists and tree diagram show that there are 12 possible breakfast bags. Instead of counting all the possible choices, students can use the Multiplication Counting Principle (also known as the Fundamental Counting Principle) as follows: Number of Beverage Choices Number of Bagel Choices Number of Fruit Choices Total Number of Choices 2 2 3 2 ∗ 2 ∗ 3 = 12 1 Teaching the Lesson NOTE The Mental Math and Reflexes problems deal with compound events. For additional practice with compound events, see www.everydaymathonline.com. ▶ Math Message Follow-Up WHOLE-CLASS DISCUSSION The probability that Tim will be late on any particular school day 1 = 0.1 = 10%. The expected number of late arrivals depends is _ 10 on the number of days being considered. 1 of Out of 30 school days, Tim is expected to be late _ 30 days, or 3 days. 10 1 of Out of 200 school days, Tim is expected to be late _ 10 200 days, or 20 days. ▶ Revisiting Probability WHOLE-CLASS DISCUSSION Tree Diagrams (Math Journal 2, p. 260; Math Masters, p. 226) The mazes shown in Problems 1 and 2 on journal page 260 are the same as those from journal pages 257 and 258 of Lesson 7-4. In Lesson 7-4, students were told the number of people who entered a maze and then they were asked how many could be expected to enter Rooms A and B. In this lesson, students calculate the probability that a person walking through the maze enters Room A and the probability that a person enters Room B. 646 Unit 7 Probability and Discrete Mathematics Teaching Master Work through Problem 1 with the class. If you have a transparency of Math Masters, page 226, use the top diagram. Go through each step carefully, making sure students understand the procedure. Name Date LESSON Time Tree Diagrams 7 4 䉬 Step 1: Calculate the probability of reaching each of the endpoints. Because the tree diagram divides into four branches at the first intersection, the probability of following any one of these 1 . Write _ 1 next to each branch. branches is 1 out of 4, or _ 4 4 B For each of the second set of intersections, the probability of 1 . Write _ 1 following any one of the branches is 1 out of 2, or _ 2 2 next to each branch. B Calculate the probabilities of reaching each of the endpoints of 1 of the time and the tree diagram. The first branch is taken _ 4 1 1 1 ∗_ 1 _ _ each of the second branches taken 2 of 4 of the time, or _ 2 4 1 ∗_ 1 =_ 1 , so the probability of reaching any one of the time. _ 2 4 8 1 . Record this probability at each endpoint. of the endpoints is _ 8 1 4 1 2 1 2 1 2 1 4 1 2 1 8 1 8 1 8 1 8 1 4 A B A B A A A A A B B Math Masters, p. 226 1 4 1 2 1 2 1 2 1 2 1 8 1 8 1 8 1 8 Students should notice that, to calculate the probability of any of the endpoints in the diagram, they multiply the probabilities of 1 ∗_ 1 , or _ 1. the branches that lead to that endpoint; in this case, _ 2 4 8 Step 2: Calculate the probabilities of entering each room. Use the tree diagram to calculate the probability of entering Room A and the probability of entering Room B. There are five different endpoints in Room A. Add the probabilities of reaching these endpoints. Since 5 , the probability of 1 +_ 1 +_ 1 +_ 1 +_ 1 =_ _ 8 8 8 8 8 8 5. entering Room A is _ 8 There are three different endpoints in Room B. Add the 3, 1 +_ 1 +_ 1 =_ probabilities of reaching these endpoints. Since _ 8 8 8 8 3 _ the probability of entering Room B is 8 . Student Page Date Suppose 80 people walk through the maze. The probability of 5 . The number of people who can be entering Room A is _ 8 5 of 80 people, or 50 people. expected to enter Room A is _ 8 3 _ Similarly, 30 people ( 8 of 80 people) can be expected to enter Room B. Probability Tree Diagrams 75 䉬 Complete the tree diagram for each maze. 154 155 Write a fraction next to each branch to show the probability of selecting that branch. Then calculate the probability of reaching each endpoint. Record your answers in the blank spaces beneath the endpoints. 1. 5 ᎏᎏ 8 3 ᎏᎏ 8 What is the probability of entering Room A? What is the probability of entering Room B? 1 4 From Step 2, we know the probability that a person will enter Room A and the probability that a person will enter Room B. If a large number of people enter the maze, the number expected to enter each room can be found. Example: Time LESSON 1 2 1 4 1 4 1 1 2 2 1 1 8 8 2. 1 4 1 1 2 2 1 1 8 8 Enter 1 1 2 2 1 1 8 8 1 2 Room A 1 1 8 8 Room B 5 ᎏᎏ 12 7 ᎏᎏ 12 What is the probability of entering Room A? What is the probability of entering Room B? 1 3 1 3 1 6 1 6 1 2 1 2 1 3 1 2 1 6 1 4 Enter 1 2 Room A Room B 1 4 Math Journal 2, p. 260 Lesson 7 5 647 Student Page Date Time LESSON Probability Tree Diagrams 75 䉬 3. Adjusting the Activity continued Josh has 3 clean shirts (red, blue, and green) and 2 clean pairs of pants (tan and black). He randomly selects one shirt. Then he randomly selects a pair of pants. 夹 a. 1 3 Shirts: Refer students to journal page 257, where they worked with numbers on the tree diagram. Relate the numbers at each step to the fraction of people walking a given path. Remind students that the whole changes as each new 1 intersection is reached. In the beginning, the whole is 80 people and _ 4 of 80 people, or 20 people, walk down each path. At the next intersection, the whole is 20 people. One-half of 20 people, or 10 people, continue down each path from there. 1 3 1 3 red 1 2 blue 1 2 1 2 Pants: tan black 1 1 6 6 b. 155 156 Complete the tree diagram by writing a fraction next to each branch to show the probability of selecting that branch. Then calculate the probability of selecting each combination. green 1 2 1 2 1 2 tan black tan black 1 1 6 6 1 1 6 6 A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L List all possible shirt-pants combinations. One has been done for you. red-tan red-black, blue-tan, blue-black, green-tan, green-black 6 combinations c. How many different shirt-pants combinations are there? d. Do all the shirt-pants combinations have the same chance of being selected? e. What is the probability that Josh will select 1 ᎏᎏ 3 1 ᎏᎏ 2 the blue shirt? the tan pants? the blue shirt and the tan pants? Yes Ongoing Assessment: Informing Instruction 1 ᎏᎏ 6 2 ᎏᎏ 3 a shirt that is not red? Watch for students who think that the correct strategy is to divide the original total by the number of exits. After working through Problem 2, students should see that this strategy is faulty. 1 ᎏᎏ 3 the black pants and a shirt that is not red? Try This Suppose Josh has 4 clean shirts and 3 clean pairs of pants. Explain how to calculate the number of different shirt-pants combinations without drawing a tree diagram. Multiply the number of clean shirts (4) by the number of clean pants (3); 4 ⴱ 3 ⫽ 12 Ask students to work Problem 2 independently. Bring the class together to discuss the answers. You might want to use the diagram at the bottom of the transparency of Math Masters, page 226 to discuss the problem. The probability of entering 7. 1 +_ 1 +_ 1 , or _ Room B is _ 6 6 4 12 Math Journal 2, p. 261 1 2 1 3 1 3 1 2 1 3 1 1 1 6 6 6 Room B Room B Room A 75 Going to Space Star masses are measured in a unit called a solar mass, which is equal to the mass of the Sun. One solar mass is equal to 2 ∗ 1033 grams. 1. Low-mass stars use hydrogen fuel so slowly that they may shine for over 100 billion years. A low-mass star has a mass of at least 0.1 solar mass but less than 0.5 solar mass. a. b. ▶ Calculating Probabilities List several possible masses that a low-mass star could have. Sample answers: 0.1 solar mass; 0.45 solar mass; 0.32 solar mass with Tree Diagrams Describe in words the set of all possible masses of a low-mass star. Sample answer: All masses between 0.1 and 0.5 solar mass, including 0.1 c. 2. (Math Journal 2, p. 261) Use an inequality or two inequalities to describe the set of possible masses. Sample answer: m < 0.5 and m ≥ 0.1 d. Graph the set of all possible masses on the number line below. e. Do the values represented on the graph make sense in the situation? Explain your answer. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 High-mass stars have short lives and sometimes become black holes. When a high-mass star dies, an explosion occurs, leaving behind a stellar core. If the stellar core has a mass of at least 3 solar masses, the star becomes a black hole. b. List several possible masses of a stellar core that will become a black hole. Sample answers: 3 solar masses; 5.5 solar masses; 3.1 solar masses Describe the set of possible masses in words. Sample answer: All masses that are 3 solar masses or greater Math Journal 2, p. 261A 261A_261B_EMCS_S_G6_MJ2_U07_576442.indd 261A 648 Unit 7 INDEPENDENT ACTIVITY PROBLEM PRO PR P RO R OB BLE BL LE L LEM EM SO S SOLVING OL O LV VING VI VIN IN IN NG G Read the introduction to the problem as a class, so students know what to do. The tree diagram has already been drawn and labeled. Students should quickly notice that the probability of taking any 1 ∗_ 1 , or _ 1. path in Problem 3 is _ 3 2 6 1.0 Sample answer: Yes. A mass can be any positive number, and the graph does not show any negative masses. a. 1 1 4 4 Room A Room B Ask: If 60 people walk through the maze, about how many would 5 of 60, or 25 people About how many would enter enter Room A? _ 12 7 of 60, or 35 people Room B? _ 12 Time LESSON 1 2 5. 1 +_ 1 , or _ The probability of entering Room A is _ 6 4 12 Student Page Date 1 2 3/4/11 10:22 AM Probability and Discrete Mathematics Student Page Ongoing Assessment: Recognizing Student Achievement Journal Page 261 Date Time LESSON Going to Space 75 2. c. Use journal page 261 to assess students’ abilities to determine expected outcomes and to use a tree diagram to calculate probabilities of chance events. Students are making adequate progress if they are able to complete the tree diagram and list all the possible shirt-and-pants combinations (Problems 3a–3d). Some students may be able to calculate all probabilities in Problem 3e. continued c≥3 Use an inequality to describe the set of possible masses. d. Graph the set of all possible masses on the number line below. e. Do the values represented on the graph make sense in the situation? Explain your answer. 0 1 2 3 4 5 6 7 8 9 10 Sample answer: Not all of the solutions make sense. For example, it is probably not possible for a stellar core to have 1,000 solar masses. But since I don’t know the maximum mass, I can’t change the graph. [Data and Chance Goal 3] In the United States, people who pilot spacecrafts or work in space are called astronauts. In Russia and other former republics of the Soviet Union, these people are called cosmonauts. 3. Links to the Future Cosmonauts began flying the Soyuz series of spacecraft in 1967. These vehicles can transport up to 3 cosmonauts. a. List the possible numbers of cosmonauts that can go on a mission in a Soyuz spacecraft. b. Describe the possible numbers of cosmonauts in words. c. Write an inequality or inequalities to represent the possible numbers of cosmonauts. d. Graph the possible numbers of cosmonauts on the number line below. e. Do the values shown on the graph make sense in the situation? Explain your answer. 1, 2, or 3 cosmonauts In this unit, students discover that they can determine the number of possible outcomes by making an organized list, drawing a tree diagram, or multiplying. Students will extend their knowledge of the Multiplication Counting Principle when they study combinatorics in future probability and statistics courses. Sample answer: At least 1 but no more than 3 cosmonauts Sample answer: n ≥ 1 and n ≤ 3 0 1 2 3 4 5 6 7 8 9 10 Sample answer: Yes. I used dots on the whole numbers because you can’t have part of a cosmonaut. 2 Ongoing Learning & Practice ▶ Writing and Graphing Inequalities Math Journal 2, p. 261B 261A_261B_EMCS_S_G6_MJ2_U07_576442.indd 261B 3/4/11 10:22 AM INDEPENDENT ACTIVITY (Math Journal 2, pp. 261A and 261B) Review with students the steps for graphing inequalities. Ask: How is graphing an inequality that represents a real-world situation different from graphing an inequality without context? If you are representing something in the real world, you need to check whether all the solutions on your graph make sense for the situation. Have students complete journal pages 261A and 261B. They will write and graph inequalities to represent various objects and situations in space. When they have finished, discuss their answers to Part e as a class. Student Page Date Math Boxes 75 䉬 1. ▶ Math Boxes 7 5 Time LESSON Darnell has 3 jackets and 4 baseball hats. Complete the tree diagram. Hats Braves Jackets Dodgers Cubs Black INDEPENDENT ACTIVITY Marlins a. (Math Journal 2, p. 262) Writing/Reasoning Have students write a response to the following: Explain why the graph in Problem 2 does not represent the set of counting numbers. Sample answer: The graph is a solid arrow pointing to the right, which means that all numbers between the counting numbers (1.5, 1.501, 1.5011, and so on) are also included in the graph. A graph of only the counting numbers would be filled-in dots on 1, 2, 3, ... . D Red 12 b. Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 7-7. The skills in Problems 4 and 5 preview Unit 8 content. B How many jacket-hat combinations are possible? Yes 2. C M Do all the jacket-hat combinations have the same chance of being selected? Which set of numbers is represented by the graph below? Choose the best answer. ⫺3 ⫺2 B D C M White ⫺1 0 1 2 3. 3 positive real numbers positive integers 155 156 Write each number in standard notation. 72,000,000,000 a. 72 billion b. 42.78 million c. 89.6 billion 89,600,000,000 d. 0.5 million 500,000 42,780,000 whole numbers counting numbers 4 244 4. Janella walks at a speed of 6.9 kilometers per hour. At this rate, how far can she walk b. 13.8 kilometers in 20 minutes? 2.3 kilometers c. in 1 hour 40 minutes? a. 5. — Express the length of JK to the length — of JL as a simplified fraction. J in 2 hours? 11.5 kilometers 6 K 110 111 JK ᎏᎏ JL ⫽ 10 8 L 3 ᎏᎏ 5 179 Math Journal 2, p. 262 Lesson 7 5 649 Study Link Master Name Date STUDY LINK Time ▶ Study Link 7 5 A Random Draw and a Tree Diagram 75 䉬 154 155 I T P O A Box 1 (Math Masters, p. 231) Home Connection Students solve a problem involving random draws. N E Box 2 INDEPENDENT ACTIVITY Boxes 1, 2, and 3 contain letter tiles. Box 3 Suppose you draw one letter from each box without looking. You lay the letters in a row—the Box 1 letter first, the Box 2 letter second, and the Box 3 letter third. Complete the tree diagram. Fill in the blanks to show the probability for each branch. 1. 1 ᎏᎏ 2 1 ᎏᎏ 2 Box 1 T 1 ᎏᎏ 3 Box 2 I 1 ᎏᎏ 2 1 ᎏᎏ 2 Box 3 N 1 ᎏᎏ 12 1 ᎏᎏ 3 1 ᎏᎏ O 3 1 ᎏᎏ 2 A 1 ᎏᎏ 2 P 1 ᎏᎏ 3 I 1 ᎏᎏ 2 1 ᎏᎏ 2 1 ᎏᎏ 2 1 ᎏᎏ 2 A 1 ᎏᎏ 2 1 ᎏᎏ 2 1 ᎏᎏ 2 1 ᎏᎏ 2 E N E N E N E N E N E 1 ᎏᎏ 12 1 ᎏᎏ 12 1 ᎏᎏ 12 1 ᎏᎏ 12 1 ᎏᎏ 12 1 ᎏᎏ 12 1 ᎏᎏ 12 1 ᎏᎏ 12 1 ᎏᎏ 12 1 ᎏᎏ 12 1 ᎏᎏ 12 2. How many possible combinations of letter tiles are there? 3. What is the probability of selecting: 1 ᎏᎏ 6 a. the letters P and I? c. 3 Differentiation Options 1 ᎏᎏ 3 1 ᎏᎏ O 3 1 ᎏᎏ 3 the letter combinations TO or PO? ▶ Adding Fractions 12 b. 3 ᎏᎏ, the letter I, O, or A? 3 d. two consonants in a row? 36.5 5. 22.6 858.8 ⫼ 38 ⫽ 0% 12.6 1,575 ⫼ 125 ⫽ 6. 5–15 Min or 100% Practice 657 ⫼ 18 ⫽ 4. SMALL-GROUP ACTIVITY READINESS Math Masters, p. 231 To provide students with more practice using fractions to calculate probability, have them add fractions. Review the quick common denominator (QCD) and least common denominator (LCD) methods for adding fractions with unlike denominators by having students find sums. Suggestions: 5 2 +_ 1 _ _ 3 1 _ 6 1 _ 5 + + 6 1 _ 4 1 _ 3 6 5 _ 12 8 _ 15 5 1 +_ 1 _ _ 2 8 8 5 2 +_ 1 _ _ 9 3 9 3 _ 19 1 +_ 2 +_ _ 4 5 10 20 EXTRA PRACTICE ▶ Experimenting with Coin Flips PARTNER ACTIVITY 15–30 Min (Math Masters, p. 232) To provide students more practice with theoretical and experimental probabilities, have them predict the results of a coin-flipping experiment with the help of tree diagrams. They analyze the results and answer questions about the experiment. Teaching Master Name Date LESSON 75 䉬 1. Time A Coin-Flipping Experiment Suppose you flip a coin 3 times. What is the probability that the coin will land a. HEADS c. HEADS e. 3 times? 1 ᎏᎏ 8 b. HEADS 3 ᎏᎏ 8 1 time and TAILS 2 times? 1 ᎏᎏ 8 3 times? d. TAILS with the same side up all 3 times (that is, all 2 times and TAILS 1 time? or all HEADS TAILS)? 3 ᎏᎏ 8 1 ᎏᎏ 4 Students can extend the activity by actually carrying out the experiment, flipping a coin 300 times and recording the results. They can then compare actual results with predicted results. Make a tree diagram to help you solve the problems. EXTRA PRACTICE H1 T1 1 ᎏᎏ 2 1 ᎏᎏ 2 2nd flip: H2 3rd flip: H3 1 ᎏᎏ 2 1 ᎏᎏ 2 T3 H3 1 ᎏᎏ 8 2. 1 ᎏᎏ 2 1 ᎏᎏ 2 T2 H2 1 ᎏᎏ 2 1 ᎏᎏ 8 ▶ 5-Minute Math 1 ᎏᎏ 2 1 ᎏᎏ 2 1st flip: 1 ᎏᎏ 8 1 1 ᎏᎏ ᎏᎏ 2 2 T3 H3 1 ᎏᎏ 8 1 ᎏᎏ 8 One trial of an experiment consists of flipping a coin 3 times. Suppose you perform 100 trials. For about how many trials would you expect to get HHH or TTT? What percent of the trials is that? T2 1 ᎏᎏ 2 1 ᎏᎏ 2 T3 H3 1 ᎏᎏ 8 1 ᎏᎏ 8 1 ᎏᎏ 2 T3 1 ᎏᎏ 8 About 25 trials 25% Math Masters, p. 232 650 Unit 7 Probability and Discrete Mathematics SMALL-GROUP ACTIVITY 5–15 Min To offer more practice applying the Multiplication Counting Principle (also known as the Fundamental Counting Principle), see 5-Minute Math, pages 130 and 209.