Mathematics Success Grade 7

Transcription

1 T894 Mathematics Success Grade 7 [OBJECTIVE] The student will find probabilities of compound events using organized lists, tables, tree diagrams, and simulations. [PREREQUISITE SKILLS] Simple probability, relative frequency [MATERIALS] Student pages S456 S472 Centimeter cubes Cups Fair number cubes (1 per student pair) [ESSENTIAL QUESTIONS] 1. What is compound probability? 2. How does compound probability differ from simple probability? 3. Explain how compound probability can be calculated. [WORDS FOR WORD WALL] compound probability, sample space, tree diagram, frequency table, list, simulation [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) *For Cooperative Pairs (CP) activities, assign the roles of Partner A and Partner B to students. This allows each student to be responsible for designated tasks within the lesson. [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer [WARM-UP](IP, WG, I) S456 (Answers on T904.) Have students turn to S456 in their books to begin the Warm-Up. Students will be working with simple probability. Give students a few minutes to complete the problems and then take a few more minutes to review and explain the answers. {Verbal Description} [HOMEWORK] Take time to go over the homework from the previous night. [LESSON][2-3 Days (1 day = 80 minutes) M, GP, WG, CP, IP]

2 Mathematics Success Grade 7 T895 SOLVE Problem (WG, CP, IP) S457 (Answers on T905.) Have students turn to S457 in their books. The first problem is a SOLVE problem. Students will complete the entire SOLVE problem as it relates to probability. The SOLVE problem will then be extended to introduce compound probability. {SOLVE, Verbal Description, Graphic Organizer} Extend the SOLVE Problem Compound Probability with a List (M, GP, IP, CP, WG) S458, S459 (Answers on T906, T907.) M, GP, CP, WG: Have students turn to S458 in their books. Use the following activity to help students build understanding of probabilities with compound events using a list. Students will build on the SOLVE problem from S457. {Verbal Description, Pictorial Representation} Extend the SOLVE Problem Compound Probability with a List Step 1: Have student pairs read and complete the paragraph at the top of S458. Partner A, how many events are involved in this scenario? (two) Record. Partner B, what are the events? (the probability of rolling a 2 or 3, and then the probability of flipping a red) Record. Partner A, explain how we determined the probability of rolling a 2 or 3. (We created a fraction with the number of favorable outcomes over the number of total possible outcomes.) Record. Partner B, how can we determine the probability of flipping the counter and having it land on red? (Create a fraction with the number of favorable outcomes over the total number of possible outcomes.) Record. Partner A, what is the probability of flipping a counter and having it land on red? (1 out of 2 or 1 2 ) Record. Step 2: What is another word for all the possible outcomes? (sample space) Record. Have student pairs work together to create a list of the possible outcomes of the first event and then the second event. 1 Red 1 Yellow 2 Red 2 Yellow 3 Red 3 Yellow 4 Red 4 Yellow 5 Red 5 Yellow 6 Red 6 Yellow Partner A, how many total outcomes are possible? (12) Record. Partner B, how many outcomes have a 2 or a 3 and red? Circle and list them. (Two: 2 Red and 3 Red) Record. What is similar about finding the probability using a list and using the fraction method? (Answers will vary.)

3 T896 Mathematics Success Grade 7 Partner A, how do we find probability? (We place the number of favorable outcomes over the total possible outcomes.) Record. Partner B, what is the probability of rolling a 2 or 3 and then the counter landing on red? ( 2 12 = 1 6 ) Record. CP, IP, WG: Have students complete Questions 1 8 on S459 in student pairs. They will be using a list to show all the possible outcomes and identify the probability of an event based on that information. Have students come back together as a whole group and review the answers from S459. {Verbal Description, Graphic Organizer} Compound Probability with a Tree Diagram (M, GP, IP, CP, WG) S460, S461 (Answers on T908, T909.) M, GP, CP, WG: Have students turn to S460 in their books. Use the following activity to help students build understanding of probabilities with compound events using tree diagrams. {Verbal Description, Pictorial Representation} Compound Probability with a Tree Diagram Step 1: Have student pairs read the word problem at the top of S460. Partner A, what do you notice about the problem at the top of S460? (It is the same problem as on S459.) Tell students that we can also determine the probability of multiple events using a second method. Partner B, identify the possible outcomes for the toss of the number cube. (1, 2, 3, 4, 5, 6) Record below Question 2. Partner A, below each of the possible outcomes for the first event, list the 3 possibilities for the second event. (Record below Question 2.)

4 Mathematics Success Grade 7 T897 Step 2: Have student pairs discuss the characteristics of this method of modeling the possible outcomes. Partner B, describe this particular diagram with the branching. (tree diagram) Record. Partner A, where can you look on the tree diagram to determine the number of possible outcomes? (at the bottom row) Partner B, how many possible outcomes are there? (18) Record. Partner A, which numbers are greater than 4? (5 and 6) Record. Partner B, how many of the branches represent a possible outcome of a number greater than 4 and a blue marble? (2 favorable outcomes) Circle the favorable outcomes of 5 B and 6 B. Record. Partner A, how do we determine the probability? (Place the number of favorable outcomes over the total possible outcomes.) Record. Partner B, what is the probability of rolling a number greater than 4 and choosing a blue marble? ( 2 18 = 1 9 ) CP, IP, WG: Have students create the tree diagram and the list for the two situations on S461. After creating the tree diagram and the list, they will determine the number of total outcomes and the probability of the favorable outcome for each. Have students come back together as a whole group and review the answers. {Verbal Description, Graphic Organizer} Comparing Simple and Compound Probability (M, GP, CP, WG) S462 (Answers on T910.) M, GP, CP, WG: Have students turn to S462 in their books. Students will be using three different scenarios from S457 - S460 to compare and contrast simple and compound probability. {Verbal Description, Graphic Organizer} Comparing Simple and Compound Probability Step 1: Have student pairs review the three events in the graphic organizer. (rolling a 2 or 3, rolling a 2 or 3 and flipping red, rolling a number greater than 4 and picking a blue marble) Partner A, in Column 1, how many events were there in the SOLVE problem? (1) Record for #1 and in the graphic organizer. Partner B, what type of probability do we have when there is only one event? (simple) Record for #2 and in the graphic organizer.

5 T898 Mathematics Success Grade 7 Step 2: Partner A, in Columns 2 and 3, how many events were there to consider? (2) Record for #3 and in the graphic organizer Partner B, when we have more than one event what do we call this? (compound probability) Record for #4 and in the graphic organizer Step 3: Refer students to the graphic organizer. Partner A, what method did we use to determine the probability in Column 1? (Create a fraction of favorable outcomes over total outcomes.) Record. Partner B, what method did we use to determine the probability in Column 2? (Create a list of possible outcomes; create a fraction of favorable outcomes over total outcomes.) Record. Partner B, what method did we use to determine the probability in Column 3? (Create a list of possible outcomes and a tree diagram; create a fraction of favorable outcomes over total outcomes.) Record. Step 4: Partner A, what is similar between the simple probability scenario and the compound probability scenario? (In both cases, we still found the favorable outcomes over the total outcomes.) Record. Partner B, what are some differences between scenarios with simple and compound probability? (Compound probabilities involve more than one event, and we found it helpful to use lists and tree diagrams to help us find all of the favorable outcomes, as well as all of the possible outcomes.) Extend the SOLVE Problem Multiplication with Compound Probability (M, GP, WG, CP, IP) S463, S464, S465 (Answers on T911, T912, T913.) M, GP, CP, WG: Have students turn to S463 in their books. They will work in student pairs to complete the SOLVE problem by using their knowledge of compound probability and tree diagrams. {SOLVE, Verbal Description, Graphic Organizer}

6 Mathematics Success Grade 7 T899 Extend the SOLVE problem- Multiplication with Compound Probability Step 1: Have students complete the SOLVE problem on S463 and then review the answers as a whole group. Step 2: Direct students attention to the top of S464. Partner A, what is the first event for Lashelle s game scenario? (Not spinning blue on the spinner) Record. Partner B, what is the number of favorable outcomes for the first event? (2) Record. Partner A, what are the favorable outcomes? (spinning yellow or red) Partner B, what is the total number of possible outcomes for the first event? (3) Record. Step 3: Partner A, what is the second event? (rolling a number 5 or greater on the 8-sided die) Record. Partner B, what is the number of favorable outcomes for the second event? (4) Record. Partner B, what are they? (5, 6, 7, or 8) Partner A, what is the total number of possible outcomes for the second event? (8) Step 4: Have students look at the SOLVE problem from S463 and record the compound probability both before and after she simplified the probability. (Probability of not spinning blue and rolling 5 or greater was 8 out of 24, or one-third.) Record in the graphic organizer. Partner A, what was the probability of not spinning blue and rolling a number of 5 or greater? ( 8 24 = 1 3 ) Record. Partner B, what do you notice about the relationship between the numerators of the individual events and the numerator of the compound probability? (Multiplying the probability of the two numerators of the single events has a product that is equal to the numerator of the compound probability before simplifying.) Partner A, what do you notice about the relationship between the denominators of the individual events and the denominator of the compound probability? (Multiplying the probability of the two denominators of the single events has a product that is equal to the denominator of the compound probability before simplifying.) Step 5: Partner B, what conclusion can we draw about the probabilities of the individual events and the probability of the compound event? (Multiplying the probability of each individual event together will give us the probability of the compound event.) Record.

7 T900 Mathematics Success Grade 7 Step 6: Have student pairs read the situation at the top of S465. Partner A, how is this situation similar to the situation on S464? (We have two events, choosing a red marble and choosing a blue marble.) Record. Partner B, how is this situation different from the situation on S464? (This situation involves only one bag of marbles, and the second choice will be made without replacing the first marble.) Record. Partner A, how will this affect the denominator of the second event? (The denominator will be one less because the marble will not be replaced.) CP, IP, WG: Have students complete the chart on S465 with a partner. After completing the chart, review the answers as a whole group and then answer Questions 4 8 together. {Verbal Description, Graphic Organizer} Exploring Compound Probability (M, GP, IP, WG, CP) S466, S467, S468 (Answers on T914, T915, T916.) M, WG, CP, GP: Have students turn to S466 in their books. Use the following activity to help students complete an experiment using the probabilities of compound events. {Concrete Representation, Verbal Description, Graphic Organizer} Exploring Compound Probability Step 1: Direct students attention to S466. Provide student pairs each with a cup containing an assortment of centimeter cubes. Students will only need to use the yellow, red, and blue centimeter cubes for this particular experiment. Partner A, how can you find the probability of each of the compound events that will occur? (Multiply the probability of choosing the first color by the probability of choosing the second color. The denominator for the second probability will be one less for all of the events because we assume that the first marble isn t replaced. Also, we must decrease the numerator by one if it is a compound event where we are choosing the same color twice.) Record. Partner B, without completing any calculations, what will be the probability of choosing two blues in a row? (The probability will be 0.) Record. Partner A, explain how you know this? (There is only one blue cube in the cup, so without replacement, there is no chance of picking a second blue cube from the cup.) Record.

8 Mathematics Success Grade 7 T901 Step 2: Using the cubes provided, students will find the probability of each compound event occurring. They will record these probabilities in the rectangles that show the cross between two colors. Students should assume that the first centimeter cube will NOT be replaced after it is chosen when figuring the probability. Review the probability of each of the compound events that the students have marked in the chart. Partner B, which color combinations do you expect to actually occur most often based on your table? (The largest fractions appear with yellow then yellow, yellow then red, and red then yellow.) Record. Step 3: Have students turn to page S467 in their books. Student pairs will conduct an experiment using the cubes. They will create a frequency table with tally marks of the actual outcomes that occurred. The table is labeled with Cube 1 and Cube 2 so that students can record the specific orders. Have students complete the probability experiment. Step 4: Direct students attention to Question 2 below the chart. What did we create by recording all of our data in the table? (A frequency table) Record. When we conduct an experiment where we are actually able to carry out the experiment this is called a (simulation). Record. How can we calculate the probability of each combination occurring? (Record the probabilities for each compound event under the Relative Frequency column. Favorable over Total: A fraction can be created for each combination by placing the total number of tally marks for that combination over the total number of trials, which was 10.) Record. Step 5: Direct students attention to S468. Now that we have found the probability and the relative frequencies of the compound events occurring, let s write them in the chart above so that we can compare. To make comparing easier, let s change all of the probabilities to decimals. Partner A, how do we change the fractional probabilities to decimals? (Divide the numerator of each fraction by the denominator and record the decimal.) Record. Students will fill in the table on S468 by writing in the decimal form of both the probability and relative frequency that they found on S466 and S467.

9 T902 Mathematics Success Grade 7 Partner B, how do your probabilities compare with the relative frequencies of the events? (Answers will vary based on actual outcomes. Expect that some students will have actual outcomes that may be close to the probability, but most will find that they are not.) Record. Partner A, how did the actual outcome of choosing red and then yellow compare to the actual outcome of your neighbor? (Answers will vary. Students should compare decimals regarding this specific event with their neighbor.) Record. Partner B, how did your outcome of choosing two blues compare with the probability? How did it compare with your neighbor s results? (All probabilities were 0. It was impossible for more than one blue to occur without replacement, so every group had the same result for this particular compound event.) Record. SOLVE Problem (GP, IP) S469 (Answers on T917.) Have students turn to S469 in their books. Students will apply their knowledge of compound probability to complete the SOLVE problem. Have students complete the SOLVE problem in pairs and then review the answers as a whole group. {SOLVE, Verbal Description, Graphic Organizer} Foldable Probability Foldable (M, WG) Have students continue to add to their foldable created in Lesson 32. {Verbal Description, Graphic Organizer, Pictorial Representation} Probability Foldable Step 1: Label the fourth flap Compound Probability. On the inside, complete the section for Compound Probability with the given information (refer to your foldable). You can create an example to model for students what should be written on each flap. Use the foldable you made in training or the one on the CD for the information. Likelihood and Probability Relative Frequency Uniform and Non-Uniform Probability Models Compound Probability

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