7.S.8 Interpret data to provide the basis for predictions and to establish


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1 7 th Grade Probability Unit 7.S.8 Interpret data to provide the basis for predictions and to establish experimental probabilities. 7.S.10 Predict the outcome of experiment 7.S.11 Design and conduct an experiment to test predictions 7.S.12 Compare actual results to predicted results Unit Notes: 6th grade probability standards are PostMarch and thus will occur on the 7 th grade State Exam The 7 th grade focus is on making predictions about an experiment, conducting the experiment, and then comparing actual to predicted results. Many of these standards overlap and can be taught through a series of experiments. After reviewing the released state questions, all of the 7 th grade samples look at making predictions about single event probabilities and not compound events. Students will need to determine probabilities of compound events, but they are not asked to make predictions and compare theoretical and experimental probabilities of compound events. Unit Vocabulary *Other probability vocabulary and definitions can be found in the units for 5 th and 6 th grades. Word Experimental Probability Theoretical Probability Definition 1) Probability based on experimental data; 2) the ratio of the total number of times the favorable outcomes happens to the total number of times the experiment is done found by repeating the experiment several times The chance of an event happening, it is a prediction
2 Aims Sequence Lesson Aims 1 SWBAT define theoretical and experimental probability SWBAT explain the difference between theoretical and experimental probability SWBAT explain that the theoretical probability will more closely match the experimental if you perform more trials 2 SWBAT predict the outcome of rolling a die number 16 SWBAT compare predicted and actual results from a probability experiment. 3 SWBAT predict the outcome of spinning an equally divided spinner SWBAT compare predicted and actual results from a probability experiment. 4 SWBAT predict the outcome of spinning a spinner that is not equally divided SWBAT compare predicted and actual results from a probability experiment. 5 SWBAT create a spinner to correspond to a set of probabilities. (This lesson will ask students to take what they learned in geometry and apply it to probability). It is an effective way to review central angles and protractor skills 6 Assessment Task SWBAT create a spinner to correspond to a set of probabilities. SWBAT predict the outcome of spinning a spinner that is not equally divided SWBAT compare predicted and actual results from a probability experiment.
3 Lesson 1 In this lesson students will define the terms theoretical and experimental and conduct a probability experiment in order to make comparisons between theoretical and experimental. I am attaching a handout for this lesson. If you use handouts in your class, then you can just make any adjustments. If you use notebooks, you can use this worksheet as the basis of their notes. In this lesson, you can make the jump to expressing probability using more formal notation. Instead of saying the probability of flipping a coin and getting a tails, you can introduce the notation P(T) = ½. This type of notation is used on Regents, but not the 7 th grade state test; so give problems using both notations. Lesson Overview: 1) Define terms and new Probability notation. Key point to make is that theoretical probabilities do not change, while experimental probabilities change from trial to trial and experiment to experiment. 2) Make predictions about outcomes based on the theoretical probability of flipping a coin. 3) Explain how to make predictions 4) Conduct experiment of flipping a single coin. Students can work with a partner or individually. Have a box top or some sort of container to flip the coin into, otherwise you will have coins all over the room. 5) Bring everyone back together. Calculate as a class the % of coins that were heads and % tails after 10 flips. Calculate as a class the % of coins that were heads and % tails after 20 flips (make sure to include data for trials 120). Calculate as a class the % of coins that were heads and % tails after all 50 flips. 6) Compare what happened. Everyone will have slightly different data, but there are easy numbers to work with and can reinforce basics of percents. 7) Use this data to lead into a discussion of how the theoretical and experimental probabilities get closer together by increasing the number of trials. They get closer because of the Law of Large Numbers. Many people think of this as the law of averages. Because flipping a coin is a completely random event, the outcomes average out over multiple flips and you get closer to 50% heads and 50% tails. It is not necessary to go into this level of depth, but this is the mathematical reasoning. 8) Which statement is a correct application of the Law of Large Numbers? a) If we toss a coin 100,000 times then 50,000 times we will get a head. b) If we have a long run of heads when we toss a coin then the next toss is more likely to be a tail. c) The difference between the numbers of heads and tails gets smaller as the number of tosses gets larger. d) The ratio of heads to tails gets closer to 1:1 as the number of tosses gets larger.
4 Student Handout Lesson #1 Lesson Vocabulary Word Definition Theoretical Probability Experimental Probability Trial What is the probability of flipping a coin and landing on a tails? What is the probability of flipping a coin and landing on a heads? Using Probability Notation P(T) = P(H) = Making Predictions 1. If we flip a coin 10 times: how many tails would we expect? how many heads would we expect? 2. If we flip a coin 50 times: how many tails would we expect? how many heads would we expect? 3. If we flip a coin 100 times: how many tails would we expect? how many heads would we expect? 4. How did we make our predictions? 5. If we actually flip a coin 50 times, will we actually get these results? Why or why not?
5 Partner Coin Flipping Directions: Flip your coin ten times. Using tally marks record your results in the chart below. Trials 110 Trials Trials Trials Trials Heads Tails After 10 trials % Heads % Tails After 20 trials After 50 trials Class Questions 1. In 5 th and 6 th grades you learned to express probabilities as a fraction or ratio. Why could we express the probability as a percent? 2. How does the experimental probability change as we increase the number of trials? Does it get closer to or further away from the theoretical probability? 3. Do you think that theoretical probability is a good prediction tool?
6 Exit Ticket 1. Explain the difference between theoretical and experimental probability. 2. If we were to flip the coin 10,000 times, how many heads would you expect? 3. If we were to flip the coin 1,000,000 times, how many heads would you expect? 4. If we were to actually perform this experiment, in which scenario (question 2 or 3), would you expect the results to more closely match the theoretical probability? Why?
7 Lessons 2, 3, and 4 Overview Over the next three lessons students will engage more complicated probability experiments to make predictions from and then compare their results to their expectations. These lessons have students go through 3 different experiments: 1) Rolling a number cube 2) Spinning an equally divided spinner 3) Spinning an unequally divided spinner. There are very specific scenarios that get students making predictions, performing experiments, and comparing results. At the same time HW problems and Quick Questions should reflect the multiple ways the question could be asked on the state exam. In lesson 4 students perform a probability experiment with another spinner, but then have some time to tackle probability problems that might be presented slightly differently. There is not a specified problem set, but the workbook does supply some great problems here. Having a check list of steps on how to attack these problems can be very helpful, especially for your low level students. A. Steps to determine the theoretical probability of an event Step 1: Add the total number of outcomes = 30 Step 2: Count the number of outcomes for a specific event 9 Step 3: Write as a fraction with the total on the bottom 9 30
8 Step 4: Check to make sure the question is not asking you for a percent B. Steps to make a prediction Step1 : Underline the event the problem is looking (P (odd number)) Step2: Write the theoretical probability as a fraction (if the probability is given as a percent, students can change it to a fraction or decimal first) 3/5 Step 2: Multiply the fraction by the number of trials 3/5 x 240 How to Make Spinners In these lessons, students will conduct two spinner experiments. See Marilyn Burns About teaching Mathematics, 2 nd edition, for instructions about how to make spinners, p. 62. However, there is one change I would suggest. Instead of using a piece of straw, just take another paper clip and put that on the other paper clip post. This paper clip is now the spinner.
9 Student Handout Lesson #2 Rolling a Number Cube Imagine that you are rolling 1 number cube. What is the probability of each of the following rolls?: 1. P(1) = 2. P(2) = 3. P(3) = 4. P(4) = 5. P(5) = 6. P (6) = 7. P(even) 8. P(odd) 9. P(prime) 10. P(composite) Now let s use these theoretical probabilities to make some predictions about what will actually happen. 12rolls 60 rolls 120 rolls 360 rolls Even Odd Prime Composite
10 Now let s see what actually happens. Roll your die 60 die times and record your results on the chart below. Number on the cube 1 Number of rolls How many times did you roll an even number? How many times did you roll an odd number? How many times did you roll a prime number? How many times did you roll a composite number? Answer the following set of questions: 1. Did you roll more or fewer 1 s than expected? 2. Did you roll more or fewer 5 s than expected? 3. Did you roll more or fewer prime numbers than expected? 4. Did you roll more or fewer even numbers than expected? 5. Did you roll more or fewer composite numbers than expected? 6. If you had rolled the die 120 times, would the results have been closer or further away from the predicted values.
11 Exit Ticket 1. Derek conducts a probability experiment for his math class. He uses the 10 cards shown below. Card 1 Card 2 Card 3 Card 4 Card 5 Black Black Black White White Card 6 Card 7 Card 8 Card 9 Card 10 White White Grey Grey Grey He randomly picks one of the 10 cards from a container, looks at the color, and replaces the card. He repeats this 100 times. How many times would you expect Derek to pick a white card? A. 20 times B. 30 times C. 40 times D. 50 times Explain on the lines below how you determined this answer.
12 Student Handout Lesson #3 Spinner Experiment 1: Use the spinner labeled Using your spinner determine the probability of the following events as a fraction and a percent. 2. What if you spun the spinner 30 times? How many of each outcome would you expect? 3. What if you spun the spinner 240 times? How many of each outcome would you expect? 4. Spin your spinner 30 times and record your results in the chart below. P(0) P(1) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(even) P(odd) P(multiple of 3) P(factor of 8) P(prime P(composite) Probability as fraction Probability (%) Prediction, 30 spins Prediction, 240 spins Experimental Results (30 total spins) Comparing results 1. Did you spin more or fewer prime numbers than you expected? 2. Did you spin more or fewer multiples of 3 than you expected? 3. Did any of your experimental results match your predicted values?
13 Exit Ticket Directions: Fill out the chart below using this spinner. A Probability as a fraction Probability as a % Prediction for 64 spins Prediction for 328 spins B C D E F G H
14 Student Handout Lesson #4 Spinner Experiment 2 Using your spinner determine the probability of the following events as a fraction and a percent. Record your answers in the first column of the chart. P(3) Probability as fraction Probability as a % Prediction for 30 spins Prediction for 324 spins Experimental results 30 spins P(4) P(6) Comparing results 1. Did you spin more or fewer 3s than you expected? 2. Did you spin more or fewer 4s than you expected?
15 Spinner for experiment #2
16 Exit Ticket D 35% A 20% C 15% B 30% A Probability as a fraction (simplified) Prediction for 20 spins Prediction for 80 spins Prediction for 860 spins B C D
17 Lesson 5 Overview Today we are going to combine what we know about probability with what we know about geometry. We will use many of the skills from creating circle graphs and using protractors. Potential quick questions: Measuring angles with a protractor Finding the part of a whole (3/5 of 500 or 10% of 40) reading data from circle graphs In this lesson students will be given a blank spinner template. Students will 1. Create a spinner that has 5 sections: A, B, C, D, and E 2. Your spinner must meet the following requirements: P(A) = 20 % P(B) = 1/3 P (C) = 25% P(D)= 1/6 P(E) = 5% 3. In order to create your spinner, you will need to know the central angle for each of the sections of your spinner. Ask students why they would need to know this in order to create their spinner. Make explicit connections to circle graphs to access student prior knowledge about central angles. Calculate the central angle for each section of the spinner. 4. Work with students on constructing spinners using their protractor. Steps 5 and 6 should be completed independently and answers collected from students as their exit ticket. 5. Writing a paragraph explanation for step 7 might be a stretch for some students. Depending on your class, you might want to bring everyone back together to work on writing a quality paragraph. Or have them write the paragraph, but don t grade that section. Share examples without names in the next class period. As a class discuss what made a good paragraph or a bad paragraph and then give students an opportunity to correct.
18 Lesson 5: Student Handout 1. Create a spinner that has 5 sections: A, B, C, D, and E 2. Your spinner must meet the following requirements: P(A) = 20 % P(B) = 1/3 P (C) = 25% P(D)= 1/6 P(E) = 5% 3. In order to create your spinner, you will need to know the central angle for each of the sections of your spinner. Calculate the central angle for each section of the spinner. Probability A 20% Size of each angle B 1/3 C 25% D 1/6 E 5% 4. Using your blank spinner template and your protractor, draw each sector of your spinner and label it with A, B, C, D, E. 5. Complete the chart below A B C D E Predict outcomes for 40 spins Predict outcomes for 100 spins Predict outcomes for 340 spins
19 6. Make your spinner. Spin your spinner 40 times and record the results below. A Experimental results B C D E 7. Write a paragraph, explaining how your experimental results compare to your predictions.
20 Blank Spinner Template
21 Performance Assessment Task: Make your own spinner 1. Create a spinner that has 4 sections: A, B, C, and D. 2. You may use the same probability for a section only once! 3. In order to create your spinner, you will need to know the central angle for each of the sections of your spinner. Calculate the central angle for each section of the spinner. Probability Size of each angle A B C D 4. Using your blank spinner template and your protractor, draw each sector of your spinner and label it with A, B, C, and E. 5. Complete the chart below (if you answer is a decimal, round to the nearest whole number). Predict outcomes Predict outcomes for 50 spins for 250 spins A B C D Predict outcomes for 420 spins 6. Make your spinner. Spin your spinner 50 times and record the results below. Experimental results A B C D 7. Write a paragraph, explaining how your experimental results compare to your predictions.
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