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Lesson 10: Marbles, Marbles…

Lesson 10: Marbles, Marbles...

Objective:

Students will understand that random events vary, so that the percentage predicted by a probability isn't exact, but varies. Students practice converting percentages to proportions.

Materials:

  1. For each student team: 50 marbles – 20 of one color, and 30 of another color

    Note: Marbles can be substituted for other materials, such as small blocks, as long as they are the same size.

Vocabulary:

proportion percentage event with replacement without replacement

Essential Concepts:

Essential Concepts:

There are two ways of sampling data that model real-life sampling situations: with and without replacement. Larger samples tend to be closer to the "true" probability.

Lesson:

  1. Remind students that, during the previous two lessons, they learned how to estimate probabilities for a population with the help of simulations to create sample data. Both lessons had nice, prepackaged functions already available in RStudio, which made the simulations fairly quick and easy – in Lesson 8, the rflip() function was used to simulate flipping a coin; and in Lesson 9, the roll_die() function was used to simulate rolling one of two dice.

  2. But what if we don’t have a nice function to perform a simulation for us? Can we create our own method? Yes! We will actually learn to create a simulation from scratch during Lab 2C.

  3. Ask students:

    1. If you have a bag of 50 marbles, where 20 of them are blue and 30 of them are red, what is the probability of drawing a red marble? 30/50 or 60%.

  4. Select a student to answer the question. Ask the class if they agree or disagree. If they agree, ask them to raise their hand. If there are students who disagree, lead a class discussion until a consensus is reached.

  5. Ask students to share their strategies on how to convert the proportion into a percentage. As strategies are being shared, students should take notes in their IDS journals. Review how to turn fractions into percentages, if necessary.

  6. Ask students:

    1. What if we actually drew out one marble, recorded its color, then replaced it back in the bag, and did this 10 times? Answers will vary by class.

    2. Would the percentage of red marbles in this sample necessarily be exactly the same as the probability? Identify that each time a marble is drawn, we are creating an event. Answers will vary by class.

  7. Distribute the bags of marbles to each team. Ask each team to:

    1. Shake the bag of marbles.

    2. Draw one marble out of the bag.

    3. Record the marble’s color in their IDS journal.

    4. Replace the marble back into the bag. Inform them that this is called sampling with replacement. Ask them to consider what “with replacement” means and discuss with the class. “With replacement” means that after you select a marble from the bag, you have to put it back into the bag (replace it) before you select another marble.

    They should draw 10 marbles from the bag and record the observed colors.

  8. Do a Whip Around to find out how many times each team drew a red marble out of their 10 draws. Have them calculate the corresponding sample proportions. For example, if one team drew 7 red marbles out of their 10 draws, their sample proportion is 7/10 = 0.70 (which is the sample as a sample percentage of 70%).

  9. Ask students why the proportions are perhaps different from each other and from the “true” probability of drawing a red marble?

  10. Have the student teams continue drawing marbles, one at a time and with replacement, until they have 50 events recorded. Discuss the following questions:

    1. How many times did they draw a red marble out of these 50? Answers will vary by class.

    2. What’s the corresponding sample proportion? Is it closer to the true probability than when you only drew 10 marbles? Answers will vary by class. But, they should notice that, as the sample size increases, the sample proportion gets closer to the true population proportion.

  11. Engage students in a discussion about how the sample size affects the sample proportion. They should see that as they draw more marbles, their sample probability gets closer and closer to the true probability. If we were to continue drawing marbles forever, in the long run, our sample proportion should equal our true probability.

  12. Have students consider what it might mean to sample without replacement. How would they do that with their bag of marbles? “Without replacement” means that after you select a marble from the bag, you never put it back into the bag (do not replace it). Instead, you simply select another marble from the bag immediately. Students should recognize that, by not replacing the marble each time, the probabilities will change. This means each draw from the marble bag is NOT independent from another draw because removing one marble impacts the next event.

  13. Exit Slip: Based on this lesson, ask students to describe a sample, an event, and replacement.

Class Scribes:

One team of students will give a brief talk to discuss what they think the 3 most important topics of the day were.

Next Day

LAB 2C: Which Song Plays Next?

Complete Lab 2C prior to Lesson 11.