Lesson 10

The purpose of this number talk is to elicit strategies and understandings students have for division, particularly when the quotient is the same for different expressions. These understandings will be helpful for students as they are finding mean in upcoming lessons. 

While four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem.

Reveal one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all previous problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”

• “Did anyone have the same strategy but would explain it differently?”

• "What did you notice about the answers to these questions?"

  • "How could we use the number we are dividing by each time to explain why the answers are all the same?"

• “Did anyone solve the problem in a different way?”

• “Does anyone want to add on to _____’s strategy?”

• “Do you agree or disagree? Why?”

In this activity, students revisit the idea of simulating real-life situations with chance experiments. In the next activity, they will design and run their own simulations for situations that involve multiple steps. Here, students are asked to use their understanding of experiments that have multiple steps to simulate a single part of a larger simulation. Namely, flipping two coins represents a single offspring from a pair of mice. Since the outcome probabilities of the simulation and the real-life situation are the same, this is another option for creating simulations that represent real-life scenarios (MP4).

The purpose of the discussion is for students to articulate why the simulation is appropriate and think about other methods of simulating the same situation.

Consider asking these questions for discussion:

• “How could we get a better estimate than what you got in your group?” (Repeat the experiment many more times or combine the data from the class.)
• Collect data from the class to find a better estimate. (For reference, the actual probability is \(\frac{47}{128} \approx 0.37\).)
• “Notice that we used a two-part experiment (flipping two coins) to represent a single thing (one offspring). Why was this ok to do?” (The probability of getting HH on two coins is the same as the probability of getting a single offspring with white fur.)
• “Can you think of another method that would work to simulate a single offspring?” (A spinner with 25% of the circle labeled “white” and 75% labeled “brown.” One white block and three brown blocks in a bag.)

In this activity, each group is assigned a situation for which they will design and perform a simulation to estimate the probability. Students will give a short presentation on the methods and results of their simulation for the class after they have designed and run the simulation. Students will need to attend to precision (MP6) as well as present arguments (MP3) for the simulation method they chose. At this stage, students have experienced a large number of simulation methods and should be able to design their own to represent the situations using the appropriate tools (MP5).

Arrange students in groups of 3. Assign each group a question slip from the blackline master. Provide access to number cubes, compasses, protractors, rulers, paper bags, colored snap cubes, scissors, and coins. Give students 15 minutes for group work followed by a whole-class discussion.

Ask each group to share their situation, their method of simulating the situation, and their results. Students should explain why their chosen method works to simulate the situation they were given. In particular, all important outcomes should be represented with the same probability as stated in the situation.

If all groups that have the same situation use the same simulation method, ask for ideas from the class about alternate methods that could be used for the situation.

For reference, the computed probabilities for each situation are:

1. \(\frac{5}{32} \approx 0.16\)
2. \(\frac{1}{64} \approx 0.02\)
3. \(\frac{59049}{100000} \approx 0.59\)
4. \(\frac{2072}{3125} \approx 0.66\)
5. \(\frac{181}{3125} \approx 0.06\)

Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students show their representations of simulations and explain their reasoning, use color and annotations to scribe their thinking on a display of each problem so that it is visible for all students.
Supports accessibility for: Visual-spatial processing; Conceptual processing