This Math Talk focuses on dividing sums. It encourages students to think about division problems with the same solution and to rely on the structure of previous solutions to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students calculate means.
To divide sums with more terms, students need to look for and make use of structure (MP7).
In describing their strategies, students need to be precise in their word choice and use of language (MP6).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
Find the value of each expression mentally.
Student Response
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Building on Student Thinking
Activity Synthesis
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because . . . .” or “I noticed , so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking, Representing
17.2
Activity
Standards Alignment
Building On
Addressing
7.SP.C.8.c
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
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: They flip two coins to represent 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).
During the Launch, explain that a simulation is an experiment with similar probabilities for outcomes that is used to approximate another experiment. For example, it would be difficult and time consuming to breed a lot of mice to look at the color of their fur, but flipping coins is much easier. Because the probability of getting two heads () matches the probability of getting a baby mouse with white fur, flipping two coins can be used to simulate the outcome of breeding mice.
Launch
Arrange students in groups of 3. Provide 2 coins for each group.
Ask students: “When flipping two coins, what is the probability of both landing heads up?” ()
For context, it might be helpful to explain that mice are often used in science experiments since they have similar genetics to humans, but are easier to maintain. Setting up a mating to work with a new generation of mice with specific combinations of genes can be costly and time-consuming, so it can help to simulate some outcomes before actually beginning the experiment. The word “offspring” refers to children.
Give students 5–7 minutes for group work, and follow with a whole-class discussion.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Check in with students to provide feedback and encouragement after 3–5 minutes of work time. Supports accessibility for: Attention, Social-Emotional Functioning
Activity
None
A scientist is studying the genes that determine the color of a mouse’s fur. When two mice with brown fur breed, there is a 25% chance that each baby will have white fur. For the experiment to continue, the scientist needs at least 2 out of 5 baby mice to have white fur.
To simulate this situation, a coin can be flipped twice for each baby mouse.
If the coin lands heads up both times, it represents a mouse with white fur.
Any other result represents a mouse with brown fur.
Each group member will simulate the mice having 5 baby mice three times. Write your own results for the fur color of the mice in the table.
mouse 1
mouse 2
mouse 3
mouse 4
mouse 5
Do at least 2
have white fur?
simulation 1
simulation 2
simulation 3
Based on the results from everyone in your group, estimate the probability that the scientist’s experiment will be able to continue.
How could you improve your estimate?
Activity Synthesis
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. “What is the probability using the class data?” (Answers vary. For reference, the actual probability is .)
“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?” (We could use a spinner with 25% of the circle labeled “white” and 75% labeled “brown,” or one white block and three brown blocks in a bag.)
17.3
Activity
Standards Alignment
Building On
Addressing
7.SP.C.8.c
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
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) 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).
Launch
Math Community
Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms or invite a student to read them out loud. Tell students that during this activity they are going to practice looking for their classmates putting the norms into action. At the end of the activity, students can share what norms they saw and how the norm supported the mathematical community during the activity.
Arrange students in groups of 3. Assign each group a question slip from the blackline master. Provide access to materials for simulation such as number cubes, compasses, protractors, rulers, paper bags, colored snap cubes, scissors, and coins. Give students 15 minutes for group work, and follow with a whole-class discussion.
Activity
None
Your teacher will give your group a situation.
Design a simulation that you could use to estimate a probability. Show your thinking. Organize it so it can be followed by others.
Explain how you used the simulation to answer the questions posed in the situation.
Student Response
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Activity Synthesis
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:
Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:
“I noticed our norm “” in action today and it really helped me/my group because .”
MLR8 Discussion Supports. Display sentence frames to support students justifying their simulation from their interpretation of the situation: “This simulation was designed so that . . . .” and “The simulation and the actual event have the same probabilities because . . . .” Advances: Speaking, Writing, Representing
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
Lesson Synthesis
Consider these questions for discussion:
“What are some things you had to consider when designing your simulation?” (Among other things, the probability of the actual portion of the event should match the probability of the associated simulated event.)
“What did you learn from the simulations the other groups did?”
“Were the results of any of the simulations surprising?”
“Why would it make sense to design and run a simulation rather than repeat the actual experiment multiple times?” (A simulation is useful when the actual experiment is costly in time or resources or cannot be controlled or repeated.)
Student Lesson Summary
Many real-world situations are difficult to repeat enough times to get an estimate for a probability. If we can find probabilities for parts of the situation, we may be able to simulate the situation using a process that is easier to repeat.
For example, if we know that each egg of a fish in a science experiment has a 13% chance of having a mutation, how many eggs do we need to collect to make sure we have 10 mutated eggs? If getting these eggs is difficult or expensive, it might be helpful to have an idea about how many eggs we need before trying to collect them.
We could simulate this situation by having a computer select random numbers between 1 and 100. If the number is between 1 and 13, it counts as a mutated egg. Any other number would represent a normal egg. This matches the 13% chance of each fish egg having a mutation.
We could continue asking the computer for random numbers until we get 10 numbers that are between 1 and 13. How many times we asked the computer for a random number would give us an estimate of the number of fish eggs we would need to collect.
To improve the estimate, this entire process should be repeated many times. Because computers can perform simulations quickly, we could simulate the situation 1,000 times or more.
Standards Alignment
Building On
5.OA.A.1
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
