The purpose of this Warm-up is to get students thinking about probabilities and spinners, which will be useful when students compute these values in a later activity. While students may notice and wonder many things about these spinners, the probabilities are the important discussion points.
Launch
Arrange students in groups of 2. Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner.
Student Lesson in Spanish
Activity
None
What do you notice? What do you wonder?
Circular spinner divided into four equal parts. The first part is red and labeled “R,” the second part is blue and labeled “B,” the third part is green and labeled “G,” and the fourth part is yellow and labeled “Y.” The pointer is in the part labeled “B.”
Circular spinner divided into five equal parts. Starting from the top right, and moving clockwise, the first part is labeled 1, the second, 2, the third, 3, the fourth, 4, and the fifth, 5. The pointer is in the part labeled “5.”
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If probability does not come up during the conversation, ask students to discuss this idea.
9.2
Activity
Standards Alignment
Building On
Addressing
7.SP.C.8.a
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
In this activity, students are reminded how to calculate probability based on the number of outcomes in the sample space, then apply that to multi-step experiments. The events are described in everyday language, so students need to reason abstractly (MP2) to identify the outcomes described. This lesson begins with students returning to a problem they have previously seen when writing out the sample space. This will save students some time if they can recall or refer back to the initial problem.
Launch
Tell students: “For sample spaces where each outcome is equally likely, recall that the probability of an event can be computed by counting the number of outcomes in the event and dividing that number by the total number of outcomes in the sample space.”
For example, there are 12 possible outcomes when flipping a coin and rolling a number cube. If we want the probability of getting heads and rolling an even number, we count that there are 3 ways to do this (H2, H4, and H6) out of the 12 outcomes in the sample space. So the probability of getting heads and an even number should be ( or , or 0.25).
Remind students that they have already drawn out the sample space for this chance experiment in a previous activity, and they may use that to help answer the questions.
Give students 5 minutes quiet work time followed by partner and whole-class discussion.
Action and Expression: Develop Expression and Communication. Give students access to physical spinners to manipulate. Supports accessibility for: Conceptual Processing, Organization
Activity
None
What is the probability of getting:
Green (G) and 3?
Blue (B) and any odd number?
Any region other than red (R) and any number other than 2?
Circular spinner divided into four equal parts. The first part is red and labeled “R,” the second part is blue and labeled “B,” the third part is green and labeled “G,” and the fourth part is yellow and labeled “Y.” The pointer is in the part labeled “B.”
Circular spinner divided into five equal parts. Starting from the top right, and moving clockwise, the first part is labeled 1, the second, 2, the third, 3, the fourth, 4, and the fifth, 5. The pointer is in the part labeled “5.”
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is for students to explain their interpretations of the questions and share methods for solving.
Some questions for discussion:
“How did you calculate the number of outcomes in the sample space?” (Counting the items in the tree, table, or list, or using the multiplication idea from an earlier lesson.)
“For each problem, how many outcomes were in the event that was described?” (There is only 1 way to get G and 3. There are 3 ways to get B and an odd number: B1, B3, B5. There are 12 outcomes in the last event because there are 3 regions that are not R and 4 regions that are not 2, and .)
9.3
Activity
Standards Alignment
Building On
Addressing
7.SP.C.8.a
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
In this activity, students continue to compute probabilities for multi-step experiments using the number of outcomes in the sample space. They are assigned either a list, table, or tree to examine the sample space for a chance experiment, then they use their understanding of the sample space to write probabilities for various events happening. Students are also reminded that some events have a probability of 0, which represents an event that is impossible. In the discussion following the activity, students are asked to think about the probabilities of two events that make up the entire sample space and have no outcomes common to both events (MP2).
Launch
Keep students in groups of 2.
Assign each group a representation for writing out the sample space: a tree, a table, or a list. Tell students that they should write out the sample space for the first problem using the representation they were assigned. (This was done for them in a previous lesson, and they are allowed to use those as a guide if they wish.)
Tell students that they should work on the first problem only and then pause for a discussion before proceeding to the next problems.
Give students 2 minutes of partner work time for the first problem followed by a pause for a whole-class discussion centered around the different representations for sample space.
After all groups have completed the first question, select at least one group for each representation, and have them explain how they arrived at their answer. As the groups explain, display the appropriate representations for all to see. Ask each of the groups how they counted the number of outcomes in the sample space as well as the number of outcomes in the event using their representation.
After students have had a chance to explain how they used the representations, ask students to give some pros and cons for using each of the representations. For example, the list method may be easy to write out and interpret but could be very long and is not the easiest method for keeping track of which outcomes have been written and which still need to be included.
Allow the groups to continue with the remaining problems, telling them they may use any method they choose to work with the sample space for these problems. Give students 10 minutes of partner work time followed by a whole-class discussion about the activity as a whole.
Activity
None
Your teacher will assign you to use either a list, table, or tree. Be prepared to explain your reasoning.
A number cube is rolled and a coin is flipped.
What is the probability of getting tails and a 6?
What is the probability of getting heads and an odd number?
Pause here so your teacher can review your work.
You may use any method you wish to answer these questions. Suppose you roll two number cubes. What is the probability of getting:
Both cubes showing the same number?
Exactly one cube showing an even number?
At least one cube showing an even number?
Two values that have a sum of 8?
Two values that have a sum of 13?
Jada flips three quarters. What is the probability that all three will land showing the same side?
Student Response
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Building on Student Thinking
Some students may not recognize that rolling a 2 then a 3 is different from rolling a 3 then a 2. Ask students to imagine the number cubes are different colors to help see that there are actually 2 different ways to get these results.
Similarly, some students may think that HHT counts the same as HTH and THH. Ask the student to think about the coins being flipped one at a time rather than all tossed at once. Drawing an entire tree and seeing all the branches may further help.
Activity Synthesis
The purpose of the discussion is for students to explain their methods for solving the problems and to discuss how writing out the sample space aided in their solutions.
Poll the class on how they computed the number of outcomes in the sample space and the number of outcomes in the event for the second set of questions given these options: list, table, tree, computed outcomes without writing them all out, or another method.
Consider these questions for discussion:
“Which representation did you use for each of the problems?”
“Do you think you will always try to use the same representation, or can you think of situations when one representation might be better than another?”
“Do you have a method for finding the number of outcomes in the sample space or event that is more efficient than just counting them?” (The number of outcomes in the sample space for the number cubes can be found using . To find the number of outcomes with at least 1 even number, I know there are 6 for each time an even is rolled first and only 3 for each time an odd number is rolled first, so the number of outcomes is .)
“One of the events has a probability of 0. What does this mean?” (It is impossible.)
“What is the probability of an event that is certain?” (1)
“Jada was concerned with having all three coins show the same side. What would be the probability of having at least one coin not match the others?” (68. There are 6 outcomes where at least one coin does not match: HHT, TTH, HTH, THT, HTT, THH.)
“How are the probability of having at least 1 coin not matching the other and the probability of having the coins match related? How does it address Jada's concern?” (Because every outcome in the sample space has the coins either matching or not, and because there is no outcome that applies to both events, together the sum of their probabilities must be 100%, or 1.)
9.4
Activity
Optional
Standards Alignment
Building On
Addressing
7.SP.C.8.a
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
In this activity, students see an experiment that has two steps where the result of the first step influences the possibilities for the second step. Often this process is referred to as doing something “without replacement.” At this stage, students should approach these experiments in a very similar way to all of the other probability questions they have encountered, but they must be very careful about the number of outcomes in the sample space (MP6).
Launch
Keep students in groups of 2. Give students 5–7 minutes of quiet work time followed by partner and whole-class discussion.
Identify students who are not noticing that it is impossible to draw the same color twice based on the instructions. Refocus these students by asking them to imagine drawing a red card on the first pick and thinking about what’s possible to get for the second card.
MLR6 Three Reads. Keep books or devices closed. Display only the problem stem, without revealing the questions. “We are going to read this paragraph 3 times.”
After the 1st read: “Tell your partner what this situation is about.”
After the 2nd read: “List the quantities. What can be counted or measured?”
For the 3rd read: Reveal and read the first few questions. Ask, “What are some ways we might get started on this?”
Advances: Reading, Representing
Representation: Access for Perception. Use colored cards to demonstrate the situation. Supports accessibility for: Conceptual Processing, Language, Memory
Activity
None
Imagine there are 5 cards. On one side they look the same and on the other side they are colored red, yellow, green, white, and black. You mix up the cards and select one of them without looking. Then, without putting that card back, you mix up the remaining cards and select another one.
What are the outcomes in the sample space? How many outcomes are there?
What structure did you use to write all of the outcomes (list, table, tree, something else)? Explain why you chose that structure.
What is the probability that:
You get a white card and a red card (in either order)?
You get a black card (either time)?
You do not get a black card (either time)?
You get a blue card?
You get 2 cards of the same color?
You get 2 cards of different colors?
Student Response
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Building on Student Thinking
Students may misread the problem and think that they replace the card before picking the next one. Ask these students to read the problem more carefully and ask them, “What is possible to get when you draw the second card while you already have a red card in your hand?”
Activity Synthesis
The purpose of the discussion is for students to compare the same context with replacement and without replacement.
Consider asking these questions for discussion:
“How would your calculations change if the experiment required replacing the first card before picking a second card?” (There would be 25 outcomes in the sample space. The probability of getting the same color twice would be . The probability of getting different colors would be . The probability of getting red and white would be . The probability of getting a black card would be , and not getting a black card would be . It would still be impossible to get a blue card, so its probability would be 0.)
“What do you notice about the sum of the probability of getting a black card and the probability of not getting a black card?” (They have a sum of 1.)
“Why might these outcomes have probabilities with this relationship?” (Since the player either gets a black card or not, together their probabilities should be 1, or 100%.)
Lesson Synthesis
These discussion questions will help students reflect on their learning:
“When the outcomes in the sample space are equally likely, how is the size of the sample space used to calculate the probability of an event?” (When the sample space has size , the probability of a single outcome should be .)
“Now that you’ve had plenty of practice, do you have a favorite method for writing out the sample space?” (I like using the tree.)
“What are some disadvantages of using lists, tables, and trees?” (Lists are hard for large sample spaces, tables are not useful if there are more than one part, and trees can get large.)
Student Lesson Summary
Suppose we have two bags. One contains 1 star block and 4 moon blocks. The other contains 3 star blocks and 1 moon block.
If we select 1 block at random from each, what is the probability that we will get 2 star blocks or 2 moon blocks?
To answer this question, we can draw a tree diagram to see all of the possible outcomes.
A tree diagram. The first choice has 5 branches, representing the 5 blocks in the bag: one branch is labeled “star,” the other 4 are labeled “moon.” Each of these branches has 4 branches, representing the 4 blocks in the second bag. 3 branches are labeled “star” and one is labeled “moon.” The word “star” in the first choice, and the 3 “star” choices branching from it are highlighted gold. From the first choice, the word “moon” is highlighted blue on each of the four remaining branches. From each of those branches, the one choice of “moon” for each is also highlighted blue.
There are possible outcomes. Of these, 3 of them are both stars, and 4 are both moons. So the probability of getting 2 star blocks or 2 moon blocks is .
In general, if all outcomes in an experiment are equally likely, then the probability of an event is the fraction of outcomes in the sample space for which the event occurs.
Standards Alignment
Building On
5.OA.A.1
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression as a product of two factors; view as both a single entity and a sum of two terms.
