standard_description: Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
problem_activity: Optional activity
In this activity, students begin by practicing their understanding of proportions and and probabilities by examining the data set they have available. In the fourth problem, students obtain a sample from the population using tools they choose (MP5)and examine the sample they selected to compare it to the expected proportions and probabilities calculated in the first 3 problems.
The problems are intended for students to use their own data set to answer. Although they are kept in pairs for the entire lesson, this activity should be done individually.
Launch
Keep students in the same groups of 2. Give students 5–7 minutes of quiet work time followed by a whole-class discussion.
If possible, allow students to use their chosen method of random sampling to obtain a sample of 10 for this activity. Have items such as paper clips, scissors, 10-sided polyhedra, and other materials available for student use. The blackline master for the first activity in this lesson contains accurate spinners that could be used to select a random sample.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. After students have solved the first 2-3 problems, check-in with either select groups of students or the whole class. Invite students to share the strategies they have used so far as well as any questions they have before continuing.
Supports accessibility for: Organization; Attention
Student Facing
Continue working with the data set your teacher gave you in the previous activity. The data marked with a star all came from students at Springfield Middle School.
When you select the first value for your random sample, what is the probability that it will be a value that came from a student at Springfield Middle School?
What proportion of your entire sample would you expect to be from Springfield Middle School?
If you take a random sample of size 10, how many scores would you expect to be from Springfield Middle School?
Select a random sample of size 10.
Did your random sample have the expected number of scores from Springfield Middle School?
Student Response
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Activity Synthesis
The purpose of this discussion is to connect the ideas of probability and random sampling from the unit.
Consider these questions for discussion:
“How is selecting a sample at random connected to probability?” (A random sample should give each value an equal chance of being chosen. Therefore, each value has a
\(\frac{1}{100}\)
probability of being chosen.)
“How could we simulate the probability of getting at least 2 values in the sample of 10 from Springfield Middle School?” (Since 20% of the values come from Springfield Middle School, we could put 10 blocks in a bag with 2 colored red to represent Springfield Middle School. Draw a block from the bag, and if it is red, it represents a score from Springfield Middle School; replace the block and repeat. Get a sample of 10 and see if the sample has at least 2 red blocks. Repeat this process many times and use the fraction of times there are at least 2 red blocks as an estimate for the probability that a random sample will have at least 2 scores from Springfield Middle School.)
Representing, Speaking: MLR7 Compare and Connect.
Invite students to create a visual display of their random sample of size 10 and response to the question: “Did your random sample have the expected number of scores from Springfield Middle School?” Invite students to investigate each other’s work and compare their responses. Listen for the language students use to describe a random sample and assign a probability of each value being chosen. This will help students connect the ideas of probability and random sampling through discussion.
Design Principle(s): Optimize output (for representation); Cultivate conversation


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