Unit 7, Lesson 3
Randomness in Groups
Integrated Math 3
3.1
Warm-up
Standards Alignment
Building On
Addressing
HSS-IC.A.1
Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
HSS-IC.B.3
Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
The mathematical purpose of this activity is to understand the importance of random selection when selecting a sample. Students evaluate 4 methods for selecting a group of people to participate in a survey, to determine the benefits and drawbacks of each.
Launch
Display the four methods for all to see.
Activity
NoneA reporter wants to know how people feel about the governor of her state. She decides to ask 100 people their opinions and thinks of several ways to ask the 100 people. For each method, explain the benefits and drawbacks, then choose the method for selecting 100 people that would best represent the people of the state.
- Go to the capital city, and find 100 people interested in politics to respond to the survey.
- Ask the 100 most politically influential people in the state to respond to the survey.
- Obtain census data for the state and select, using a random process, 100 people from the list to survey.
- Ask 50 registered voters who voted for the governor and 50 registered voters who did not vote for the governor to respond to the survey.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is for students to understand the importance of random selection when selecting a sample. Ask students, “What are the benefits and drawbacks of each method?” and record their thinking for all to see. During the discussion, students should recall the term “sample” from earlier grades to mean a subset of the population. Define the term random selection as a selection process in which each item in a set has an equal probability of being selected.
- “Which method is most likely to represent the people of the state?” (Selecting people with a random process. With random selection, the selection of the people of the state who participate in the survey is due to chance rather than other factors.)
- “What group is being sampled in each method?” (The first method samples only people interested in politics in the capital. The second method samples only the 100 most politically influential people. The third method samples from the state’s population who took part in the census. The fourth method samples registered voters.)
- “Why is random selection important when selecting a sample?” (Random selection allows the sample to better represent the whole population.)
3.2
Activity
Instructional Routines
MLR1: Stronger and Clearer Each TimeMaterials
NoneActivity Narrative
The mathematical purpose of this activity is to understand the importance of randomness when dividing a sample into groups for an experimental study. Students classify the study, determine which method is best for splitting subjects into two groups in order to compare them, and discuss the importance of randomness when assigning groups.
Launch
Conduct a short demonstration similar to the activity students will complete in this task.
Divide the class into 2 halves based on where students are in the room. Tell students that half the class will try to memorize as many words in a list as they can, in silence, and the other half of the class will try to memorize as many words in a list as they can while students are clapping.
For the first half of the class, tell students to be silent and to put down their pencils. Display the word list for 3 seconds:
- ant
- car
- lot
- jug
- hat
- ball
- mat
- lap
- art
- cup
- pit
- sip
After 3 seconds, ask students to write all the words they remember. Display the word list again, and ask students to count the number of words they recalled correctly. Record the number of words each student recalled correctly for the first half of the class. Display the data for all to see.
For the second half of the class's turn, ask the first half of the class to start clapping. Display the words for 3 seconds:
- art
- tar
- cot
- mug
- cat
- mall
- bat
- cap
- ask
- pup
- lit
- dip
After 3 seconds, ask students to write all the words they remember. Display the word list again, and ask students to count the number of words they recalled correctly. Record the data for the second half of the class.
Compare the data for each group. Ask, “Is there evidence to suggest one of the conditions made it easier to recall the words?” (The mean for the silent group is slightly greater, but only by a little bit, so I’m not sure if it’s enough to conclude that it matters.) Students will explore how to analyze data from experiments in a future lesson.
Ask students how the experiment could be improved. (Using a random process to divide the class rather than according to where people are sitting. A better experiment would have used the same list of words and separated the two groups to make sure that all the conditions were the same except for the clapping.)
Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies, such as a computer for students to type the words they remember instead of handwriting them. If this task would be stressful for the student because of dyslexia, dysgraphia, or short-term memory challenges, consider finding another role for the student, such as recording data (using technology, if appropriate).
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing, Organization
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing, Organization
3.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
To Gather
Random number generatorActivity Narrative
The mathematical purpose of this activity is to investigate how finding the mean using different sampling methods relates to the mean for the whole population. Students should recognize the importance of using a true random process for selecting a sample rather than having a person attempt to select items randomly. Students are asked to select items for a sample using several methods that may seem random and one that is truly random, then compare the mean from each sample to the true mean.
Launch
NoneActivity
NoneLesson Synthesis
Here are some questions for discussion.
- “Why is it important to use random selection when selecting a sample to study?” (Using a random selection to get samples means that every item in the population has an equal chance of being selected for the sample.)
- “What is an example of selecting a sample without using random selection? What are the drawbacks of using a sample selected this way?” (If I wanted to know the favorite 11th-grade student council candidate, I might sample only my friends. The drawbacks are that my friends might not represent the whole 11th grade, and my friends might be more likely to vote for the same person as me.)
Display this statement for all to see: “Kiran has a group of 50 people that he is going to study in an experimental study, and he needs them to sign a form to participate. He creates two groups of 25 people. The first group consists of the first 25 people who turned in the signed form. The second group consists of the other 25 people who signed the form.”
- “Are there any potential problems with Kiran’s study? Explain your reasoning.” (Yes, Kiran went through the effort of getting 50 participants, but when he made his groups, he did not choose them randomly. This could be a problem because maybe the first 25 people to sign the form have a different personality or are different in some other way from the other 25 people. By using random selection, Kiran could have avoided this issue.)
- “Why is it important to randomly assign people to groups in an experimental study?” (Using a random process to create the groups helps reduce the likelihood of grouping subjects into groups that may differ on some characteristic that is related to the response of interest.)
Student Lesson Summary
To get information about a characteristic of a population, people often measure that characteristic on a sample of individuals chosen from a population of interest. The idea is to draw conclusions about the population based on data collected from only the sample, a group that is selected from the whole population. To correctly generalize from the sample to the population, the researcher needs to know that the sample is representative of the population as a whole.
Suppose the engineer counts the pretzels only in the last 25 bags of trail mix that are produced on a particular day, and she finds that they contain too many pretzels. Should the engineer conclude that all the bags of trail mix produced that day contain too many pretzels? Not necessarily. Something might have happened late in the day that affected the number of pretzels in the bags. The last 25 bags of trail mix may not be a representative sample of the population.
So how do we get a representative sample? The best way is to let chance select the sample. For example, you might randomly select 25 different times throughout the day to remove the next bag of trail mix from the conveyor belt and count its pretzels. Using a process based on chance, in which each individual in the population is equally likely to be selected, is called random selection of the sample.
In experimental studies, it is often necessary to assign the individual participants in the sample to one or more groups. It is also best to assign individuals to groups using a random process.
For example, say that you were studying the effect of students turning off electronic devices while doing homework. After a representative sample is selected, you need to assign the individuals in the sample to two groups: one group makes no changes to the conditions in which they normally do homework, and another group that turns off electronic devices while doing homework. Examples of assignment processes that are not random include:
- Assigning students whose names start with A–L to one group and M–Z to the other group
- Assigning students who play a musical instrument to one group and the rest to the other group
- Asking for volunteers to be part of the group that turns off electronic devices
In order to assign individuals randomly to groups, every individual must have an equal chance of being assigned to either group. Examples of assignment processes that are random include:
- Writing each participant’s name on a slip of paper and mixing the slips well in a bag. Drawing half of the names from the bag and assigning these participants to one group, and the rest to the other group.
- Flipping a coin for each participant, and placing them in one group if the result is heads and the other group if the result is tails
- Writing a list of participants and numbering the list, then a random number generator to select participants for one group
When subjects are not assigned to experimental groups using a random process, other factors may influence the results from the experimental study so that the data does not answer the initial question. In this example, if the groups are split by volunteering, the impact of turning off the devices may be impacted by similar traits of the subjects who volunteer, such as their not using electronic devices much already or having a personality that is willing to volunteer to try something new. These traits may influence the results so the data from the experimental study do not accurately address the question about the impact of electronic devices on student homework.