Statistics and Probability
Material Type:
Lesson Plan
Middle School
  • 7th Grade Mathematics
  • Sampling
    Creative Commons Attribution Non-Commercial
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    Self Check Exercise

    Self Check Exercise


    Students critique and improve their work on the Self Check, then work on additional problems. Students revise the Self Check problem from the previous lesson and discuss their strategies.

    Key Concepts

    Students apply what they have learned to date to solve the problems in this lesson.

    Goals and Learning Objectives

    • Apply knowledge of sampling and data analysis to solve problems.
    • Determine a random, representative sample that is nonbiased and of adequate sample size.
    • Generalize about a population based on sampling.
    • Compare data sets.


    Lesson Guide

    Return students' solutions to the assessment task. Students can then select questions appropriate to their own work.

    Have students spend 10 working individually to answer questions.

    Mathematical Practices

    Mathematical Practice 1: Make sense of problems and persevere in solving them.

    During this lesson, students review their work to make sure that they understand how to solve the problem. They also extend their knowledge by applying it to additional problems.

    Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

    In addition to reviewing their own work, students support others and help with their understanding of the concepts.

    SWD: Make sure you provided clear feedback to students as they attempt to solve problems or articulate concepts. This type of feedback guides students explicitly as they develop their thinking about mathematics.



    Review your work on the Self Check problem and think about these questions:

    1. Did you find the five-number summary? How can these values help you analyze the data?
    2. What is the sum of the data?
    3. How many data points are in each quartile?

    Math Mission

    Lesson Guide

    Discuss the Math Mission. Students will apply what they know about sampling and data analysis.


    Apply what you know about sampling and data analysis.

    Revise Your Work

    Lesson Guide

    Organize the class into small groups of two or three students and have them revise the problem together.

    ELL: When forming groups, be aware of your ELLs and ensure that they have a productive learning environment. Different types of partnerships include:

    You can also pair them up based on their math proficiency.

    • Pairing them up with English speakers so they can learn language skills.
    • Pairing them up with students who are at the same level of language skills, so they can take a more active role and they can work things out together.
    • Pairing them up with students whose proficiency level is lower, so they play the role of the ‘supporter’.


    While students work in small groups, note:

    • Different student approaches to the task
    • Students' chosen problem-solving approaches
    • Students' choice of math
    • Students' choice of resources

    Attend in particular to the students' mathematical decisions:

    • Do they track their progress as they use their chosen mathematics?
    • Do they notice if they have chosen a strategy that does not seem to be productive? If so, what do they do?
    • Do students calculate the measures of center and/or spread correctly?
    • Do students set the proportion correctly to generalize for the population?
    • How do they justify their solutions?

    Try not to make suggestions that move students toward a particular approach to this task. Instead, ask questions that help students to clarify their thinking.

    If students find it difficult to get started, these questions might be useful:

    • What do you already know? What do you need to know?
    • How can you summarize the information in the line plots in a single number?
    • What measures can you use?
    • What do you already know about line plots and measures of center and spread?
    • SWD:Make sure that all students have access to and can comprehend the information in the rubric so that they can accurately interpret your assessment of their work. Students with disabilities may benefit from support in understanding the expectations. Allow multiple means of representing the information in the rubric (visual presentation of text, visual supports, etc.). Create and provide an enhanced version of the rubric with embedded text structures (labels, highlights, words in bold) to cue students to pay closer attention to particular terms.


    Student does not understand the idea of a random, representative sample.
    • How many seventh grade students are in the school?
    • How many students should be in the sample?
    • How should the sample be chosen?
    • How should each class be represented?

    Student does not understand how to analyze the data.
    • What measures can you use to decide what is typical for the data?
    • What graph could you draw to represent the data visually?

    Student does not understand how to use ratios to generalize for the population.
    • How many students are in the sample? How many students are in the population?
    • How many students in the sample think the water weighs between 7 and 9 lbs?
    • Using your knowledge of the sample, how can you estimate the number of students in the entire seventh grade who think a gallon of water weighs between 7 and 9 lbs?

    Student provides a poor explanation. For example, the student explains calculations rather than gives mathematical reasons.
    • How can you convince a student in another class that your answer is correct?

    Student provides adequate solutions to all questions.
    • Can you find a different way to solve the problem?


    Answers will vary. Possible answers:

    1. Lucy needs to select a random sample for her data collections. One way to do this would be to write the numbers 1 to 300 on slips of paper, attaching each number to a student, then drawing a few randomly to generate a sample. Each student should be surveyed separately so their answers don't influence other students' answers.
    2. Based on the sample, a typical seventh grade student thinks that a gallon of water weighs 8 lbs.
    3. The median and mode are 8 lbs, and the mean (7.8) rounds to 8 lbs. The range is 4, from 6 to 10, with the data clustered around 8 lbs.
    4. 210 students, or 70% of the students. 710 of students in the sample think a gallon of water weighs between 7 and 9 lbs. The whole population is 300 (10 classes of 30 students each). 710=x300x=210

    Step 1: Work Time

    Revise Your Work

    Work with your partner to revise your work on the Self Check problem based on the previous questions and feedback from your partner.

    Self Check Problem

    A school has 10 seventh grade classes, each with 30 students. Lucy wants to find out how well the students can estimate the weight of a gallon of water. She realizes that she cannot survey everyone, but she isn’t sure how to proceed.

    1. How could Lucy go about collecting her data? Lucy decides to start with a random sample of 10 seventh grade students. The students made the following estimates for the weight of a gallon of water (in pounds): 6, 7, 9, 8, 7, 10, 8, 6, 9, 8
    2. Based on this sample, what does a typical seventh grade student at the school think a gallon of water weighs?
    3. How did you reach your conclusion?
    4. Based on the sample, how many seventh grade students at the school think that a gallon of water weighs between 7 and 9 lbs?

    Challenge Problem


    • Answers will vary. Possible data sets: {3, 4, 5, 6, 8, 8, 9, 10, 11, 12} Mean = 7.6, Range = 9 {2, 2, 2, 2, 8, 8, 10, 10, 10, 10} Mean = 6.4, Range = 8
    • Answers will vary, depending on data sets generated for the previous question. Possible answer, based on the data sets given as examples:
      • Both data sets have a larger range than the data that Lucy collected. The first data set is very spread out, while the second has two peaks at the ends of the range.

    Step 2: Work Time

    Challenge Problem

    • Create a data set for a different group of 10 students such that the data set has a lower mean, a greater range, but the same median.
    • Compare your data set with that of another student. How are the data sets the same? How are they different?

    Make Connections

    Lesson Guide

    Organize a whole class discussion to consider issues arising from the work students did to revise their work. You may not have time to address all these issues, so focus the discussion on the issues most important for your students.

    • Have students share their work and talk about how they approached the problem.
    • Have students share the questions that they addressed and how they addressed those questions.
    • Have students ask questions and make observations as they view each other's work.


    • What part of the task was most difficult?
    • Did you and your partner ever disagree about a set of data values? Did you resolve your difference of opinion? How?

    ELL: Encourage students to use the academic vocabulary they are learning. When ELLs participate in the discussion, monitor for knowledge of the topic. Follow up on statements that seem unclear. When ELLs contribute, focus on content and don't allow grammar difficulties to distract you from understanding the meaning (as much as possible). Help ELLs who make grammar mistakes by rephrasing, ensure your rephrasing does not interrupt or interfere with their thinking.

    Performance Task

    Ways of Thinking: Make Connections

    Take notes about your classmates’ approaches to revising the Self Check problem.

    As your classmates present, ask questions such as:

    • What part of the task was the most difficult?
    • Did you and your partner ever disagree about some aspect of the problem? Did you resolve your difference of opinion? How?
    • How can you summarize the information in the line plots?
    • What measures of center did you use to determine a typical student?

    Reflect on Your Work

    Lesson Guide

    Have each student write a brief reflection before the end of class. Review the reflections to find out how students would revise their work if they were to do the Self Check problem again.

    Work Time


    Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

    Something I would do differently if I started the Self Check problem now is…