Grade Level(s): 9, 10
Computer for Presenter, Scientific Calculator
Keywords: Standard deviation, normal distribution, histogram, mean, Empirical Rule
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
- Students will be able to calculate the values in a data set that are one, two, and three standard deviations above and below the mean.
- Students will be able to estimate population percentages for a normally distributed data.
- Students will use the empirical rule for a normal distribution to draw conclusions about the performance of the participant in a math competition.
Prior Knowledge: What prior knowledge should students have for this lesson?
- In order to be actively engaged in the lesson activities and discussions students need to know how the mean and standard deviation are determined and interpreted in the context of a given data set.
- Students need to know how histograms are constructed and interpreted. In order to elicit prior knowledge students will be provided with two histograms representing a symmetric and a skewed distribution and asked to describe the differences in the two distributions.
- To avoid common errors in estimating the population percentages students may be reminded that they must find the sum of all values that satisfy the given condition.
Guiding Questions: What are the guiding questions for this lesson?
- How do we measure success or failure in a math competition?
- How does the Empirical Rule help us evaluate scores?
- What results can be considered top performance?
Teaching Phase: How will the teacher present the concept or skill to students?
The attached PowerPoint (answers are found on the notes section of the powerpoint) is to be used to conduct the following lesson:
- The teacher will present the "Problem of the Day: Brian's Dilemma" describing a real-life situation – the final scores of a participant in the MathCounts competition.
- The teacher will initiate a brief discussion about whether the information presented in the opening question indicates success or failure.
- The teacher will explain that while the perception of success or failure is often subjective, certain objective information gathered from the presented data can help to determine the participant’s standing in the competition.
- The teacher will ask students to recall the procedure for calculating the mean and standard deviation of set of data.
- The teacher will ask students to compare the spread of two sets with the same mean but different standard deviations.
- The teacher will ask students to recall how to construct histograms and how to interpret the shape of the distribution of the data preparing them to the discussion of a normal distribution.
- The teacher will use the PowerPoint slides to introduce the Empirical Rule for a Normal Distribution and provide examples of normally distributed data.
- The students will complete Guided Practice and Independent Practice assignments on the PowerPoint finding the percentages in a normally distributed data that satisfies given conditions as well as finding the values in a given set that fall within certain numbers of standard deviations of the mean.
- The students will answer questions on the 5-minute check and turn in.
- In the final stage of the lesson students will revisit the opening Question of the Day to estimate what percent of the participants of the MathCounts Competition scored below the student whose data was presented in the question.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
The problems in the Guided Practice presented on the PowerPoint (answers are found on the notes section of the PowerPoint) will be first discussed and solved in small groups and then checked in a brief whole class discussion. For each problem, clarifications will be made and misconceptions will be corrected before students proceed to the next problem.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?
Working individually, students will solve the Independent Practice problems presented on the PowerPoint (answers are found on the notes section of the PowerPoint) that involve the concepts developed in this lesson. After independent work, answers will be discussed.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Students will revisit the opening "Problem of the Day: Brian's Dilemma" about the participant of the MathCounts Competition and write a short paragraph about the conclusion they have reached. They will have to justify their conclusion by determining the percent of all participants whose scores were below than Brian's.
- A 5-minute check at the end of the lesson, found on the attached PowerPoint, consists of three questions that will require students to estimate population percentages for a normally distributed data. Students should complete this check at end of class and turn it in.
- After presenting the Question of the Day (found on the attached PowerPoint) about a student's competition score at Mathcounts, the teacher will help reactivate student prior knowledge about the procedures for computing mean and standard deviation and the method of constructing histograms in order to prepare students to learn about a normal distribution.
- During the initial discussion of the Question of the Day, the teacher will ask students to share their intuition about whether the participant's final score in the competition appears to indicate that he did better than a few or the majority of the competitors.
- Throughout the teaching phase, the teacher will ask leading questions to ensure student comprehension of each topic of the discussion.
- Using questioning, the teacher will check that each step of the procedure for estimating the population percentages and calculating data values that fall within certain number of standard deviations of the mean are understood and performed accurately.
- Upon completion of the Guided Practice, the teacher will ask students to recall and review steps of the above procedures in order to prepare them for Independent Practice and 5-minute check at the end of the lesson.
Feedback to Students
- In the wrapping-up stage of the lesson, the teacher will summarize the discussions that took place in the lesson and ask students to write a short paragraph about the conclusions they have reached regarding the Problem of the Day: Brian's Dilemma. Some of the paragraphs will be read in class.
- The solutions to the Summative Assessment – the 5 minute check – may be revealed on the following day, or, time permitting, at the end of the lesson.
Accommodations & Recommendations
- The teacher may provide copies of the histograms representing normally distributed data for students who have difficulty in understanding the bell-shape of a normal curve and ask them to mark the midpoint of the top of each bar. The teacher will have them connect the marked points with a smooth curve where the midpoint of the bar with the highest frequency represents the mean, median, and mode of the set.
- For the initial guided example, the teacher may present a data set consisting of 100 items to make the concept easier to understand.
- If time permits, students can be given examples of how knowing percentages of data under sections of a normal curve allows to make predictions about the larger population for which a normally distributed sample of data is given.
- Some students may ask questions about what happens when a certain value in a set is not an integral number of standard deviations away from the mean. An introduction of z-scores may be appropriate for some advanced students.
Suggested Technology: Computer for Presenter, Scientific Calculator
Special Materials Needed:
The lesson should be presented with a PowerPoint. However, students may be provided with printed copies of the PowerPoint Slides if computer projection is not available.
The teacher should be familiar with the PowerPoint presentation before the lesson is taught.
MAFS.K12.MP.1 Make sense of problems and persevere in solving them.
MAFS.K12.MP.2 Reason abstractly and quantitatively.
MAFS.K12.MP.3 Construct viable arguments and critique the reasoning of others.
Source and Access Information
Name of Author/Source: Margarita Melkumova
District/Organization of Contributor(s): Miami-Dade
Access Privileges: Public
* Please note that examples of resources are not intended as complete curriculum.