Grade Level(s): 9, 10, 11, 12
Document Camera, Graphing Calculators, Computer for Presenter
Resource supports reading in content area:Yes
Freely Available: Yes
Keywords: Mean, Median, Central Tendencies, Normal Distribution, Standard Deviation, Range, Interquartile Range
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 identify which measure of center is appropriate for a data set or graph.
- Students will identify which measure of spread is appropriate for a data set or graph.
- Students will compare two or more data sets or graphs using the appropriate measures of center and shape.
Prior Knowledge: What prior knowledge should students have for this lesson?
The prior knowledge students should have for this lesson is the basic understanding of measures of central tendency and spread.
Students should also understand the application of central tendency and spread concepts with respect to each type of graphical display.
Guiding Questions: What are the guiding questions for this lesson?
The following will act as guiding questions for this lesson: (Answers and instructional information will be included in parentheses and italics.)
- Explain to your partner what is meant by the following statement: "If Bill Gates walked into the faculty meeting this afternoon, the average income of the people in the room will increase by about two million dollars."
(The income of billionaire Bill Gates is so much higher than the salary of the typical teacher that when his income is combined with that of the teachers in the room, the average salary, as represented by the mean, will dramatically increase.)
- If you were the bargaining agent for the Los Angeles basketball players association, would you use the mean or median to describe the "average" salary to the news media
(Consider that Kobe Bryant's salary is approximately $28 million and others on the team receive substantially less.) Why? (The bargaining agent for the basketball players association would want to use the median to emphasize how little the "average" player gets. The mean will be higher than the median because Kobe Bryant's salary will artificially inflate the "average" salary. The median is not impacted extremely large or extremely small amounts. The median represents a location in a data line-up.)
- If you were the owner of the Los Angeles Lakers, would you use the mean or median to describe the "average" salary when you were speaking to the press about contract negotiation? Why?
(The owners would represent the "average" by arguing that the mean salary should be examined because the owners have to pay the entire sum of the salaries and the sum of the salaries is expensive.)
- Are we able to choose between the mean and median on any given data set if one measure of center represents our position better than another measure?
(We are NOT able to chose which measure to use. The mean is appropriate for normally distributed data, while the median is appropriate for data that is not normally distributed. When the data is a perfect example of a normal distribution, the mean and the median will be identical, but this does not mean that it is appropriate to chose which measure of central tendency to use. There is a link between the appropriate measure of central tendency and spread. The shape of the distribution dictates what measures of central tendency and spread to use. If a data set is normally distributed, the mean should be used to describe the center and the standard deviation should be used to measure the spread. If the data is skewed or is not normally distributed, the median is the appropriate measure of central tendency and the range or inter-quartile range would be appropriate to use to describe the spread).
- Are you able to determine if the data which created a box plot is normally distributed? (NO!!!! If a data set is normally distributed, the box plot will appear symmetric.
However, when the box plot appears symmetric, there is insufficient information to suggest that the data is normally distributed).
- If the condition for using the mean and standard deviation is that the data is normally distributed, and your data is displayed on a box plot, how could you determine whether or not it is appropriate to describe the distribution by using the mean and standard deviation?
(The mean and standard deviation are not part of the graph of the box plot, but might be listed somewhere in the problem. If you don't have the raw data, you should not use the mean and standard deviation. The standard deviation finds the typical distance from the mean; therefore, it is always linked to the mean).
- Student Challenge: Define the word "mean" without using the algorithm to find the mean. The word "average" is not a sufficient definition.
(Most students will not be able to describe this without telling you how to find the mean. The students will know what a mean is without understanding it enough to define. The next couple questions will lead them to discover the answer. The answer is that the mean is a balancing point of a set of data. However, if the students don't tell you it is the balance point, use the next few examples to elicit this from them).
- If a 100 pound student sits on one end of a see-saw, and an 800 pound math teacher sits on the other end of the see saw, what is going to happen?
(The 100 pound student will be stuck in the air because s/he does not weigh enough to bring the see saw down to the ground.)
- How do you resolve the dilemma posed by the last illustration so that these people can enjoy the see saw?
(Put seven more 100 pound students on the see saw)
- If this see saw is viewed as a number line, how would we refer to the point at which the see saw will function properly?
(Those who have been exposed to the science concept of "fulcrum" will probably use this vocabulary term. Others may call this a "balancing point." This is the concept that is crucial. The diagram on the Mean Illustration attachment will show that the fulcrum is pulled toward the extremely large or extremely small observation. This illustrates why the mean is so dramatically different from the median. Logic dictates that only one of these values could be the correct answer).
- Take the data used in the last illustration and create a histogram from this data. On the histogram, locate the mean and the median.
(Notice that the mean is influenced by the large observation, yet the median is not. The diagram is included in the attachment entitled Box plot and Histogram).
Teaching Phase: How will the teacher present the concept or skill to students?
- The teacher will pose the question about Bill Gates and the average salary of people in the faculty meeting.
- The teacher will use the guiding questions concerning the salary negotiations for the Los Angeles Lakers and the ways in which people misuse statistics (out of ignorance or deliberate manipulation.)
- The students will be asked to describe WHAT an "average" is. The teacher will remind the students that they are not being asked for the mathematical procedure to find an average. Most students will be stumped at this question, so the teacher will use the illustration of a see-saw with an 800 pound teacher on one side and 8 children who each weigh 100 pounds on the other side. The discussion of "balancing point" vs. middle of data set will be fostered by the teacher.
- The students will be asked to come up with a scenario in which a specific measure of central tendency and spread must be used and they will identify a counter-group who would want to use the wrong measures to misrepresent their side of the issue.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
The activities that the students will complete with teacher guidance are the discussions of the appropriate representation of the Los Angeles Lakers' salary, and the discovery learning process of what an average is. This will all connect to allow students to see that there must be a right and wrong way to represent data set using central tendency, or averages.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?
The exercises that the students will complete to reinforce the concepts and skills developed in the lesson will be questions that lead them forward and backward in the concept of the comprehension of what we are describing to others when we say that we have an "average".
- Students will be given data sets to determine whether it is appropriate to have an average for the data set at all.
- Students will be asked to select five classmates and list their zip code. Mathematically, they will be asked to calculate the "average" zip code, which they will be able to do. Then they will be asked what the meaning of the average zip code is.
- They will be asked to examine whether there are some data sets that should not have an "average" associated with the data.
This practice will allow them to gain deeper insight into describing data sets and understanding that data must be quantitative for an average to have any meaning. Even though zip code is a number, the number is not measuring anything and therefore, it is not quantitative (but is categorical.) This is bringing them back to the understanding of why we have different measures for center and spread, and why it may be appropriate to have no measure of either.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
The teacher will ask the students to name five examples of data which appears to be numerical (like zip codes) but is really categorical data in disguise.
The teacher will then relate all of the conditions that have been found through discovery learning in summary form for the students to apply.
The teacher will determine if the students have achieved the learning targets for this lesson by asking some follow-up questions. The follow-up questions will be based on some common misconceptions related to the concept(s) addressed. Some sample questions are:
- On the first day of class, if you were given the choice to have your grade in this class calculated by the mean or the median, which measure would you chose? Why?
(This is typically a challenging question because the students don't know in advance how they would do on the tests in this class, but many believe that the median is appropriate because a low score won't impact them as severely. However, students neglect to take into consideration that the high scores that they receive will not be able to pull their grade up either.)
- After seeing your grade on the first quiz in this class, would you choose to have your grade calculated by the mean or the median? Why?
(If students do poorly on the first quiz, they may opt to select the median, but if they do well, they may not realize that they can't tell which measure to select. All they can go on is the variability in previous courses. These questions address an understanding of the relationship of extreme observations on the mean or median. However, the issue of variability and its impact on the mean or the median also occurs.)
The teacher will gather information about student understanding and prior knowledge throughout the lesson through the use of questioning techniques. Sample questions include:
- Which measure of central tendency should the Los Angeles Lakers' owners release to the media to gain public support concerning salary negotiations?
- Which measure of central tendency should the Lakers players release to the media to gain public support concerning a pay increase?
Feedback to Students
The students will get feedback about their performance or understanding during the lesson with each question as this is an guided inquiry lesson.
Each question should lead to a sharper, more precise understanding of a concept that appears simple at first glance.
Accommodations & Recommendations
Students with special needs would be given a special selection of problems which include all of the key concepts provided in the work for the class as a whole. The special selection would have a more structured sequence so that the key concepts are more evident through the progression of the assignment.
This lesson could be extended to the more complex concept of ANOVA. Take two pairs of box-plots, with both pairs having the same relationship between the first and second box-plot, but the ranges are very different. This will establish the idea that an ANOVA compares the means of both (or all) groups, with respect to the standard deviation of the data sets involved.
Suggested Technology: Document Camera, Graphing Calculators, Computer for Presenter
Special Materials Needed:
This lesson requires a graphing calculator for each student. The lesson is enhanced by the use of a class set of graphing calculators and the accompanying WiFi system, but the WiFi system is not a necessity for the lesson. Although any graphing calculator will allow students to perform the actual graphs identified in the lesson, the WiFi system allows the teacher to determine if the group of students understand the concepts, both individually and as a whole. The WiFi system stores the data for each question posed by the teacher for examination after the lesson has been completed.
The concepts in this lesson would be enhanced by having half the class create a data set and use the graphing calculator to create a box-plot with the data and then create a histogram with the data.
- Using the five number summary from the box-plot as a guideline, the student will create three more histograms that have similar features (such as the same range and median, but different quartiles from the original data set).
- Each student would display the original graph (using the doc cam) and ask classmates to identify which of the histograms was created from the same data set as the original graph.
This lesson deals with a concept that sounds very easy and very basic (mean and median) and deepens student knowledge by allowing them to have meaningful discussions about topics that are posed.
The lesson may align with the following standards of math practice:
MAFS.K12.MP.1.1 - Make sense of problems and persevere in solving them.
MAFS.K12.MP.2.1 - Reason abstractly and quantitatively.
MAFS.K12.MP.3.1 - Construct viable arguments and critique the reasoning of others.
MAFS.K12.MP.4.1 - Model with mathematics.
MAFS.K12.MP.5.1 - Use appropriate tools strategically.
MAFS.K12.MP.7.1 - Look for and make use of structure.
Source and Access Information
Name of Author/Source: Marie Causey
District/Organization of Contributor(s): Seminole
Is this Resource freely Available? Yes
Access Privileges: Public
* Please note that examples of resources are not intended as complete curriculum.