<p>In this lesson, students will determine which measure of central tendency best represents a given data set. Students will:&nbsp;<br>- determine the mean, median, and mode of a dataset.&nbsp;<br>- differentiate between the mean, median, mode as approaches to represent what is common in a data set.&nbsp;<br>- determine real-world circumstances in which a certain measure of central tendency would be a more appropriate descriptor for that data set.&nbsp;<br>- explain why the mean, median, and mode are measures of the "center" of the data.&nbsp;</p>

<p>- How do we use the mean, median, mode, and range to describe a set of data? Why do we need three different measurements of central tendency?&nbsp;<br>- How can we use mathematics to create models that help us analyze data, make predictions, and better comprehend the world in which we live, and what are the limitations of these models?&nbsp;</p>

<p>- Mean: Average; the number found by dividing the sum of a set of numbers by the number of addends.&nbsp;<br>- Median: The middle number in an ordered set of data, or the average of the two middle numbers when the set has two middle numbers.&nbsp;<br>- Mode: The number(s) that occurs most often in a set of data.&nbsp;<br>- Descriptor: A term or phrase used to describe or identify something.&nbsp;<br>- Skewed: Distribution of data that is not symmetrical.&nbsp;<br>- Outlier: A value far away from most of the rest in a set of data.</p>

<p>- calculators&nbsp;<br>- chart paper&nbsp;<br>- colored pencils (optional for Routine activity)&nbsp;<br>- index cards (optional for Quick Write)&nbsp;<br>- Review of Data Displays (M-6-5-1_Optional Review of Data Displays), <i>optional</i>&nbsp;<br>- Line Plot Practice (M-6-5-1_Line Plot Practice and KEY), <i>optional</i>&nbsp;<br>- Measures of Central Tendency Match Game Cards (M-6-5-1_Match Game Cards and KEY)&nbsp;<br>- Sports Stadium Capacity Data (M-6-5-1_Sports Stadium Data and KEY)&nbsp;<br>- Pick the Best Measure of Central Tendency activity sheet (M-6-5-1_Pick the Best Measure and KEY)&nbsp;<br>- Three Circle Venn Diagram (M-6-5-1_Venn Diagram and KEY)&nbsp;<br>- Lesson 1 Exit Ticket (M-6-5-1_Lesson 1 Exit Ticket)&nbsp;<br>- Measures of Central Tendency Sort (M-6-5-1_Central Tendency Sort and KEY)&nbsp;<br>- Measures of Central Tendency Review (M-6-5-1_Central Tendency Review and KEY)&nbsp;<br>- Misleading Data Displays worksheet (M-6-5-1_Misleading Data Displays)</p>

<p>- Students' responses to the Sports Stadium Data activity sheet (M-6-5-1_Sports Stadium Data and KEY) will assist guide instruction.<br>- Monitor and assess student responses to questioning during the Sports Stadium Data activity to measure their level of knowledge.&nbsp;<br>- The Lesson 1 Exit Ticket (M-6-5-1_Lesson 1 Exit Ticket) can be used to evaluate student mastery.</p>

<p>Scaffolding, Active Engagement, Modeling, and Explicit Instruction&nbsp;<br>W: The lesson starts with an interactive matching of terms and definitions for central measures of tendency. Once students have identified their matches, a class discussion reinforces prior concepts and prepares students for the lesson.&nbsp;<br>H: Students use stadium capacity statistics to analyze measures of central tendency. The incredible number of total seats will entice students to apply mean, median, and mode to large numbers. The use of calculators is strongly recommended.&nbsp;<br>E: Students will analyze statistics about National League Central stadiums. They will develop measures of central tendency using previously collected data and then compare all stadium capacity data. This is followed by a discussion of possible best applications of mean, median, and mode.&nbsp;<br>R: The teacher presents scenarios for small groups of students to investigate. Each group examines the selected data set and considers when the mean, median, or mode may be the best representation of that data. The activity concludes with a Quick Write, which gives students a few minutes to record their observations and justifications.&nbsp;<br>E: Students investigate the possibility that certain measures of central tendency can be used to present data more or less favorably for a certain perspective. The use of an Exit Ticket is suggested for assessing student comprehension.&nbsp;<br>T: Use extension recommendations to personalize the lesson to students' needs. The small-group activity is appropriate for students who require more assistance, while the expansion is appropriate for students who have demonstrated proficiency. Additional exercises are suggested for classroom stations, as well as the use of technology.&nbsp;<br>O: Students should understand the concepts of mean, median, mode, and range. In this lesson, mean is defined as average. Sometimes the term "average" refers to all three measures of central tendency: mean, median, and mode. When deciding which measure of central tendency best depicts a set of data, students should be able logically support their choice. Because the goal of this lesson is for students to be able to defend whatever measure of central tendency best represents a data set, the use of calculators is recommended.&nbsp;</p>

<p>Note: An extra review is provided for students who require a review on different types of data displays (M-6-5-1_Optional Review of Data Displays). If students need to practice using line plots, an optional worksheet is available (M-6-5-1_Line Plot Practice and KEY).&nbsp;<br><br>Before the lesson, print and cut the Measures of Central Tendency Match Game cards (M-6-5-1_Match Game Cards and KEY). Distribute cards to six different students. Have these students come to the front of the room to find their match. Check for accuracy, and then display the matched cards for students to use as a visual reminder of the meanings of each measure of central tendency. Have a brief discussion on these terms. <strong>"In today's lesson we are going to look at data and determine which measure of central tendency, mean, median, or mode, is the best descriptor of the data set."</strong><br><br>Give each student a copy of the Sports Stadium Capacity Data (M-6-5-1_Sports Stadium Data and KEY). If available, allow students to use a calculator during this part of the lecture. <strong>"Let's look at the data for North American stadiums. What are your observations on the data? What observations can you make?"</strong> Allow students time to think-pair-share their observations. Then, have students share and record their responses on chart paper. Encourage students to utilize mathematical language whenever possible. Guide students through the many measurements of central tendency by thinking aloud. <strong>"Is there a mode for the data set? As we discussed before in our Match Game, mode is the value that occurs most frequently. There is no mode because the data set contains no repeated values. Mode would not be an appropriate descriptor for this data set.</strong> (Note this information on the board.) <strong>"Would a calculator help me determine the median value? No. The median is the value that divides the dataset in half. I need to arrange the items in the data set in sequence order."</strong> (Note this information on the board.)</p><figure class="image"><img style="aspect-ratio:588/40;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_27.png" width="588" height="40"></figure><p><strong>“Then I determine the middle value.”</strong> Record on the board.</p><figure class="image"><img style="aspect-ratio:586/38;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_26.png" width="586" height="38"></figure><p><strong>"I average the two middle values. I add them together and divide by 2."</strong><br><br>87,000 + 92,000 = 179,000 ÷ 2 = 89,500<br><br><strong>"The median value in this data set is 89,500. This may be an appropriate descriptor for this data set."&nbsp;</strong><br><br><strong>"If I needed to find the mean value for this data set, a calculator would be useful. First, I must add all of the numbers in the data set together. I could also add these values together using paper and pencil."</strong> Record on the board.<br><br>65,000 + 80,000 + 87,000 + 92,000 + 102,000 + 110,000 = 536,000<br><br><strong>"Then I need to divide the total by the number of items in the data set. The total is 536,000, with six items in the data set."</strong> Record on the board.<br><br>536,000 ÷ 6 = 89,333<br><br><strong>"The mean value in this dataset is 89,333. This value is similar to the median."&nbsp;</strong><br><br><strong>"Now, let's have a look at the South American sports stadium capacity data. How does the data compare to North American sports stadium capacity data, and how does it differ? Take a few moments with a partner to discuss the data."</strong> Allow students to share their observations in pairs before asking them to share them aloud. Add responses to the chart paper provided at the start of the the lesson. <strong>"Calculate the measures of central tendency for the capacities of South American sports stadiums with your partner. When you've completed computing the values, check with another set of partners to determine if your calculations agree. If there is a discrepancy, recalculate the values together. Then determine which measure of central tendency best represents this data set and justify your decision with mathematical logic. Answer the questions on the Sports Stadium activity sheet."</strong> When students have completed this part of the task, share their results. While students are working, check their performance and assess their understanding with questions similar to those mentioned below.<br><br>How did you start this task?&nbsp;<br>What do you think will be the best measure of central tendency for this data set? Why are you thinking this way?&nbsp;<br>How can outliers in data affect a measure of central tendency?&nbsp;<br>How do you determine the mode of a data set?&nbsp;<br>Why don't you think there is a mode for this dataset?&nbsp;<br>How does calculating the median value in this dataset differ from calculating the median value for North American sports stadium capacities?&nbsp;<br>What factors should you consider while calculating the median value? (Organize the numbers sequentially. If there is an even number of elements in a data set, calculate the mean of the two middle numbers. This will be the median for that data set.)&nbsp;<br>How do you find the mean of a data set?&nbsp;<br>Is the mean value you calculated an actual value in the data set?<br>Why are the mean, median, and mode considered measures of the "center" of data? (The median is literally the center, as it is the value that divides a dataset in half. The mean can be regarded the center of the data because it balances the highs and lows. The mode can be regarded the core of the data because it is the value that occurs the most frequently.)&nbsp;<br>In your opinion, which measure of central tendency best represents this data set, and why?<br><strong>"Depending on the data set, a certain measure of central tendency may be more appropriate than others. When we use mode as a measure of central tendency, it signifies that the value appears the most frequently in a data set. In what situations could we want to know a value that occurs most frequently?"</strong> Allow time for students to think-pair-share their ideas aloud. Consider the validity of each response. Then, on chart paper, label valid student responses: When to Use Mode, When to Use Median, When to Use Mean. Repeat the process for the median and mean. Guide student thinking and provide verbal prompting as needed. The chart below shows instances for each measure of central tendency.</p><figure class="image"><img style="aspect-ratio:602/309;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_28.png" width="602" height="309"></figure><p>Post problems similar to those listed on the Pick the Best Measure of Central Tendency activity sheet (M-6-5-1_Pick the Best Measure and KEY). In groups, have students calculate the mean, median, and mode. The usage of calculators is suggested. Then ask them to decide which measure of central tendency would be the best descriptor. Have students write the measure of central tendency they choose on an index card and display it after all students have finished. This can assist provide fast feedback and identify which students demonstrate proficiency and which students need further educational support. Encourage students to discuss their reasoning for their decision and offer probing questions to verify they are using mathematical reasoning and math language to defend their answers.<br><br><strong>"In your opinion which measure of central tendency best represents this data set and why?"&nbsp;</strong><br><strong>"Is there a different perspective where you would choose one of the other measures of central tendency to best represent the data?"&nbsp;</strong><br><strong>"Why might the mean, median, and mode be considered measures of the 'center' of the data?"</strong> (The median is literally the center, as it is the value that divides a data set in half. The mean can be regarded the center of the data because it balances the highs and lows. The mode can be regarded the core of the data because it is the value that occurs the most frequently.<br><br>Following the activity, have students complete a Quick Write to summarize their understanding of when a certain measure of central tendency may be a better descriptor than another for a data set. A Quick Write is a three to five-minute practice that allows students to reflect on their learning. Students can do their Quick Writes in a math journal or on an index card.<br><br><strong>"In this lesson, we looked at data sets and determined which measure of central tendency would best represent a data set. Different measures of central tendency are often used on purpose to describe data in a way that favors one point of view over another."</strong> Have students fill out an exit ticket (M-6-5-1_Lesson 1 Exit Ticket). Give the exit ticket to students with about 5 minutes left in class; they must complete it and turn it in before they leave. You can rapidly review the students' responses. The information provided by the exit ticket will indicate who needs more practice and who has mastered the skill.<br><br><strong>Extension:</strong><br><br>Use the ideas and activities below to satisfy your students' needs during the lesson and throughout the year.<br><br><strong>Routine:</strong> To recap the concept of measures of central tendency, give each student a Three Circle Venn Diagram (M-6-5-1_Venn Diagram and KEY) and ask them to use different colored pencils (if available). Students can fill up their own Venn diagram with as much information as possible in 30 seconds before moving on to the person on their right. Students can then read what the student(s) before them recorded and contribute additional information to the Venn diagram over the next 45 seconds. The usage of different colors for each student will aid with accountability and evaluations. Encourage students to record real-world scenarios where a specific measure of central tendency may also be the best description for that data. Repeat the process three times more. Then, after four rotations, students should return the Venn diagram to its original owner. Students can then review the new information that was recorded and correct any errors. Then, have each student contribute one piece of information that was added to his/her Venn diagram that he/she had not considered previously.<br><br><strong>Small Group:</strong> Have students complete a Measures of Central Tendency Sort (M-6-5-1_Central Tendency Sort and KEY) if they are having difficulty choosing which measure of central tendency may best describe a piece of data. These are taken from the chart that appeared early in the lesson. Allow students to enter their own ideas in the blank spaces. Then, students should switch with a partner and re-sort for fluency. Students can also use a Three Circle Venn Diagram (M-6-5-1_Venn Diagram and KEY) to organize information in a different structure. Students can get more practice with the Measures of Central Tendency Review (M-6-5-1_Central Tendency Review and KEY).<br><br><strong>Expansion:&nbsp;</strong></p><p><strong>Activity 1:</strong> Students who demonstrate proficiency to calculate measures of central tendency for data in current events. Have students look for information on the Internet or in newspapers. Once a valid set of data has been identified, students might collaborate to determine measures of central tendency, or they can work alone and compare results. Ask students to draw conclusions about the data set based on their observations.&nbsp;</p><p><strong>Activity 2:</strong> Distribute the Misleading Data Displays worksheet (M-6-5-1_Misleading Data Displays) and have students work on it together or individually. The purpose is to practice detecting what makes a data display misleading. If you have time and resources, this could be an interesting activity to attempt with current events.</p>