<p>In this lesson, students will interpret and analyze data, as well as their distribution in a graphic representation. Students will:&nbsp;<br>- create stem-and-leaf plots and histograms to represent data.&nbsp;<br>- describe the shape of a data set.&nbsp;<br>- identify ways for estimating and determining the mean, median, and mode of data in a stem-and-leaf plot and histogram.&nbsp;<br>- compare the measures of central tendency.</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>- Central Tendency: The degree of clustering of the values of a statistical distribution that is usually measured by the arithmetic mean, mode, or median.&nbsp;<br>- Cluster: Numbers which tend to crowd around a particular point in a set of values.&nbsp;<br>- Data: Collection of information, usually gathered by observation, questioning, or measurement, often organized in graphs or charts for analysis, may include facts, numbers, or measurement.&nbsp;<br>- Mean: Average; the number found by dividing the sum of a set of numbers by the number of items of data.&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>- Outlier: Data that is unusually large or small in comparison to the others.&nbsp;<br>- Range: The difference between the greatest and least numbers in a set of data.&nbsp;<br>- Symmetric: When the data values are distributed in the same way above and below the middle of the sample (median).</p>

<p>- sticky notes&nbsp;<br>- chart paper (optional)&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>- Stem-and-Leaf Plot for High Temperatures (M-6-5-2_Stem-and-Leaf Plot for High Temperatures)&nbsp;<br>- Stem-and-Leaf Plot Template (M-6-5-2_Stem-and-Leaf Plot Template)&nbsp;<br>- Pick a Number Stem-and-Leaf Plot (M-6-5-2_Pick a Number Stem-and-Leaf Plot)&nbsp;<br>- How-to Reference Sheet (M-6-5-2_How-to Reference Sheet)&nbsp;<br>- Data to Use for Review (M-6-5-2_Data for Review)&nbsp;<br>- Admit Ticket (M-6-5-2_Admit Ticket and KEY)&nbsp;<br>- Double Stem-and-Leaf Plot (M-6-5-2_Double Stem-and-Leaf Plot)&nbsp;<br>- Double Stem-and-Leaf Plot Activity (M-6-5-2_Double Stem-and-Leaf Plot Activity and KEY)</p>

<p>- The Quick Whip Around activity can be used to assess students' level of comprehension.&nbsp;<br>- The admit ticket is a good way to assess student readiness and mastery before beginning the course. This will allow you to customize the lesson to match the needs of the students.&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling, Explicit Instruction, Nonlinguistic Representation, Differentiated Learning, Auditory, and Visual/Spatial&nbsp;<br>W: Explain the meaning of data. Discuss with students the fact that data is all around them—in their own test scores, in sports scores or averages of points per season a team earns, and in historical data such as the average age of previous Presidents of the United States. Also, introduce the terms stem-and-leaf plots and histograms.&nbsp;<br>H: Provide context for the sample data set presented in the introduction. Teacher and students will collaborate to analyze a stem-and-leaf plot for the data set and evaluate its utility as a data organization method. Students will also review the graphic and discuss what information it does and does not describe.&nbsp;<br>E: As a class, the teacher will guide students through the creation of additional stem-and-leaf plots, using these plots as an effective way to organize data, and to present the data in various ways, including histograms. Throughout the discussion, the teacher will guide students through characteristics of the data such as range, clusters, and measures of central tendency.&nbsp;<br>R: Student pairs apply their knowledge to produce a double stem-and-leaf plot. They will transfer data from a table and then calculate the range, clusters, and central tendency measures of the data.&nbsp;<br>E: During the Quick Whip Around activity, students will discuss one significant concept from the lesson to demonstrate their understanding. This will give the teacher a general level of understanding of concept mastery. Students who demonstrate proficiency may collect their own set of data and select one or more methods for displaying and describing it.&nbsp;<br>T: Use extension suggestions to personalize the lesson to match student requirements. The small-group activity provides a review of lesson concepts for students who may require more practice. The expansion activity is appropriate for students who are ready to face a greater challenge.&nbsp;<br>O: The lesson begins with an introduction of vocabulary specific to data displays and data analysis. Students learn how to understand and create stem-and-leaf plots and histograms. They also display an understanding of determining the range, clusters, mean, median, and mode of data sets.&nbsp;</p>

<p>Note: If students require a review of the various types of data displays, an optional review is offered (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).<br><br>Write the following numbers on the board for discussion: 74, 90, 85, 86, 78, 82, 95. <strong>"What conclusions can you make if I give you this set of data 74, 90, 85, 86, 78, 82, 95? What other information would you like to know to help you better comprehend the data? Is the data effectively organized?"</strong> Allow students a few moments to think-pair-share before eliciting their comments. The goal here is just to get students thinking about data. <strong>"There are two ways to represent data: stem-and-leaf plots and histograms. In this lesson, we will learn how to create stem-and-leaf plots and histograms."</strong><br><br><strong>"The data on the board represents the daily high temperatures for a week in a Pennsylvania city. How might we represent the daily high temperatures for a week in any given city?"</strong> (Possible responses: <i>table, chart, bar graph, line graph</i>) <strong>"Data can be represented and organized using a variety of techniques. Let's look at this data as a stem-and-leaf plot."</strong> Display the Stem-and-Leaf Plot for High Temperatures chart (M-6-5-2_Stem-and-Leaf Plot for High Temperatures) so that all students can see the data. Ask students to figure out how this stem-and-leaf plot was created using the original data set. Give students some time to think-pair-share. Monitor student discussion to measure their reasoning. Ask students to share their predictions and observations.<br><br><strong>"The stem-and-leaf plot can represent every number in a data set. Notice that all of our data in the original data set we examined for daily high temperatures throughout a week is a two-digit value."</strong> (Make sure the data set is still visible to students.) <strong>"This is a key consideration when creating a stem-and-leaf plot. You want to identify the range of numbers so that you know which stems to include in the stem-and-leaf plot. Let's take another look at our data 74, 90, 85, 86, 78, 82, 95. I notice three stems that could indicate the tens place: 7, 8, 9. In our data set, stems are represented by digits in the tens place. I can organize this data set by grouping these numbers based on their common stems."</strong> While thinking aloud, write the following on the whiteboard for students to see.<br><br><u>stem 7</u>: 74, 78 &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<u>stem 8</u>: 85, 86, 82 &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<u>stem 9</u>: 90, 95<br><br><strong>“I can now create a T-chart and start my stem-and-leaf plot</strong> (M-6-5-2_Stem-and-Leaf Plot Template). <strong>I need to record the stems in the first column like this.”</strong></p><figure class="image"><img style="aspect-ratio:137/128;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_29.png" width="137" height="128"></figure><p><strong>"Now I need to record the leaves on each stem. Notice how my stems are arranged in consecutive sequence. This is important to remember. You now arrange the corresponding leaves for each stem. What numbers would appear in the first row with a stem of 7?"</strong> (<i><u>7</u>4, <u>7</u>8</i>) <strong>"Notice how both of these numbers have a 7 in the tens place, so the leaves are the numbers that follow 7 tens."</strong> (<i>4 and 8</i>) <strong>"What numbers would be represented in the second row with a stem of 8?"</strong> (<i><u>8</u>5, <u>8</u>6, <u>8</u>2</i>) <strong>"Notice how these three numbers have an 8 in the tens place and leaves in the ones place. When recording the leaves in a stem-and-leaf plot, it is better to record them in sequential order, regardless of where they appear in the dataset. Can I have a volunteer show us how to record the numbers that would be in the third row with a stem of 9?"</strong> Monitor student responses and explain the process once the student has completed it.<br><br>Stem-and-Leaf Plot for Daily High Temperatures</p><figure class="image"><img style="aspect-ratio:137/120;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_30.png" width="137" height="120"></figure><p><strong>"I can add a key to the bottom of the stem-and-leaf plot like this so that other people who look at the data know what data is being represented."</strong>&nbsp;<br><br>Record on the board: 7|4 = 74 degrees Fahrenheit.<br><br><strong>"By looking at the data shown in the stem-and-leaf plot what do we know?"</strong> Ask questions like the ones listed below.&nbsp;<br><br>"What is the lowest temperature for the week?"&nbsp;<br>"What is the highest temperature for the week?"&nbsp;<br>"Were there any repeated temperatures? How do you know?"&nbsp;<br>"Were there more 70, 80, or 90 degree days?"&nbsp;<br>"Suppose I wanted to include the following two temperatures in the stem-and-leaf plot: 82 and 88. How would I do that?"<br>Students may benefit from working together to create a line plot using the same data. Note that if the stem-and-leaf plot was flipped counterclockwise a quarter turn, it would have the same shape as the line plot. Of course, it would have sideways numbers instead of x's, but the shapes of the data would be the same in both displays.&nbsp;<br><br>For additional practice, give each student in your class a sticky note. Have students write a two-digit number on their sticky notes. Then, have students get up and divide themselves into groups based on related stems. In this scenario, the stem would be the tens place of the number they recorded. Check for comprehension. Then, on the board or on chart paper, instruct students to organize their data by stems by adding sticky notes in sequential order. After student data has been posted, indicate if any of the following occur:<br><br>"Notice how there are several numbers repeated. We'll need to make sure we include this in our stem-and-leaf plot."&nbsp;<br>"Notice how we don't have any numbers in the 50s or 80s. We'll need to make sure we include this in our stem-and-leaf plot."&nbsp;<br>"Notice how the lowest number is 22 and the highest number is 99."&nbsp;<br>Use a template, such as the Stem-and-Leaf Plot template (M-6-5-2_Stem-and-Leaf Plot Template), to have students collaborate with others to produce a stem-and-leaf plot. Monitor student interaction and performance while they are working. When required, use vocal prompts to guide comprehension. As students finish their stem-and-leaf plots, ask them to compare them to their classmates. Encourage students to discuss and resolve any differences they may have. This will provide students with immediate feedback. Allow a volunteer to share his/her stem-and-leaf plot, review accuracy, and correct any misunderstandings. Then, have students look at the stem-and-leaf plot and ask questions like the ones given below.<br><br><strong>"How many pieces of data should be in the stem-and-leaf plot?"</strong> (<i>the number of students involved in this activity</i>)<br><strong>"What is the lowest number in our data set? What is the highest number? What is the range of this data set?"</strong> (The range is calculated by subtracting the lowest number from the highest numbers in the data set. Knowing the lowest and highest numbers in a data set also assists in determining the range of stems required in a stem-and-leaf plot.)&nbsp;<br><strong>"Are there any repeated numbers? If so, which ones?"</strong>&nbsp;<br><strong>"Is there a number repeated most frequently?"</strong> (This would be the mode. The mode is the number that appears most frequently in a data set.)&nbsp;<br><strong>"Does our data set contain any stems without leaves, or in other words, 'gaps'? What does this tell us?"</strong><br><strong>"Where do you find the largest 'cluster' of numbers?"</strong> (A cluster is a specific spot in a data set where numbers tend to group together.)<br><br>Give each student a copy of Pick a Number Stem-and-Leaf Plot (M-6-5-2_Pick a Number Stem-and-Leaf Plot). Ask students to discuss how they can estimate and determine the mean of the data. Allow time for students to think-pair-share. Monitor the discussion and, as needed, guide understanding. Think aloud to demonstrate essential understandings. <strong>"Mean is a measure of central tendency. While monitoring, I noticed students discussing how to get the mean of a data set by adding all of the numbers and then dividing by the total number of items in the data set. How could we estimate the mean?"</strong> Ask for volunteers to give their ideas. <strong>"Let's look at the first row: 22, 25, and 26. Let's pretend the average for this row is around 25. There are three numbers in this row, so the total is 75 (25 × 3 = 75). Let's look at the second row: 31, 33, 33. Let's say the average of this row is 33; remember, we're estimating. This row contains three numbers, thus the total is 99 (33 × 3 = 99), which is close to 100. I'm going to repeat this method for the remaining rows."</strong> While thinking aloud, write the work on the board for students to see.<br><br><strong>"Because I want an estimated mean, I can add all the numbers and get 1,285 (75 + 100 + 45 + 190 + 300 + 575 = 1,285). I know I need to divide by 20, so I'll round 1,285 to 1,280. My estimated mean is 64 (1280 ÷ 20 = 64). Looking at the stem-and-leaf plot, does this appear to be a reasonable estimated mean?"</strong> Allow students to think-pair-share before proceeding to a discussion. If available, have a student use a calculator to calculate the actual mean. (Actual mean: 1,273 ÷ 20 = 63.65). Then compare the results. <strong>"Does anyone else have a strategy we can use to estimate mean?"</strong> Allow time for discussion.<br><br><strong>"Data comparison can also be done using a stem-and-leaf plot. For example, to compare two sets of data, I can create a stem-and-leaf plot with data on the left and right sides of the stem. Let's look at another instance."</strong> Give students a copy of the Double Stem-and-Leaf Plot (M-6-5-2_Double Stem-and-Leaf Plot).&nbsp;<br><br>Ask students to look at the example and talk about their observations with a partner. Ask each student to write one observation on a sticky note. Then, encourage volunteers to share their findings and write valid responses on chart paper for other students to see. If students do not make these observations on their own, highlight points like the ones listed below.<br><br>The tens column is in the middle, while the ones column is to the right and the left of the middle.&nbsp;<br>Two sets of test results can be compared.&nbsp;<br>Each test yields twenty-three scores.&nbsp;<br>Scores with a stem of 7 for Test #1 would be 71%, 71%, 76%; for Test #2 would be 72%, 73%, 75%, 77%, 78%, 79%, respectively.&nbsp;<br>The range of the scores for Test #1 is 100 - 64 = 36. The range of the scores for Test #2 is 100 - 66 = 34. Range is calculated by subtracting the lowest from the highest scores.&nbsp;<br>On Test #1, more students scored in the 90s, whereas Test #2 saw more students score in the 80s.&nbsp;<br>The mode for Test #1, that score which appears the most, is 95. The modes for Test #2 are 69, 83, and 89. Remind students that a data set can have zero modes, one mode, or more than one mode.<br>There appears to be a cluster of data for Test #1 in the 90s, and a cluster of data for Test #2 in the 80s.&nbsp;<br>Knowing that each data set has 23 pieces of data on each side allows us to estimate/find the median. Median is the middle value of a sequentially ordered set of data. Because the median is the middle value and we have 23 pieces of data, we would look for the median in the 12th piece of data, because the 12th piece of data is the middle number in the set of 23. Because the data is ordered sequentially in the stem-and-leaf plot, the median for Test #1 can be calculated as 90. Test #2 would be 82. Half of the scores would above the median, while the other half would be below the median.<br><br><strong>"We can also use a histogram to describe this data. A histogram is similar to a bar graph, except with no spaces between the bars. A histogram also has intervals. If I look at the test scores for #1 from the stem-and-leaf plot, the intervals can be 61–70, 71–80, 81–90, 91–100. The intervals are spaced consistently, so the bars are of equal width. I also need to determine which value to use for my y-axis. I've labeled both my x- and y- axes. The greatest number of items in each interval is 10, therefore I'll move up by one on the y-axis."</strong> While creating the histogram, think aloud. Students can consult the How-to Reference Sheet (M-6-5-2_How-to Reference Sheet). The completed histogram should look like the one below.<br><br>Test Scores for 6th Grade Class ~ Test #1</p><figure class="image"><img style="aspect-ratio:318/284;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_31.png" width="318" height="284"></figure><p><strong>"My stem-and-leaf plot shows that there are two values in the 60-69 range, so I fill in the bar accordingly. When I fill in the bar for the range of 70-79, I leave no space. Who can tell me how many values are in the interval based on the stem-and-leaf plot? Yes, there are three values, so I fill in the bar up to 3. I continue the same procedure for the other intervals."</strong> If more experience is required, students might collaborate with a partner to create a histogram using Test #2 data from the stem-and-leaf plot (M-6-5-2_Double Stem-and-Leaf Plot).<br><br><strong>"Now, you and a partner will complete a Double Stem-and-Leaf Plot. The data is presented in a data table. Transfer the data accurately and respond to the related questions."</strong> Before beginning the activity, ensure that each student has a copy of the Double Stem-and-Leaf Plot (M-6-5-2_Double Stem-and-Leaf Plot Activity and KEY). While students are working, check their performance and provide guidance. To test student understanding, ask questions similar to those mentioned below.&nbsp;<br><br>How many pieces of data should be in the stem-and-leaf plot?<br>What is the lowest recorded temperature in our data set? Highest temperature? Are there any repeated numbers? If so, which ones?&nbsp;<br>Is there a number that is repeated the most? This would be the mode.<br>What temperatures are represented in this row? (Point to any particular row.)&nbsp;<br>Are there any stems with no leaves, or in other words, "gaps" in our data set? What is this telling us?&nbsp;<br>Where can you discover the largest "cluster" of numbers?&nbsp;<br>What comparison can you make about the high temperatures between the two cities?&nbsp;<br>What conclusions can you take based on this data?&nbsp;<br>Which city should have a higher median temperature? Why?&nbsp;<br>If we collected average high temperatures from a city like Washington, D.C., which side of the double stem-and-leaf plot would the data most closely resemble? Why are you thinking this way?<br><strong>"Please select another pair of students to compare your double stem-and-leaf plot to and explain the differences. Make any necessary changes."</strong><br><br>Use a Quick Whip Around task to help students process what they have learned about stem-and-leaf plots. Each student should present one essential concept gained from stem-and-leaf plots and/or histograms. The Quick Whip Around activity is intended to involve every student and to gain a general understanding of the group. Pose a question/statement like these:<br><br>What did you learn about stem-and-leaf plots and/or histograms that you did not previously know?&nbsp;<br>Explain how to estimate or find the mean, median, and mode of a data set found in a stem-and-leaf plot.<br>Then give students a few moments to think; they can jot down their ideas first to see if this change will be helpful. Then, beginning with one student, have the remaining students answer rapidly in a wave-like fashion until all students have shared a thought. There should be no discussions or comments that interfere with the Whip. The goal for students is to avoid repeating what someone else has said, though this may be allowed at the teacher's discretion. Students can elaborate or "piggy-back" on what another student has said. Allowing a student to pass is an option, but at this point in the lesson, all students should be able to participate. Once all students have spoken, explain any misunderstandings you may have heard. This activity is a fast formative assessment to determine where there are gaps in understanding. For students who may require additional clarification or practice, the small-group activity below will provide additional reinforcement of key concepts.&nbsp;<br><br>Students who demonstrate proficiency might focus on collecting data and creating a graphic depiction. Give students the Data to Use for Review sheet (M-6-5-2_Data for Review) and assign them to create separate graphical representations. Students should describe the shape of their data using mathematical terms such as clusters, gaps, and outliers. In addition, when possible, have students describe the following: mean, median, mode, and range. If students have demonstrated proficiency, they can also practice on this skill independently using the Expansion Activity indicated in the Extension section. <strong>"In this lesson, we discussed how to create and evaluate data from a stem-and-leaf plot and a histogram. We can estimate or identify the mean, median, and mode of a data set in a graphical representation by examining clusters and gaps."</strong>&nbsp;<br><br><strong>Extension:</strong>&nbsp;<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> Give students an Admit ticket to ensure they comprehend (M-6-5-2_Admit Ticket and KEY). There are two choices. Students are given two options: interpret a stem-and-leaf plot or construct one. An admit ticket is a way for reinforcing previously taught skills. As students enter the classroom, hand them an Admit ticket to complete at the start of class. This will provide you with fast feedback on which students demonstrate mastery and which may require extra instruction.<br><br><strong>Small-Group Activity:</strong> Remind students that a stem-and-leaf plot helps to arrange data, while a histogram is similar to a bar graph but without spaces between the bars. Provide students with a copy of the How-to Reference Sheet (M-6-5-2_How-to Reference Sheet). Review the steps required to create a stem-and-leaf plot and a histogram. Using a step-by-step approach, walk students through the creation of a graphical representation using the Data to Use for Review (M-6-5-2_Data for Review). Limiting the amount of data to begin with can assist students in constructing these graphical representations more efficiently. The small-group setting will allow for appropriate scaffolding and feedback. Once students have completed the graphic representations, ask questions similar to those mentioned below.<br><strong>"How many pieces of data should be in the stem-and-leaf plot?"</strong> (<i>the number of students involved in this activity</i>)&nbsp;<br><strong>"What is the least (minimum) number in our data set? What is the greatest (or biggest) number? What is the range of this data set?"</strong> (The range is calculated by subtracting the lowest from the highest numbers in the data set. Knowing the lowest and highest numbers in a data set also assists in determining the range of stems required in a stem-and-leaf plot.)&nbsp;<br><strong>"Are there any repeated numbers? "If so, which ones?"&nbsp;</strong><br><strong>"How can you describe this data set?"</strong> Encourage students to use terms like data shape, range, median, mean, clusters, gaps, outliers, and data distribution.<br>Data distribution is a description of a collection of data. <strong>"What is the distribution of the data?"</strong>&nbsp;<br><br><strong>Expansion:</strong> Students can collect temperature data from newspaper or an online website such as <a href="http://www.weather.com/">www.weather.com</a>. Students can use double stem-and-leaf plots to compare the city they live in to a city of their choice over a ten-day period; or the city's high temperature to its low temperature over the same ten-day period. Students might then present their findings to the class. Students should be asked to discuss the data's shape, including clusters, gaps, and/or outliers, either verbally or in writing. Another alternative is for students to collect high temperatures from at least 25 different cities using one of the sources indicated above. Then, students can examine the data and create a histogram to depict the temperature data. Students might then present their findings to the class. Students should be asked to discuss the data's shape, including clusters, gaps, and/or outliers, either verbally or in writing.</p>