• The Quick Whip Around activity may be used to get a sense of students’ level of understanding.
• The admit ticket is a good way to determine student readiness and to ascertain the level of mastery students have before beginning the lesson. This will help you adjust the lesson to meet the needs of the learners.

Note: If students need a review in types of data displays, an optional review is available (M-6-5-1_Optional Review of Data Displays.docx). If students need practice with line plots, an optional worksheet is available (M-6-5-1_Line Plot Practice and KEY.docx).

Write the following numbers on the board for discussion: 74, 90, 85, 86, 78, 82, 95. “If I give you this set of data 74, 90, 85, 86, 78, 82, 95, what conclusions can you draw? What additional information might you want to know to help you to better understand the data? Is the data effectively organized?” Allow students a few moments to think-pair-share and then elicit student responses. The purpose here is to simply get students thinking about data. “Stem-and-leaf plots and histograms are two ways we can represent data. In this lesson we are going to learn how to construct stem-and-leaf plots and histograms.”

“The data on the board actually represents the daily high temperatures for a week in a city in Pennsylvania. If we wanted to represent the daily high temperatures for a week for any given city, how might we do that?” (Possible answers: table, chart, bar graph, line graph) “Data can be represented and organized using various methods. Let’s look at this data represented in a stem-and-leaf plot.” Display the Stem-and-Leaf Plot for High Temperatures chart (M-6-5-2_Stem-and-Leaf Plot for High Temperatures.doc) so that all students can see the data. Ask students if they can determine how this stem-and-leaf plot was constructed using the original set of data. Give students some time to think-pair-share. Monitor student dialogue to assess student thinking. [IS.5 - All Students] Have students share their predictions and observations.

“The stem-and-leaf plot allows you to represent every number in a data set. Notice that all of our data in the original data set we looked at for the daily high temperatures for a week is a two-digit number.” (Be sure data set is still visible to students.) “This is important to notice when creating a stem-and-leaf plot. You want to identify the range of numbers so you know what stems need to be included in the stem-and-leaf plot. Let’s look at our data again: 74, 90, 85, 86, 78, 82, 95. I see three stems that would represent the tens place: 7, 8, 9. For our data set, the stems are represented by digits in the tens place. To help me organize this data set, I can group these numbers by their common stems.” While thinking aloud, write the following on the board for students to see. [IS.6 - Struggling Learners and ELL Students]

stem 7: 74, 78              stem 8: 85, 86, 82                    stem 9: 90, 95

“Now I can construct a T-chart and begin my stem-and-leaf plot (M-6-5-2_Stem-and-Leaf Plot Template.doc). I need to record the stems in the first column like this.”

“Now for each stem, I need to record the leaves. Notice that my stems are in sequential order. This is important to remember. Now you organize the corresponding leaves for each stem. What numbers would be represented in the first row with a stem of 7?” (74, 78) “Notice how both of these numbers have a 7 in the tens place, so the leaves are the numbers that follow 7 tens.” (4 and 8) “What numbers would be represented in the second row with a stem of 8?” (85, 86, 82) [IS.7 - All Students] “Notice how these three numbers have an 8 in the tens place and leaves in the ones place. When we record the leaves in a stem-and-leaf plot, it is best to record them in sequential order regardless of where they occur in a data set. Can I have a volunteer come and show us how we should record the numbers that would be represented in the third row with a stem of 9?” Monitor student response and summarize process when student is completed.

Stem-and-Leaf Plot for Daily High Temperatures

“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.” [IS.8 - Struggling Learners and ELL Students]

Record on board: 7|4 = 74 degrees Fahrenheit

“By looking at the data shown in the stem-and-leaf plot what do we know?” Ask questions similar to the ones listed below. [IS.9 - All Students]

• “What is the lowest temperature for the week?”
• “What is the highest temperature for the week?”
• “Were there any repeated temperatures? How do you know?”
• “Were there more 70, 80, or 90 degree days?”
• “Suppose I wanted to add the following two temperatures to the stem-and-leaf plot: 82 and 88. How would I do that?”

Students may benefit from working as a class to make a line plot for the same data. Point out that if the stem-and-leaf plot were turned 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 in both displays would be the same.

For additional practice give each student in your class a sticky note. Have students record a two digit number on their sticky notes. Then have students get up and organize themselves into groups according to similar stems. [IS.10 - All Students] In this case, the stem would be the tens place of the number they recorded. Check for understanding. Then on the board or on chart paper have students organize their data by stems by placing their sticky notes in sequential order. After student data has been posted point out if any of the following occurs:

• “Notice how there are several numbers repeated. We will need to be sure that we represent this in our stem-and-leaf plot.”
• “Notice how we do not have any numbers in the 50s or 80s. We will need to be sure that we represent this in our stem-and-leaf plot.”
• “Notice how the lowest number is 22 and the highest number is 99.”

Using a template, such as the Stem-and-Leaf Plot template provided (M-6-5-2_Stem-and-Leaf Plot Template.doc), have students collaboratively work with others around them to create a stem-and-leaf plot. While students are working, monitor student dialogue and performance. When necessary provide verbal prompts to guide understanding. Once students begin to finish their stem-and-leaf plots, have them compare with other classmates. [IS.11 - All Students] Encourage students to discuss and modify any differences they have. This will provide students with immediate feedback. Have a volunteer share his/her stem-and-leaf plot and discuss accuracy and correct any misunderstandings. Then have students look at the stem-and-leaf plot and ask questions similar to those listed below. [IS.12 - All Students]

• “How many pieces of data should be in the stem-and-leaf plot?” (the number of students participating in this activity)
• “What is the lowest number in our data set? Highest number? What would the range of this data set be?” (The range is calculated by subtracting the lowest number from the highest number in the data set. Knowing the lowest and the highest number in a data set also helps to set the range of stems that will be needed in a stem-and-leaf plot.)
• “Are there any repeated numbers? If so, which ones?”
• “Is there a number repeated most frequently?” (This would be the mode. Mode is the number that appears most frequently in a data set.)
• “Are there any stems with no leaves, or in other words, ‘gaps’ in our data set? What does this tell us?”
• “Where do you find the largest ‘cluster’ of numbers?” (A cluster is a particular point in a data set where numbers tend to group together.)

Provide students with copies of Pick a Number Stem-and-Leaf Plot (M-6-5-2_Pick a Number Stem-and-Leaf Plot.doc). Ask students to discuss ways they can estimate and find the mean of the data. Allow time for students to think-pair-share. Monitor dialogue and guide understanding where necessary. Model key understandings with a think aloud. “Mean is a measure of central tendency. While monitoring, I noticed students explaining that to find the mean of a data set you add up all the numbers and then divide by the total number of items in a data set. If we wanted to estimate the mean, how could we do that?” Ask for volunteers to share their ideas. “Let’s look at the first row: 22, 25, and 26. Let’s say the average of this row is about 25. Since there are three numbers in this row the total would be 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 are estimating. Since there are three numbers in this row the total would be 99 (33 × 3 = 99), or close to 100. I am going to repeat this process for the other rows.” While thinking aloud, record the work on the board for students to see.

“Since I want an estimated mean I can add up 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 am going to round 1,285 to 1,280. My estimated mean would be 1280 ¸ 20 = 64). By looking at the data in the stem-and-leaf plot does this look like a reasonable estimated mean?” Allow students to think-pair-share and then allow for some discussion. If available have a student calculate the actual mean using a calculator. (Actual mean is 1,273 ¸ 20 = 63.65) Then compare results. “Does anyone else have a strategy we can use to estimate mean?” Allow time for discussion.

“A stem-and-leaf plot also can be used to compare data. For example, if I wanted to compare two sets of data, I can construct a stem-and-leaf plot with data on the left and the right of the stem. Let’s look at another example.” Provide students with copies of the Double Stem-and-Leaf Plot (M-6-5-2_Double Stem-and-Leaf Plot.doc).

Ask students to look at the example and discuss their observations with a partner. Ask students to write one observation on a sticky note. Then ask for volunteers to share their observations and record valid responses on chart paper for other students to see. Be sure to highlight points similar to those listed below if students do not make these observations on their own.

• The tens column is in the middle and the ones column is to the right and the left of the middle.
• Two sets of test scores can be compared.
• Twenty-three scores are recorded for each test.
• For Test #1 scores with a stem of 7 would be 71%, 71%, 76%; for Test #2 scores with a stem of 7 would be 72%, 73%, 75%, 77%, 78%, 79%.
• The range of the scores for Test #1 would be 100 – 64 = 36. The range of the scores for Test #2 would be 100 – 66 = 34. Range is the lowest score subtracted from the highest score.
• More students scored in the 90s on Test #1; more students scored in the 80s on the Test #2.
• 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 set of data can have zero modes, one mode, or more than one mode.
• There seems to be a cluster of data for Test #1 in the 90s; there seems to be a cluster of data for Test #2 in the 80s.
• We can estimate/find the median of each data set by knowing that there are 23 pieces of data on each side. Median is the middle value of a set of data arranged sequentially. Since the median is the middle value and we have 23 pieces of data, we would find the median as the 12^th piece of data, because the 12^th piece of data is the middle number in a set of 23. Since the data is organized sequentially in the stem-and-leaf plot it can be determined that the median for Test #1 would be 90. For Test #2 would be 82. Half of the scores would fall above the median; half of the scores would fall below the median. [IS.13 - Struggling Learners and ELL Students]

“We also can construct a histogram to represent this data. A histogram is similar to a bar graph 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. Notice the intervals are consistent in spacing, and thus the bars are of equal width. I also have to decide on what value to use for my y-axis. I have labeled both my x- and y- axes. The greatest number of items in any interval is 10 so I will go up by ones on my y-axis.” While constructing the histogram think aloud. Students can refer to the How-to Reference Sheet (M-6-5-2_How-to Reference Sheet.doc). [IS.14- All Students] The completed histogram should look like the one below.

Test Scores for 6^th Grade Class ~ Test #1

“From my stem-and-leaf plot I see 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 interval of 70–79, I do not leave a space. Who can tell me how many values are in that interval by looking at the stem-and-leaf plot? Yes, there are three values so I fill in the bar up to 3. I repeat the process the same way for the other intervals.” If additional practice is needed, students can work with a partner and construct a histogram using Test #2 data from the stem-and-leaf plot (M-6-5-2_Double Stem-and-Leaf Plot.doc).

“Now you and a partner are going to complete a Double Stem-and-Leaf Plot. The data is included in a data table. Transfer the data accurately and answer the related questions.” Be sure each student has a copy of the Double Stem-and-Leaf Plot before beginning the activity (M-6-5-2_Double Stem-and-Leaf Plot Activity and KEY.doc). While students are working monitor student performance and guide understanding. Use questions similar to those listed below to assess student understanding.

• How many pieces of data should be in the stem-and-leaf plot?
• What is the lowest temperature in our data set? Highest temperature? Are there any repeated numbers? If so, which ones?
• Is there a number repeated most often? That would be the mode.
• What are the temperatures represented in this row? (Point to any given row.)
• Are there any stems with no leaves, or in other words, “gaps” in our data set? What does this tell us?
• Where do you find the largest “cluster” of numbers?
• What comparison can you make about the high temperatures between the two cities?
• What conclusions can you draw by looking at this data?
• Which city should have the higher median temperature? Why?
• If we collected the average high temperatures from a city like Washington, D.C., which side of the double stem-and-leaf plot would the data resemble more closely? Why do you think this way?

“Please find another pair of students to compare your double stem-and-leaf plot with and discuss any differences. Make any necessary modifications.” [IS.15- All Students]

Have students process what they have learned about stem-and-leaf plots by doing a Quick Whip Around activity. Each student should share one key idea s/he learned about stem-and-leaf plots and/or histograms. The Quick Whip Around activity is designed to have every student participate and to get a general level of understanding of the group. Pose a question/statement similar to these:

• What is one thing you learned about stem-and-leaf plots and/or histograms that you did not already know?
• Describe how to estimate or find the mean, median, mode of a data set found in a stem-and-leaf plot.

Then give students a few moments of think time; students can jot down their ideas first if this modification will be helpful. Then start with any student and have the remaining students quickly respond in a wave-like fashion until all students have shared a thought. No discussion or comments should interfere with the Whip. The goal for students is to not repeat what someone else has said although this may be allowed at the teacher’s discretion. Students can elaborate or “piggy-back” off what another student says.. Allowing a student to pass is an option, but at this point in the lesson all students should be able to contribute. Once all students have given input, clarify any misunderstandings you may have heard. This activity is a quick formative assessment to show if there are gaps in understanding. For those students who may need further clarification or practice see the small-group activity below to provide further reinforcement of key concepts. [IS.16- All Students]

Those students who demonstrate proficiency can work on gathering data and constructing a graphic representation. Provide students with the Data to Use for Review sheet (M-6-5-2_Data for Review.doc) and ask them to construct independent graphical representations. Students should describe the shape of their data using math language like clusters, gaps, and outliers. Also have students describe the following when possible: mean, median, mode, and range. Using the Expansion Activity listed in the Extension section is another option to have students independently work on this skill if they have demonstrated proficiency. “In this lesson we looked at the how to construct and analyze data found in a stem-and-leaf plot and a histogram. We can estimate or find the mean, median, and mode of a data in a graphical representation by looking at clusters and gaps in data.”

Extension:

Use the following strategies and activities to meet the needs of your students during the lesson and throughout the year. [IS.17- All Students]

• Routine: Give students an Admit ticket to check for student understanding (M-6-5-2_Admit Ticket and KEY.doc). There are two options. One option asks students to interpret a stem-and-leaf plot; the other option asks students to construct a stem-and-leaf plot. An Admit ticket is a strategy that can be used to reinforce skills previously taught. As students enter the classroom, give them an Admit ticket to complete at the beginning of class. This will give you immediate feedback on which students show mastery and which students may need additional instruction.
• Small-Group Activity:Remind students that a stem-and-leaf plot helps to organize data; a histogram is similar to a bar graph with no spaces in between the bars. Give students a copy of the How-to Reference Sheet (M-6-5-2_How-to Reference Sheet.doc). Review the steps necessary to construct a stem-and-leaf plot and a histogram. Using a step-by-step format guide students through the construction of a graphical representation using the Data to Use for Review (M-6-5-2_Data for Review.doc). Limiting the amount of data to start can help students create fluency of construction of these graphical representations. The small-group setting will allow for necessary scaffolding and feedback. Once students construct the graphic representations, ask questions similar to the ones listed below.
  • “How many pieces of data should be in the stem-and-leaf plot?” (the number of students participating in this activity)
  • “What is the least (minimum) number in our data set? Greatest (maximum) number? What is the range of this data set?” (The range is calculated by subtracting the lowest number from the highest number in the data set. Knowing the lowest and the highest number in a data set also helps to set the range of stems that will be needed in a stem-and-leaf plot.)
  • “Are there any repeated numbers? If so, which ones?”
  • “How can you describe this data set?” Encourage students to use terms such as shape of data, range, median, mean, clusters, gaps, outliers, and distribution of data.
  • Distribution of data: a description of a set of data. “What is the distribution of the data?”

Expansion: Students can collect temperature data either from the newspaper or an online site like www.weather.com. Students can create double stem-and-leaf plots to compare the city they live in to a city of their choice for a ten-day period; or the high temperature of the city they live in to the low temperature of the city they live in for a ten-day period. Students then can share their findings with the class. Students should be asked to discuss the shape of the data including clusters, gaps, and/or outliers in either verbal or written form. Another idea can be for students to collect the high temperatures for a minimum of 25 different cities from one of the sources listed above. Then students can look at the data and construct a histogram to represent the temperature data. Students then can share their findings with the class. Students should be asked to discuss the shape of the data including clusters, gaps, and/or outliers in either verbal or written form.