The purpose of this Warm-up is to elicit the idea that it can be difficult to describe a distribution from the data alone, which will be useful when students visualize data using a box plot in a later activity. While students may notice and wonder many things about these data, the usefulness of data displays is the important discussion point.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language that they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Student Lesson in Spanish
Launch
Arrange students in groups of 2. Display the table for all to see. Give students 1 minute of quiet time to look at the data set and to identify at least one thing they notice and at least one thing they wonder about the distribution of the data. Ask students to give a signal when they have noticed or wondered about something. When the minute is up, give students 1 minute to discuss their observations and questions with their partner. Follow with a whole-class discussion.
Activity
None
Here are the birth weights, in ounces, of all the puppies born at a kennel in the past month.
What do you notice? What do you wonder?
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Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share the things that they noticed and wondered. Record and display their responses for all to see, without editing or commentary. If possible, record the relevant reasoning on or near the data. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If a visual display of the distribution does not come up during the conversation, ask students to discuss that idea.
Display the dot plot of the same data.
Ask students:
“How would a data display like a dot plot help describe the distribution?” (It would be easier to see what is typical, estimate the center, and describe the spread.)
“Is it easier to find the mean from the data list or the dot plot? What about the median?” (It is easier to estimate the mean from the dot plot, but maybe a little easier to calculate it using the list. The median is probably a little easier to find from the list because this list is already organized.)
16.2
Activity
Standards Alignment
Building On
Addressing
6.SP.B.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
Previously, students learned to identify the median, quartiles, and five-number summary of data sets. They also calculated the range and interquartile range of distributions. In this activity, students rely on those experiences to make sense of box plots. They explore this new representation of data kinesthetically: by creating a human box plot to represent class data on the lengths of student names, which they collected in the “Finding the Middle” activity in an earlier lesson.
Launch
Before the lesson, use thin painter’s tape to make a number line on the ground. If the floor is tiled with equal-sized tiles, consider using the tiles for the intervals of the number line. Otherwise, mark off equal intervals on the tape. The number line should cover at least the distance between the least data value (the fewest number of letters in a student's name) and the greatest (the most number of letters).
Provide each student with a copy of the data on the lengths of students’ names from the “Finding the Middle” activity. If any students were absent then, add their names and numbers of letters to the data set.
Give students 4–5 minutes to find the quartiles and write the five-number summary of the data. Then, invite several students to share their findings and come to an agreement on the five numbers. Record and display the summary for all to see.
Explain to students that the five-number summary can be used to make another visual representation of a data set called a “box plot.” Tell students that they will create a human box plot in a way similar to the way they found the median.
Return to students the index cards from the lesson on finding the median. If any students were absent when the cards were made, give them each an index card and ask them to record on the card their full name and the number of letters in their name. If any student who made a card is absent, have another student with the same number of letters in their name hold the card of the absent student.
Ask students to stand up, holding their index card in front of them, and place themselves on the point on the number line that corresponds to their number. (Consider asking students to do so without speaking at all.) Emphasize: Students who have the same number of letters should stand one in front of the other.
Hold up the index card that has been labeled with “minimum.” Ask students who should claim the card, then hand the card to the appropriate student. Do the same for the other labels of a five-number summary. If any quartile falls between two students' numbers, write that number of the index card and have both students hold that card together.
Now that the five numbers are identified and each is associated with one or more students, use wide painter's tape to construct a box plot.
Form a rectangle on the ground by affixing the tape around the group of students between Q1 and Q3. If a quartile is between two people, put the tape down between them. If a quartile has the value of a student's number, put the tape down at that value and have the student stand on it.
Put a tape segment at Q2, from the top side of the rectangle to the bottom side, to subdivide the rectangle into two smaller rectangles. If Q2 is a student's number, have the student stand on the tape.
For the left whisker, affix the end of the tape to the Q1 end of the rectangle, and extend it to where the student holding the “minimum” card is standing. Do the same for the right whisker, from Q3 to the maximum.
Tape the five-number summary cards and students’ cards that correspond to them in the correct locations.
This image shows an example of a completed human box plot.
Explain to students that they have made a human box plot. Consider taking a picture of the box plot for reference and discussion later.
Activity
None
Your teacher will give you the data on the lengths of names of students in your class. Write the five-number summary by finding the data set's minimum, Q1, Q2, Q3, and the maximum.
Pause for additional instructions from your teacher.
Student Response
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Building on Student Thinking
Activity Synthesis
Tell students that a box plot is a representation that shows the five-number summary of a data set. Discuss:
“Where can the median be seen in the box plot? What about the first and third quartiles?” (The median is the line inside the box. The left and right sides of the box represent the first and third quartiles.)
“Where can the IQR be seen in the box plot?” (It is the length of the box.)
“The two segments of tape on the two ends are called ‘whiskers.’ What do they represent?” (The lower one-fourth of the data and the upper one-fourth of data.)
“How many people are part of the box, between Q1 and Q3? Approximately what fraction of the data set is that number?” (About half. Note that the number of people that are part of the box may not be exactly one half of the total number of people, depending on whether the number of data points is odd or even, and depending on how the values are distributed.)
“Why might it be helpful to summarize a data set with a box plot?” (It could help us see how close together or spread out the values are, and where they are concentrated.)
16.3
Activity
Standards Alignment
Building On
Addressing
6.SP.B.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
In this activity, students learn to draw a box plot, and they explore the connections between a dot plot and a box plot of the same data set. Then they compare the representations by commenting on what information can be quickly understood from each, based on the structure of the representations (MP7).
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Arrange students in groups of 2. Give students 8–10 minutes to complete the questions, and then follow with a whole-class discussion.
Tell students that they will now draw a box plot to represent another set of data. For their background information, explain that scientists believe people blink their eyes to keep the surface of the eye moist and also to give the brain a brief rest. On average, people blink between 15 and 20 times a minute.
Display the box plot for all to see. Tell students that their box plot will have all of these features, but will not look exactly like this because their data is different from the one used to make this box plot.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to compare the representations. Display words and phrases such as “median,” “quartiles,” “five-number summary,” “estimate,” and “exact.”
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Within the first 3–5 minutes, check in with students to provide feedback and encouragement. Check to see if they were able to find Q1, Q2, and Q3. If necessary, remind students how to find each quartile. Supports accessibility for: Attention, Social-Emotional Functioning
Activity
None
Twenty people participate in a study about blinking. The number of times each person blinked while watching a video for one minute is recorded. The data values are shown here, in order from smallest to largest.
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Here is a dot plot showing these data.
Find the median (Q2) and mark its location on the dot plot.
Find the first quartile (Q1) and the third quartile (Q3). Mark their locations on the dot plot.
What are the minimum and maximum values?
A box plot can be used to represent the five-number summary graphically. Let’s draw a box plot for the number-of-blinks data. Above the dot plot:
Draw a box that extends from the first quartile (Q1) to the third quartile (Q3). Label the quartiles.
At the median (Q2), draw a vertical line from the top of the box to the bottom of the box. Label the median.
From the left side of the box (Q1), draw a horizontal line (a whisker) that extends to the minimum of the data set. On the right side of the box (Q3), draw a similar line that extends to the maximum of the data set.
Compare the information that can be quickly understood from each representation.
Student Response
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Building on Student Thinking
Activity Synthesis
Display the dot plot and the box plot for all to see.
Direct students' attention to the reference created using Collect and Display. Ask students to share their comparison of the representations. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. (For example, “The box plot shows the five-number summary easily, but the exact data are lost. The dot plot shows the shape of the distribution better, but there are no calculated values shown.”)
Discuss:
“How many data values are included in each part of the box plot?” (5 data values in each part.)
“What percentage of data is included in the boxed portion of the box plot?” (50%)
The focus of this activity is on constructing a box plot and understanding its parts, rather than on interpreting it in context. If students seem to have a good grasp of the drawing process and what the parts entail and mean, consider asking them to interpret the plots in the context of the research study. Ask: “Suppose you are the scientist who conducted the research and are writing an article about it. Write 2–3 sentences that summarize your findings, based on your analyses of the dot plot and the box plot.” (Half of the participants blink between 12 and 21 times per minute. A person blinked as few as 3 times and another as many as 51 times, but these values were unusual in this group.)
Lesson Synthesis
The purpose of this discussion is for students to clarify their understanding of box plots and how they are useful.
Continue to add to the display created in an earlier lesson. Include the five-number summary and a box plot diagram to illustrate the vocabulary as well. Invite students to suggest additional language or diagrams that will support their understanding.
Review with students:
“How is a box plot made?” (The box is a rectangle with the left side at Q1 and the right side at Q3. The line inside the box is the median. The whiskers on the sides extend to the minimum and maximum values of the data set.)
“What does a box plot tell you about the shape, center, and spread of a distribution?” (The median is the line in the middle, which tells you about the center. The IQR is the width of the box in the middle, which tells you about the spread. You can also tell if the distribution is roughly symmetrical.)
“Why is it useful to use a dot plot and a box plot together?” (The dot plot shows the actual data values while the box plot tells the story of the data in fourths.)
“How can the box plot be helpful in comparing two data sets?” (You could compare the minimum and maximum values, where the median falls, and how the data is distributed among the four quarters.)
Student Lesson Summary
A box plot represents the five-number summary of a data set.
It shows the first quartile (Q1) and the third quartile (Q3) as the left and right sides of a rectangle, or a box. The median (Q2) is shown as a vertical segment inside the box. On the left side, a horizontal line segment, sometimes called a whisker, extends from Q1 to the minimum value. On the right, a whisker extends from Q3 to the maximum value.
The rectangle in the middle represents the middle half of the data. Its width is the IQR. The whiskers represent the bottom quarter and the top quarter of the data set.
Here are data about pug and beagle weights represented as both dot plots and box plots.
Box plots and dot plots for two sets of data: "pug weights in kilograms” and "beagle weights in kilograms". The numbers 6 through 11 are indicated and there are tick marks midway between each indicated number. Each box plot is above it's corresponding dot plot. The approximate data for the box plot for "pug weights in kilograms" are as follows: Minimum value, 6. Maximum value, 8. Q1, 6.5. Q2, 7. Q3, 7.3. The approximate data for the dot plot for "pug weights in kilograms" are as follows: 6 kilograms, 1 dot. 6.2 kilograms, 2 dots. 6.4 kilograms, 2 dots. 6.6 kilograms, 2 dots. 6.8 kilograms, 2 dots. 7 kilograms, 3 dots. 7.2 kilograms, 3 dots. 7.4 kilograms, 1 dot. 7.6 kilograms, 2 dots. 7.8 kilograms, 1 dot. 8 kilograms, 1 dot. The approximate data for the box plot for "beagle weights in kilograms" are as follows: Minimum value, 9. Maximum value, 11. Q1, 9.6. Q2, 10. Q3, 10.5. The approximate data for the dot plot for "beagle weights in kilograms" are as follows: 9 kilograms, 1 x. 9.2 kilograms, 2 x's. 9.4 kilograms, 1 x. 9.6 kilograms, 3 x's. 9.8 kilograms, 1 x. 10 kilograms, 3 x's. 10.2 kilograms, 3 x's. 10.4 kilograms, 1 x. 10.6 kilograms, 2 x's. 10.8 kilograms, 2 x's. 11 kilograms, 1 x.
We can tell from the box plots that, in general, the pugs in the group are lighter than the beagles. The median weight of pugs is 7 kilograms and the median weight of beagles is 10 kilograms. Because the two box plots are on the same scale and the rectangles have similar widths, we can also tell that the IQRs for the two breeds are very similar. This suggests that the variability in the beagle weights is very similar to the variability in the pug weights.
Standards Alignment
Building On
Addressing
6.SP.B.5.c
Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
