The purpose of this Warm-up is to elicit the idea that there are multiple ways to describe variability, which will be useful when students learn about range and interquartile range in a later activity. While students may notice and wonder many things about these dot plots, variability and how to measure it are the important discussion points.
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
Launch
Arrange students in groups of 2. Display the dot plots for all to see. Ask students to think of at least one thing that they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss with their partner the things that they notice and wonder about.
Activity
None
Here are dot plots that show the ages of people at two different parties. The mean of each distribution is marked with a triangle.
Data set A
<p>A dot plot from 5 to 45 by 5’s. Age in years, labeled data set A. There is a red triangle indicated at 15, and the data set are as follows: 8 years, 12 dots. 10 years, 3 dots. 12 years, 1 dot. 15 years, red triangle. 36 years, 1 dot. 42 years, 1 dot. 44 years, 2 dots.</p>
Data set B
<p>A dot plot from 5 to 45 by 5’s. Age in years, labeled data set B. The data are as follows: 7 years, 1 dot. 8 years, 1 dot. 9 years, 1 dot. 10 years, 2 dots. 15 years, 1 dot. 16 years, 1 dot. 20 years, 2 dots and 1 red triangle. 22 years, 1 dot. 23 years, 1 dot. 24 years, 1 dot. 28 years, 1 dot. 30 years, 1 dot. 33 years, 1 dot. 35 years, 1 dot. 38 years, 1 dot. 42 years, 1 dot.</p>
What do you notice and what do you wonder about the distributions in the two dot plots?
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share the things 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 dot plots. 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 the idea that the MAD does not describe the variability of these two sets well does not come up during the conversation, ask students to discuss that idea.
Two key ideas to uncover here are:
The MAD is a way to summarize variation from the mean, but the single number does not always tell us how the data are distributed.
The same MAD could result from very different distributions.
7.2
Activity
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.
This activity introduces students to the five-number summary and the process of identifying the five numbers. Students learn how to partition the data into four sets: using the median to decompose the data into upper and lower halves, and then finding the middle of each half to further decompose it into quarters. They learn that each value that decomposes the data into four parts is called a quartile, and the three quartiles are the first quartile (Q1), second quartile (Q2, or the median), and third quartile (Q3). Together with the minimum and maximum values of the data set, the quartiles provide a five-number summary that can be used to describe a data set without listing or showing each data value.
Students reason abstractly and quantitatively (MP2) as they identify and interpret the quartiles in the context of the situation given.
To save time, find the median together with the whole class. Then assign half of the class to find the first quartile and the other half to find the third quartile. After 2 minutes of group work time, ask a group from each half to share their results with the class.
During the Activity Synthesis, introduce the terms range and interquartile range (IQR). The range is the difference between the maximum and minimum values in the data, and the IQR is the difference between the third and first quartiles. Demonstrate how to compute the range () and interquartile range () of the data.
Launch
Arrange students in groups of 2. Give groups 8–10 minutes to complete the activity. Follow with a whole-class discussion.
Ask students if they have attended different kinds of parties with different age ranges. If a few different possibilities don’t come up, share some examples:
A birthday party for a sixth grader may have most people there around the same age.
A family get-together may have a wide range of ages fairly evenly distributed.
A dinner party with mostly your parents’ friends may have mostly older people.
Remind students that they previously summarized variability by finding the MAD, which involves calculating the distance of each data point from the mean and then finding the average of those distances. Explain that we will now explore another way to describe variability and summarize the distribution of data. Instead of measuring how far away data points are from the mean, we will decompose a data set into four equal parts and use the markers that partition the data into quarters to summarize the spread of data.
If necessary, remind students how to find the median, especially when there are an even number of values in the data set.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Have students complete the first question, then the next 2, and finally the last one. Check in with students to provide feedback and encouragement after each chunk. Ensure that students understand how the data are being divided. Supports accessibility for: Attention, Social-Emotional Functioning
Activity
None
Here are the ages of the people at one party, listed from least to greatest.
7
8
9
10
10
11
12
15
16
20
20
22
23
24
28
30
33
35
38
42
Split the data into 4 equal parts using 3 values called quartiles.
Find the median of the data set and label it Q2 for second quartile. This splits the data into an upper half and a lower half.
Find the middle value of the lower half of the data, without including the median. Label this value Q1 for first quartile
Find the middle value of the upper half of the data, without including the median. Label this value Q3 for third quartile.
Label the least value in the set “minimum” and the greatest value “maximum.”
The values you have identified make up the five-number summary for the data set. Record them here.
The median of this data set is 20. This tells us that half of the people at the party were 20 years old or younger, and the other half were 20 or older. What do each of these other values tell us about the ages of the people at the party?
The third quartile
The minimum
The maximum
Activity Synthesis
Ask a student to display the data set that they have decomposed and labeled, or display the diagram for all to see.
Focus the conversation on students' interpretation of the five numbers. As students discuss their solutions, color code or annotate the five-number summary on the data set and diagram. Discuss:
“In this context, what do the minimum and maximum values tell us?” (The ages of the youngest and oldest partygoers.)
“In this context, what does Q1 (10.5) tell us?” (That a quarter of the partygoers are 10.5 years old or younger.)
“What does Q3 (29) tell us?” (That three quarters of the partygoers are 29 years old or younger.)
“How do the five numbers help us to see the distribution of the data?” (It divides the values in the data into sections containing one-fourth of the values each. This gives us an idea about the distribution of the data by looking at how varied each section is.)
7.3
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.
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.
This activity is optional. It provides an opportunity for students to learn kinesthetically as they line up and create a box plot based on the number of letters in their names.
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.)
7.4
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.
3
6
8
11
11
13
14
14
14
14
16
18
20
20
20
22
24
32
36
51
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 quartiles, interquartile range, and box plots.
Review with students:
“What are the quartiles for a numerical data set?” (Numbers that show where we split the data up so it is in quarters.)
“What is the Interquartile range (IQR)? What does it mean?” (The IQR is the difference between the third and first quartile. It is a measure of the variability or spread of the data. It tells us how much “space” the middle half of the data occupies.)
“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.)
Student Lesson Summary
Earlier we learned that the mean is a measure of the center of a distribution and the MAD is a measure of the variability (or spread) that goes with the mean. There is also a measure of spread that goes with the median. It is called the interquartile range (IQR).
Finding the IQR involves splitting a data set into fourths. Each of the three values that splits the data into fourths is called a quartile. For example, here is a data set with 11 values.
12
19
20
21
22
33
34
35
40
40
49
Q1
Q2
Q3
The median, or second quartile (Q2), splits the data into two equal halves. For this data set, the median is 33.
The first quartile (Q1) is the middle value of the lower half of the data. For this data set, the first quartile is 20.
The third quartile (Q3) is the middle value of the upper half of the data. For this data set, the third quartile is 40.
The difference between the maximum and minimum values of a data set is the range. For this data set, the range is 37 because .
The difference between Q3 and Q1 is the interquartile range (IQR). For this data set, the IQR is 20 because . Because the distance between Q1 and Q3 includes the middle two-fourths of the distribution, the values between those two quartiles are sometimes called the middle half of the data.
The bigger the IQR, the more spread out the middle half of the data values are. The smaller the IQR, the closer together the middle half of the data values are. This is why we can use the IQR as a measure of spread.
A five-number summary can be used to summarize a distribution. It includes the minimum, first quartile, median, third quartile, and maximum of the data set. For the previous example, the five-number summary is 12, 20, 33, 40, and 49. These numbers are marked with diamonds on the dot plot.
A dot plot. The numbers 10 through 50, in increments of 5, are indicated. There are diamonds indicated at 12, 20, 33, 40 and 49. The data are as follows: 12, 1 dot; 19, 1 dot; 20, 1 dot; 21, 1 dot; 22, 1 dot; 33, 1 dot; 34, 1 dot; 35, 1 dot; 40, 1 dot; 49, 2 dots.
A box plot represents the five-number summary of a data set.
A box plot. The numbers 10 through 50, in increments of 5, are indicated. The data are as follows: Minimum value, 12. Maximum value, 49. Q1, 20. Q2, 33. Q3, 40.
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.
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.
