The purpose of this Warm-up is to connect the analytical work that students have done with dot plots in previous lessons with statistical questions. This activity reminds students that we gather, display, and analyze data in order to answer statistical questions. This work will be helpful as students contrast dot plots and histograms in subsequent activities.
Launch
Arrange students in groups of 2. Give students 1 minute of quiet work time, followed by 2 minutes to share their responses with a partner. Ask students to decide, during a partner discussion, if each question proposed by their partner is a statistical question that can be answered using the dot plot. Follow with a whole-class discussion.
If students have trouble getting started, consider giving a sample question that can be answered using the data on the dot plot (for example, “How many dogs weigh more than 100 pounds?”)
Student Lesson in Spanish
Activity
None
Here is a dot plot showing the weights, in pounds, of 40 dogs at a dog show.
A dot plot, weight in pounds, labeled 60 to 180 by tens. Starting at 68 with intervals of 2, the number of dots above each increment is 1, 1, 2, 0, 1, 1, 0, 2, 1, 0, 0, 4, 1, 3, 0, 0, 4, 0, 0, 1, 0, 0, 1, 5, 0, 3, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1.
Write two statistical questions that can be answered using the dot plot.
What would you consider a typical weight for a dog at this dog show? Explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share questions that they agreed were statistical questions that could be answered using the dot plot. If there is time, consider asking students how they would find the answer to some of the statistical questions.
Display the dot plot for all to see. Ask students to share a typical weight for a dog at this dog show and why they think it is typical. Mark their answers on the displayed dot plot. After each student shares, ask the class if they agree or disagree.
6.2
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.
This activity introduces students to histograms. By now, students have developed a good sense of dot plots as a tool for representing distributions. They use this understanding to make sense of a different form of data representation. The data set shown on the first histogram is the same one from the preceding Warm-up, so students are familiar with its distribution. This allows them to focus on making sense of the features of the new representation and comparing them to the corresponding dot plot.
At this point students do not yet need to see the merits or limits of histograms and dot plots. Students should recognize, however, how the structures of the two displays are different (MP7) and start to see that the structural differences affect the insights we are able to glean from the displays.
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
Explain to students that they will now explore histograms, another way to represent numerical data. Give students 3–4 minutes of quiet work time, and then 2–3 minutes to share their responses with a partner. Follow with a whole-class discussion.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to discuss how dot plots and histograms are alike and different. Display words and phrases such as “precise,” “frequency,” “distribution,” “center,” and “spread.”
Activity
None
Here is a histogram that shows some dog weights in pounds.
Each bar includes the left-end value but not the right-end value. For example, the first bar includes dogs that weigh 60 pounds and 68 pounds but not 80 pounds. An 80-pound dog would be included in the second bar with a frequency of 11.
Use the histogram to answer these questions.
How many dogs weigh between 100 and a little less than 120 pounds?
How many dogs weigh exactly 70 pounds?
How many dogs weigh at least 120 pounds?
How much does the heaviest dog at the show weigh?
What would you consider a typical weight for a dog at this dog show? Explain your reasoning.
Discuss with a partner:
If you used the dot plot to answer the same five questions you just answered, how would your answers be different?
How are the histogram and the dot plot alike? How are they different?
Student Response
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Building on Student Thinking
Activity Synthesis
Ask a few students to briefly share their responses to the first set of questions to make sure that students are able to read and interpret the graph correctly.
Then direct students' attention to the reference created using Collect and Display. Ask students to share their comparison of dot plots and histograms. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. (For example, “The histogram is less precise, but you can still see the distribution. The center and spread appear similar in both.”)
If not already mentioned by students, highlight that, in a histogram:
Data values are grouped into intervals or “bins” and represented as vertical bars.
The height of a bar reflects the combined frequency of the values in that bin.
A histogram uses a number line.
Representation: Develop Language and Symbols. Maintain a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of displays of data. Terms may include "dot plot" and "histogram." Supports accessibility for: Conceptual Processing, Language
6.3
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 continue to develop their understanding of histograms. They begin to notice that a dot plot may not be best for representing a data set with a lot of variability or when a data set has a large number of different values. Histograms may help one visualize a distribution more clearly in these situations. Students organize a data set into bins and draw a histogram to display the distribution.
As students work and discuss, listen for explanations for why certain questions might be easy, hard, or impossible to answer using each graphical display.
Launch
Give students a brief overview of census and population data because some students may not be familiar with them. Display, for all to see, the dot plot and problem stem, “Every ten years, the United States conducts a census, which is an effort to count the entire population. The dot plot shows the population data from the 2010 census for each of the fifty states and the District of Columbia (DC).” Discuss questions such as:
“How many total dots are there?” (51)
“What's the population of the state with the largest population? Do you know what state that is?” (Between 37 and 38 million. It's California.)
“Look at the leftmost dot. What state might it represent? Approximately what is its population?” (The leftmost dot represents Wyoming, with a population of around half a million.)
“Do you know the approximate population of our state? Where do you think we are in the dot plot?”
Explain to students that they will now draw a histogram to represent the population data. Remind them that histograms organize data values into “bins” or groups. In this case, the bin sizes are already decided for them. Next, arrange students in groups of 3–4. Provide access to straightedges. Give students 10–12 minutes to complete the activity. Encourage them to discuss their work within their group as needed.
Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, have students focus on the first question. Next, have them focus on the frequency table and histogram creation. Finally, have them return their focus to the initial table. Supports accessibility for: Organization, Attention
Activity
None
Every ten years, the United States conducts a census, which is an effort to count the entire population. The dot plot shows the population data from the 2010 census for each of the fifty states and the District of Columbia (DC).
A dot plot, population of states in millions, labeled 0 to 40 by twos. A cluster of approximately 30 dots have a population between point six and six point 4. There is one dot above each of the tick marks for 8, 8 point 7, 9, 9 point five, 11, 12, 12 point 1, 18 point 8, 19 point 2, 25, 37 point 2.
Here are some statistical questions about the population of the fifty states and DC. How difficult would it be to answer the questions using the dot plot?
In the middle column, rate each question on its difficulty to answer based on this dot plot as either easy, hard, or impossible. Be prepared to explain your reasoning.
statistical question
using the dot plot
using the histogram
a. How many states have populations greater than 15 million?
b. Which states have populations greater than 15 million?
c. How many states have populations less than 5 million?
d. What is a typical state population?
e. Are there more states with fewer than 5 million people or more states with between 5 and 10 million people?
f. How would you describe the distribution of state populations?
Here are the population data for all states and the District of Columbia from the 2010 census. Use the information to complete the table.
A table of population of states in millions. Alabama, 4 point 7 8, Alaska, zero point 7 1, Arizona, 6 point 3 9, Arkansas, 2 point 9 2, California, 37 point 2 5, Colorado, 5 point 0 3, Connecticut, 3 point 5 7, Delaware, 0 point 9 0, District of Columbia, 0 point 6 0, Florida, 18 point 8 0, Georgia, 9 point 6 9, Hawaii, 1 point 3 6, Idaho, 1 point 5 7, Illinois, 12 point 8 3, Indiana, 6 point 4 8, Iowa, 3 point 0 5, Kansas, 2 point 8 5, Kentucky, 4 point 3 4, Louisiana, 4 point 5 3, Maine, 1 point 3 3, Maryland, 5 point 7 7, Massachusetts, 6 point 5 5, MIchican, 9 point 8 8, Minnesota, 5 point 3 0, Mississippi, 2 point 9 7, Missouri, 5 point 9 9, Montana, 0 point 9 9, Nebraska, 1 point 8 3, Nevada, 2 point 7 0, New Hampshire, 1 point 3 2, New Jersey, 8 point 7 9, New Mexico, 2 point 0 6, New York, 19 point 3 8, North Carolina, 9 point 5 4. North Dakota, 0 point 6 7, Ohio, 11 point 5 4, Oklahoma, 3 point 7 5, Oregon, 3 point 8 3, Pennsylvania, 12 point 7 0, Rhode Island, 1 point 0 5, South Carolina, 4 point 6 3, South Dakota, 0 point 8 1, Tennessee, 6 point 3 5, Texas, 25 point 1 5, Utah, 2 point 7 6, Vermont, 0 point 6 3, Virginia, 8 point 0 0, Washington, 6 point 7 2, West Virginia, 1 point 8 5, Wisconson 5 point 6 9, Wyoming, 0 point 5 6.
population (millions)
frequency
0–5
5–10
10–15
15–20
20–25
25–30
30–35
35–40
Now, use the grid and the information in your table to create a histogram.
Return to the statistical questions at the beginning of the activity. Which ones are now easier to answer?
Complete the table, rating the difficulty of answering each question—this time using your histogram. Be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
Much of the discussion about how to construct histograms should have happened in small groups. Address unresolved questions about drawing histograms if they are relatively simple. Otherwise, consider waiting until students have more opportunities to draw histograms in upcoming lessons.
Focus the discussion on comparing the effectiveness of dot plots and histograms to help us answer statistical questions.
Select a few students or groups to share how their ratings of easy, hard, and impossible, changed when they transitioned from using dot plots to using histograms to answer statistical questions about populations of states. Then discuss and compare the two displays more generally. Solicit as many ideas and observations as possible regarding these questions:
“When might histograms be preferable to dot plots?” (It can be easier to see general trends when grouping similar values together when the data have a large number of different values.)
“When might dot plots be preferable to histograms?” (When it is important to remain precise about the actual values of the data.)
By grouping similar values together, a histogram can sometimes show general trends better than dot plots can. The individual precision is lost in a histogram, so if that information is important, it may be worth using a dot plot.
MLR8 Discussion Supports. Display a sentence frame to help students make decisions about the type of graph to use to display different types of data sets: “Histograms are easy (or hard or impossible) to use when _____, because . . . .” Advances: Conversing, Representing
Lesson Synthesis
A histogram is a visual representation of data that groups values together in intervals, or “bins,” to combine their frequency. This histogram, for instance, represents the distribution for the weights of some dogs.
“What could the smallest dog weigh? The largest?” (10 kilograms; up to almost 35 kilograms)
“What does the bar between 25 and 30 tell you?” (5 dogs weigh between 25 and just under 30 kilograms.)
“What can you say about the dogs who weigh between 10 and 20 kg?” (There are 16 total dogs in this range, including 7 dogs between 10 and 15 kg and 9 between 15 and 20 kg.)
“In general, what information does a histogram allow us to see? How is it different from a dot plot?” (A bigger picture of the distribution is shown in the histogram, but some of the detail is lost when compared to a dot plot. For example, this histogram does not show the weight of any individual dogs.)
“What are some advantages of using a histogram instead of a dot plot?” (A histogram is useful for looking at general trends by grouping a range of values together into a single frequency. When the data are very spread out, when there are not very many data points with the same value, or when an overall idea of the distribution is more important than a detailed view, a histogram may be more useful than a dot plot.)
Student Lesson Summary
In addition to using dot plots, we can also represent distributions of numerical data using histograms.
Here is a dot plot that shows the weights, in kilograms, of 30 dogs, followed by a histogram that shows the same distribution.
A dot plot, the numbers 10 through 35, in increments of 5, are indicated. The 30 data values are as follows: 10 kilograms, 1 dot. 11 kilograms, 1 dot. 12 kilograms, 2 dots. 13 kilograms, 1 dot. 15 kilograms, 1 dot. 16 kilograms, 2 dots. 17 kilograms, 1 dot. 18 kilograms, 2 dots. 19 kilograms, 1 dot. 20 kilograms, 3 dots. 21 kilograms, 1 dot. 22 kilograms, 3 dots. 23 kilograms, 1 dot. 24 kilograms, 2 dots. 26 kilograms, 2 dots. 28 kilograms, 1 dot. 30 kilograms, 1 dot. 32 kilograms, 2 dots. 34 kilograms, 2 dots.
A histogram, the horizontal axis is labeled “dog weights in kilograms” and the numbers 10 through 35, in increments of 5, are indicated. On the vertical axis the numbers 0 through 10, in increments of 2, are indicated. The data represented by the bars are as follows: Weight from 10 up to 15, 5. Weight from 15 up to 20, 7. Weight from 20 up to 25, 10. Weight from 25 up to 30, 3. Weight from 30 up to 35, 5.
In a histogram, data values are placed in groups, or “bins,” of a certain size, and each group is represented with a bar. The height of the bar tells us the frequency for that group.
For example, the height of the tallest bar is 10, and the bar represents weights from 20 to less than 25 kilograms, so there are 10 dogs whose weights fall in that group. Similarly, there are 3 dogs that weigh anywhere from 25 to less than 30 kilograms.
Notice that the histogram and the dot plot have a similar shape. The dot plot has the advantage of showing all of the data values, but the histogram is easier to draw and to interpret when there are a lot of values or when the values are all different.
Here is a dot plot showing the weight distribution of 40 dogs. The weights were measured to the nearest 0.1 kilogram instead of the nearest kilogram.
A dot plot for “dog weights in kilograms”. The numbers 8 through 36, in increments of 2, are indicated. There is 1 dot on each of the following values: 10 kilograms,, 11 kilograms, 11.3 kilograms, 12 kilograms, 12.1 kilograms, 13 kilograms, 14.7 kilograms, 15 kilograms, 15.1 kilograms, 16 kilograms, 16.5 kilograms, 17 kilograms, 18 kilograms, 18.5 kilograms, 19 kilograms, 19.1 kilograms, 20 kilograms, 20.2 kilograms, 20.4 kilograms, 21 kilograms, 21.5 kilograms, 22.6 kilograms, 22.7 kilograms, 22.8 kilograms, 23.2 kilograms, 23.4 kilograms, 24 kilograms, 24.9 kilograms, 26 kilograms, 26.1 kilograms, 26.7 kilograms, 28 kilograms, 28.4 kilograms, 30 kilograms, 31.5 kilograms, 32 kilograms, 32.1 kilograms, 33.5 kilograms, 34 kilograms, 34.4 kilograms.
Here is a histogram showing the same distribution.
In this case, it is difficult to make sense of the distribution from the dot plot because the precision of the measurement means the dots are distinct and so close together. The histogram of the same data set does a much better job showing the distribution of weights by grouping similar values to show an overall trend, even though we can’t see the individual data values.
Standards Alignment
Building On
Addressing
6.SP.A.1
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students' ages.
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
