Unit 8, Lesson 7
Using Histograms to Answer Statistical Questions
Grade 6
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Learn how to use instructional routines to facilitate problem-based learning!
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In this Warm-up, students use the terminology they have learned in the unit so far to describe questions. Students are reminded to talk about statistical questions, numerical and categorical data, and typical values. Students have the chance to practice being precise in their mathematical language (MP6).
Arrange students in groups of 2–4. Display mathematical vocabulary related to questioning and data already discussed in the unit such as:
Here are four questions about the population of Alaska.
Describe the questions as precisely as you can.
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For each question, invite a group to share how they described the question. After the group shares, invite other groups to add any additional information. If necessary, direct students to the displayed list of words to help students use more formal language.
MLR7: Compare and Connect
Learn how to use instructional routines to facilitate problem-based learning!
To Gather
Rulers marked with centimeters
In this activity, students practice drawing a histogram for a given data set and using it to answer statistical questions. To help students understand the lengths involved in the data set, students are asked to draw various lengths used to group the worms in the first histogram.
Monitor for groups who identify a typical length as:
Any of these choices are valid due to the vagueness of histograms, but should be backed by a reasoning that makes sense with the distribution.
Arrange students in groups of 2. Provide access to centimeter rulers.
Consider giving students a brief overview of the context for the problems in the activity. Tell students that there are nearly 6,000 species of earthworms in the world. Some earthworms help the environment, while others (generally not native to the region in which they are found) may harm the environment. Earthworms that are native to a particular region of the world are often raised, by farmers, in terrariums (a container or bin similar to an aquarium but it contains soil and leaves). The terrarium-raised earthworms provide bait for people who fish, provide food for various wildlife, and decompose food waste into soil. Food waste and water are added to the terrariums as food for raising and growing worms. Soil produced by the worms as they eat the food waste is often used as fertilizer.
Explain that the lengths of the worms in the terrariums provide information about the ages of the worms, which can be useful for the farmer. In this activity, students will organize the lengths of the earthworms in several terrariums.
Give students 8–10 minutes of quiet work time, and then 3–4 minutes to discuss their work and to complete the activity with a partner.
Select work from students with different strategies, such as those described in the activity narrative, to share later.
An earthworm farmer sets up several containers of a certain species of earthworms so that he can learn about their lengths. The lengths of the earthworms provide information about their ages. The farmer measures the lengths, in millimeters, of 25 earthworms in one of the containers.
Using a ruler, draw a line segment for each length:
20 millimeters
40 millimeters
60 millimeters
80 millimeters
100 millimeters
Here are the lengths, in millimeters, of the 25 earthworms.
Complete the table for the lengths of the 25 earthworms.
| length | frequency |
|---|---|
| 0 millimeters to less than 20 millimeters | |
| 20 millimeters to less than 40 millimeters | |
| 40 millimeters to less than 60 millimeters | |
| 60 millimeters to less than 80 millimeters | |
| 80 millimeters to less than 100 millimeters |
Use the grid and the information in the table to draw a histogram for the worm length data. Be sure to label the axes of your histogram.
Write 1–2 sentences to describe the spread of the data. Do most of the worms have a length that is close to your estimate of a typical length, or are they very different in length?
Ask one or two students to display their completed histograms for all to see and briefly describe the overall distribution.
Display 2–3 ways of estimating the center of the distribution from previously selected students for all to see. If time allows, invite students to briefly describe their reasoning for their choice. Then, use Compare and Connect to help students compare, contrast, and connect the different estimates. Here are some questions for discussion:
Focus the discussion on how identifying the center and spread using a histogram is different from doing so using a dot plot. Discuss:
Depending on how the description of the distribution using center and spread are being used, it may be okay that the description is vague. If you don’t know about these worms, it might be ok to know “most worms are around 20 to 40 millimeters long, but can get as large as about 100 millimeters.” If you are a worm expert or are comparing 2 worm farms, you might want more detail and need to know more precisely how long the worms typically are at this particular farm.
MLR1: Stronger and Clearer Each Time
Learn how to use instructional routines to facilitate problem-based learning!
None
In this activity, students use histograms to compare two groups by studying the shape, center, and spread of each distribution. Although histograms are not precise, often they can be enough to make a general comparison of groups.
Arrange students in groups of 2. Give students 4–5 minutes of quiet work time and 1–2 minutes to share their responses with a partner.
Professional basketball players tend to be taller than professional baseball players.
Here are two histograms that show height distributions of 50 professional baseball players and 50 professional basketball players.
A histogram, height in inches, 66 through 90 by twos. Beginning with 70 up to but not including 72, height of bar for each interval is 1, 0, 4, 12, 10, 12, 4, 2, 0, 1.
A histogram, height in inches, 66 through 90 by twos. Beginning with 66 up to but not including 68, height of bar for each interval is 2, 3, 14, 19, 7, 5, 1.
Select a few students to share their descriptions about basketball players and baseball players. After each student shares, ask others if they agree with the descriptions and, if not, how they might revise or elaborate on them. In general, students should recognize that the distributions of the two groups of athletes are different and be able to describe how they are different.
Highlight the fact that students are using approximations of center and different adjectives to characterize a distribution or a typical height and that, as a result, there are variations in our descriptions. In some situations, these variations might make it challenging to compare groups more precisely.
If time allows, remind students that this type of analysis uses trends to compare groups, not individuals. There are some baseball players that are taller than some basketball players in these groups, so we cannot determine which sport each person plays based on their height.
In this lesson, we learn how to draw a histogram and how to use it to describe characteristics of a data set.
Once we have a histogram drawn, we can use it to answer some questions about a data set.
Here are the weights, in kilograms, of 30 dogs.
Before we draw a histogram, let’s consider a couple of questions.
What are the smallest and largest values in our data set? This gives us an idea of the distance on the number line that our histogram will cover. In this case, the minimum is 10 and the maximum is 34, so our number line needs to extend from 10 to 35 at the very least.
(Remember the convention we use to mark off the number line for a histogram: We include the left boundary of a bar but exclude the right boundary. If 34 is the right boundary of the last bar, it won't be included in that bar, so the number line needs to go a little greater than the maximum value.)
What group size or bin size seems reasonable here? We could organize the weights into bins of 2 kilograms (10, 12, 14, . . .), 5 kilograms, (10, 15, 20, 25, . . .), 10 kilograms (10, 20, 30, . . .), or any other size. The smaller the bins, the more bars we will have, and vice versa.
Let’s use bins of 5 kilograms for the dog weights. A bin size of 2 would show more precision, but would have a lot of bars to consider. A bin size of 10 might be too big and lose the shape of the distribution with only 3 bars. The boundaries of our bins will be: 10, 15, 20, 25, 30, 35. We stop at 35 because it is greater than the maximum value.
Next, we find the frequency for the values in each group. It can be helpful to organize the values in a table.
| weights in kilograms | frequency |
|---|---|
| 10 to less than 15 | 5 |
| 15 to less than 20 | 7 |
| 20 to less than 25 | 10 |
| 25 to less than 30 | 3 |
| 30 to less than 35 | 5 |
Now we can draw the histogram.
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.
The histogram allows us to learn more about the dog weight distribution and describe its center and spread.
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When determining frequencies of data values, students might lose track of their counting. Suggest that they use tally marks to keep track of the number of occurrences for each bin.
When drawing the histogram, students might mistakenly use bar graphs as a reference and leave spaces between the bars. Ask them to look at the bars in other histograms they have seen so far and to think about what the gaps might mean considering that the bars are built on a number line.
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