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).
Launch
Arrange students in groups of 2–4. Display mathematical vocabulary related to questioning and data already discussed in the unit such as:
Numerical data.
Categorical data.
Statistical question.
Typical value.
Spread.
Variability.
Student Lesson in Spanish
Activity
None
Here are four questions about the population of Alaska.
Describe the questions as precisely as you can.
In general, at what age do Alaska residents retire?
At what age can Alaskans vote?
What is the age difference between the youngest and oldest Alaska residents with a full-time job?
Which age group is the largest part of the population: 18 years or younger, 19–25 years, 25–34 years, 35–44 years, 45–54 years, 55–64 years, or 65 years or older?
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
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.
7.2
Activity
Standards Alignment
Building On
Addressing
6.SP.A.2
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
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:
An interval.
A value within a bin.
A boundary value for a bin.
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.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Launch
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.
Action and Expression: Provide Access for Physical Action. Support effective and efficient use of tools and assistive technologies. To use a ruler, some students may benefit from a demonstration. Supports accessibility for: Organization, Memory, Attention
Activity
None
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.
6
11
18
19
20
23
23
25
25
26
27
27
28
29
32
33
41
42
48
52
54
59
60
77
93
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.
Based on the histogram, what value could be given as the center as a typical length for these 25 earthworms? Explain how you know.
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?
Student Response
Loading...
Building on Student Thinking
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.
Activity Synthesis
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:
“What do the estimates have in common? How are they different?”
“How does the shape of the distribution show up in each method?”
Focus the discussion on how identifying the center and spread using a histogram is different from doing so using a dot plot. Discuss:
“In a histogram, are we able to see clusters of values in the distribution?” (In a way, yes. The values that are close together are probably grouped together into the same bin.)
“Can we see the largest and smallest values? Can we tell the overall spread?” (We cannot get exact values for the largest and smallest values. We can only vaguely describe the spread from the histogram.)
“How do we identify the center of a distribution?” (It is somewhere near the middle of the distribution, but not necessarily halfway between the greatest and least values.)
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.
7.3
Activity
Standards Alignment
Building On
Addressing
6.SP.A.2
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
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.
Launch
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.
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example: “Histogram ___ represents the heights of _______ players because . . . .” and “I agree/disagree because . . . .” Supports accessibility for: Language, Social-Emotional Functioning
Activity
None
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.
Describe the distribution of each histogram. Comment on the center and spread in your description.
Decide which histogram shows the heights of baseball players and which shows the heights of basketball players. Be prepared to explain your reasoning
A
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.
B
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.
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
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.
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to the questions about describing the distribution of heights of basketball and baseball players. Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback that they receive. Advances: Writing, Speaking, Listening
Lesson Synthesis
In this lesson, we learn how to draw a histogram and how to use it to describe characteristics of a data set.
“What are some decisions we should think about and make before drawing a histogram?” (We should consider how large to make the axes and how large the bins should be.)
“Does the width of each bar have to represent a distance of 5 units, or can it represent another number of units?” (They can be any size that balances the precision of smaller bins with the convenience of combining data to reveal the distribution shape. All of the bars must be the same bin size once it is decided, though.)
“What does the horizontal axis of a histogram tell us? What about the vertical axis?” (The horizontal axis tells us the range of values from the data that are included in the bar. The vertical axis is the frequency of values in the data that fall in that bin.)
“How do we know how tall to make each bar?” (Count the number of values in the interval or bin in the data set to find the frequency. That is the height of the bar.)
Once we have a histogram drawn, we can use it to answer some questions about a data set.
“How would you describe a typical weight for this group of dogs?” (The center of the distribution seems to be around 20–25 kilograms. I might choose 22 because it’s in that interval and there seem to be a few more smaller dogs than larger ones.)
“What can we say about the spread of the dog weights based on this histogram?” (The dogs range in weight from around 10 kilograms up to almost 35 kilograms. It is hard to be precise with a histogram, but those are the outer edges of the bars shown.)
Student Lesson Summary
Here are the weights, in kilograms, of 30 dogs.
10
11
12
12
13
15
16
16
17
18
18
19
20
20
20
21
22
22
22
23
24
24
26
26
28
30
32
32
34
34
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.
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.
