The purpose of this Warm-up is for students to become familiar with a bar graph by noticing and wondering things about it. While reading a bar graph is a review of a previous grade's work, it is an important for students to look for patterns of association in categorical data.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language 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 table and images for all to see. Ask students to think of at least 1 thing they notice and at least 1 thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner.
Activity
None
What do you notice? What do you wonder?
Two-way table
has
cell phone
does not have
cell phone
total
10 to 12 years old
25
35
60
13 to 15 years old
40
10
50
16 to 18 years old
50
10
60
total
115
55
170
Bar graph
Segmented bar graph
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 without editing or commentary for all to see. If possible, record the relevant reasoning on or near the table and images. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If a relationship between the 2 variables does not come up during the conversation, ask students to discuss this idea.
Tell students that, in this course, bar graphs are assumed to have a vertical axis representing the frequency of the categories and segmented bar graphs have a vertical axis representing percentage of the category.
9.2
Activity
Standards Alignment
Building On
Addressing
8.SP.A.4
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
In this activity students become familiar with two-way tables, clustered bar graphs, and segmented bar graphs by matching different situations.
Students sort different representations of data during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).
Monitor for different ways groups choose to categorize the representations, but especially for categories that distinguish between two-way tables, bar graphs, and segmented bar graphs.
As students work, encourage them to refine their descriptions of each kind of representation using more precise language and mathematical terms (MP6).
This activity uses the Collect and Display math language routine to advance students in developing their mathematical language.
Launch
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language students use to describe two-way tables, bar graphs, and segmented bar graphs. Display words and phrases such as “total,” “stacked,” “legend” or “key,” “percentage,” and “categories.”
Arrange students in groups of 2 and distribute pre-cut cards. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the representations, they should work with their partner to come up with categories.
Distribute 1 set of the pre-cut cards from the blackline master to each group.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches. Supports accessibility for: Conceptual Processing, Organization, Memory
Activity
None
Your teacher will give you a set of cards. Each card contains a representation of some data.
Sort the systems into groups that represent the same situation. Be prepared to explain how you know where each representation belongs.
One of the groups does not have a two-way table. Make a two-way table for the situation described by the graphs in the group.
Label the bar graphs and segmented bar graphs so that the categories represented by each bar are indicated.
Describe in your own words the kind of information shown by a segmented bar graph.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this activity is to help students understand the connections among the 3 representations of data. It also helps students see the importance of labeled visual representations.
Direct students’ attention to the reference created using Collect and Display. Ask students to share their matches from the Card Sort. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond. Keep the display for the next activity.
9.3
Activity
Standards Alignment
Building On
Addressing
8.SP.A.4
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
In this activity, students create two-way tables displaying relative frequency. The relative frequency table converts the actual frequency data to percentages which can be useful when comparing groups that include different totals. Finally, students use the relative frequencies to look for a pattern in the data. In the following lesson, students will work with associations in categorical data more explicitly. In this activity, students should use an informal understanding of association to think about whether one is present in the data based on the relative frequency table.
This activity uses the Collect and Display math language routine to advance students in developing their mathematical language.
Launch
Keep students in groups of 2–3. After an introduction to relative frequency tables, allow students 3 minutes quiet work time followed by partner discussion and whole-class discussion.
Display the table for all to see.
watch the news daily
does not watch the news daily
total
younger than 18
30
80
110
18 or older
10
5
15
total
40
85
125
Ask students, "Based on this data, who is more likely to watch the news daily: someone who is younger than 18 or someone who is 18 or older?"
Tell students that, based on the numbers in the table, there are more younger people who watch the news (30) than older (10). On the other hand, the survey reached out to 110 young people and only 15 older people. Without looking at the whole table, that information may have been missed.
In cases like this, finding a relative frequency including percentages can be more helpful than looking at the actual frequency, which is what they are going to do now.
Display the relative frequency table:
watch the news daily
does not watch the news daily
total
younger than 18
27%, because
73%
100%
18 or older
67%
33%
100%
Tell students that we would say there is an association between these variables because the relative frequencies of the columns are very different. This means that knowing information about one of the variables, like the age range, would be helpful in making a prediction about the other variable, like whether that person watches the news. If the relative frequencies had been close, then knowing information about one of the variables would not be helpful in making a prediction about the other variable and there would not be an association between the variables.
Use Collect and Display to direct attention to words collected and displayed from an earlier activity. Ask students to suggest ways to update the display: “Are there any new words or phrases that you would like to add?” “Is there any language you would like to revise or remove?” Display additional words or phrases such as “actual frequency” or “relative frequency.” Encourage students to use the display as a reference.
Activity
None
Here is a two-way table that shows data about cell phone usage among children aged 10 to 18.
has cell phone
does not have cell phone
total
10 to 12 years old
25
35
60
13 to 15 years old
40
10
50
16 to 18 years old
50
10
60
total
115
55
170
Complete the table. In each row, the entries for “has cell phone” and “does not have cell phone” should have the total 100%. Round entries to the nearest percentage point.
has cell phone
does not have cell phone
total
10 to 12 years old
42%
13 to 15 years old
100%
16 to 18 years old
17%
This is still a two-way table. Instead of showing frequency, this table shows relative frequency.
Two-way tables that show relative frequencies often don’t include a “total” row at the bottom. Why?
Is there an association between age and cell phone use? How does the two-way table of relative frequencies help to illustrate this?
Student Response
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Building on Student Thinking
Students may not calculate relative frequencies correctly. Look to see if they are dividing into the total for each row, instead of some other number in the row, or the total for the entire table.
Students may struggle to decide if there is an association or not. Tell students that an association means that knowing information about one variable makes it easier to predict the other variable. For example, if they wanted to predict whether a person has a cell phone, would their prediction change if they knew which age category the person fit into?
Activity Synthesis
The purpose of this discussion is to help students see the usefulness of two-way tables that display relative frequency.
Direct students’ attention to the reference created using Collect and Display. Ask students to share their solutions to the questions. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond.
Some questions for discussion:
“What does the 42% in the table mean?” (42% of people 10-to-12 years old have a cell phone.)
“What is the total of the percentages in the first column? What's wrong with that answer?” (205%. It doesn't make sense since it is greater than 100%. It is combining percentages of different wholes, so it doesn't make sense to add them.)
“Did you say there was or was not an association between age and cell-phone use? Explain your reasoning.” (There is an association since the percentages for the 10 to 12 year olds is very different from the other 2 age groups.)
“What did the relative frequency table show you that was harder to see in the original two-way table?” (It was much easier to see the association with the percentages than the numbers. For example, the number of people without a cell phone for the older 2 age groups was the same, but the percentages were a little different.)
Lesson Synthesis
Display the graphs and tell students to refer to the data from the tables in the "Building Another Type of Two-Way Table" activity.
Double bar graph in blue and yellow stripes. Horizontal labeled 10 to 12 years old, 13 to 15 years old and 16 to 18 years old. Vertical labeled 0 to 60, by 15's. Blue section means has cell phone. Yellow stripes means has no cell phone.
Stacked bar graph in blue and yellow stripes. Horizontal labeled 10 to 12 years old, 13 to 15 years old and 16 to 18 years old. Vertical labeled 0 to 100, by 25's. Blue section means has cell phone. Yellow stripes means has no cell phone.
Consider asking these discussion questions to emphasize the main ideas from the lesson:
"Which graphical representation do you prefer for the data?" (It is easier to see the association with the segmented bar graph, but the actual values are lost.)
"In the original table, what did the number 40 represent? How is that group of people represented in the other table and the two graphs?" (There are 40 people who are 13 to 15 years old that have a cell phone. In the relative frequency table, this is represented by the 80%. In the bar graph, this is represented by the taller, blue bar for the set of bars labeled 13–15 years old. In the segmented bar graph, this is represented by the top, blue segment of the bar labeled 13–15 years old.)
"What values in the tables represent the same information as the tallest, yellow-striped bar in the bar graph?" (The tallest yellow-striped bar represents the number of people who are 10–12 years old who do not have cell phones. In the original table, this is the value 35, and in the relative frequency table this is about 58%.)
Survey students about whether they play a musical instrument or not and whether they play a sport or not. These data will be needed for the next lesson.
Student Lesson Summary
When we collect data by counting things in various categories, like red, blue, or yellow, we call the data “categorical data,” and we say that color is a “categorical variable.”
We can use two-way tables to investigate possible connections between two categorical variables.
For example, this two-way table of frequencies shows the results of a study of meditation and state of mind of athletes before a track meet.
meditated
did not meditate
total
calm
45
8
53
agitated
23
21
44
total
68
29
97
If we are interested in the question of whether there is an association between meditating and being calm, we might present the frequencies in a bar graph, grouping data about those who meditated and those who did not meditate, so we can compare the numbers of calm and agitated athletes in each group.
Double bar graph in blue and red. Horizontal labeled meditated and did not meditate. Vertical labeled 0 to 50, by 10's. Blue represents calm. Red represents agitated.
Notice that the number of athletes who did not meditate is small compared to the number who meditated (29 as compared to 68, as shown in the table).
If we want to know the proportions of calm meditators and calm non-meditators, we can make a two-way table of relative frequencies and present the relative frequencies in a segmented bar graph.
meditated
did not meditate
calm
66%
28%
agitated
34%
72%
total
100%
100%
Stacked bar graph in blue and red. Horizontal labeled meditated and did not meditate. Vertical labeled 0 to 100, by 25's. Blue represents calm. Red represents agitated.
Standards Alignment
Building On
Addressing
Building Toward
8.SP.A.4
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
