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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

Student Task Statement

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

Are You Ready for More?

24.2

Activity

15 mins

Card Sort: Matching Representations

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR2: Collect and Display

Materials

To Copy (from Blackline Masters)

Matching Representations Cards

Activity Narrative

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

Student Task Statement

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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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.

24.3

Activity

15 mins

Building Another Type of Two-Way Table

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR2: Collect and Display

Materials

None

Activity Narrative

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 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.)

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.

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%

Student Task Statement

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?

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Lesson 24

Looking for Associations

Warm-up

Notice and Wonder: Bar Association

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Activity

Card Sort: Matching Representations

Standards Alignment

8.SP.4

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Activity Synthesis

Activity

Building Another Type of Two-Way Table

Standards Alignment

8.SP.4

Instructional Routines

Materials

Activity Narrative

Launch

Activity Synthesis

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Standards Alignment

8.SP.4

Building on Student Thinking

Are You Ready for More?

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?