Lesson 23

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. 

Tell students you are going to show them a graph and that their job is to find at least one thing they notice and one thing they wonder about the graph. Give students 1 minute of quiet think time followed by a whole-class discussion. 

Ask students to share things they noticed and wondered. Record and display their responses for all to see. If students wondered who the data represents, tell them the data was collected from a survey of students. They were asked whether they played a sport or not and whether they watched more or less than 1 hour of TV each night.

If they did not wonder these things, tell them this information and then ask, “Based on the graph, do you think playing a sport and the amount of time watching TV are related?” (Yes. Among people who watch more than an hour of TV, there are a lot more who do not play a sport than people who do. For people who play a sport, more do not watch TV than do and this is reversed for those who do not play a sport.)

In this activity students become familiar with two-way tables, clustered bar graphs, and segmented bar graphs by matching different situations. They label the diagrams to match the data given and create a table to match the data shown in one of the bar graphs.

Arrange students in groups of 2–3. Display the two-way table from the activity for all to see. Use MLR 2 (Collect and Display) to gather student language used to describe two-way tables, bar graphs, and segmented bar graphs to be referenced and revised throughout later activities.

Ask students:

• “Why do you think this is called a 'two-way table'?” (The table represents 2 variables. You can think of the people who were surveyed either by age group or by whether or not the person has a cell phone.)
• “What does the 25 mean?” (25 people 10 to 12 years old have cell phones.)
• “What does the 55 mean?” (55 of the people who were surveyed do not have a cell phone.)
• “How many people are represented in this table?” (170 people.)

Display the bar graph from the activity for all to see.

Ask students:

• “Why do you think this is called a 'bar graph?'” (Each bar represents how many people are in each group named on the bottom.)
• “What do the blue bars represent? How do you know?” (The blue bars represent how many people in each group have cell phones. I can tell by the key on the right.)
• “Where does the 25 from the table show up in this bar graph?” (The first blue bar shows that 25 10–12 year olds have cell phones.)

Display the segmented bar graph from the activity for all to see.

Ask students:

• “Why do you think this is called a 'segmented bar graph?'” (Each bar is segmented into pieces.)
• “What do you think the vertical axis represents?” (The percentage in each section.)

Distribute 1 set of the pre-cut cards from the blackline master to each group.

The purpose of this activity is to help students understand the connections among the three representations of data. It also helps students see the importance of labeled visual representations.

To highlight the connections, ask:

• “What were some strategies you used to match the cards?” (The book table had the same number for those who liked the movie and those who did not like the movie among those who did not read the book, so I knew the two graphs would show equal sized bars.)
• “Were some cards easier to match than others? Explain.” (The cards showing the actual numbers were easier to match than the cards showing the percentages.)
• “What strategy did you use to create the two-way data table?” (I started by thinking of two yes or no questions that could be asked and use those to label the rows and columns of the table.)

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.

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. 

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.

Use MLR 2 (Collect and Display) to co-create a graphic organizer with students comparing and contrasting the definitions and characteristics of “actual frequency” and “relative frequency” to be referenced and revised throughout this and later activities.

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.

The purpose of this discussion is to help students see the usefulness of two-way tables that display relative frequency.

Some questions for discussion:

• “What does the 42% in 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 two 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.)