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Unit 3, Lesson 2

Relative Frequency Tables

Algebra 1

2.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Notice and Wonder

Materials

None

Activity Narrative

The purpose of this Warm-up is to elicit the idea that two-way tables can be used to think about relative frequency, which will be useful when students create relative frequency tables in a later activity. While students may notice and wonder many things about these images, the ways in which the values in the two-way tables relate to the totals are the important discussion points.

Launch

Arrange students in groups of 2. Display the two-way table for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.

Activity

None

Several adults in a school building were asked about their highest degree completed and whether they were a teacher.

What do you notice? What do you wonder?

	teacher	not a teacher
associate degree	4%	16%
bachelor’s degree	52%	64%
master’s degree or higher	44%	20%

Student Response

to access Student Response.

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. 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.

It is not important to answer the questions that students are wondering about at this time. Many of their questions will be addressed later in the lesson. If the concept of relative frequency does not come up during the conversation, ask students, “What does the 44% represent?” (44% of the teachers interviewed have a master’s degree or higher.)

Display the actual values used in the relative frequency table in the task for all to see.

	teacher	not a teacher
associate degree	2	4
bachelor’s degree	26	16
master’s degree or higher	22	5

Point out that, although there are more teachers with bachelor’s degrees than nonteachers with bachelor’s degrees, the percentage is lower. This is because there are more teachers who responded to this survey overall, and the table shows the relative percentage of adults for each group. That is, the 26 teachers with bachelor’s degrees are considered among the 50 teachers, while the 16 nonteachers with bachelor’s degrees are only considered among the 25 nonteachers.

Ask students:

“What is the categorical variable for the categories that label the rows?” (It is the type of degree earned. It likely resulted from a survey question like “What is the highest degree that you hold?”)
“What is the categorical variable for ‘teacher’ and ‘not a teacher’?” (It is the person’s occupation.)

2.2

Activity

Instructional Routines

MLR8: Discussion Supports

Materials

None

Activity Narrative

The mathematical purpose of this activity is for students to gain an understanding of the relationship between a two-way frequency table and a two-way relative frequency table. Students have the same information in a two-way table and a segmented bar graph, then are asked to interpret the information given in context. In some cases, it makes sense to view the raw data in the table, and in other cases, it can make sense to understand the data as percentages of various larger groups. Students examine the meaning of data when they are presented as a percentage of the entire group or either of the two subgroups.

Launch

Arrange students in groups of 2.

Help orient students by having them notice:

The sum of all 4 percentages in the second table is 100%.
For each of the next two tables, ask students to find percentages that sum to 100%. (The third table has columns that sum to 100%, and the fourth table has rows that sum to 100%.)

Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, instead of presenting tables and graphs all at once, provide four different pieces. Ask students to place the table with the actual values at the top of their workspace and line up the other tables and graphs in a new row below. This allows students to see all of the information while reading and answering the questions without flipping back and forth. Encourage students to move or cover tables to focus on one table at a time.
Supports accessibility for: Memory, Organization

2.3

Activity

Instructional Routines

None

Materials

None

Activity Narrative

The mathematical purpose of this activity is for students to create and interpret different relative frequency tables in context. Identify students who record the totals for the columns and rows on the first table.

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Arrange students in groups of 2. Read the directions for the activity to the class. Ask students if they have heard of a placebo or the placebo effect. If necessary, explain that a placebo is a pill that looks just like the one used in the study, but without the active ingredient in it. This is to remove from the study the mental influence of the subject believing that taking the pill might have an effect on them.

Give students 5 minutes to work through the problems, then have a whole-group discussion.

Activity

None

Lesson Synthesis

The mathematical purpose of this lesson is for students to understand how to create and interpret relative frequency tables. Here are some questions for discussion.

“How do you create a relative frequency table?” (There are several different ways to do it. In short, the numbers in a two-way table are divided by a total or totals. The total can be for the entire table, or the totals can be for each row or column, depending on what relationship I am trying to find.)
“What are some categorical variables you could use to make a frequency table? Share them with your partner.” (Here are some examples: Categorize bird species by the dominant color they display on their feathers. Categorize students from two different sports teams and whether or not they are in band. Categorize students by their type of cell phone and what type of social media they use.)

to access Cool-down.

Student Lesson Summary

Converting two-way tables to relative frequency tables can help reveal patterns in paired categorical variables. Relative frequency tables are created by dividing the value in each cell in a two-way table by the total number of responses in the entire table, or the total responses in a row or a column. Depending on what patterns are important, different types of relative frequency tables are used. To examine how individual combinations of the categorical variables relate to the whole group, divide each value in a two-way table by the total number of responses in the entire table to find the relative frequency.

For example, this two-way table displays the condition of a certain textbook and its price for 120 of the books at a college bookstore.

	$10 or less	more than \$10 but less than \$30	$30 or more
new	3	9	27
used	33	36	12

A two-way relative frequency table is created by dividing each number in the two-way table by 120 because there are 120 values (3+9+27+33+36+12) in this data set. The resulting two-way relative frequency table can be represented using fractions, decimals, or percents. so inexpensive and that 10% of the books in the bookstore are both expensive and in used condition.

	$10 or less	more than \$10 but less than \$30	$30 or more
new	0.025	0.075	0.225
used	0.275	0.300	0.100

This two-way relative frequency table allows you to see what proportion of the total is represented by each number in the two-way table. The number 33 in the original two-way table represents the number of used books that also sell for $10 or less, which is 27.5% of all the books in the data set. Using this two-way relative frequency table, we can see that there are very few (2.5%) new books that are al

In other situations, it makes sense to examine row or column proportions in a relative frequency table. For example, to convert the original two-way table to a column relative frequency table using column proportions, divide each value by the sum of the column.

	$10 or less	more than \$10 but less than \$30	$30 or more
new	0.083	0.2	0.692
used	0.917	0.8	0.308

This shows that about 91.7% () of the books that are sold for $10 or less are in used condition. Notice that each column of this column relative frequency table reveals the proportions of the books in each price category that are in each condition and that the relative frequencies in each column sum to 1. In particular, this shows that most of the inexpensive and moderately priced books are used, and most of the expensive books are new.

200 people were asked if they prefer dogs or cats and whether they live in a rural or urban setting.

The actual values collected from the survey are in the first table.

	urban	rural	total
cat	54	42	96
dog	80	24	104
total	134	66	200

The next table shows what percentage of the 200 total people are represented by each combination of categories. The segmented bar graph represents the same information graphically.

	urban	rural
cat	27%	21%
dog	40%	12%

<p>A segmented bar graph. Vertical scale from 0 to 100 by 25’s. 0 to 27 is urban and cat. 27 to 65 is urban and dog. 65 to 88 is rural and cat. 88 to 100 is rural and dog.</p>

The next table is a column relative frequency table and shows the percentage of people represented in each column that have a certain pet preference. The segmented bar graph represents the same information graphically.

	urban	rural
cat	40%	64%
dog	60%	36%

<p>Graph with 2 segmented bars. Vertical scale from 0 to 100 by 25’s. Bars labeled urban, and rural. For urban bar, 0 to 40 is cat. 40 to 100 is dog. For rural bar, 0 to 63 is cat. 63 to 100 is dog.</p>

The last table is a row relative frequency table and shows the percentage of people represented in each row that live in a certain area. The segmented bar graph represents the same information graphically.

	urban	rural
cat	56%	44%
dog	77%	23%

<p>Graph with 2 segmented bars. Vertical scale from 0 to 100 by 25’s. Bars labeled cat, and dog. For cat bar, 0 to 60 is urban. 60 to 100 is rural. For dog bar, 0 to 77 is urban. 77 to 100 is rural.</p>

For each relative frequency table, select a percentage, and explain how numbers from the original table were used to get the percentage.
What percentage of those surveyed live in an urban area and prefer dogs?
Among the people surveyed who prefer dogs, what percentage of them live in an urban setting?
What percentage of people surveyed who live in an urban setting prefer dogs?
How many of the people responded that they prefer dogs and live in an urban setting?
Among the people surveyed, are there more people who prefer dogs or cats?
Your pet food company has access to a billboard in a rural setting. Would you recommend advertising dog food or cat food on this billboard? Which table did you use to make this decision? Explain your reasoning.

In an experiment to test the effectiveness of vitamin C on the length of colds, two groups of people with colds are given a pill to take once a day. The pill for one of the groups contains 1,000 mg of vitamin C, while the other group takes a placebo pill. The researchers record the results in a table.

	group A	group B
cold lasts less than a week	16	27
cold lasts a week or more	17	53

First, the researchers want to know what percentage (to the nearest whole percent) of people are in each combination of categories. 14% of all the participants had a cold that lasted less than a week and were in group A. What percentage of all the participants had a cold that lasted less than a week and were in group B? Complete the rest of the relative frequency table with the corresponding percentages.

	group A	group B
cold lasts less than a week	14%
cold lasts a week or more

Next, the researchers notice that among participants who had colds that lasted less than a week, 37% were in group A. Among participants who had colds that lasted a week or more, what percentage were in group B? Complete the table with the corresponding percentages.

	group A	group B
cold lasts less than a week	37%
cold lasts a week or more

Finally, the researchers notice that among the participants in group A, 48% had colds that lasted less than one week. Among the participants in group B, how many had colds that lasted a week or more? Complete the table with the corresponding percentages.

	group A	group B
cold lasts less than a week	48%
cold lasts a week or more

To understand the results, the researchers want to know the following: Among people whose colds lasted less than a week, what percentage are in group B? Explain your reasoning.
If the researchers believe that vitamin C has a small effect on the length of a cold, which group most likely got the pills containing vitamin C? Explain your reasoning.

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Lesson 2

Relative Frequency Tables

Warm-up

Standards Alignment

Building On

Addressing

HSS-ID.B.5

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSS-ID.B.5

Building Toward

Standards Alignment

Building On

Addressing

HSS-ID.B.5

Building Toward