• Use performance on the Two-Way Table worksheet to measure concept mastery.
• Observe students’ creation of their own questions, two-way table, and questions and answers analyzing the data presented in their two-way table to gain a sense of student fluency with the lesson concept.
• Use the Lesson 1 Exit Ticket to ascertain student comprehension levels.

“Suppose that the same 100 high-school students were surveyed for both questions: whether they have jobs and whether they have cars. We can see that most have jobs and most have cars. What is the relationship between high-school students who have jobs and high-school students who have cars?” Students will most likely suppose that most students with jobs have cars (because they have money to purchase a car and also probably need the car to get to their job). Point out to students that while these conclusions could be correct, they aren’t supported by the data.

“The two tables are presented as the results of two unrelated surveys. There may be a connection between the two sets of results, but there is no indication of that based just on this table. The table below is called a two-way table. It displays the same data from the previous two tables in one single table.” Show the following table:

Survey of 100 High-School Students

“How is this survey different than the other two surveys?” Students should note that it shows how many students, for example, have both a car and a job. In other words, it shows the relationship between the two survey questions by showing how each of the 100 surveyed students answered both questions at once.

“Does it display the same data as the previous two tables?” Students should note that it does. If they don’t see that it does, cover up the “Has a Car / Does Not Have a Car” columns and ask students, based on the table, how many of those surveyed have jobs. Students should realize that a total of 77 students surveyed have jobs, the same as in the previous table. This method works equally well when covering up the “Has a Job / Does Not Have a Job” rows and asking students how many of those surveyed do and do not have cars.

Point out to students that the two-way table cannot be constructed solely from the two individual tables—it contains information that is not represented in either of the two individual tables. However, the two-way table can be deconstructed into two individual tables.

Give each student a copy of the Two-Way Table worksheet (M-8-7-1_Two Way Table Worksheet.docx). In pairs, have each student create a two-way table representing 100 students.

“Your table should have the same column and row headers as the one we have been looking at, but you can make up your own numbers. Remember, though, that it represents surveying 100 students, so when you add up all four numbers in your table, the total should be exactly 100.”

Once all students have created a two-way table, they should switch worksheets. Students should then “deconstruct” the given two-way table to create two separate “one-way” tables that represent the same data as the two-way table they were given.

Students should then pass their papers back and check each other’s work.

Activity 2

This activity will continue working with the original two-way table presented above. Erase the one-way tables so students have to consult the data as presented in the two-way table.

“In the two-way table, what percentage of students have cars?” It may be useful to write this question out, word for word, on the board. A review of creating percentages might be helpful for some students.

“When we create a percentage, we can think of it, first of all, as a fraction with a numerator and denominator. The denominator always represents the ‘whole,’ the population we’re sampling from. That’s usually indicated beginning with the word ‘of’ in our statement.” Underline the word of in the question, and then underline students surveyed.

“Here, our population is all the students we surveyed. How many students were surveyed?” (100) Create a fraction with a denominator of 100.

“And we’re interested in how many of those students have cars. Based on the table, how many have cars?” Make sure to point out that both rows need to be added. The first column represents how many students have cars. The fact that the data is broken into two parts (based on whether the student has a job or not) doesn’t change the fact that every student represented in the first column has a car. Students should determine that 65 students have cars and put 65 in the numerator.

“Here, it’s straightforward to convert  to a percentage, since percentages are always out of 100. So, 65 out of 100 is simply 65%. So, 65% of the students surveyed have cars.”

“What percentage of students surveyed do not have jobs?” Students should determine that 23% do not have jobs. Again, remind them they need to sum the entire row.

“This question is a bit different than the other two: What percentage of students with cars have jobs?” Again, writing this on the board is probably helpful to emphasize the word of and help students identify the whole.

“What number goes in our denominator; what number represents the population we’re interested in?” Students may be tempted to just say it’s all the students surveyed, since the question reads “of students.” Point out that it’s not “of students surveyed.” The entire clause is “of students with cars.”

“How many students have cars?” (65) “So, 65 will be our denominator. And, of those students with cars, what do we want to know about them?” (How many of them have jobs?) “From the table, how many of the students with cars have jobs?”

Here, students may be tempted to add up the entire “jobs” row, but point out the difference in this question versus the previous questions. Here, we’re only interested in those students who have cars. For the time being, we aren’t interested in students without cars. Students should determine that, of the students with cars, 58 of them have jobs.

“So, 58 is our numerator and our entire fraction is . That’s tougher to view as a percentage because it doesn’t have 100 in the denominator. Here, we’ll just use a calculator to divide 58 by 65.” Write 0.892… on the board. “The decimal keeps going, but we’ll just use the first three digits. Remember, to convert a decimal like 0.892 to a percentage, we need to multiply by 100 or, essentially, just move the decimal point two places to the right. So, what percentage of students with cars have jobs?” (89.2%) Write this value on the board.

“Now, determine what percentage of students with cars do not have jobs.” Students should determine that 10.8% of students with cars do not have jobs.

“So, based on this population, if you have a car, you are more than 8 times as likely to have a job as to not have a job. This two-way table shows a clear relationship between having a car and having a job. This is the kind of data a two-way table contains that one-way tables do not.”

Have students complete the second page of the Two-Way Table worksheet using the data they created. (Alternatively, students can work in pairs and complete the second page for one set of data.)

Activity 3

“Two-way tables can be created whenever a single population, like our 100 fictional high-school students, is surveyed about two different questions. Such a table is useful for showing relationships between the responses to the two questions. For example, our surveys dealt with having a car and having a job, two topics that are likely to be related.” Ask students what other topics might be related to having a job. Possible answers include having money to spend, earning an allowance, having free time on the weekends, and so on.

“Come up with two questions you can ask your classmates for which you think there might be some relationship. For example, you might ask them if they have more than two siblings and if they have to share a room with a sibling. Remember, your questions for this exercise should be yes or no questions or questions with only two possible responses.”

As students are writing down their questions, help students brainstorm and make sure their questions are appropriate; it isn’t important whether you think there actually will be a relationship between the responses for the two questions.

Once students have come up with questions, have them ask at least 15 people each question and then compile the results in a two-way table. Depending on time, students can write questions similar to the questions on page 2 of the Two Way Table worksheet and answer (or have a classmate answer) the questions to examine their data. Students can also do this as homework and write a brief paragraph summarizing what the data reveals about the relationship (or lack thereof) between their questions.

The Lesson 1 Exit Ticket (M-8-7-1_Lesson 1 Exit Ticket and KEY.docx) can be used as a quick check for mastery of lesson concepts.

Extension:

Use the following suggestions to modify the lesson as needed.

• Routine: Periodically throughout the school year, find two-way tables in the newspaper or in a magazine. Post the table or hand out copies to students and offer extra credit to students who can decompose the table and draw some valid conclusions about the data.
• Small Group: Students who need additional learning and practice opportunities can be assigned to work on the problems at this Web site:

http://www.bbc.co.uk/bitesize/ks3/maths/handling_data/collecting_recording/revision/5/

• Expansion: Have students search the Internet, newspapers, and magazines to locate actual two-way tables. Have students explore two-way tables that cover more than just yes or no questions; for example, a table that shows how many siblings students have and whether they share a room with a sibling or whether they have cable and how many TVs are in their house.