<p>The lesson focuses on generating two-way tables and evaluating the data represented in two-way tables. Students are going to:<br>- read data from a two-way table.<br>- analyze how one-way and two-way tables relate to each other.<br>- using the data in the two-way tables to calculate percentages.<br>- create a two-way table with the data from surveys they perform on their own.</p>

<p>- What does it mean to analyze or estimate numerical quantities?&nbsp;<br>- What qualifies a tool or approach as suitable for a certain task?&nbsp;<br>- How may data be arranged and portrayed to reveal the link between quantities?&nbsp;<br>- What impact does the type of data have on the display option?&nbsp;<br>- How can predictions be made using data analysis and probability?&nbsp;<br>&nbsp;</p>

<p>- Two-Way Table Worksheet (M-8-7-1_Two Way Table Worksheet)&nbsp;<br>- Lesson 1 Exit Ticket (M-8-7-1_Lesson 1 Exit Ticket and KEY)</p>

<p>- Assess the extent of concept proficiency through the use&nbsp;of performance on the Two-Way Table worksheet.&nbsp;<br>- Observe students' production of their&nbsp;questions, two-way table, and questions and answers assessing the data supplied in their two-way table to gauge their understanding of the lesson subject.&nbsp;<br>- Utilize the Lesson 1 Exit Ticket to determine the levels of student comprehension.&nbsp;<br>&nbsp;</p>

<p>Modeling, Explicit Instruction, Formative Assessment.&nbsp;<br>W: This lesson teaches students about two-way tables and how to compare the frequencies and percentages of different populations.&nbsp;<br>H: The lecture starts with two common one-way tables and demonstrates how a two-way table, which is accessible, contains the same information and more.&nbsp;<br>E: In a teacher-guided activity, students will initially investigate a two-way table within the context of one-way tables. Subsequently, they will construct their own two-way tables and collaborate with a companion to investigate the connections and data that are revealed. Subsequently, students generate their own questions and collect their own data.&nbsp;<br>R: Using two-way tables, students will initially practice reading and calculating relative frequencies with the teacher. They will rehearse in pairs, exchanging ideas with classmates and receiving feedback from the teacher.&nbsp;<br>E: Through worksheet exchanges with a partner, students will assess their own work. Students can confer with the teacher for clarification or to ask any questions after grading each other's work.&nbsp;<br>T: The Extension section allows you to modify the lesson to better fit the needs of your students. The Routine section provides recommendations for systematically revisiting course concepts throughout the academic year. The Small Group section offers additional learning and practice opportunities for students who would benefit from them. The Expansion section offers a challenging task for students who are willing to exceed the standard standards.&nbsp;<br>O: The lesson is set up such that, before going on to the following topic, students are gradually introduced to the idea of relative frequencies and two-way tables through guided instructor instruction. (For instance, students build two-way tables independently before dividing them into one-way tables and doing percentage and frequency calculations.) Percentages and frequencies are conveyed both mathematically and linguistically.&nbsp;</p>

<p><strong>Activity 1</strong><br><br>On the board, display the following two tables:&nbsp;<br><br><strong>High-School Students with Jobs</strong></p><figure class="image"><img style="aspect-ratio:218/131;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_12.png" width="218" height="131"></figure><p><strong>High-School Students with Cars</strong></p><figure class="image"><img style="aspect-ratio:215/132;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_13.png" width="215" height="132"></figure><p><br><br><strong>"Suppose the same 100 high school students were surveyed on whether they have jobs and whether they have cars. We can observe that most people have jobs and own cars. What is the relationship between high school students who have jobs and high school students who have cars?"</strong> Students will most likely assume that most students with jobs own 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 may be correct, they are not supported by the&nbsp;data.&nbsp;<br><br><strong>"The two tables represent the results of two unrelated surveys. There may be a connection between the two sets of results, but this table provides no indication of it. The table below is known as a two-way table. It shows the same information from the previous two tables in a single table."</strong> Display the following table.<br><br><strong>Survey of 100 High-School Students</strong></p><figure class="image"><img style="aspect-ratio:437/197;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_14.png" width="437" height="197"></figure><p><br><strong>"How does this survey differ from the other two surveys?"</strong> Note that it displays the number of students who, for instance, have both&nbsp;a car and a job. In other words, it demonstrates the relationship between the two survey questions by displaying how each of the 100 surveyed students responded to&nbsp;both questions at once.<br><br><strong>"Does it present the same data as the two tables we have previously reviewed?"</strong> Students need to recognize that it does. If they are unable to see that it does, cover up the "Has a Car / Does Not Have a Car" columns and ask of the students, based on the table, how many of the surveyed have jobs. Students need to be aware that the previous table indicates that 77 students surveyed have jobs. This method works equally effective when covering up the "Has a Job / Does Not Have a Job" rows and asking of students about the number of respondents who do or do not have cars.<br><br>Explain to students that the two-way table cannot be constructed only from the two individual&nbsp;tables because it contains information not found in either of the two individual tables. However, the two-way table can be divided into two individual tables.&nbsp;<br><br>Give each student a copy of the Two-Way Table Worksheet (M-8-7-1_Two Way Table Worksheet). In pairs, each student should design a two-way table that represents 100 students.&nbsp;<br><br><strong>"Your table should have the same column and row headers as the one we've been looking at, but you can make up&nbsp;your own numbers. Remember, however, that it represents surveying 100 students, therefore when you add up all four numbers&nbsp;in your table, the total should be exactly 100."</strong>&nbsp;<br><br>After all students&nbsp;have&nbsp;created a two-way table, they should switch worksheets. Students should then "deconstruct" the given two-way table, resulting in two different "one-way" tables that reflect the same data as the two-way table they were given.&nbsp;<br><br>Students should then return their papers and review each other's work.&nbsp;<br><br><strong>Activity 2</strong>&nbsp;<br><br>This activity will continue using the original two-way table shown above. Remove the one-way tables, forcing students to review the data as presented in the two-way table.&nbsp;<br><br><strong>"In the two-way table, what percentage of students have cars?"</strong> It may be beneficial to write this question word for word on the board. Some students may benefit from reviewing how to calculate percentages.&nbsp;<br><br><strong>"When we calculate a percentage, we can conceive of it as a fraction with a numerator and a denominator. The denominator always represents the 'whole,' or population from which we are sampling. That is frequently expressed by the word 'of' in our statement."</strong> Underline the word <i>of</i> the question, and then underline <i>students surveyed</i>.&nbsp;<br><br><strong>"Here, our population consists of all the students we surveyed. How many students were surveyed?"</strong> (<i>100</i>) Create a fraction with a denominator of 100.&nbsp;<br><br><strong>"And we're curious about how many of those students own cars. Based on the table, how many people own cars?"</strong> Make sure to remind out that both rows must be added. The first column shows how many students have cars. The fact that the data is divided into two parts (depending on whether the student has a job&nbsp;or not) does not change the fact that every student reflected in the first column has a car. Students&nbsp;should determine that 65 students have cars and put 65 in the numerator.<br><br><strong>"In this case, it is simple to convert \(65 \over 100\)&nbsp; to a percentage, as percentages are always expressed as a fraction of 100. Therefore, 65 out of 100 is equivalent to 65%. Therefore, 65% of the students who participated in the survey have cars."</strong><br><br><strong>"What is the percentage of students who do not have jobs?"</strong> Students should calculate that 23% do not have jobs. Once more, remind them that they must add up the entire row.<br><br><strong>"This question is somewhat different from the other two: What percentage of students with cars have jobs?"</strong> Writing this on the board may be beneficial to emphasize the word <i>of</i> and help students identify the whole.<br><br><strong>"What number goes in our denominator; what number represents the population we're interested in?"</strong> Because the question says "of students," students may be tempted to just state that it is all of the students questioned. Note that it is not "of students surveyed." The entire clause reads, "of students with cars."&nbsp;<br><br><strong>"How many students have cars?"</strong> (<i>65</i>) <strong>"So 65 will be the </strong><i><strong>denominator</strong></i><strong>. And what do we want to know about the students&nbsp;who have cars?"</strong> (<i>How many of them have jobs?</i>) "<strong>From the table, how many of the students with cars have jobs?"</strong>&nbsp;<br><br>Students may be tempted to add up the entire "jobs" row, but emphasize the difference between this question and the preceding questions. We're only interested in students who have cars. For the time being, we are not interested in students who do not have a car. Students should determine that 58 of the students with cars have jobs.&nbsp;<br><br><strong>"So, 58 is the numerator, and our total fraction is \(58 \over 65\)&nbsp;. That is more difficult to interpret as a percentage because the denominator does not include 100. We'll just use a calculator to divide 58 by 65."</strong> Write 0.892... on the board. <strong>"The decimal keeps going, but we'll only use the first three digits. Remember that to convert a decimal like 0.892 to a percentage, we must multiply by 100 or, more simply, move the decimal point two places to the right. So, what percentage of students with cars have jobs?"</strong> (<i>89.2%</i>) Write this figure on the board.&nbsp;<br><br><strong>"Now, determine what percentage of students with cars do not have jobs."</strong> Students should note that 10.8% of students with&nbsp;cars do not have jobs.&nbsp;<br><br><strong>"So, based on this population, if you have a car, you are more than 8 times more likely to have a job&nbsp;as to not have a job. This two-way table demonstrates a clear connection between having a car and having a job. This is the type of data that two-way tables include but one-way tables do not."</strong>&nbsp;<br><br>Students should complete the second page of the Two-Way Table worksheet with the data they generated. (Alternatively, students can work in pairs to finish the second page for one set of data.)<br><br><strong>Activity 3</strong><br><br><strong>"Two-way tables can be generated whenever a single population, such as our 100 fictional high school students, is surveyed on two different questions. Such a table is beneficial for displaying the relationships between the responses to the two questions. For example, our surveys addressed having a car and having a job, which are likely to be associated."</strong> Ask students what other topics are relevant to having a job. Possible responses include having money to spend, receiving an allowance, having free time on weekends, and so on.&nbsp;<br><br><strong>"Come up with two questions to ask your classmates&nbsp;about which you think there may be a relationship. For example, you could ask if they have more than two siblings and if they have to share a room with a sibling. Remember that your questions for this exercise should be either </strong><i><strong>yes</strong></i><strong> or </strong><i><strong>no</strong></i><strong>, or have only two possible answers."</strong>&nbsp;<br><br>As students are writing down their questions, assist them in brainstorming and ensuring that their questions are appropriate; it makes no difference whether you think there will be a relationship between the responses to the two questions.&nbsp;<br><br>After students have brainstormed questions, have them ask at least 15 people each, and then put the results in a two-way table. Depending on time, students can write questions similar to those on page 2 of the Two Way Table worksheet and answer them (or have a classmate answer them) to analyze their data. Students can also complete this as homework by writing a brief paragraph summarizing the data's findings concerning the relationship (or lack thereof) between their questions.&nbsp;<br><br>Lesson 1 Exit Ticket (M-8-7-1_Lesson 1 Exit Ticket and&nbsp;KEY) can be used to quickly assess understanding of lesson concepts.&nbsp;<br><br><strong>Extension:</strong>&nbsp;<br><br>Use the recommendations below to adapt the lesson as needed.&nbsp;<br><br><strong>Routine:</strong> Throughout the school year, look for two-way tables in the newspaper or in a&nbsp;magazine. Post the table or distribute copies to students, and award extra credit to students who can decompose the table and make reasonable conclusions from the data.<br><br><strong>Small Group:</strong> Students in small groups can work on problems at <a href="http://www.bbc.co.uk/bitesize/ks3/maths/handling_data/collecting_recording/revision/5/">http://www.bbc.co.uk/bitesize/ks3/maths/handling_data/collecting_recording/revision/5/</a> for further learning and practice.&nbsp;<br><br><strong>Expansion:</strong> Have students search the Internet, newspapers, and magazines to locate actual&nbsp;two-way tables. Allow students to examine two-way tables that include more than simply yes or no questions, such as a table that indicates how many siblings students have, whether they share a room with a sibling, whether they have cable, and how many TVs they have in their home.</p>