8.1 Two-Way Frequency Tables

Transcription

1 2 Locker LESSON 8.1 Two-Way Frequency Tables Common Core Math Standards The student is expected to: S-ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Mathematical Practices MP.7 Using Structure Language Objective Distinguish between quantitative data and categorical data. Name Class Date 8.1 Two-Way Frequency Tables Essential Question: How can categorical data for two categories be summarized? Explore Categorical Data and Frequencies Data that can be expressed with numerical measurements are quantitative data. In this lesson you will examine qualitative data, or categorical data, which cannot be expressed using numbers. Data describing animal type, model of car, or favorite song are examples of categorical data. Circle the categorical data variable. Justify your choice. temperature weight height color Temperature, weight, and height are measured on a numerical scale, so they are quantitative data. Color cannot be expressed numerically. Identify whether the given data is categorical or quantitative. large, medium, small categorical 120 ft 2 2 2, 130 ft, 140 ft quantitative Resource Locker ENGAGE Essential Question: How can categorical data for two categories be summarized? You can summarize categorical data for two categories in a two-way frequency table. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online, and then take a quick, show-of-hands survey to determine the most popular sports among the students in your class. Discuss whether the survey results might have been different had you surveyed the boys and the girls separately. Then preview the Lesson Performance Task. A frequency table shows how often each item occurs in a set of categorical data. Use the categorical data listed on the left to complete the frequency table. Reflect Ways Students Get to School bus car walk car car car bus walk walk walk bus bus car bus bus walk bus car bus car Way Frequency bus 8 1. How did you determine the numbers for each category in the frequency column? You can determine the numbers for each category by counting the number of times each category is listed in the data. 2. What must be true about the sum of the frequencies in a frequency table? The sum should equal the total number of items in the data set. Module Lesson 1 Name Class Date 8.1 Two-Way Frequency Tables Essential Question: How can categorical data for two categories be summarized? S-ID.5 For the full text of these standards, see the table starting on page CA2. Explore Categorical Data and Frequencies car walk Data that can be expressed with numerical measurements are quantitative data. In this lesson you will examine qualitative data, or categorical data, which cannot be expressed using numbers. Data describing animal type, model of car, or favorite song are examples of categorical data. Circle the categorical data variable. Justify your choice. temperature weight height color Temperature, weight, and height are measured on a numerical scale, so they are quantitative data. Color cannot be expressed numerically. Identify whether the given data is categorical or quantitative. large, medium, small 120 ft 2 2, 130 ft, 140 ft A frequency table shows how often each item occurs in a set of categorical data. Use the categorical data listed on the left to complete the frequency table. Ways Students Get to School Way Frequency categorical quantitative bus car walk car car car bus walk walk walk bus bus car bus bus walk bus car bus car bus 8 car walk 7 5 Resource 7 5 HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. Reflect 1. How did you determine the numbers for each category in the frequency column? You can determine the numbers for each category by counting the number of times each category is listed in the data. The sum should equal the total number of items in the data set. 2. What must be true about the sum of the frequencies in a frequency table? Module Lesson Lesson 8.1

2 Explain 1 Constructing Two-Way Frequency Tables If a data set has two categorical variables, you can list the frequencies of the paired values in a two-way frequency table. Example 1 Complete the two-way frequency table. A high school s administration asked randomly selected students in the 9th and 10th grades about what fruit they like best. Complete the table. Preferred Fruit Grade Apple Orange Banana 9th th Row totals: Column totals: Grand total: 9th: = 54 10th: = 46 Apple: = 41 Orange: = 21 Sum of row totals: = Sum of column totals: = Banana: = 38 Both sums should equal the grand total. Preferred Fruit Grade Apple Orange Banana 9th th EXPLORE Categorical Data and Frequencies INTEGRATE TECHNOLOGY Students have the option of completing the Explore Activity online or in the book. CONNECT VOCABULARY Make sure that students understand the distinction between quantitative and categorical data. Quantitative data can be measured on a numbered scale. Categorical data involves either/or choices between two or more descriptive categories. Ask students to provide some examples of quantitative and categorical data. Jenna asked some randomly selected students whether they preferred dogs, cats, or other pets. She also recorded the gender of each student. The results are shown in the two-way frequency table below. Each entry is the frequency of students who prefer a certain pet and are a certain gender. For instance, 8 girls prefer dogs as pets. Complete the table. Preferred Pet Gender Dog Cat Other Girl Boy Row totals: Girl: = 16 Boy: = Column totals: Dog: = 18 Cat: = 12 Other: = Grand total: Sum of row totals: = 40 Sum of column totals: = 40 Both sums should equal the grand total. Module Lesson 1 EXPLAIN 1 Constructing Two-Way Frequency Tables INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 To check students understanding of the information presented in a two-way frequency table, have one student ask a question about the data in the table. Have another student answer the question and provide an explanation of how to use the table to arrive at the answer. PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunity to use Mathematical Practice MP.7, which asks students to look for and make use of structure. In this lesson, students use the structure of two-way frequency tables to analyze data and understand a problem-solving situation. QUESTIONING STRATEGIES What is the difference between a two-way frequency table and an ordinary frequency table? In a two-way frequency table, each value (other than row and column totals) indicates the number of items that fit two data categories rather than one, such as the number of boys who prefer cats. In an ordinary frequency table, data values correspond to only one data category. Two-Way Frequency Tables 348

3 EXPLAIN 2 Reading Two-Way Frequency Tables INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Remind students that there may be more than one process for filling in a particular table. Discuss different ways to fill cells in a table, such as using addition to find the total for a column or row, and working backward to find a missing value that is not a total. Reflect 3. Look at the totals for each row. Was Jenna s survey evenly distributed among boys and girls? Explain. No, there were 24 boys and 16 girls. 4. Look at the totals for each column. Which pet is preferred by the most students? Justify your answer. Dogs; the total for Dog is greater than the total for Cat and the total for Other. Your Turn Complete the two-way frequency table. 5. Antonio surveyed 60 of his classmates about their participation in school activities and whether they have a part-time job. The results are shown in the two-way frequency table below. Complete the table. Activities Job Clubs Only Sports Only Both Neither Yes No Jen surveyed students about whether they like baseball or basketball. Complete the table. AVOID COMMON ERRORS Remind students to use the row totals and column totals to check that their work is correct. The sum of the row totals should be the same as the sum of the column totals. If the sums are different, students need to review their work to look for errors. Like Basketball Like Baseball Yes No Yes No Explain 2 Reading Two-Way Frequency Tables You can extract information about paired categorical variables by reading a two-way frequency table. Example 2 Read and complete the two-way frequency table. Suppose you are given the circled information in the table and instructed to complete the table. Eat Cereal for Breakfast Gender Yes No Girl Boy Find the total number of boys by subtracting: - 54 = 46 Find the number of boys who do eat cereal by subtracting: = 36 Add to find the total number of students who eat cereal and the total number of students who do not eat cereal. Module Lesson Lesson 8.1 COLLABORATIVE LEARNING Peer-to-Peer Activity Have students work in pairs. Challenge one in each pair to describe a data set for which the partner must create a two-way frequency table. Students should provide as little direct information as possible, while still making it possible to complete the table. For example, instead of saying, Twenty boys prefer baseball, they might say, The number of boys who prefer baseball is 15 less than the number of girls who prefer softball. After students complete the frequency tables, have them compare results with their partners. If the students disagree, have them discuss and determine which is correct.

4 B One hundred students were surveyed about which beverage they chose at lunch. Some of the results are shown in the two-way frequency table below. Complete the table. Lunch Beverage Gender Juice Milk Water Girl Boy QUESTIONING STRATEGIES To complete a two-way frequency table that is missing information, is it necessary to know the total of all the values in the table? Explain. No; if there is enough other information given in the table, the total can be determined using other information. Find the total number of girls by subtracting: - 60 = 40 So, the total number of girls is 40. The number of girls who do not choose milk is = 27. Find the number of girls who chose milk by subtracting: = 13 Reflect 7. Which lunch beverage is the least preferred? How do you know? Juice: the total for Juice is less than the total for Milk and the total for Water. Your Turn Read and complete the two-way frequency table. 8. students were asked what fruit they chose at lunch. The two-way frequency table shows some of the results of the survey. Complete the table. Lunch Fruit Gender Apple Pear Banana Girl Boy high school teachers were asked whether they prefer to use the chalkboard or projector in class. The two-way frequency table shows some of the results of the survey. Complete the table. Preferred Teaching Aid Gender Chalkboard Projector Female Male Module Lesson 1 DIFFERENTIATE INSTRUCTION Cognitive Strategies When students are completing two-way frequency tables that are missing information, discuss how to determine which cells in the table can be completed first. Students should understand that in order for a value to be determined, it must be the only missing value in either its row or its column. By completing a row or column in which only one value is missing, students move one step closer to completing other rows or columns. Two-Way Frequency Tables 350

5 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Have students discuss why the order of words is important in identifying a cell in a two-way frequency table. For example, students should understand that people who watch TV but not movies,, refers to a different cell than people who watch movies but not TV.,, Elaborate 10. You are making a two-way frequency table of 5 fruit preferences among a survey sample of girls and boys. What are the dimensions of the table you would make? How many entries would you need to fill the table with frequencies and totals? You would make a 2-by-5 table for the frequencies. Adding totals would increase the dimensions to 3-by-6. You need 3 times 6, or 18, entries to fill the table. 11. A 3 categories-by-3 categories two-way frequency table has a row with 2 numbers, and no row or column totals. Can you fill the row? This table would be a 4-by-4 table, including the row of totals. You cannot fill the row because you need 3 numbers to figure out the last one. 12. Essential Question Check-In How can you summarize categorical data for 2 categories? You can use a two-way frequency table. SUMMARIZE THE LESSON What information is provided by a two-way frequency table, and how is that information organized? A two-way frequency table provides categorical data for two categorical variables, such as color and size or gender and job. Data for one variable is organized in rows, and data for the other variable is organized in columns. The value in each cell of the table identifies the number of items that fit the intersection of the two categories. Evaluate: Homework and Practice 1. Identify whether the given data is categorical or quantitative. gold medal, silver medal, bronze medal categorical m, 200 m, 400 m quantitative 2. A theater company asked its members to bring in canned food for a food drive. Use the categorical data to complete the frequency table. Cans Donated to Food Drive peas corn peas soup corn corn soup soup corn peas peas corn soup peas corn peas corn peas corn soup corn peas soup corn corn Complete the two-way frequency table. Cans soup peas corn 3. James surveyed some of his classmates about what vegetable they like best. Complete the table. Preferred Vegetable Frequency Online Homework Hints and Help Extra Practice Grade Carrots Green Beans Celery 9th th Module Lesson 1 LANGUAGE SUPPORT Connect Vocabulary The word quantitative in the phrase quantitative data shares a root with the word quantity, meaning an amount. The word categorical in the phrase categorical data shares a root with the word category, meaning a descriptive grouping. Categorical data can also be referred to as qualitative data because these data describe qualities or characteristics. The word frequency in the phrase frequency table shares a root with the word frequent, meaning often. Each value in a frequency table shows how often that value falls into a given category. 351 Lesson 8.1

6 4. A high school s extracurricular committee surveyed a randomly selected group of students about whether they like tennis and soccer. Complete the table. Like Tennis Like Soccer Yes No Yes No After a school field trip, Ben surveyed some students about which animals they liked from the zoo. Complete the table. Preferred Animal at a Zoo Grade Lion Zebra Monkey 11th th Jill asked some randomly selected students whether they preferred blue, green, or other colors. She also recorded the gender of each student. The results are shown in the two-way frequency table below. Complete the table. Preferred Color Gender Green Blue Other Girl Boy Kevin surveyed some students about whether they preferred soccer, baseball, or another sport. He also recorded their gender. Complete the table. Preferred Sport Gender Soccer Baseball Other Girl Boy EVALUATE ASSIGNMENT GUIDE Concepts and Skills Explore Categorical Data and Frequencies Example 1 Constructing Two-Way Frequency Tables Example 2 Reading Two-Way Frequency Tables Practice Exercises 1 2, 23 Exercises 3 12 Exercises 13 22, INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Have students try to find the smallest number of filled cells in a two-way frequency table that makes it possible to complete the table. Work backward from a completed frequency table by removing values one at a time, making sure it is possible to do so while leaving a row or column with only one missing value. Students may discover that completing a frequency table requires that the table have at least as many filled cells as the number of interior cells. Module Lesson 1 Exercise Depth of Knowledge (D.O.K.) Mathematical Practices Recall of Information MP.7 Using Structure Skills/Concepts MP.3 Logic Skills/Concepts MP.4 Modeling 23 1 Recall of Information MP.7 Using Structure 24 2 Skills/Concepts MP.3 Logic Strategic Thinking MP.3 Logic Two-Way Frequency Tables 352

7 MULTIPLE REPRESENTATIONS Have students create a two-question survey whose results can be summarized in a two-way frequency table. Students should understand that the questions they choose for their surveys should not allow for open-ended answers. AVOID COMMON ERRORS Remind students that in a two-way frequency table, the total of the values in each row and the total of the values in each column must add up to the same number. Students can check their work for errors by comparing the sum of the row totals to the sum of the column totals. 8. A school surveyed a group of students about whether they like backgammon and chess. They will use this data to determine whether there is enough interest for the school to compete in these games. Complete the table. Like Backgammon Like Chess Yes No Yes No Hugo surveyed some 9th and 10th graders in regard to whether they preferred math, English, or another subject. The results of the survey are in the following table. Complete the table Preferred Subject Grade Math English Other 9th th Luis surveyed some middle school and high school students about the type of music they prefer. Complete the table. Image Credits: Dragon Images/Shutterstock Preferred Music School Level Country Pop Other Middle School High School Natalie surveyed some teenagers and adults on whether they prefer standard cars, vans, or convertibles. Her results are in the following table. Complete the table. Preferred Car Type Age Standard Van Convertible Adults Teenagers Module Lesson Lesson 8.1

8 12. Eli surveyed some teenagers and adults on whether they prefer apples, oranges, or bananas. His results are in the following table. Complete the table. Preferred Fruit Age Apple Orange Banana Adults Teenagers students were asked to name their favorite science class. The results are shown in the two-way frequency table. Use the table for the following questions. Favorite Science Class Gender Biology Chemistry Physics Girl Boy How many boys were surveyed? Explain how you found your answer. 96 boys: 104 of the 200 students were girls, so = 96 of them were boys. 14. Complete the table. How many more girls than boys chose biology as their favorite science class? Explain how you found your answer = 23, so 23 more girls than boys chose biology. The results of a survey of 150 students about whether they own an electronic tablet or a laptop are shown in the two-way frequency table. Device Gender Electronic tablet Laptop Both Neither Girl Boy Complete the table. Do the surveyed students own more laptops or more electronic tablets? The number of boys is = 62. Girls who own both electronic devices: = 10 Boys who own an electronic tablet: = 14; Students own more laptops. 16. Which group had more people answer the survey, boys or students who own an electronic tablet only? Explain. Boys: 62 boys answered the survey, which is more than the 29 people who own an electronic tablet only. Module Lesson Two-Way Frequency Tables 354

9 17. The table shows the results of a survey about students preferred frozen yogurt flavor. Complete the table, and state the flavors that students preferred the most and the least. Preferred Flavor Gender Vanilla Mint Strawberry Girl Boy Students preferred mint the most and vanilla the least. 18. Teresa surveyed students about whether they like pop music or country music. Out of the students surveyed, 42 like only pop, 34 like only country, 15 like both pop and country, and 9 do not like either pop or country. Complete the two-way frequency table. Like Pop Like Country Yes No Yes No Forty students in a class at an international high school were surveyed about which non-english language they can speak. Complete the table. Foreign Language Gender Chinese Spanish French Girl Boy Luis surveyed students about whether they like soccer. The number of girls and the number of boys completing the survey are equal. 20. Complete the table. Likes Soccer Gender Yes No Girl Boy Twice as many girls like soccer as the number that like tennis. The same number of students like soccer as like tennis. Construct a table containing the tennis data. Likes Tennis Gender Yes No Girl Boy Module Lesson Lesson 8.1

10 22. A group of 200 high school students were asked about their use of and text messages. The results are shown in the two-way frequency table. Complete the table. Text Messages Yes No Yes No Circle the letter of each data set that is categorical. Select all that apply. A. 75, 79, 77, 85 B. apples, oranges, pears C. male, female D. blue, green, red E. 2 feet, 5 feet, 12 feet F. classical music, country music G. 1 centimeter, 3 centimeters, 9 centimeters 24. Explain the Error Find the mistake in completing the two-way frequency table for a survey involving 50 students. Then complete the table correctly. Favorite Foreign Language Class Gender Russian German Italian Girl Boy Correct table: 50 Possible answer: 50 was used as the total whose favorite is Russian instead of as the grand total. boys: = 26 Boys who prefer Russian: = 10 Favorite Foreign Language Class Gender Russian German Italian Girl Boy Module Lesson 1 Two-Way Frequency Tables 356

11 JOURNAL Have students explain the steps they take to complete a two-way frequency table that has missing values. Students should be sure to include descriptions of how to determine column totals and row totals. H.O.T. Focus on Higher Order Thinking 25. Justify Reasoning Charles surveyed boys about their favorite color. Of the boys surveyed, 44 preferred blue, 25 preferred green, and 31 preferred red. a. Explain why it is not possible to make a two-way frequency table from the given data. Charles only surveyed boys, so there is only one categorical variable: color. You need two categorical variables to make a two-way frequency table. b. Suppose Charles also surveyed some girls. Of the girls surveyed, 30 preferred blue and 43 preferred green. Can Charles make a two-way frequency table now? Can he complete it? He can make a table, but he cannot complete it. There are now two categorical variables, color and gender, but there is not enough information about the girls preferences. In addition to the information provided, you need to know how many girls were surveyed or how many prefer red to complete the table. 26. Persevere in Problem Solving Shown are two different tables about a survey involving students. Each survey had a few questions about musical preferences. All students answered all questions. Complete the tables. What type of music do the students prefer? Likes Classical Music Gender Yes No Girl Boy Likes Blues Music Gender Yes No Girl Boy All students answered all the questions, so total girls in both is 49, and the total students in both is. Overall, the students prefer blues music: 70 students like blues, 50 like classical. Module Lesson Lesson 8.1

12 Lesson Performance Task Two hundred students were asked about their favorite sport. Of the 200 students surveyed, 98 were female. Some of the results are shown in the following two-way frequency table. Favorite Sport Gender Football Baseball Basketball Soccer Female Male a. Complete the table. number of males: = 102 Females who chose football: = 26 Females who chose baseball: = 24 Males who chose soccer: = 24 Males who chose basketball: = 21 baseball: = 43 basketball: = 57 b. Which sport is the most popular among the students? Which is the least popular? Explain. Most popular is football; 64 is the greatest total. Least popular is soccer; 36 is the least total. c. Which sport is most popular among the females? Which sport is most popular among the males? Explain. Most popular among females is basketball; 36 is the greatest number of choices. Most popular among males is football; 38 is the greatest number of choices. QUESTIONING STRATEGIES In what order would you fill in the empty spaces in the table? Explain why. Possible answer: I would first fill in the numbers given in the text of the problem (200 students in total, and 98 female students). Then I would look for a row or column with only one empty cell, and find the missing number (such as the total number of males, the number of females who chose football, or the number of males who chose soccer). After I fill in those numbers, there would still be other rows and columns with only one empty cell, so I would repeat the process until the table is completed. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Discuss with students how the sample of students chosen for the survey could have influenced the survey results. For example, consider whether age, country of origin, participation in sports, or other characteristics of survey participants might influence the results. Have students describe a target population for which soccer is likely to be the most popular sport among those surveyed. Module Lesson 1 EXTENSION ACTIVITY Have students create and conduct a survey of 20 students and then record the results in a two-way frequency table. Students should begin by selecting two categorical variables to record, such as gender and favorite subject, or grade level and music preference. Remind them that quantitative and free-response answers are not appropriate for recording in a frequency table. Encourage students to select a sample of students that is not expected to skew the survey results. Have students share their frequency tables and discuss their findings. Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Two-Way Frequency Tables 358