Legos, lunch time, and lollipops Two-way Frequency Tables Image used with Permission from Easyvectors.com Taking a first look at related data

Let’s start with a little magic This is a Magic Square The challenge is to insert the numbers 1 – 9 in the squares, using each number only once. ***and*** Every column, row, and diagonal must add up to the same total: 15. Like magic! 6 2 9 The “Magic Square” puzzle is a fun way to introduce concepts that will be used with two-way frequency tables. Students can develop and share strategies for determining what numbers go in each square. Those same strategies will be used in the lesson in completing a two-way frequency table. The teacher can encourage and prompt the students to use the fact that each row and column (and diagonal) must have a sum of 15. If a row or column is only missing one number, then that missing number can be found by simply subtracting the sum of the two given numbers in that row or column from 15. The solution to this magic square is: 6 7 2 1 5 9 8 3 4 Encourage the students to check the sum of every row, column, and diagonal. Try it on your own!!!

Understanding the magic How did you figure out where to put each of the numbers? 6 2 9 The solution to this magic square is: 6 7 2 1 5 9 8 3 4 Encourage the students to check the sum of every row, column, and diagonal.

Try to complete this magic square on your own: 8 4 7 Again, encourage the students to explain their thinking and the process they used. The solution to this magic square is: 3 4 1 5 9 6 7 2 How did you decide where to put each number?

Connecting magic squares to “real-world” mathematics Many times in real life, we need to look at a set of numbers and analyze how they are related to one another, similar to how we solved magic squares. Using a table to organize the information is very helpful….

Rainy Day Fun The Summer Camp for Kids staff is planning an indoor activity for the campers to do on rainy days. They are considering a Lego activity and a finger painting activity. They send out a survey to all the kids who will be coming to the camp to find out their preferences.

37 8 45 26 14 40 63 22 85 A table like this that displays data The staff organized the results of the survey in this table: Legos painting total boys 37 8 45 girls 26 14 40 63 22 85 A table like this that displays data in rows and columns is called a two-way frequency table.

What this two-way frequency table tells us Legos painting total boys 37 8 45 girls 26 14 40 63 22 85 How many boys preferred Legos? How many girls preferred Legos? How many campers preferred painting? How many girls responded to the survey? How many campers responded to the survey? The students should number their papers 1 – 5, record the parameters of each example (i.e. boys & Legos, or total campers), and write their responses to each question on their papers. Ask the students to put down their pencils when they are finished, so you will know when to begin the discussion. Before discussing the answers to the questions, first ask the students what they observe about the data in the table. They should respond that the numbers in the right column give the total for each row, for instance 37 + 8 = 45. They may also notice that the bottom row gives the total for each column; for instance the column “painting”: 8 + 14 = 22. These characteristics will be important in the next example. Rather than asking for the answers, ask students to raise their hand if they would like to explain how they found the answer to a question. Ask them to use the words “row” and “column” in their explanation. Sample answers: To find the number of boys who preferred Legos, look in the column for Legos and the row for boys. The number of boys choosing Legos is 37. To find the number of girls who preferred Legos, look in the column for Legos and the row for girls. The number of girls choosing Legos is 26. To find the number of campers choosing painting, look at the bottom of the “painting” column, which is the total: 22. To find the number of girls responding, look at the row for girls, and the column for total: 40. To find the total number of campers who replied to the survey, you can either add the column on the right, which is the number of boys + girls, or you can add the bottom row, which is votes for Legos + painting. Both totals = 85.

Vocabulary used with two-way frequency tables legos painting total boys 37 8 45 girls 26 14 40 63 22 85 The data on the edges of the table are called the marginal frequencies. Point out to the students that “Marginal Frequencies” appear on the margins (edges) of the table and indicate totals for the rows and columns, and the grand total. “Joint Frequencies” appear at the intersection of a row and column, and tell 2 data facts, i.e. the number of girls who prefer Legos. Direct the students to: Look at the totals for each row. Did the survey include about the same number of boys and girls? (Answer: yes, 45 and 40 are close values.) Look at the totals for each column. Which activity is preferred most? Is this a strong preference? (Ans: the legos are very strongly preferred by a wide margin.) The data in the middle of the table are called the joint frequencies.

Making a two-way frequency table The Math Club is going to sell candy as a fund raiser. They surveyed 80 students about their favorite candy. The results are shown in the two-way frequency table. Fill in the missing information: lollipop Peanut butter cups total boys 19 girls 43 35 This is the beginning of the guided practice. The teacher can introduce the example on this slide so that all students, regardless of reading level, can understand the premise of the example and the directions. The teacher can hand out the document: “Lollipops, anyone?” after this slide has been introduced. The students can be directed to work individually or with a partner. They may use calculators. Students should be encouraged to think about what strategies and methods they are using, and to find ways to check their results. The teacher may refer back to the magic squares at the beginning of the lesson if students are unsure how to proceed. After the class has completed the table, it is important to discuss how they did so. Important to note and emphasize: Important information was given in the introduction: the total number of people was 80. There may be more than one way to complete the table; students can explain or demonstrate their steps and reasoning and then compare with one another. Regardless of the steps a student used to fill in the empty cells, the table has only one correct final solution when it is completely filled out. Ask the students how they can check their work. They should check each row total and each column total.

Lollipops, anyone??? 19 18 37 26 17 43 45 35 80 lollipop Peanut butter cups total boys 19 18 37 girls 26 17 43 45 35 80 After students have completed their tables, and compared their results and techniques, the answer slide can be displayed for verification and final checking.