Lesson 4

The purpose of this warm-up is to elicit the idea that two-way tables can be used to determine relative frequencies, which will be useful when students use relative frequencies as estimates of probabilities in a later activity. While students may notice and wonder many things about these images, the relative frequency is the important discussion point. This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that by dividing the values in the cells by a total allows them to calculate the relative frequency.

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the table for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification or point out contradicting information. If relative frequency does not come up during the conversation, ask students to discuss this idea.

Here are some questions for discussion.

• “What is the relative frequency of dogs?” (\(\frac{104}{200}\))
• “How do you change this two-way table into a relative frequency table?” (You divide each of the values by 200)
• “If a person surveyed is chosen at random, what is the probability that their home environment is urban and their pet preference is cat?” (\(\frac{54}{200}\) or 27%)

The mathematical purpose of this activity is to construct and interpret two-way frequency tables of data, and to use the table as a sample space to estimate probabilities. Monitor the groups to determine when to tell the students to stop rolling. When each group has rolled at least 50 times tell the students to stop rolling.

The purpose of this discussion is to make the connection between a two-way table, relative frequencies, and estimated probabilities. Here are some questions for discussion.

• “How would you draw a similar table showing the estimated probabilities for each cell in the table?” (Divide each cell by the total number of rolls.)
• “Which roll had the highest estimated probability?” (My group rolled 60 times and 3,4 showed up 7 times.)
• “Which roll had the lowest estimated probability?” (The lowest estimated probability for my groups was 2,2 and 6,1. Each of these only appeared once. The estimated probability for each was \(\frac{1}{70}\) since my group rolled 70 times.)
• “How are relative frequencies and estimated probability related?” (The relative frequency for the total number rolls can be used to estimate the probability.)

The mathematical purpose of this activity is to use a two-way table to estimate probabilities for some events. Monitor for students who struggle with the questions in which a person is selected from only a subgroup of the population.

Help students to imagine someone going to the office in Copenhagen and selecting someone at random while there. Ask them, “How many people did they have to choose from? How many of those people rode a bike to work that day?”

Some students may struggle finding the probability for the questions in which a person is selected from only a subgroup of the population. Prompt students to look at the table of the students in a school district and ask, "How many students are in grade 9?" (4,565) and then ask, "If a grade 9 student is selected at random, what is the probability that the student takes a car to school?" (\(\frac{1,141}{4,565}\)) . Emphasize that the denominator in the probability is the total number of students in grade 9 (4,565) and the numerator is the number of students in grade 9 who take a car to school (1,141).

The goal of this discussion is to make sure students understand how to estimate probabilities using information in a two-way table.

Here are some questions for discussion.

• “How do you use a two-way table to estimate probabilities?” (You look at the cell determined by the question being asked and divide it by the appropriate total).
• “How do you know when to use the total for the whole table, or a total for a row or column to estimate the probability?” (You really have to look at how the question is worded. For example, for the second table, if it is a high school student selected at random then you would use the total for the whole table, but if is a student from one grade being selected at random the you would just use the total for that grade.)
• “Did any of the probability questions stand out to you as more difficult? Explain your reasoning.” (I found that the questions that used row or column totals were more difficult because I could not just use the total for the table by default. I really had to think about what the question was asking for.)