4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

It's the most common dilemma any amateur baker faces: now that we have chocolate cake batter mixed and ready to go, do we make individual cupcakes or one big cake for the room to share? Of course, by the time you've gotten to this step, you've already solved the problem of listing ingredients, doubling the recipe for the ginormous party, and being careful not to mistake the teaspoon of salt for a tablespoon. (Whew!)

When it comes to statistics, we need a way to list the ingredients of our sample population. Two-way frequency tables give us a clean way to list statistics and solve problems that will feed the masses with the knowledge they desire.

When constructing two-way frequency tables, we are just reporting raw data. Students probably won't find it difficult. Just fill in the numbers as they are reported. Construct a matrix where one set of comparison statistics, say yellow, chocolate, strawberry, and angel food, are on one axis, and another set, cupcake and three-tiered, are on the other axis. It's useful to include total columns and rows as well, where they are just sums of all the outcomes of each comparison statistic. If they are not there, teach the students to do the arithmetic themselves before they use the table.

Sample Space of the Fourth Grade's Preferences for Cake

What we've just done is construct another version of the sample space. Like the Venn diagram, this is just another tool to represent the data in a way we can make use of it. It's the cupcake of statistics, where the Venn diagram is the three-tiered chocolate volcano cake. Students should understand that they are both useful in their own ways. 

With our recipe we need to bake up some statistics. Ask the students some simple questions about your example table. Here, we could ask, "What is the probability that a fourth grader prefers cupcakes over three-tiered cakes?" or, "What is the probability that a student prefers chocolate cupcakes from over anything else?" Mixing up the questions for students to see the different ways the table can be used will help the students see the utility of the tables. 

The students have constructed two-way tables, and they know how to calculate probabilities from the tables in every way possible. The last step is to apply the formula they already know for conditional probability. From this formula, they can easily determine the independence or dependence of two events, for instance, whether chocolate cake preference affects cupcake preference. 

One of the most difficult things about mathematics is how to determine when a formula is applicable and when it is not. Students struggle all of the time in mathematically-based sciences and in mathematics with this concept. They have determined probabilities, so all they need to do is to apply the formula to the situation. Reinforcing this concept with a Venn diagram for your chosen example could prove helpful. 

Students should be able to recognize that, regardless of visualization, the formula is applicable to probabilities. Like baking, it doesn't matter what tools they use to mix the batter. As long as they used the right ingredients, they should still end up with a delicious treat.