Best Friends Forever: Two-Way Tables and Venn Diagrams
Math Medic
Two-Way Table
Two-way tables provide a clear and concise method for organizing probabilities for two events. I have discovered that students develop deeper understanding of two-way tables if I have them create one from a scenario, rather than putting a completed one in front of them and then asking a bunch of questions.
Scenario: 80% of students at East Kentwood High School have an Instagram account. 60% have a Twitter account and 45% have both Instagram and Twitter. Fill in the table:
Most often students will correctly start with the 45% who have both Instagram and Twitter. The struggle is where to put the 80%. With some thinking they realize that the 80% includes both Instagrammers with Twitter and Instagrammers without Twitter. The 80% belongs in the Total column for Instagram. This realization will make them less likely to make the 4-hour detention mistake in the Venn Diagram (see below).
Venn DiagramUsing the same Instagram, Twitter context from above, I have students create a Venn Diagram. I am sure to point out that the Venn Diagram has 4 different spaces, which correspond to the 4 values in the two-way table (not including the totals).
Space #1 ā The left pacman. Instagram and no Twitter. 35%
Space #2 ā The center football. Instagram and Twitter. 45%.
Space #3 ā The right pacman. No Instagram and Twitter. 15%.
Space #4 ā The outside. No Instagram and no Twitter. 5%.
4-hour detention
The most common student error on the Venn Diagram is to put 80% in the left pacman and 60% in the right pacman. An immediate red flag for this decision is that now the sum of the probabilities is well over 100%. Hopefully by having to construct the two-way table first, they will be less likely to make this mistake. I model this mistake in front of the class several times, each time waiting for students to call me out. If students make this mistake on a quiz or a test, they are asked to serve a 4-hour detention with me after school.
About the Author
Matt McBurney
Matt is a teacher at West Allegheny High School in Pittsburgh, PA, who’s certified to teach math, physics, and computer science. He works alongside the Math Medic team to field community questions, create new resources, and take on any other tasks. Matt was the 2018 recipient of the Pennsylvania Council of Mathematics New Teacher Award and has two published mathematics papers. Matt loves that he gets to help students from around the nation and feels truly passionate about teaching in the EFFL style, always asking the question, “How can we make this fun?” A lifelong learner and accomplished limbo-er, Matt spends his free time having fun and learning new things ā which are really the same thing to him.