Welcome back to Thursday Tool School! I apologize for the delay this week. I had a minor surgery on Wednesday that turned out to be a bit more involved than my doctor thought it might be, so I have been off my feet for the last two days. I’m feeling much better today and am ready to get back on track, so once again, welcome back.

For the month of September, I will be highlighting graphic organizers for Thursday Tool School. We tend to use graphic organizers in reading and language arts but not much in math. Graphic organizers are great visuals for students and we can use them to help students make sense of mathematics and infuse a little critical thinking too. If you’ve been following this blog for a while, you know that I really enjoy challenging students with critical thinking and problem solving activities. With that in mind, today’s activity highlights how Venn diagrams can be used to challenge your students.

Most students are used to using Venn diagrams to compare and contract stories, characters, or even science concepts, so they generally understand that the outside sections of the circles are used to show the contrasting elements and the over-lapping circles are used to show similarities. Today’s math challenge builds on this idea. It requires students to use the characteristics of a Venn diagram to complete a math problem. See the task below.

At the time that I posted the challenge, I didn’t anticipate how truly challenging the task would be for some students. So, while I have decided that it may not be accessible to many students as a Solve It! Friday challenge, it is an excellent problem solving and critical thinking task.

Considering the task above, there are several components that must be attended to before students begin to complete the task. First, students must understand the structure of the Venn diagram– the non-overlapping parts of the circle indicate having brothers or sisters and the overlapping part indicates having a mix of brothers and sisters. Second, students must determine how many students were surveyed. In this case, the task states that five students were surveyed. Third, students must take note of the number of students who reported having brothers and the number of students who reported having sisters.

The task then becomes to determine how many students to place in the non-overlapping parts and the overlapping part so that there are five students represented but with four students having brothers and three students having sisters. Once students have had an opportunity to work through the challenge, they will discover that two students have only brothers, one student has only sisters, and two students have both brothers and sisters.

After students have determined a possible solution, have them double-check their answer by checking to make sure that there are four students who have brothers (either just brothers or a mix of brothers and sisters) and three students who have sisters (either just sisters or a mix).

To make the task more accessible to a wider array of students, using small objects to help find the solution will support the understanding of some of your more tactile or lower-level students. The action of moving the objects may also help students who are still developing the understanding of how Venn Diagrams work. Try using centimeter cubes in two colors. In the example above, students could use four blue cubes and three pink cubes. Students would then separate the cubes into the parts of the Venn diagram, noting that when one blue cube is placed in the overlapping section, a pink cube must also be placed there. The pair then represents one student. Shuffling the cubes around until only five cubes and/or pairs are represented, generates a correct solution for the student.

Initially, students will assume that all students made a selection– there are no students outside of the diagram. However, for an extra challenge, ask students to re-solve the problem assuming a specific number of students did not make a selection– they choose neither option. For example, what if there were seven students surveyed but some of them did not have any siblings.

Perplexed? Imagine the power of using Venn diagrams to challenge students in this way on a regular basis in class. You may even want to try collecting some class data and presenting it as a two-choice tally chart. Then have students create a Venn diagram to represent it. Cool, right?

I’ve included two more challenge below. Give them a try!

**Sound Off!** How might you use Venn diagrams to challenge your students?

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