Shaun Errichiello Middle School 255 SALK SCHOOL OF SCIENCE, NEW YORK, NY
8th Grade Math : Unit #9 - Sets, Subsets and the Universe : Lesson #5

Venn Diagrams

Objective: SWBAT construct and analyze a Venn Diagram with up to three sets.
Standards: HSS-ID.A.2 HSS-CP.A.1 MP5
Subject(s): Math
60 minutes
1 Start Up - 15 minutes

As a final analysis of sets, we look at different examples of a Venn Diagrams dealing with three sets. The intro problem included in Venn Diagrams asks students to identify one intersection. I will expand the question by asking for other intersections, unions, complements, etc. My goal is to extract as many types of set questions from the one example as possible

Rethinking the Start Up
Lesson Planning
2 Partner Work - 30 minutes

As we begin this section of the lesson, I coach the students on how to create their own two-set and three-set Venn diagrams. I ask students to pick numbers for sets and then populate the diagrams with the numbers.

For example, a student might make the following sets:

set A = {Prime Numbers}

set B = {Even Numbers}

set C = {x|-2<x<3, where x is an integer}

Students could then write the numbers that are at the intersection of all three sets first (this helps in the design and layout of a 3 set Venn Diagram). In this case that would include only one number: 2. Then, they could continue from there.

Or they might go for a more discrete set, where

A = {1,2,3,4} B = {-9,-7,1}  C = {0.1}

Here the only element in all three sets is 1. If they create  three sets with nothing in common, I use it as an opportunity to introduce an empty set (if the intersections are empty) and to show when circle don’t always intersect. I also sometimes offer this as a challenge, where students might design two circles that intersect and a third that doesn’t touch at all (breaking the misconception that all circles must intersect each other in Venn Diagrams). 

3 Summary - 15 minutes

Today's Summary is always fun and challenging. Students share the Venn diagrams they created and then we analyze them as a class. 

One interesting challenge is to make sure students think about the universe in which their sets live. When they set up there sets and analyze all the intersections and unions formed, it is important to also ask them about the universe in which those numbers live. 

For example, if they created A = (1,2,3), B = (2,3,4) and C = (-1,0,1), they could live in the universe of Integers. If we put an interval on that universe in set builder notation (tying back to earlier lessons) we could quantify the amount of numbers not in any set, thus asking them essentially about the complement of the union of sets A, B and C. If we defined our variable x as {x|-5<x<6, where x is an integer}, we could say that the complement of the union of A, B and C has the numbers -4, -3, -2, 5. 

We could then build off of this question and ask, "how is this different from the complement of the intersection of A,B, and C? the intersection of A,B? The union of A,B?" These types of questions will really help the class learn from each presented example.