# Complements

## Objective

SWBAT identify the complement of a set

#### Big Idea

Complements of a set represent everything that is not in the set.

## Start Up

15 minutes

After we have introduced the notation for the intersection, union, set builder notation and subset notation, students are ready to work with the complement of a set. I could have introduced the complement earlier, but many of the problems we deal with attach previous concepts to the complement questions.

I start with the symbols associated with the complement of a set. As I mentioned in previous lessons, I think this is a powerful way of hooking students into the world of set notation. They will begin to see the value of mathematics as a language as well as a problem solving tool.

After we establish a basic working definition of complement, we ask students to write their own version of the definition. Students might write something like, “A~ represents all the stuff not in the set A.” Then we continue to develop the definition of a complement using a Venn Diagram with two sets and another with 4. I leave the diagrams blank and fill them in with numbers as we teach. I highlight the significance of certain overlapping regions and stress that the complement of one set is simply not another because many sets live in a universe of numbers beyond their own.  For example, A might equal {1,2,3} and B might equal {3,4,5}. The complement of A might just by 4 and 5, but what if A and B were subsets of an entire number universe including all natural numbers. Then the complement of A would {4,5,6,7,8,…}. By thinking carefully about the universe of numbers in which each set lives changes a problem with a small answer to one with an infinite answer.

We then support this with a Venn diagram representing two sets and then another representing 3.

## Partner Work

30 minutes

The variety of questions from this lesson combine work from the previous lessons. Students navigate through questions using set notation, intersection etc.

I partner students up and have them work on the complement problems from our Set Builder Template: Set Builder Template

My goal is to gather misconceptions for the summary and scoop up interesting algorithms and strategies. However, the way I circulate so fast around the room is to quickly engage students on the concept of a complement by drawing several types of Venn Diagrams and asking them questions to follow up and assess their understanding of the start up. For example, I might draw two intersecting circles, label one as A and the other as B and then make a comment, "Sue told me that the complement of A is everything in B. What do you think?" This question is meant to help students realize that the intersection of A and B is not included in the complement. To follow up, I might ask them to create a scenario where Sue is correct. "Could you draw to circles so that the complement of A is everything in B?" Here the goal would be to draw two mutually exclusive circles. "Is it possible to draw a diagram so that the complement of A is nothing in B?" Here their only option is to make B a subset of A or exactly equal to A."

These conversations and their responses really help me think about the class discussion at the end.

## Summary

15 minutes

I review any question that students had unique approaches or questions around, but I like make sure I cover Problem_15. This question is confusing because students find the complement to a set and get {1,2,5,7,9,12}. However the question asks students to list their answer in set notation. Students attempt to use set builder notation, but that is quite difficult as there is no easy way to define these numbers. It is important to help students realize that they can just list the numbers in a pair of brackets in these types of situations.