Subsets

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Objective

SWBAT understand the relationship between a set and a subset.

Big Idea

An intuitive idea of a subset is accessible to all students.

Start Up

15 minutes

In this lesson we focus on the meaning of a subset and help students develop an elementary sense of set closure. I start the lesson by showing a brief presentation of the symbols associated with subsets. I show big, clear slides to help them focus on how cool these symbols are. Mathematics is a beautiful language and we should embrace it as such. I used to be nervous about showing the symbols, but now realize that we should embrace the cool way in which mathematics is written. In fact, the symbols are part of the hook. 

I keep a blank slide displayed after I show the symbols to create some type of simple introduction, something like A = (1,2,3) and B is a subset of A, what possible sets could represent B? Then we list out the seven possibilities:

 

(1,2,3)

(1,2)

(2,3)

(1,3)

(1)

(2)

(3)

 

Some students also think we should list out (3,2,1) and other permutations, but the class discussion around this helps students realize that the set notation isn’t focusing on order, but what is inside the brackets. With set notation, the order doesn’t count, but the content does.

I use this opportunity to name the items in a set as elements and try to use that language throughout the lesson.

After giving one or two quick number examples, I introduce the idea that sets and subsets can represent anything. I like to give a Donut Store example, where jelly donuts are a subset of the local bakery.

In the context of school they are each a subset of the class, grade, school, etc. Also, each partnership, group, etc is also a subset of the class, grade, school and so forth.

We then move to number classification. I like to introduce the symbols for integers, real numbers, irrational and rational numbers and then ask students to draw the universe of numbers. The sketch is tough at first, but we work together to list the correct subset order of the number sets:

  • Natural numbers are subset of Whole numbers , whole numbers are a subset of rational numbers and rational numbers are a subset of real numbers.
  • Irrational numbers are mutually exclusive of rational numbers but are also a subset of real numbers.

Students always ask if there are ways to expand this number universe. I can’t help but bring up the square root of a negative and encourage them to think how they might invent their own numbers with their own imagined properties.

 

Partner Work

30 minutes

I ask students to work on the ten problems in this worksheet subset worksheet.pptx and circulate to gather common misconceptions and interesting ideas about problems. 

Summary

15 minutes

Problem_2 focuses on the idea of breaking a subset by adding an element not from the given universe. The idea of breaking or fixing (as shown in question 3) can be a valuable way to build intuition around the concept of subsets. It challenges students focus in on the importance of each element and carefully weight their options in the problem solving process of defining a subset.

Problem_6 is a fun question because D only has one element. D = {0}

Problems 7-10 introduce the idea of closure. Students have fun finding way to use operations that reach beyond the capacity of a set or universe. For example, they quickly realize that we can subtract a larger quantity from a smaller one to create a negative number. Of course, the original set only includes positive values and is those broken (not closed) under subtraction.