Sets, quantifiers, and Venn Diagrams
Helpful resources:
Trefor Bazett, “Defining Numbers and Functions Using Set Theory”: https://www.youtube.com/watch?v=fDncO6zkosM
Optional materials:
For young kids, having physical circles representing categories in Venn Diagrams (e.g. using yarn or hoops) could be really helpful for practice with understanding overlapping categories
——————————————————————————————————————
Tying this into the previous week’s content, sets are really useful for describing the vertices and edges that make up a graph! Sets don’t have order, meaning they are essentially “bags” of elements that are in some way related. Since the vertices and edges in graphs are typically unordered, it makes more sense to use sets to group them together rather than a structure like a list. Additionally, sets include each object only once, so the set {3, cat, cat} is exactly equivalent to the set {3, cat} or {cat, cat, cat, 3} in the usual definition of sets.
Set theory uses lots of notation that may seem like a lot at first but greatly helps in performing set operations later on. It is worth spending a chunk of the lesson simply practicing with these notations, especially if the kids get to make up the sets and thus be more invested in them. Suppose we have sets A and B. The main notations that are useful include:
Element of A
Subset of A
Size/cardinality of a set
Power set of A (What’s the set of all the possible subsets we can make out of the elements in A? Don’t forget the subset that includes all of the elements, and the empty set that includes none! Also, a great tangent from here is to figure out what the size of the power set will be for different sizes of the original set A. Kids can be pretty quick to figure out that each time one element is added to the set, the size of the power set gets doubled. The sizes of power sets are powers of 2. There is an intuitive explanation of this: each element gets to decide whether to “go” or “not go” into the subset. That’s 2 choices per element, for each of n elements, so there are 2^n total results!)
Discussion of infinite sets (note that this can very easily go down a long tangent about infinities of different sizes, countable vs uncountable infinities, etc. It’s an interesting discussion for sure, and allows for introducing symbols such as R for the reals and Z for the integers.)
Simple introduction to set builder notation (where the bar represents “such that…”)
Intersections and unions —> this is most intuitively done using Venn Diagrams
As an overall note, I turn to several common techniques to improve kids’ engagement during these lessons. One is allowing students to choose the direction of the material (did they find power sets particularly interesting, for example? Is there more that can be said branching off of them? Can we save the rest for a separate lesson, or is there a need to fit it all into one?). Another is, on occassion, holding “kids be the teachers” sessions where it’s up to them to choose several topics that they were most fond of over the past several lessons, and teach those topics to the instructor as well as their peers. It can be quite short, just a few minutes per student, but still leads to a more engaged class and is an opportunity for kids to practice expressing their understanding.
