Sets
A set is simply a group of objects — numbers, people, or toys. We can talk about the set of even numbers, the set of people who are at a given house right now, or the set of blocks in a box. Some of the blocks might look exactly the same, but they still belong to the same set.
The inventor of set theory, Georg Cantor, described it beautifully:
“By a set we mean any collection of objects gathered together into a whole.”
Why Does It Matter for Kids?
Jean Piaget showed that before kids fully understand numbers, they first develop the ability to order objects and to grasp relationships such as part–whole and class inclusion.
Modern standards echo this idea. The Principles and Standards for School Mathematics (NCTM, Pre-K–2) highlight the importance of sorting and classifying objects by their attributes. The National Research Council report also emphasizes that comparing, grouping, and organizing data form the foundation of early mathematical thinking.
The Learning Trajectories research program describes a step-by-step path of development in classification and data analysis — from a Simple Sorter to a Multiple Attribute Classifier and then to a Hierarchical Classifier. The Building Blocks program, which includes the line Data Analysis and Classification, has shown significant gains in key areas of early math compared with control groups.
How Do We Teach?
A set can appear in many ways — as a real box, a picture, a list of objects, or a shared feature that ties items together. Kids start forming sets naturally, long before they learn the word itself. “This is my family,” they might say, or “I’ll put all the red cars here,” or “These are the animals that can fly.” Our role is to help them notice what connects those things and put it into words.
In a game where kids catch fish with certain features, they cast a net and collect only those that match a given rule. At first, the rule is simple: yellow fish, spotted fish, or fish with a symmetrical tail. Soon the challenges become more interesting — kids need to choose fish whose tails are not of a certain kind, or those that meet several properties at once, like red or yellow fish with scales and a specific tail shape.
Then comes the idea of identical sets. If you put a few toys in a transparent bag and shake it, the toys move, but the set stays the same. The contents haven’t changed, even if the order has. Kids explore this idea through playful challenges: which of several identical bags lost a toy? What needs to be added to make all the given bags the same? Which bags hide matching sets? And when there are several bags to compare, the questions get even more interesting: which toy appears in every bag, and which one isn’t in any of them?
In the timed challenge where kids work with sets and subsets, they look at different groups of shapes and decide which set each one belongs to. Then they identify which larger set includes all of them as a subset.
In some tasks, we use Venn diagrams to show how different properties relate to each other. Each circle represents one property — for example, color or shape. On such a diagram, kids can see how these properties overlap. If one area contains blue shapes and the other contains triangles, the overlapping part shows the blue triangles — the figures that have both properties and belong to both groups.
First Steps
Sets and Attributes
A set is simply a collection of things. You can describe it by listing what it includes: two eggs, a toast, and a glass of orange juice — a cheerful breakfast set kids might make for Mother’s Day.
Sets and subsets
Same and different sets of objects
In everyday life, kids constantly create sets without noticing. Picking out only blocks from a toy box means forming a subset of blocks within the larger set of toys.
Properties can be simple or complex. Among all birds, kids might point to those in red hats, or choose all fruits except bananas.
In the Logical Operators game, kids learn to work with rules and properties step by step. At first, the tasks are simple:: catch the yellow fish or those with a particular tail shape. Step by step, the tasks become trickier. Kids might be asked to catch all the green fish that are not spotted and have symmetrical tails. Each new rule adds a bit more logic and curiosity to the challenge.
Classify and sort
Choose all objects that fit
Classify and sort
Choose all objects that fit
Logical operators
Classify objects by color
Logical operators
Logical operators AND and NOT
Deep Understanding
Identical Sets
Two sets are identical if they contain the same elements, no matter how those elements are arranged. A dinosaur and a party popper may sit in different spots inside a bag, but the gift is still the same.
Sets and subsets
Equal sets
In some tasks, kids search for a missing piece: every child was supposed to get the same gift bag, but one toy disappeared — which one? In others, they figure out how many identical building kits can be made from the blocks that remain.
Some challenges require imagination and focus: which sets would become identical if all the pinecones were removed? Or what tool needs to be added to make all the sets identical?
Equal sets
Making sets
Sets and subsets
Equal sets
Sets and subsets
Equal sets
Sets and subsets
Equal sets
Confident Mastery
Multiple Sets
When there are several bags of toys, new questions appear:
Which gift is in every gift bag?
Which gift is not in either bag?
Which gift is just in one bag?
Sets and subsets
Equal sets
When sets are grouped by features, kids decide which groups a particular object belongs to. A purple round button, for instance, fits both the set of purple buttons and the set of round ones.
On a Venn diagram, these relationships become visible. Each circle stands for a property, and where the circles overlap, we find the objects that share both. If one circle shows pink shapes and the other triangles, the intersection holds pink triangles.
In the Sets & Subsets game, a timed race where kids work with sets, they first determine which group each figure belongs to, and then find the larger set that includes them all — discovering how subsets fit within bigger sets.
Sets and subsets
Properties and attributes
Sets and subsets
Sort objects using Venn diagrams
Sets & subsets
Find objects by shape and color
Sets and subsets
Equal sets
Big Ideas
Set theory was created to bring order to the foundations of mathematics, and it succeeded. The language of sets now runs through almost every branch of math. Later in school, students use symbols for membership, intersection, and union as naturally as they use numbers.
In computer graphics, a “selected area” is really a set of pixels. Masks, alpha channels, and fill tools all work by performing operations on sets of points.
Understanding how OR expands a set, while AND narrows it, is a key part of logical and linguistic thinking. In probability, this connects to the idea of an event as a subset of a sample space and to formulas for conditional probability.
The same reasoning appears in everyday life. When we apply filters while shopping online — “sneakers or sandals, but not red” — we’re performing set operations. Marketing teams do the same: “Customers who bought toys and books, but not gadgets.”
Doctors use similar logic when analyzing data during an epidemic: “Who became ill and had contact, but wasn’t vaccinated?” In ecology, scientists ask, “Which animals live both in the forest and near the water?” or “Which ones are not predators?”
Even in computer systems, administrators rely on these ideas when assigning roles and access rights: an accountant should have access to financial data, but not to employee records or trade secrets.
Venn diagrams also lead to fascinating puzzles connected with the inclusion–exclusion principle, which helps calculate how large combined sets really are when they overlap.
