The significant mathematical activity represented in this section is the reasoning involved in understanding mathematical objects in terms of their properties.

In any branch of mathematics, the mathematical object is the "thing" under consideration. The mathematical object is defined clearly and unambiguously, and it is straightforward to verify whether something is one of those objects or not.
The characteristics that an object has are its properties. Mathmatical objects can be classified and sub-classified according to the properties that they do and don't possess, and this is how much knowledge about them is gained.

The foundational four standards of the NCTM (problem solving, communcation, reasoning, and making mathematical connections) are carried out by mathematicians worldwide as they study mathematical objects in terms of their properties. When students begin to see how the study of objects and their properties is common across all branches of mathematics, and when they develop tools for doing this themselves, they can become confident that they can do any kind of mathematics, too. Seeing mathematics in this way can help them to understand how the mathematical results that they learn about may have been discovered, and to go ahead and discover things on their own rather than waiting to be "taught".

Why Graphs?

Graphs are facinating mathematical objects in the classroom because it is easy to discover things about them, yet you are unlikely to ever run out of things to discover. Because they are made of lines with dots on each end, children can draw graphs long before they can draw numbers, and they can discover many things about graphs without being able to count.

The relationships between the parts of a graph are a treasure-trove for seeking patterns and relationships and for spatial reasoning.

Why Games?

There is a direct relationship between the success or failure of the strategies that one uses to play a game on a graph and the properties that the graph has. Games become easy or hard as the graphs change; certain rules become essential or irrelevant when you try the same game on different graphs. Articulating this is an essential part of communicating about the problem solving process.

When games are associated with stories, the stories show how an abstract mathematical object can be connected to a situation in everyday life in the real world.