Students understand that proof involves showing that something must be (or is always) true. It is different from showing a few examples that seem to verify your claim, and often involves reasoning from known facts or definitions.
Students understand that similar figures have proportional relationships, and they use these relationships to reason about and solve a variety of problems (solving for unknown parts, derive trig ratios, scaled 3D figures, etc.).
Students understand the connections between multiple representations of quadratic functions (table, graph, rule, context) and can use these representations to reason about and solve a variety of real-world problems.
Students understand that probability is a way to quantify chance and predict what might happen in the long run. They build and use a variety of representations (area model, tree diagram, two-way table) to reason about a variety of probabilistic situations.
Students use definitions and known facts to reason about various features of shapes in 2D (in the plane and otherwise) and 3D (angle, area, volume, etc).