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In this task, you must solve a problem about circles inscribed in squares.

This incomplete Core Resource supports students in making connections between the Pythagorean theorem and the equation of circles.

Connect visuals to arc length formulas involving the size of an angle.

The slider allows the interior angles of the quadrilateral to change.  the 4 interior angles of the quadrilateral are also organized with the same vertex (like a circle).  As they interior angles of the quadrilateral change, the same 4 angles change in the circular format.

A Classroom Challenge (aka formative assessment lesson) is a classroom-ready lesson that supports formative assessment. The lesson’s approach first allows students to demonstrate their prior understandings and abilities in employing the mathematical practices, and then involves students in resolving their own difficulties and misconceptions through structured discussion.

This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle. It also shows that there cannot be more than one circumcenter. 

This is a fun and inspiring description of someone's first conversation about the use of compasses - I think it could be a nice conversation at other early points in students exposure to compasses as well.
A note on math circles: Math Circles are a particular talk protocol for facilitating a discovery based conversation with a group of learners. Typically, there's a starting question and the facilitator responds to each student's contribution to the conversation with another question. It is common in math circles that the end result of the conversation is both more sophisticated and powerful than what happens in a typical math classroom, however it is hard to facilitate and requires that you be willing to go with what your students are thinking about. It also helps to have a great deal of mathematical knowledge, not to mention a very engaged group of learners.

3 Acts activity about the area of a sector.

This task provides a context for connecting an angle in radians to the arc length intercepted by the angle. 

This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle with one side of the triangle a diameter: the fact that these triangles are always right triangles is often referred to as Thales' theorem. It does not have a lot of formal prerequisites, just the knowledge that the sum of the three angles in a triangle is 180 degrees.

The result here complements the fact, presented in the task ''Right triangles inscribed in circles I,'' that any triangle inscribed in a circle with one side being a diameter of the circle is a right triangle. A second common proof of this result rotates the triangle by 180 degrees about M and then shows that the quadrilateral, obtained by taking the union of these two triangles, is a rectangle. 

The goal of this task is to work on showing that all circles are similar using these two different methods, the first visual and the second algebraic.