Embedded Files
Lesson Video
Lesson Video
Focus Standards
Focus Standards
Learning Focus
Learning Focus
Additional Resources
Additional Resources
A Develop Understanding Task
Develops a strategy for locating the center of rotation between an image and pre-image. Leads to observation that the center of rotation lies on the perpendicular bisectors of the segments joining image and pre-image points.
NC.M2.G-CO.9
Find the center of rotation for a given pair of pre-image/image figures.
How can you locate the center of a given rotation using only the figures given as the pre-image and final image?
To use a compass to draw a circle, you start with the center point and radius, and then construct the circle. But what if the circle has already been drawn? How do you determine the location of its center and its radius?
A Solidify Understanding Task
Provides two different transformation strategies that map one circle onto another, proving that all circles are similar.
NC.M2.G-SRT.2.a
Prove that all circles are similar.
It seems intuitively obvious that circles are similar, but how do we prove it?
What key features of circles are revealed by the fact that all circles are similar?
Video: All circles are similar
A Solidify Understanding Task
Examines relationships between central angles, inscribed angles, circumscribed angles, and their arcs by constructing inscribed and circumscribed circles for triangles.
Examine the relationship between inscribed angles and the intercepted arc.
We have seen that circles can be inscribed in triangles so that the circle touches all three sides, and that triangles can be inscribed in circles, so that the circle passes through all three vertices. What other types of polygons can be inscribed in circles, or can contain inscribed circles that touch all of their sides?
What relationships exist between central angles and inscribed angles that intercept the same arc?
A Practice Understanding Task
Provides practice of all circle relationships and theorems developed in the previous lessons.
Apply circle geometry theorems in various contexts.
How do I examine a complex geometric figure for the structures and features that will support my reasoning?
How do I use the properties and theorems of circle geometry to model real-world contexts?
A Solidify Understanding Task
Provides real-world context in which students use proportional reasoning to calculate arc lengths and area of sectors.
Find formulas for arc length and area of a sector of a circle.
We all like pie, but how can you figure out the portion of the total pie you are eating when you take the biggest slice?
A Develop Understanding Task
Uses formulas from the previous lesson to find the ratio of arc length to radius and develop radians as a way of measuring angles.
Develop a new unit for measuring angles.
We are very familiar with measuring angles in degrees. But why are there 360 degrees in a circle? Why not 100 degrees, or 1,000?
Are there other ways of measuring angles that might be more natural to the features of the circle and maybe wouldn’t require additional tools like a protractor?
A Solidify Understanding Task
Establishes and practices a method for converting between degree measure and radian measure of an angle.
NC.M3.G-C.5
Measure angles in radians.
How do we convert between degree and radian measurement?
How can I visualize the size of an angle when its measure is given in radians instead of degrees?
A Develop Understanding Task
Derives the equation of a circle using right triangles and the Pythagorean theorem.
NC.M3.G-GPE.1
Find the equation of a circle.
How do circles and right triangles relate?
Review: Pythagorean Theorem
A Solidify Understanding Task
Uses completing the square to find the center and radius of a circle given by an equation.
NC.M3.G-GPE.1
Write and graph the equation of a circle.
Find the center and radius of a circle in general form.
Do translations work on circles like they do on functions?
How are different forms of the equation of a circle related?
A Practice Understanding Task
Practices writing the equation of a circle given various information to find the radius and center.
Apply understanding of circles and their equations to new situations.
How can we use algebra to find relationships within and between circles?
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