Prove that all circles are similar.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Circles
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3 - Prove (that all circles are similar)
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3 - I can prove that all circles are similar.
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A. All circles are similar.
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A.1 Why are all circles similar?
A.2 How can it be proven that all circles are similar?
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Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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2 - Identify (relationships among central, inscribed, and circumscribed angles)
2 - Identify (the right angle relationship of inscribed angles on a diameter)
2 - Identify (the perpendicular relationship between a tangent and the radius of a circle at the point of tangency)
2 - Describe (relationships among central, inscribed, and circumscribed angles)
2 - Describe (the right angle relationship of inscribed angles on a diameter)
2 - Describe (the perpendicular relationship between a tangent and the radius of a circle at the point of tangency)
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2 - I can identify relationships among central, inscribed, and circumscribed angles.
2 - I can identify the right angle relationship of an inscribed angle on a diameter.
2 - I can identify the perpendicular relationship between a tangent and the radius of a circle at the point of tangency.
2 - I can describe relationships among central, inscribed, and circumscribed angles.
2 - I can describe the right angle relationship of an inscribed angle on a diameter.
2 - I can describe the perpendicular relationship between a tangent and the radius of a circle at the point of tangency.
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A. In a circle there exist specific relationships among angles, radii, and chords.
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A.1 How is the measure of a central angle found?
A.2 How is the measure of an inscribed angle found?
A.3 How is the measure of a circumscribed angle found?
A.4 How are the measures of central angles, inscribed angles, and circumscribed angles related?
A.5 What is the measure of an inscribed angle on a diameter?
A.6 What is the spatial relationship between the radius of a circle to a tangent where the radius intersects the circle?
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Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Quadrilateral
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3 - Construct (inscribed circle of a triangle)
3 - Construct (circumscribed circle of a triangle)
3 - Prove (properties of a quadrilateral inscribed in a circle)
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3 - I can construct a circle inscribed in a triangle.
3 - I can construct a circle circum-scribed about a triangle.
3 - I can prove the properties of a quadrilateral inscribed in a circle.
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A. Constructions can be performed to create inscribed or circumscribed circles in triangles.
B. A quadrilateral inscribed in a circle has certain properties which can be proven.
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A.1 How can inscribed or circumscribed circles be constructed?
B.1 What are the properties of a quadrilateral inscribed in a circle?
B.2 How can the properties of a quadrilateral inscribed in a circle be proven?
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(+) Construct a tangent line from a point outside a given circle to the circle.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Circle
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3 - Construct (tangent line from a point outside a given circle to the circle)
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3 - I can construct a tangent line from a point outside a given circle to the circle.
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A. For a given circle, a tangent line can be constructed from a point outside the circle.
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A.1 What is a tangent line?
A.2 How can a tangent line to a circle be constructed from a point outside the circle?
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Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Similarity
Radian Measure
Formula
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3 - Use (similarity)
6 - Derive (the fact that in a circle the arc length intercepted by an angle is proportional to the radius)
2 - Define (the constant of proportionality as the radian measure of the angle)
6 - Derive (formula for area of a sector)
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3 - I can use similarity to derive.
6 - I can derive the fact that in a circle the arc length intercepted by an angle is proportional to the radius by using similarity.
2 - I can define the constant of proportionality as the radian measure of the angle.
6 - I can derive the formula for area of a sector.
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A. Properties of similarity can be used to derive the relationship between an arc length and the angle that intercepts it.
B. Radian measure is defined to be the constant of proportionality between the arc length and the angle that intercepts it.
C. The formula for area of a sector can be derived using mathematical concepts.
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A.1 How can properties of similarity be used to derive the relationship between an arc length and the angle that intercepts it?
B.1 What is the definition of radian measure?
C.1 How can the formula for area of a sector be derived?
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