Translate between the geometric description and the equation for a conic section
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Pythagorean Theorem
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6 - Derive (the equation of a circle given its center and radius, by using the Pythagorean Theorem)
3 - Find (the center and radius of a given circle’s equation by completing the square)
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6 - I can derive the equation of a circle given its center and radius, by using the Pythagorean Theorem.
3 - I can find the center and radius of a given circle’s equation by completing the square.
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A. Equations of circles are related to and can be derived using the Pythagorean Theorem, if the center and radius is known.
B. The center and radius of a circle can be found by performing mathematical processes with the circle’s equation.
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A.1 How is the equation of a circle, its center, and its radius, related to the Pythagorean Theorem?
A.2 How can the Pythagorean Theorem be used to derive the equation of a circle when its radius and center are given?
B.1 What mathematical processes can be used to find the center and radius of a circle given by an equation?
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Derive the equation of a parabola given a focus and directrix.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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6 - Derive (the equation of a parabola given a focus and directrix)
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6 - I can derive the equation of a parabola given a focus and directrix.
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A. The equation of a parabola can be derived when given a focus and directrix.
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A.1 What is the definition of a parabola?
A.2 What is a focus?
A.3 What is a directrix?
A.4 How can the equation of a parabola be derived given a focus and directrix?
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(+) Derive the equations of ellipses and hyperbolas given foci and directrices.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Ellipses
Hyperbolas
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6 - Derive (the equation of an ellipse given the foci)
6 - Derive (the equation of a hyperbola given the foci)
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6 - I can derive the equation of an ellipse given the foci and using the fact that the sum of distances from the foci to the ellipse is constant.
6 - I can derive the equation of a hyperbola given the foci and using the fact that the difference of the distances from the foci to the hyperbola is constant.
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A. The equations of ellipses can be derived when given the foci, and using the relationship of distance from the foci to the curve is a constant sum.
B. The equations of hyperbolas can be derived when given the foci, and using the relationship of distance from the foci to the curve is a constant difference.
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A.1 What is the definition of an ellipse?
A.2 What are foci of an ellipse?
A.3 How are the distances from the foci to the ellipse related?
A.4 How can the equation of an ellipse be derived if foci are known?
B.1 What is the definition of a hyperbola?
B.2 What are foci of a hyperbola?
B.3 How are the distances from the foci to the hyperbola related?
A.4 How can the equation of a hyperbola be derived if foci are known?
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Use coordinates to prove simple geometric theorems algebraically
Use coordinates to prove simple geometric theorems algebraically.
For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Geometric Theorems
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3 - Use (coordinates and algebra)
3 - Prove (geometric theorems)
3 - Disprove
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3 - I can use coordinates to prove geometric theorems algebraically.
3 - I can use coordinates to disprove geometric
1 - tatements algebraically.
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A. Some geometric theorems can be proven algebraically using coordinates.
B. Some geometric statements related to figures can be disproved algebraically using coordinates.
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A.1 How can coordinates be used to prove simple geometric theorems algebraically?
B.1 How can coordinates be used algebraically to disprove statements related to figures?
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Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Geometric problems
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3 - Prove (slope criteria for parallel lines)
3 - Prove (slope criteria for perpendicular lines)
3 - Use (slope criteria for parallel and perpendicular lines to solve problems)
4 - Find (the equation of a line parallel to a given line through a given point)
4 - Find (the equation of a line perpendicular to a given line through a given point)
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3 - I can prove slope criteria for parallel lines.
3 - I can prove slope criteria for perpendicular lines.
3 - I can use slope criteria for parallel and perpendicular lines to solve problems.
4 - I can find the equation of a line parallel to a given line through a given point.
4 - I can find the equation of a line perpendicular to a given line through a given point.
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A. Slope criteria for parallel and perpendicular lines can be demonstrated through proof.
B. If the slope of a line is known or can be found, the equation of another line parallel to it or perpendicular to it (that passes through a given point) can be found.
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A.1 What is the slope criteria for parallel lines, and how can it be proven?
A.2 What is the slope criteria for perpendicular lines, and how can it be proven?
B.1 How can the equation of a line parallel to another line, and passing through a given point, be found?
B.2 How can the equation of a line perpendicular to another line, and passing through a given point, be found?
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Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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3 - Find (point on a directed line segment between two points that partitions the segment in a given ratio.)
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3 - I can find the point on a directed line segment between two points that partitions the segment in a given ratio.
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A. On a directed line segment there exists, between two points, an identifiable point that partitions the segment in a given ratio.
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A.1 How can a point on a directed line segment (between two points) that partitions the segment in a given ratio be located and identified?
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Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Perimeters
Areas
Distance formula
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3 - Use (coordinates)
3 - Use (distance formula)
3 - Compute (perimeters of polygons)
3 - Compute (areas of triangles and rectangles)
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1 - I can use coordinates.
1 - I can use coordinates in the distance formula.
3 - I can compute perimeters of polygons using the distance formula.
3 - I can compute areas of triangles and rectangles using the distance formula.
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A. Perimeters of polygons can be found using their coordinates with the distance formula.
B. Areas of triangles and rectangles can be found using their coordinates with the distance formula.
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A.1 What coordinates are used in the distance formula?
A.2 How can perimeters of polygons be found using the distance formula?
B.1 How can areas of triangles and rectangles be found using the distance formula?
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