Level: High School: Geometry. Domain: Expressing Geometric Properties with Equations G-GPE

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1 1. 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. Translate between the geometric description and the equation for a conic section MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. Use the Pythagorean Theorem to derive the equation of a circle, given the center and the radius. Given an equation of a circle, complete the square to find the center and radius of a circle. G.4.D Apply the Pythagorean Theorem to find unknown side lengths in right triangles. Find the length of the hypotenuse of a right triangle given the three vertices as ordered pairs. Inscribe a right triangle given the three vertices as ordered pairs in a circle; find the lengths of all sides when the hypotenuse is a radius of the circle. Derive the equation of the circle, given the center and the radius. Recall the process of completing the square algebraically. Given an equation of a circle, complete the square to find the center and radius of a circle. Distance Formula Area of Circle Pythagorean Theorem Legs Hypotenuse Coordinate Plane Ordered Pair Center of Circle Radius Complete the

2 Square

3 2. Derive the equation of a parabola given a focus and directrix. Translate between the geometric description and the equation for a conic section M.P.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively MP.3 Construct viable arguments and critique the reasoning of others. Given a parabola, identify the vertex, focus, directrix, and axis of symmetry, noting that every point on the parabola is the same distance from the focus and the directrix. Given a focus and directrix, derive the equation of a parabola. NONE Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (High School Functions) Construct parabolas and identify the vertex, focus, axis of symmetry, and directrix. Interpret the pattern between the focus and directrix in a parabola. Given an equation of a parabola and the distance between the vertex and the focus, find the focus and directrix. Given a focus and directrix, derive the equation of a parabola Parabola Axis of Symmetry Vertex Equidistant Focus Directrix

4 3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Translate between the geometric description and the equation for a conic section MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. Given the foci, derive the equation of an ellipse, noting that the sum of the distances from the foci to any fixed point on the ellipse is constant, identifying the major and minor axis. Given the foci, derive the equation of a hyperbola, noting that the absolute value of the differences of the distances form the foci to a point on the hyperbola is constant, and identifying the vertices, center, transverse axis, conjugate axis, and asymptotes. Graph and label ellipses given an equation, identify major and minor axis and foci. Given the foci, derive the equation of an ellipse, noting that the sum of the distances from the foci to any fixed point on the ellipse is constant, identifying the major and minor axis. Graph and label hyperbolas given an equation, identify vertices, center, transverse axis, conjugate axis, asymptotes, and foci. Given the foci, derive the equation of a hyperbola, noting that the absolute value of the differences of the distances form the foci to a Absolute Value Ellipses Foci Hyperbolas Transverse axis NONE

5 point on the hyperbola is constant, and identifying the vertices, center, transverse axis, conjugate axis, and asymptotes. Conjugate axis Asymptotes Major Axis Minor Axis

6 4. 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). Use coordinates to prove simple geometric theorems algebraically MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.6 Attend to precision. Use coordinate geometry to prove geometric theorems algebraically; such as 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). G.4.B G.4.C Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. (6 th grade Math) Explore geometric theorems given information and misinformation for specific examples. Use coordinate geometry to prove geometric theorems algebraically. Polygons Coordinate Plane Ordered Pairs Vertices Distance Formula Midpoints Formula Algebraically

7 Domain: 5. 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). Use coordinates to prove simple geometric theorems algebraically MP.2 Reason abstractly and quantitatively MP.3 Construct viable arguments and critique the reasoning of others. MP.7 Look for and make use of structure. Using slope, prove lines are parallel or perpendicular. Find equations of lines based on certain slope criteria such as; finding the equation of a line parallel or perpendicular to a given line that passes through a given point. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. (High School Functions) Find the slope of a line given an equation or points on a coordinate plane. Recall the relationship between the slopes of parallel and perpendicular lines. Using slope, prove lines are parallel or perpendicular. Recall and use the various formulas for equations of a line (standard, Slope Parallel Lines Perpendicular Lines Slope-Intercept form Standard Form Point-Slope Form G.4.A

8 slope-intercept, point-slope form). Find equations of lines based on certain slope criteria such as; finding the equation of a line parallel or perpendicular to a given line that passes through a given point.

9 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinates to prove simple geometric theorems algebraically MP G.4.B Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. (7 th grade Math) Recall the midpoint and distance formula for a line segment given two points on the coordinate plane. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Distance Formula Midpoint Formula Ratio Proportion

10 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. * Use coordinates to prove simple geometric theorems algebraically MP.5 Use appropriate tools strategically. MP.6 Attend to precision. MP.7 Look for and make use of structure. Use coordinate geometry and the distance formula to find the perimeters of polygons and the areas of triangles and rectangles. G.4.B Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Recall and apply the distance formula given two points on the coordinate plane. Define, illustrate and notate perimeters of various polygons. Define, illustrate and notate areas for triangles and rectangles. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. * Coordinate Plane Distance Formula Perimeter Area of Triangle Area of Rectangle

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