Essential Outcomes for this course:
1.)Experiment with transformations in the plane.
(performance indicator) Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
(p.i.) Represent transformations in the plane using transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs.(x+2,Y-3 is a shift up 2 units and left 3 units) Compare transformations that preserve distance and angle to those that do not (translation vs. horizontal stretch)
2.)Understand congruence in terms of rigid motions.
(p.i.) Use the definition of congruence in terms of rigid motion (size stays constant) to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
(p.i.) Explain how the criteria for triangle congruence (AAS, ASA, SSS, SAS) follow from the definition of congruence in terms of rigid motion.
3.)Prove geometric theorems (formal proofs) specifically theorems involving similarity.
(p.i.)Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
(p.i.)Prove theorems about lines and angles.
(p.i.)Prove theorems about quadrilaterals.
4.) Make geometric constructions.
(p.i.) Construct geometric figures using the proper tools. These constructions include: copying a segment, copying an angle, bisecting a segment, bisecting an angle, constructing perpendicular lines, constructing parallel line through a given point.
(p.i.) Construct an equilateral triangle, a square and a regular hexagon inscribed in a circle.
See standard 7.
5) Understand similarity in terms of similarity transformations.
(p.i.) Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
(p.i) Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
(p.i.) Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
(p.i.) Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
6.) Define trigonometric ratios and solve problems involving right triangles.
(p.i.) Use trigonometric rations and the Pythagorean Theorem to solve right triangles in applied problems.
(p.i.) Explain and use the relationship between the sine and cosine of complementary angles.
7.) Understand and apply theorems about circles.
(p.i.) 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.
(p.i.) Construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle.(see #4)
8.) Find arc lengths and areas of sectors of circles
9.)Use coordinates to prove simple geometric theorems algebraically.
Developing – no performance indicators needed.
10.) Explain volume formulas and use them to solve problems.
(p.i.) Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
11.) Visualize relationships between two-dimensional and three-dimensional objects.
(p.i.) Identify the shapes of two-dimensional cross-sections of three- dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
12.) Apply geometric concepts in modeling situations.