Geometry
Course Overview
The primary focus of Geometry is on straight lines and objects created from straight lines – triangles and other polygons, prisms, and pyramids. Circles and parabolas are introduced as well. Other key features are linear motion, in both two and three dimensions, and questions of optimization. Vectors and parametric equations are used to represent movement along straight-line paths. Problems sets include applications and other novel problem-solving situations, with continuous spiral review incorporated throughout the year, which guides students to discovery of the major concepts of the course.
Course Content
Unit 1: The Coordinate Plane
Students will explore the concepts of midpoint and other points that are equidistant from two given points, leading them to midpoint formula and perpendicular bisectors of line segments. They will learn the term “lattice points” to describe all points in a given area with integer coordinates, as part of an introduction to the concepts of discrete and continuous solutions. Students will begin their exploration of parametric equations for an object moving along a linear path. The Pythagorean Theorem will be used as a right triangle test.
Unit 2: Problem-solving in the Coordinate Plane, and Congruent Triangles
Students use the properties of linear equations, parallel lines, triangles, and distance to analyze figures plotted on a coordinate plane. This leads to determining congruence of triangles by corresponding parts. Students expand their ability to solve problems with parametric equations as well.
Unit 3: Distance Formula & Two Column Proofs
Students will use the distance formula to find the distance between two points and from a point to a line. They will also solve rate problems, related to the D=rt formula. They will write two column proofs of triangle congruence. Properties of common right triangles such as 30-60-90 are developed.
Unit 4: Unit Vectors and Properties of Isosceles Triangles
This unit introduces unit vectors and proofs about isosceles triangles. Other geometric properties that have been included in the last two units are angle properties of regular polygons (interior, exterior), as well as transversals across parallel lines.
Unit 5: Finding Lengths of Sides and Angle Measures of triangles.
In this unit, students will employ all previous properties and figures, including altitudes, perimeter, area, and medians, to solve for missing parts of triangles, and by extension- polygons composed of triangles.
Unit 6: Centroids and Angle Bisector Theorem
Using the definition of centroid students will explore patterns in Geometry. Students will solve problems using the properties they have discovered about centroids, as well as properties of angle bisectors. There is a special emphasis in this unit of connecting previous learning with new patterns.
Unit 7: Transformations of Figures
Students apply transformation rules, such as T(x,y) = (2x/3, 2y/3), to various polygons. They learn to test for isometry and recognize dilations of figures. Review problems include area, and finding ratios of areas from ratios of corresponding linear measurements.
Unit 8: Circles and Parabolas
Students use a variety of given information to write equations of circles and parabolas. They also begin to explore the properties of cyclic figures. Calculator skills for graphing and calculating residuals for scatter plots are mastered.
Unit 9: Chords, Arcs, and Central Angles of Circles
Students solve problems involving intersections of circles and lines. Work on producing equations of parabolas and circles continues. Right triangle work includes calculating sine, cosine, and tangent of angles. Students master all properties of similar figures.
Unit 10: Cyclic Figures, Common Chords, and Tangent Lines
In the final unit, students study characteristics of intersecting figures. This includes circumscribed and inscribed circles, as well as overlapping circles and their common chords. Students will also use completing the square to find the centers and radii of circles, given equations in general form.