G.CO.A.1 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.
G.CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
G.CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
COMMON CORE OBJECTIVES G.CO.A.4 TO G.CO.B.6
G.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
G.CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
COMMON CORE OBJECTIVES G.CO.B.7 TO G.CO.C.9
G.CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G.CO.C.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
COMMON CORE OBJECTIVES G.CO.C.10 TO G.CO.D.13
G.CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G.CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
G.CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
G.CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
(1) The first week is so crazy... introduction/organization day, handing out textbooks (what textbooks?), a pre-test, etc... Once you are able to begin - lay out the basic elements of geometry - the undefined terms of point, line and plane. Jump right into constructions next as a way to generate a place for the discussion of the basic terms, relationships and notation of geometry . Perform the basic constructions using a variety of tools such as: compass/straightedge, patty paper, geometer's sketchpad - emphasizing the new definitions and their accompanying notation.
(2) Continue constructions this week. The constructions of: copy a segment, copy an angle, bisect a segment, bisect an angle, construct perpendicular lines, construct the perpendicular bisector of a segment and construct parallel lines all lay the foundations for the transformations. Near the end of the week you can now ask them to put many of these construction skills together to make new shapes such as rectangles, squares, equilateral triangles, inscribed shapes (G.CO.D.13) or any number of other shapes. This is a nice way synthesize their understanding.
(3) Discuss and investigate different types of transformations - those that are isometries and those that are not. This objective is about introducing the transformations of the plane, having students recognize them and then informally understand their impact on objects. Link transformations to the coordinate grid, coordinate rules and to functions (input/output). Discuss mapping, one to one correspondence and transformations and how they connect to functions. Also determine the symmetries of certain shapes.
(4) Define the isometric transformations of reflection, rotation, & translation. Develop familiarity with each of them by clearly defining them, discussing their properties and constructing them. Move from the general plane to the coordinate plane and have students investigate the coordinate rules and patterns found with each of these motions in the plane. Have them experience these relationships in a number of ways: compass/straightedge, patty paper, geometer's sketchpad.
(5) In preparation for congruence, we will look at how composite isometric transformations are still isometries. We will look at the composite transformations such as double reflections over parallel lines (translation), double reflections over intersecting lines (rotation), glide reflections and others. We will also discuss how the order of the composite transformations often changes the result. This links nicely to composite functions.
(6) This week we will continue to look at composite functions. What will be new in this week is that we will be working backwards through the process. Given a pre-image and an image determine the composite transformations that have taken place between them. This is a strong connection to congruence.... We also do things a bit out of order this week; we will jump to G.CO.C.9 about angles; vertical angles, pairs of angles, and angles formed by parallel lines. These relationships will be needed to do much of the proof that will follow in G.CO.B.8 when we prove triangle congruence.
(7) In this week we will continue to establish the angle relationships found with a transversal and parallel lines. This relationship will come as an angle is translated along one of its rays, forming congruent corresponding angles. These relationships unlock many problems in the future. We will also begin to work with congruence - this will come easily because of our previous work with isometric transformations.
(8) Finally we get to something we recognize from the old methodology... proving triangles to be congruent. Through discovery we will establish the criteria for triangle congruence such as SSS, SAS, ASA, AAS, HL and ASS (in special cases). I mention ASS here because if you are teaching honors geometry you will later establish the Sine Law (and its ambiguous case) and knowing about the cases of ASS will make that moment easier.
(9) Proof continues but focused on relationships found in a triangle such as: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. We prove these relationships then apply them. Some of these proofs do not have to be handled now; they may be better done during the coordinate geometry objectives.
(10) Proof continues but focused on relationships found in quadrilaterals (specifically parallelograms) such as: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. These properties get used throughout the year in many places. I also feel that it is important to also establish the conditions necessary to prove something is a parallelogram. This knowledge will make some proofs later much easier.
