Download Common Core Geometry

Common Core Geometry Course Objectives
Unit 1- CONGRUENCE, PROOF AND CONSTRUCTION
Undefined termsLay out the basic elements of geometry - the undefined terms of point, line and plane.
Introductory DefinitionsDefine bisector, vertex, polygons, and the relationships of parallel and perpendicular.
Basic ConstructionsPerform the basic constructions using a variety of tools such as: compass/straightedge, patty
paper, geogebra software - emphasizing the new definitions and their accompanying notation.
More ConstructionsThe 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. Then
students will put many of these construction skills together to make new shapes such as rectangles,
squares, equilateral triangles, inscribed shapes (G.CO.13) or any number of other shapes.
IsometryDiscuss and investigate different types of transformations - those that are isometries (rigid motions) 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.
Mapping and FunctionsLink 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.
SymmetryDetermine the rotational symmetries of certain shapes.
Transformations (reflection, rotation, translation)Define the isometric transformations of Reflection, Rotation, & Translation. Develop familiarity with each
of them by clearly defining them, discussing their properties (essential for explain proof later), 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, geogebra.
Composite TransformationsIn preparation for congruence, investigate 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. In addition we look at
transformations working backwards. Given a pre-image and an image determine the composite
transformations that have taken place between them.
Proof (lines/angles)
Relationships regarding angles; vertical angles, pairs of angles, and angles formed by parallel lines.
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 are
necessary in order to do the congruence proofs.
Congruence Criteria Proof (triangles)
Through discovery we will establish the criteria for triangle congruence such as SSS, SAS, ASA, AAS,
and HL. We will also focus 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.
Proof (quadrilaterals)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.
UNIT #2 – SIMILARITY AND RIGHT TRIANGLES
Dilation and Dilation PropertiesExplore the dilation transformation -- its properties, graphing, determining scale factors, looking at
different centers and applying dilations to real world situations.
SimilarityHere we transition from the dilation properties and graphing to connecting that to similarity and what it
means to be similar. A new group of transformations is introduced – the similarity transformations of
Reflection, Rotation, Translation, and Dilation. A similarity transformation preserves the shape, which
means proportionality of sides and congruence of angles.
Similarity ProofWe establish the minimum criteria for similarity (AA, SAS & SSS) and then begin proving triangles to be
similar. We also prove two triangles to be similar so that we can talk about angle congruence and
proportionality of sides. We will use similarity to establish two new relationships the Side Splitting
Theorem and the Angle Bisector Theorem.
Geometric Mean and Special Right TrianglesApply similarity to right triangles and investigate the geometric mean and special right triangle
relationships.
Trigonometry and Sine/Cosine RelationshipUsing similar triangles we will determine the ratios of sides and compare them to build the conceptual
understanding of trigonometry. From there we will study patterns found in these ratios. One of the many
patterns that we will discuss will be the relationship between sine and cosine as co-functions.
Trigonometry ProblemsWe extend our knowledge of trigonometry by applying the ratios to problems in multiple environments.
We will apply the terms of angle of elevation and depression and other specific descriptors to help
students establish the relationships in simplified ‘real world’ problems.
Area Formula and Law of SinesWe look at how the sine ratio helps us determine lengths and angles in oblique triangles. We will derive
the relationship and then apply them in multiple situations. One of the major discussions will be the
ambiguous case of the Law of Sines.
Cosine LawWe look at the Cosine Law and the cases that it handles so that we can solve for values of all oblique
triangles.
UNIT #3 – MEASUREMENT AND VOLUME
Developing formulas- We develop the formulas for circumference of a circle, and areas of rectangles,
parallelograms and triangles. We also learn about Pi.
Calculating area- More area relationships are developed, including formulas for trapezoids and regular
polygons.
Calculating Volume- We work with prisms and Cavalieri’s Principle. We also calculate, the volume of
cylinders, pyramids cones and spheres.
Cross sections- We examine cross sections and visualizing 2-D forms in 3-D shapes.
Rotating Volume- We work with rotating cross sections to form volume.
UNIT #4 – COORDINATE GEOMETRY
Circle Equation- Here we will connect circle geometry to the coordinate grid, and we look at the
equations of circles. A major part of this objective is to learn completing the square to place the equation
into vertex and radius form.
Parabolic Equation- We will focus on constructing a parabola, deriving the parabolic equation,
understanding basic parabolic equation concepts and graphing parabolas.
Coordinate Problems- We begin with a review of slope, distance formula and midpoint. We will apply
the slope, distance formula and midpoint to coordinate geometry. We also review equations of lines and
systems of equations and we apply of systems of equations
Parallel/Perpendicular- We extend the study of lines to parallel and perpendicular lines. We will be
solving parallel and perpendicular coordinate problems, and also problems dealing with proof of shapes
through their properties. We also look at directed line segments.
Partitioning Segments- We will use coordinates to partition directed line segments.
Perimeter and Area- We will use coordinates to find the perimeter and area of various shapes.
Unit #5- CIRCLE GEOMETRY
Circle Similarity- To continue our connectivity to transformation we begin talking about the similarity of
circles. This idea is essential to understanding radians later. We also review the basic circle terms and
relations, arcs of circles and tangent lines.
Circle Properties- We will work with tangent lines, chord properties and inscribed angles.
Circle Constructions- We look at the other angles that can be formed inside, on and out of the circle.
We also look at the length relationship found with chords and secants. We also use construction to find
the circumcenter and incenter of the circle, and to find a tangent line given a point.
Radian Measure- We will discuss and define what a radian is, and visualize radians. We also convert
degrees to radians and radians to degrees. We will use arc length formula in radians as well as the
application of arc length formula, and we will use the area formula in radians
Unit #6- PROBABILITY
Sample Spaces- We will investigate sample spaces, uniform probability and the fundamental counting
principle.
Venn Diagrams- Venn diagramming of events (subsets), and using venn diagrams to solve word
problems.
Independence- We look the basic independence rule for multiplication and then look at independence of
in conditional statements.
Two-Way Tables- . Using two way tables we will determine basic probabilities, conditional probabilities
and independence.
Everyday Probability- We will look at probability in everyday life, specifically data gathering.
Conditional Probability- We look at how to calculate conditional probabilities.
Addition Rule/Multiply Rule- We investigate the addition rule of probability (Union), and the general
multiplication rule of probability. We will also determine the type of problem, and the appropriate strategy
needed to solve it.
Permutations and Combinations-We look into permutation and combinations to calculate more complex
sample spaces and we will calculate probabilities with permutations and combinations.