Standards: Connecticut has adopted the Common Core State Standards (CCSS) as the Connecticut State Standards (CSS). The relevant Common Core State/Connecticut State Standard is listed after each content bullet (example: CC.9-12.G.CO.1).

Course Levels: Intermediate Geometry, College Prep Geometry, Honors Geometry - Topics are explored in greater depth at higher course levels.

ESSENTIAL QUESTIONS

• What is a transformation and what is the difference between congruence and similarity?
• How can slope be used to prove parallel and perpendicular lines in geometric figures?
• How can the distance formula be used to prove geometric figures?

PREREQUISITE SKILLS AND UNDERSTANDING

• Writing an equation of a line
• Simplifying radicals
• Ordered pairs and coordinate plane
• Geometric figures (basic knowledge and understanding)
  • Area and Perimeter
  • Properties of polygons - rectangle, square, etc.

CONTENT AND STANDARDS

• Students will be able distinguish between defined terms and undefined notions (CC.9-12.G.CO.1)
  • Understand and represent precise definitions of angle, circle (center and radius), perpendicular line, parallel line and line segment (CC.9-12.G.CO.1)
  • Understand and represent undefined notions of point, line, distance along a line, and distance around a circular arc. (CC.9-12.G.CO.1)
• Students will develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (CC.9-12.G.CO.4)
  • Students will observe patterns in the development of definitions using geometry software and/or manipulatives. (CC.9-12.G.CO.2)
  • Model and compare transformations that preserve distance and angle to those that do not (similarity vs. congruence) (CC.9-12.G.CO.2)
  • Students will describe transformations as functions taking points in the plane as inputs and give other points as outputs. (CC.9-12.G.CO.2)
  • Describe the rotations and reflections that carry a rectangle, parallelogram, trapezoid, or regular polygon onto itself. (CC.9-12.G.CO.3)
  • Draw a transformed geometric figure using graph paper, tracing paper or software. (CC.9-12.G.CO.5)
  • Specify the sequence of transformations that will carry a given figure onto another. (CC.9-12.G.CO.5)
  • Students will experimentally verify the properties of dilations (CC.9-12.G.SRT.1)
    • Students will understand how a scale factor changes a figure(CC.9-12.G.SRT.1)
• Students will prove the slope criteria for parallel and perpendicular lines. (CC.9-12.G.GPE.5)
  • Find the equation of a line parallel or perpendicular to a given line that passes through a given point (CC.9-12.G.GPE.5)
  • Construct a parallel or perpendicular line using any method (CC.9-12.G.CO.12)
  • Use slope, midpoint, and distance formulas (CC.9-12.G.GPE.4)
    • Example: To prove or disprove a given point lies on a circle centered at the origin and passing through a point (x, y) or that four given points form a rectangle (CC.9-12.G.GPE.4)
    • Example: To find a point on a directed line segment between two given points, that partitions the segment in a given ratio (midpoint and other ratios) (CC.9-12.G.GPE.6)
    • Example: To compute perimeters of polygons and areas of rectangles and triangles (CC.9-12.G.GPE.7)

ESSENTIAL QUESTIONS

• What is congruence and how is it proved?
• Why are ASA, SAS, and SSS criteria for triangle congruence?
• How are lines and angles related?

PREREQUISITE SKILLS AND UNDERSTANDING

• Understand transformations
• Understand parallel and perpendicular

CONTENT AND STANDARDS

• Students will show that two triangles are congruent if and only congruent pairs of sides and corresponding pairs of angles are congruent. (CC.9-12.G.CO.7)
  • Use rigid motions to transform figures (CC.9-12.G.CO.6 ) (using any tools, i.e. pencil and paper, software)
  • Predict the effect of a given rigid motion on a figure (CC.9-12.G.CO.6)
  • Use the definition of congruence in terms of rigid motion to decide congruence (CC.9-12.G.CO.6)
• Students will explain what criteria is necessary to prove triangles congruent (CC.9-12.G.CO.8 )
  • ASA, SAS, SSS (CC.9-12.G.CO.8 )
• Students will prove theorems about lines and angles using any method (explorations may include computer software, graphing calculator, pencil and paper) (CC.9-12.G.CO.9)
  • Vertical angles are congruent (CC.9-12.G.CO.9)
  • When a transversal crosses parallel lines: (CC.9-12.G.CO.9)
    • Alternate interior angles are congruent (CC.9-12.G.CO.9)
    • Corresponding angles are congruent (CC.9-12.G.CO.9)
  • Points on a perpendicular bisector of a line segment are equidistant from the segment’s endpoints (CC.9-12.G.CO.9)
• Students will make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, geometric software) (CC.9-12.G.CO.12)
  • Copying: segments and angles (CC.9-12.G.CO.12)
  • Bisecting: segments and angles (CC.9-12.G.CO.12)
  • Constructing perpendicular lines, including perpendicular bisectors of line segments (CC.9-12.G.CO.12)
  • Constructing a line parallel to a given line through a point not on the line (CC.9-12.G.CO.12)
  • Examples:
    • Construct a triangle given SAS
    • Construct the circumcenter of a triangle

VOCABULARY - congruence, corresponding parts, vertical angles, transversal, alternate interior angles, equidistant, bisector, perpendicular bisector, ASA, SAS, SSS, included angle/side

ESSENTIAL QUESTIONS

• How do you prove geometric theorems relating to triangles and parallelograms?

PREREQUISITE SKILLS AND UNDERSTANDING

• Parallel lines
• Congruence
• How to find a midpoint

CONTENT AND STANDARDS

• Students will prove and apply theorems about triangles (may explore using geometric software or graphing calculator) (CC.9-12.G.CO.10)
  • Sum of interior angles equals 180 (CC.9-12.G.CO.10)
  • Base angles of isosceles triangle are congruent (CC.9-12.G.CO.10)
  • Mid-segment of a triangle is parallel and half the length of the 3^rdside (CC.9-12.G.CO.10)
  • The medians of a triangle meet at the centroid (CC.9-12.G.CO.10)
• Students will prove and apply theorems about parallelograms (may explore using geometric software or graphing calculator) (CC.9-12.G.CO.11)
  • Opposite sides and angles are congruent (CC.9-12.G.CO.11)
  • The diagonals of a parallelogram bisect each other (CC.9-12.G.CO.11)
  • Understand how rectangles, rhombi, and squares relate to parallelograms (i.e., rectangles are parallelograms with congruent diagonals) (CC.9-12.G.CO.11)
  • Apply properties of parallelograms to coordinate geometry problems*( from unit 1) (CC.9-12.G.GPE.4)
• Supporting standard – construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle (CC.9-12.G.CO.13 )

VOCABULARY - median, isosceles, equilateral, regular, interior angles, base angles, inscribed, mid-segment, diagonal, bisector

ESSENTIAL QUESTIONS

• What is similarity and how is it proved?
• How can proportions, similarity, and trigonometric ratios be used to solve application problems?
• How are the sides of a right triangle related to its angles?

CONTENT AND STANDARDS

• Students will be able to use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures (CC.9-12.G.SRT.5)
  • Students will apply concepts of similar triangles when altitude is drawn to hypotenuse
  • Students will understand and use the proportionality of corresponding sides of similar triangles to solve problems (CC.9-12.G.SRT.4 )
  • Students will be able to determine whether two figures are similar and why, including the SSS, SAS, and AA criteria for triangles (CC.9-12.G.SRT.2 and CC.9-12.G.SRT.3 )
  • Students will use similar triangles to prove the Pythagorean theorem (CC.9-12.G.SRT.4 )
• Students will understand how to use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems (CC.9-12.G.SRT.8)
  • Students will use similar triangles to understand why a given angle always has the same trigonometric ratios. (CC.9-12.G.SRT.6)
    • Students need to be able to use right triangle trigonometry to model situations and solve problems using angle of elevation and angle of depression (CC.9-12.G.SRT.8)
    • Students will be able to calculate the six trigonometric ratios – sine, cosine, tangent, cosecant, secant and cotangent (CC.9-12.G.SRT.6)
  • Students will explain and use the relationship between the sine and cosine of complementary angles (CC.9-12.G.SRT.7)
  • Students will use Pythagorean Theorem to obtain exact trigonometric ratios for angles
• Students will be able to apply trigonometry to general triangles (nat’l standards vs. CT standards) (CC.9-12.G.SRT.11)
  • Derive or prove:
    • Area formula (CC.9-12.G.SRT.9)
    • Law of sines (CC.9-12.G.SRT.10)
    • Law of cosines (CC.9-12.G.SRT.10)

VOCABULARY - trigonometric ratios (sine, cosine, tangent, cosecant, secant and cotangent), angle of elevation, angle of depression, pythagorean theorem, right triangle, ratios and proportions, dilation

ESSENTIAL QUESTIONS

• How are the angles, arcs, segments of a circle related?
• How do you represent a circle algebraically and graphically?

PREREQUISITE SKILLS AND UNDERSTANDING

• Completing the square
• Area and circumference of a circle
• Proportions
• Pythagorean theorem
• Distance and midpoint formulas
• Ordered pairs and coordinate plane

CONTENT AND STANDARDS

• Students will be able to identify and describe relationships among angles and segments in a circle:
  • Students will understand and solve problems using the relationship between central angles, inscribed angles, and circumscribed angles and their intercepted arcs (CC.9-12.G.C.2).
  • Students will understand and solve problems using the relationship between chords, secants, tangents and their intercepted arcs and segment measures (CC.9-12.G.C.2).
  • Students will understand and solve problems using the relationship between a tangent segment and the radius of the circle (point of tangency) (CC.9-12.G.C.2).
  • Students will be able to construct the inscribed and circumscribed circles of a triangle (CC.9-12.G.C.3).
  • Students will be able to prove properties of angles for an inscribed quadrilateral (CC.9-12.G.C.3).
• Students will be able to apply proportional reasoning to derive (CC.9-12.G.C.5).:
  • The length of an arc
  • The radian measure
  • The area of a sector
• Students will be able to derive the equation of a circle given the center and radius using the Pythagorean Theorem
  • Complete the square to find the center and radius of a circle given by an equation (CC.9-12.G.GPE.1).
  • Prove all circles are similar (CC.9-12.G.C.1).
  • Examples: Write an equation of a circle given a radius and center, write an equation for a circle given the endpoints of the diameter
  • Example: To prove or disprove a given point lies on a circle centered at the origin and passing through a point (x, y)
• Students will be able to derive the equation of a parabola given a focus and directrix (CC.9-12.G.GPE.2)

VOCABULARY - central angles, inscribed angle, circumscribed angles, radii, tangent, chord, diameter, sector, arc (intercepted arc), radian, arc length

ESSENTIAL QUESTIONS

• What is volume and how is it connected to two dimensional figures and their measurements?
• How can you use volume to model real world applications?

PREREQUISITE SKILLS AND UNDERSTANDING

• Identify three dimensional figures
• Find areas of circles and polygons
• Appropriate units of measurement

CONTENT AND STANDARDS

• Students will be able to explain volume formulas and use them to solve problems (CC.9-12.G.GMD.3).
  • Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, sphere, and cone (CC.9-12.G.GMD.1).
  • Identify the shapes of two dimensional cross sections of three dimensional objects (CC.9-12.G.GMD.4).
  • Identify three dimensional objects generated by rotations of two dimensional objects (CC.9-12.G.GMD.4).
• Students will be able to apply geometric concepts and modeling situations
  • Use geometric shapes, their properties, and their measure to describe objects (CC.9-12.G.MG.1).
  • Example: modeling a tree trunk or a human torso as a cylinder
  • Apply concepts of density based on area and volume in modeling situations (CC.9-12.G.MG.2).

VOCABULARY - slant height, altitude, radius, diagonal, volume, cross section

ESSENTIAL QUESTIONS

• How do you use the concepts of independence and conditional probability to interpret data?
• How do you find probabilities of compound events?

PREREQUISITE SKILLS AND UNDERSTANDING

• Fractions and ratios
• Venn Diagrams
• Simple probability

CONTENT AND STANDARDS

• Students will be able to describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or”, “and”, “not”) (CC.9-12.S.CP.1).
• Students will be able to understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B (CC.9-12.S.CP.3).
  • Students will be able to understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent (CC.9-12.S.CP.2).
  • Students will use Venn diagrams and two way frequency tables to analyze sample spaces (CC.9-12.S.CP.4).
• Find the conditional probability of A given B as a fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model (CC.9-12.S.CP.).
• Apply the Addition Rule, P(A or B) = P(A) +P(B) – P(A and B), and interpret the answer in terms of the model (CC.9-12.S.CP.7).

VOCABULARY - events, subsets, sample space, unions, intersections, complements, probability, conditional probability, independent events, universal set