Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana
Curriculum Map
Course Title: Geometry / Quarter: 2 / Academic Year: 2011-2012Essential Questions for this Quarter:
1. How do you prove that two lines are parallel or perpendicular?2. What is the sum of the measures of the angles of a triangle?
3. How do you write and equation of a line in the coordinate plane?
4. How do you indentify corresponding parts of congruent triangles?
5. How do you show that two triangles are congruent?
6. How can you tell whether a triangle is isosceles or equilateral?
7. How do you simplify a radical expression
Unit/Time Frame / Standards / Content / Skills / Assessment / Resources
Chapter 3 (Part A) Parallel and Perpendicular Lines
3.1 Lines and Angles
3.2 Properties of Parallel Lines
3.3 Proving Lines Parallel
3.4 Parallel and Perpendicular Lines
3.5 Parallel Lines and Triangles
Chapter 3 (Part B) Parallel and Perpendicular Lines
3.7 Part 1 Equations of Lines in the Coordinate Plane
3.7 Part 2 Equations of Lines in the Coordinate Plane
3.8 Slopes of Parallel and Perpendicular Lines
Congruent Triangles
4.1 Congruent Figures
4.2 Triangle Congruence by SSS and SAS
4.3 Part 1 Triangle Congruence by ASA and AAS
4.3 Part 2 Triangle Congruence by ASA and AAS
4-4 Using Corresponding Parts of Congruent Triangles
4-5 Isosceles and Equilateral Triangles
4-6 Congruence in Right Triangles
Radical Review / State Standards
G.1.1
G.1.3
G.1.4
G.4.1
G.1.5
G.1.6
G.1.7
G.5.4
G.2.3
G.2.7
G.2.12
G.2.14
Common Core Standards
CC G.CO.1
CC G.CO.9
CC G.CO.10
CC G.CO.12
CC G.MG.2
CC G.MG.3
CC G.SRT.5
Standards for Mathematical Practice
SMP1
SMP2
SMP3
SMP4
SMP5
SMP6
SMP7
SMP8 / Angle Relationships Given Parallel Lines
Slope of Parallel and Perpendicular Lines
Connecting Algebra and Geometry
Proving Triangles Congruent
Proving Right Triangles Congruent
Overlapping Triangles
Simplifying Radical Expressions / To identify relationships between figures in space To identify angles formed b two lines and a transversal
To prove theorems about parallel lines To use properties of parallel lines to find angle measures
To determine whether two lines are parallel
To relate parallel and perpendicular lines
To use parallel lines to prove a theorem about triangles To find measures of angles of triangles
To find slope and graph linear equations
To write linear equations
To relate slope to parallel and perpendicular lines
To recognize congruent figures and their corresponding parts
To prove two triangles congruent using the SSS and SAS Postulates
To prove two triangles congruent using the ASA Postulate
To prove two triangles congruent using the AAS theorem
To simplify radical expressions / Textbook assignments
Worksheet assignments
Section Quizzes
Quizzes
Tests
Oral Responses
Observations / Textbook Prentice-Hall Geometry Foundation Series 2011 Edition
Power point
Presentations
Intro to Geometry Frank Schaffer Publications
Teacher generated worksheets
Notebooks
Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana
COMMON CORE AND INDIANA ACADEMIC STANDARDS
State StandardsG.1: Students find lengths and midpoints of line segments. They describe and use parallel and perpendicular lines. They find slopes and equations of lines.
G.1.1: Find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems.
G.1.2: Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straight edge and compass, explaining and justifying the process used.
G.1.3: Understand and use the relationships between special pairs of angles formed by parallel lines and transversals.
G.1.4: Use coordinate geometry to find slopes, parallel lines, perpendicular lines, and equations of lines.
G.2: Students identify and describe polygons and measure interior and exterior angles. They use congruence, similarity, symmetry, tessellations, and transformations. They find measures of sides, perimeters, and areas.
G.2.2: Find measures of interior and exterior angles of polygons, justifying the method used.
G.2.3: Use properties of congruent and similar polygons to solve problems.
G.2.4: Apply transformations (slides, flips, turns, expansions, and contractions) to polygons in order to determine congruence, similarity, symmetry, and tessellations. Know that images formed by slides, flips and turns are congruent to the original shape.
G.2.5: Find and use measures of sides, perimeters, and areas of polygons, and relate these measures to each other using formulas.
G.2.6: Use coordinate geometry to prove properties of polygons such as regularity, congruence, and similarity.
G.3: Students identify and describe simple quadrilaterals. They use congruence and similarity. They find measures of sides, perimeters, and areas.
G.3.1: Describe, classify, and understand relationships among the quadrilaterals square, rectangle, rhombus, parallelogram, trapezoid, and kite.
G.3.2: Use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas.
G.3.3: Find and use measures of sides, perimeters, and areas of quadrilaterals, and relate these measures to each other using formulas.
G.3.4: Use coordinate geometry to prove properties of quadrilaterals such as regularity, congruence, and similarity.
G.4: Students identify and describe types of triangles. They identify and draw altitudes, medians, and angle bisectors. They use congruence and similarity. They find measures of sides, perimeters, and areas. They apply inequality theorems.
G.4.1: Identify and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular.
G.4.2: Define, identify, and construct altitudes, medians, angle bisectors, and perpendicular bisectors.
G.4.3: Construct triangles congruent to given triangles.
G.4.4: Use properties of congruent and similar triangles to solve problems involving lengths and areas.
G.4.5: Prove and apply theorems involving segments divided proportionally.
G.4.6: Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles.
G.4.7: Find and use measures of sides, perimeters, and areas of triangles, and relate these measures to each other using formulas.
G.4.8: Prove, understand, and apply the inequality theorems: triangle inequality, inequality in one triangle, and hinge theorem.
G.4.9: Use coordinate geometry to prove properties of triangles such as regularity, congruence, and similarity.
G.5: Students prove the Pythagorean Theorem and use it to solve problems. They define and apply the trigonometric relations sine, cosine, and tangent.
G.5.1: Prove and use the Pythagorean Theorem.
G.5.2: State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.
G.5.4: Define and use the trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) in terms of angles of right triangles.
G.5.5: Know and use the relationship sin²x + cos²x = 1.
G.5.6: Solve word problems involving right triangles.
G.6: Students define ideas related to circles: e.g., radius, tangent. They find measures of angles, lengths, and areas. They prove theorems about circles. They find equations of circles.
G.6.2: Define and identify relationships among: radius, diameter, arc, measure of an arc, chord, secant, and tangent.
G.6.3: Prove theorems related to circles.
G.6.5: Define, find, and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents).
G.6.6: Define and identify congruent and concentric circles.
G.6.7: Define, find, and use measures of circumference, arc length, and areas of circles and sectors. Use these measures to solve problems.
G.6.8: Find the equation of a circle in the coordinate plane in terms of its center and radius.
G.7: Students describe and make polyhedra and other solids. They describe relationships and symmetries, and use congruence and similarity.
G.7.2: Describe the polyhedron that can be made from a given net (or pattern). Describe the net for a given polyhedron.
G.7.4: Describe symmetries of geometric solids.
G.7.5: Describe sets of points on spheres: chords, tangents, and great circles.
G.7.6: Identify and know properties of congruent and similar solids.
G.7.7: Find and use measures of sides, volumes of solids, and surface areas of solids, and relate these measures to each other using formulas.
G.8: Mathematical Reasoning and Problem Solving
G.8.6: Identify and give examples of undefined terms, axioms, and theorems, and inductive and deductive proof.
G.8.8: Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, and two-column and indirect proofs.
Common Core Standards
Congruence G-CO
Experiment with transformations in the plane
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.
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).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
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.
Understand congruence in terms of rigid motions
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.
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.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Prove geometric theorems
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.
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.
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.
Make geometric constructions
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.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Similarity, Right Triangles, and Trigonometry G-SRT
Understand similarity in terms of similarity transformations
1. Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
2. 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.
3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity
4. 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.
5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles
6. 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.
7. Explain and use the relationship between the sine and cosine of complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Apply trigonometry to general triangles
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Circles G-C
Understand and apply theorems about circles
1. Prove that all circles are similar.
2. 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.
3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
4. (+) Construct a tangent line from a point outside a given circle to the circle.
Find arc lengths and areas of sectors of circles
5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Expressing Geometric Properties with Equations G-GPE
Translate between the geometric description and the equation for a
conic section
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.