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Transcript
Essentials of Geometry
Piscataway High School
Teacher:
Your Name
Email:
[email protected]
Course Title: Geometry
Textbook:
Geometry (2014), HMH (Kanold, Burger, et al.)
Course Overview
Full year course: 5.0 credits (Honors Geometry has a 5 point weight added to the final grade)
Prerequisite: Algebra 1
Description: Geometry is a college prep course that uses constructions and transformations to explore
geometric relationships and figures. Students will use formal logic and various forms of proof in order
to explore topics such as congruence and similarity, properties of geometric shapes, measurement of
plane and solid figures, and trigonometry.
The Geometry course is structured as follows:
Unit
Topic
Length
Unit 1
Constructions and Introduction to Geometry
6 Days
Unit 2
Isometry and Transformations
14 Days
Unit 3
Introduction to Proof
12 Days
Unit 4
Triangles and Triangle Congruence
17 Days
Unit 5
Quadrilaterals and Coordinate Proofs
10 Days
Unit 6
Similarity
12 Days
Unit 7
Right Triangles and Trigonometry
17 Days
Unit 8
Circles
20 Days
Unit 9
Modeling in Three Dimensions
12 Days
1
Geometry Scope and Sequence
Unit
Timing
Topic
(1 day = 1 hour)
Concepts and Skills
Assessment for Units 1 and 2 administered by the end of Cycle 4
Unit 1
6 Days
Constructions and
Introduction to
Geometry
[1.1, 1.2, & 1.4]
Unit 2
14 Days
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


Mandatory Performance
Tasks:
1.4 #28 (p.56)




Isometry and
Transformations

[1.3, Modules 2 & 3]




Identify basic geometry vocabulary (points, lines, planes, intersections, etc.)
Use undefined terms such as points, lines and planes to create axiomatic system to build
other geometric terms (i.e. line segment, ray, angle, circle, etc.)
Draw geometric diagrams and name/label geometric figures using the proper notation and
symbols
Constructions (with compass and straightedge, string, reflective devices, paper folding, and
technology): copy segment, bisect segment, copy angle, bisect angle – in conjunction w/
book
Use the distance, midpoint, and slope formulas
Use the Segment Addition Postulate and Angle Addition Postulate
Introduce conditional statements and counterexamples
Complete simple proofs and algebraic proofs using properties and postulates to justify
statements
Rigid Transformations
o Translations (using vectors), reflections, and rotations
o Perform transformations both on and off the coordinate plane
o Represent transformations mapping points from preimage to image using coordinate
notation
o Use slope and midpoint formulas to find the equation of the line of reflection
o Use tracing paper investigations with ruler and protractor – should we find a specific
activity?
Define congruence through rigid transformations on the coordinate plane and validate using
distance formula and midpoint formula
Define compositions of transformations to map congruent figures onto one another
Write congruence statements for congruent figures and identify corresponding parts of
congruent figures
Identify lines of symmetry and angles of rotational symmetry for figures
Assessment for Unit 3 administered by the end of Cycle 6
Unit 3
12 Days
Parallel &
Perpendicular
Lines



[Module 4]


Mandatory Performance
Tasks:
Lesson 4.2 (p. 189) – Do
after Lesson 4.3


Review properties of angles (complements, supplements, vertical angles, linear pairs)
Prove vertical angles are congruent
Review parallel line properties and angle relationships (corresponding, alternate interior,
alternate exterior, consecutive interior)
o Prove angle theorems related to parallel lines cut by a transversal
Discuss converse statements
o Prove lines are parallel
Construct a line parallel to a given line
o Prove the construction using the Converse of the Corresponding Angles Theorem
Construct perpendicular bisectors
o Prove all points on a perpendicular bisector are equidistant from the endpoints of the
segment
Graph, find the slope of, and write equations for perpendicular and parallel lines
Geometry Scope and Sequence
Unit
Timing
Topic
(1 day = 1 hour)
Concepts and Skills
Assessment for Unit 4 administered by the end of Cycle 9
Unit 4
17 Days

Triangles and
Triangle
Congruence

[Module 7, 5, 6, 8]
Mandatory Performance
Tasks:
6.2 (p. 204)

Triangle Theorems [Module 7]
o Triangle Angle Sum Theorem (using auxiliary lines), Polygon Angle Sum Theorem
(using auxiliary diagonals), Exterior Angle Theorem, Isosceles Triangle Theorem
(Base Angle Theorem), Equilateral Triangle Theorem
Triangle Congruence Theorems [Modules 5 & 6]
o AAS, ASA, SSS, SAS, HL congruence theorems
o Prove triangles congruent using two-column, paragraph, and coordinate proofs
o Use rigid motions to map congruent figures onto one another
o Corresponding Parts of Congruent Figures are Congruent
o Justify constructions using triangle congruence
Inscribed and Circumscribed Triangles [Module 8]
o Construct circumscribed and inscribed circles of a triangle
o Locate circumcenter and incenter of a triangle on the coordinate plane
o Application of incenter and circumcenter
o Theorem: Medians of a Triangle meet at a point (prove using construction)
o Airport Problem
Assessment for Unit 5 administered by the end of Cycle 11
Unit 5
10 Days
Quadrilaterals and
Coordinate Proofs
[Module 9, 10]


Review properties of quadrilaterals
o Parallelogram, rectangle, rhombus, square, trapezoid, isosceles trapezoid, kite
o Discuss sufficient criteria to classify a quadrilateral
Coordinate Proofs
o Use distance formula and midpoint formula
o Review slopes of parallel and perpendicular lines
o Use distance, midpoint, and/or slopes to classify quadrilaterals in the coordinate plane
o Position figures appropriately in the plane and use general coordinates to prove
properties
o Calculate area and perimeter of triangles and quadrilaterals in the coordinate plane
Assessment for Unit 6 administered by the end of Cycle 14
Unit 6
12 Days
Similarity


[Module 11, 12]



Define similarity and create similarity statements
Discuss similarity through the use of dilations:
o Dilations completed from the origin and other points on coordinate plane
o Scale factor and center of dilation to explain similarity
o List corresponding parts (congruent or proportional)
o Finding the center of dilation given a dilation on a coordinate plane
o Prove all circles are similar
o Subdivide a Segment in a Given Ratio (partition a line using different ratios)
Establish Triangle Similarity Postulates
o AA, SAS, SSS (prove using above discussions/activities)
o Prove Triangle Proportionality Theorem (include midsegments & coordinate proofs)
Modeling: Using triangle similarity to solve indirect measurement problems
Similarity in right triangles using altitude (Lesson 12.4)
Geometry Scope and Sequence
Unit
Timing
Topic
(1 day = 1 hour)
Concepts and Skills
Assessment for Unit 7 administered by the end of Cycle 17
Unit 7
17 Days




Trigonometry
[Module 13]


Review & prove Pythagorean Theorem
Prove the Converse to the Pythagorean Theorem
Special right triangles and Pythagorean triples
Use similar triangles to define trigonometric ratios and their inverses in right triangles
o Demonstrate and apply relationship and sine and cosine of complementary angles
o Discuss range of trig ratios (e.g., certain ratios cannot have a value over 1)
Apply trigonometry and Pythagorean Theorem to solve word problems
o Include angles of elevation and depression
Calculate area and perimeter of triangles, using trig to find altitudes
Assessment for Unit 8 administered by the end of Cycle 21
Unit 8
20 Days


Circles

[Lessons 17.1, 15.1-15.3,
15.4 (chords only),
15.5???, 16.2, 16.3]



Review definition of circle and associated vocabulary
Use definition of circle to derive equation
o Complete the square to find radius and center
Review Constructions of inscribed and circumscribed polygons
o Prove properties of angles for a quadrilateral inscribed in a circle (opposite angles are
supplementary)
Identify Circle relationships
o Relationship between inscribed, central, and circumscribed angles
o Inscribed angles on diameter
o Radius is perpendicular to tangent at point of intersection
Find arc length and sector area
o Include application problems
o Define radians
Include informal discussion of area and perimeter of circumscribed polygons (limits, as the
number of sides increases what happens?)
Assessment for Unit 9 administered by the end of Cycle 23
Unit 9
12 Days
Modeling in Three
Dimensions
[Lesson 17.2, Module 18,
19, 20]



Relationships between 2D and 3D Shapes
o Identify nets and review surface area
o Discuss cross sections of conic
o Derive equation of parabola (with directrix & focus)
o Identify 3-D objects created by rotating 2-D objects
Discuss volume using Cavalieri’s Principle
o Informal limit process to find formulas
o Volume of prisms, pyramids, cones and spheres
o Volume of compound figures
Complete application problems with 3-D shapes regarding surface area and volume
o Density; Designing packaging to maximize volume or minimize surface area