Download Geometry Overview

PCTI Mathematics: Geometry Mapping OVERVIEW
Geometry Overview
Moving towards formal mathematical arguments, the standards presented in this high school geometry course
are meant to formalize and extend middle grades geometric experiences. Transformations are presented early
in the year to assist with the building of conceptual understandings of the geometric concepts.
In unit 1 various formats will be used to prove theorems about angles, lines, triangles and other polygons.
In unit 2 triangle congruence conditions are established using analysis of rigid motion and formal constructions
will build on the students understanding of dilations and proportional reasoning to develop a formal
understanding of similarity.
The standards included in unit 3 extend the notion of similarity to right triangles and the understanding of right
triangle trigonometry. In developing the Laws of Sine and Cosines, the students are expected to find missing
measures of triangles in general, not just right triangles.
Work in unit 4 will focus on circles and using the rectangular coordinate system to verify geometric properties
and to solve geometric problems. Concepts of similarity will be used to establish the relationship among
segments on chords, secants and tangents as well as to prove basic theorems about circles. The standards in
this unit will extend previous understandings of two- dimensional objects in order to explain, visualize, and
apply geometric concepts to three-dimensional objects. Informal explanations of circumference, area and
volume formulas will be analyzed.
PCTI Mathematics: Geometry Mapping OVERVIEW
Common Core State Standards Codes addressed in each Quarter
Q1
Q2
Q3
Q4
G.CO.1
G.CO.2
G.CO.3
G.CO.4
G.CO.5
G.CO.6
G.CO.7
G.CO.8
G.CO.9
G.CO.10
G.CO.12
G.C.3
G.GPE.4
G.SRT.1
G.SRT.2
G.C.1
G.CO.2
G.GPE.6
G.GPE.7
G.MG.3
G.SRT.1
G.SRT.2
G.SRT.3
G.SRT.4
G.SRT.5
G.SRT.6
G.SRT.7
G.SRT.8
G.SRT.9
G.SRT.10
G.SRT.11
G.C.1
G.C.2
G.C.3
G.C.4
G.C.5
G.CO.1
G.CO.2
G.CO.3
G.CO.5
G.CO.10
G.CO.11
G.CO.13
G.GMD.1
G.GMD.3
G.GMD.4
G.GPE.1
G.GPE.4
G.GPE.5
G.GPE.7
G.MG.1
G.MG.2
G.MG.3
G.SRT.2
G.SRT.5
G.SRT.8
G.CO.6
G.CO.9
G.CO.12
G.GMD.4
G.GPE.5
G.GPE.7
G.MG.1
G.MG.3
G.SRT.5
PCTI Mathematics: Geometry Mapping OVERVIEW
UNIT #1 : MARKING PERIOD 1 (Essentials of Geometry, Congruence, Proof and Constructions)
TECHNOLOGY STANDARDS
Chapter 1
• http://www.regentsprep.org/Regents/math/geometry/GG1/indexGG1.htm
1.1 GEOMETRY: POINTS, LINES, PLANES
• http://www.wiziq.com/tutorial/74817-GEOMETRY
• http://www.youtube.com/watch?v=JAG1zsIt93U
• http://www.youtube.com/watch?v=GK3h7LzqsUg
1.2 SEGMENTS AND CONGRUENCE
• http://www.youtube.com/watch?v=C6Iacm9scm8
• http://www.youtube.com/watch?v=SoIDx-u3h4Q
1.3 USE MIDPOINT AND DISTANCE FORMULAS
• http://www.youtube.com/watch?v=I7LZg_oTRfo
• http://www.youtube.com/watch?v=86ueEtYRj4c
• http://www.brightstorm.com/math/precalculus/vectors-and-parametricequations/the-midpoint-and-distance-formulas-in-3d/
1.4 MEASURE AND CLASSIFY ANGLES
• http://www.mathplanet.com/education/geometry/points,-lines,-planes-andangles/measure-and-classify-an-angle
1.5 DESCRIBE ANGLE PAIR RELATIONSHIPS• http://www.youtube.com/watch?v=HhphCLlonC8
• http://www.youtube.com/watch?v=yoYeUguWqnc
1.6 CLASSIFY POLYGONS
• http://freevideolectures.com/Course/3176/Introduction-to-Geometry/29
• http://www.youtube.com/watch?v=4NIVYLeGjHQ
Chapter 2
• http://www.regentsprep.org/Regents/math/geometry/GP3/indexGP3.htm
• http://www.regentsprep.org/Regents/math/geometry/GP3b/indexGP3b.htm
• http://mathbits.com/MathBits/MathMovies/ResourceList.htm (use Alice in
Wonderland)
2.1 Use Inductive Reasoning
• www.phschool.com/...demo/ph-346s.html
• www.brightstorm.com/.../reasoning.../inductive...
• www.virtualnerd.com/.../reasoning.../inductive/i...
2.2 Analyze Conditional Statements
KEY VOCABULARY
Chapter 1:
• undefined terms
point, line, plane
• collinear,
coplanar points
• defined terms
• line segment,
endpoints
• ray, opposite
rays
• intersection
postulate, axiom
• coordinate
• distance
• between
• congruent
segments
• midpoint
• segment
bisector
• angle- sides,
vertex, measure
• acute, right,
obtuse, straight
• congruent
angles
• angle bisector
• construction
• complementary
angles
• supplementary
angles
• adjacent angles
• linear pair
• vertical angles
Chapter 2:
• conjecture
• inductive
reasoning
• counterexample
• conditional
statementconverse,
inverse,
contrapositive
• if-them form
hypothesis,
conclusion
• negation
• equivalent
statements
• perpendicular
lines
• biconditional
statement
• deductive
reasoning
• line
perpendicular to
a plane
• proof
• two-column
proof
• theorem
Chapter 3:
• parallel lines
• skew lines
• parallel planes
• transversal
• corresponding
angles
• alternate
interior angles
• alternate
exterior angles
• consecutive
interior angles
• paragraph proof
• slope
• slope-intercept
form
• standard form
• distance form a
point to a line
• www.youtube.com/watch?v=e8atToiXkuA
• www.phschool.com/atschool/.../ph-365s.html
2.3 Deductive Reasoning
• http://www.youtube.com/watch?v=ZTfVIMPV8KY
• http://www.youtube.com/watch?v=g92lP_VdTac
2.4 Use Postulates and Diagrams
• http://www.youtube.com/watch?v=NrSaagr43t0
• http://www.youtube.com/watch?v=c9MVGlFIxog
• http://www.schooltube.com/video/2a9416b403cb4b2bbda5/
2.5 Reason Using Properties from Algebra
• http://www.youtube.com/watch?v=dZH3nNFDYdk
• http://www.youtube.com/watch?v=Ac3aFNtv9AE
2.6 Prove Statements about Segments and Angles
• http://www.youtube.com/watch?v=qp3tvVLqttk
• http://www.youtube.com/watch?v=NZ90lsy0mjE
• http://www.youtube.com/watch?v=GJh13H8-1jk
• http://www.ohschools.k12.oh.us/userfiles/225/classes/72/5per26day2sept28.pdf
2.7 Prove Angle Pair Relationships
• http://www.youtube.com/watch?v=bUwDkGr9Myk
• http://www.youtube.com/watch?v=Bhh5PbQqjCw
• http://www.youtube.com/watch?v=VTN03Q5FtUY
Chapter 3
• http://www.regentsprep.org/Regents/math/geometry/GP8/indexGP8.htm
• http://mathbits.com/MathBits/GSP/ParallelAngles.htm
• polygon side,
vertex
• convex, concave
• n-gon
• equilateral,
equiangular,
regular
PCTI Mathematics: Geometry Mapping OVERVIEW
CLUSTER WITH
INSTRUCTIONAL NOTES
CCSS codes
G.CO.1
Essentials of Geometry
G.CO.12
G.GPE.7
G.MG.1
G.SRT.5
G.GMD.4
Reasoning and Proof
G.CO.9
G.CO.1
G.CO.9
Parallel and Perpendicular
Lines
G.CO.12
G.GPE.5
CCSS Description
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.
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.
Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.★
Use geometric shapes, their measures, and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a cylinder). ★
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and
identify three-dimensional objects generated by rotations of two- dimensional objects.
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.
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.
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.
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
Prove the slope criteria for parallel and perpendicular lines and use them to solve
geometric problems. (e.g. find the equation of a line parallel or perpendicular to a given
line that passes through a given point.
PCTI Mathematics: Geometry Mapping OVERVIEW
UNIT #2: Marking Period 2
TECHNOLOGY STANDARDS
Chapter 9:
• image
• preimage
• isometry
• vector-initial
point, terminal
point, horizontal
component,
vertical
component
• component form
• matrix
• element
• dimensions
• line of reflection
• center of
rotation
• angle of rotation
• glide reflection
• composition of
Chapter 4
transformations
• Teacher Resources:
• line symmetry
o http://education.ti.com/en/timathnspired/us/geometry/triangles
• line of symmetry
o http://www.mathopenref.com/constcopytriangle.html
o http://exchange.smarttech.com/search.html?q=%22congruent%20tria • rotational
symmetry
ngles%22
•
center of
• Student Resources:
symmetry
o http://www.mathopenref.com/congruenttriangles.html
o http://www.regentsprep.org/Regents/math/geometry/GP4/preprooft • scalar
riangles.htm
multiplication
Chapter 9
• http://www.mathsisfun.com/geometry/transformations.html (9.1-9.7
Transformations)
• http://www.mathsisfun.com/algebra/vectors.html (9.1 Vectors)
• http://www.mathwarehouse.com/transformations/dilations/dilations-inmath.php (9.7 Dilations)
• http://www.regentsprep.org/Regents/math/geometry/GT1a/indexGT1a.ht
m
• http://www.regentsprep.org/Regents/math/geometry/GT1/indexGT1.htm
• http://www.regentsprep.org/Regents/math/geometry/GT2/indexGT2.htm
• http://www.regentsprep.org/Regents/math/geometry/GT3/indexGT3.htm
• http://www.regentsprep.org/Regents/math/geometry/GT5/indexGT5.htm
• http://www.regentsprep.org/Regents/math/geometry/GT6/indexGT6.htm
• http://www.regentsprep.org/Regents/math/geometry/MultipleChoiceRevi
ewG/Transformations.htm
• http://mathbits.com/MathBits/TISection/Geometry/Transformations.htm
• http://mathbits.com/MathBits/TISection/Geometry/Transformations2.ht
m
• http://mathbits.com/MathBits/MathMovies/ResourceList.htm (use The
Matrix Revolutions)
o
http://www.regentsprep.org/Regents/math/geometry/GP4/indexGP4
.htm
KEY VOCABULARY
Chapter 4:
• triangle- scalene,
isosceles,
equilateral,
acute, right,
obtuse,
equiangular
• interior angles
• exterior angles
• corollary to a
theorem
• congruent
figures
• corresponding
parts
• rigid motions
• right triangle
legs, hypotenuse
flow proof
isosceles
triangle- legs,
vertex angle,
base, base legs
• transformation
• image
• congruence
transformation
translation,
reflection,
rotation
Chapter 5:
• midsegment of a
triangle
• coordinate proof
• perpendicular
bisector
• equidistant
• concurrent
• point of
concurrency
• circumcenter
• incenter
• median of a
triangle
• centroid
• altitude of a
triangle
• orthocenter
• indirect proof
Chapter 5
• http://www.regentsprep.org/Regents/math/geometry/GP10/indexGP10.h
tm
• http://www.regentsprep.org/Regents/math/geometry/GC3/indexGC3.htm
• http://www.regentsprep.org/Regents/math/geometry/MultipleChoiceRevi
ewG/Constructions.htm
• http://www.regentsprep.org/Regents/math/geometry/GP7/indexGP7.htm
• http://mrlarkins.com/geometry/InteractiveTextbook/Ch05/0501/PH_Geom_ch05-01_Tech.pdf
• http://mrlarkins.com/geometry/InteractiveTextbook/Ch05/0501/PH_Geom_ch05-01_Obj1.html
PCTI Mathematics: Geometry Mapping OVERVIEW
CLUSTER WITH
INSTRUCTIONAL
NOTES
CCSS codes
G.CO.2
G.CO.3
G.CO.4
G.CO.5
Properties of
Transformations
G.CO.6
G.CO.7
G.SRT.1
G.SRT.2
CCSS Description
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).
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
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.
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.
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.
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.
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.
PCTI Mathematics: Geometry Mapping OVERVIEW
G.CO.10
Congruent Triangles
G.CO.7
G.SRT.5
G.CO.8
G.CO.10
G.CO.9
Relationships
Within Triangles
G.C.3
G.GPE.4
G.CO.12
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.
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.
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
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.
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.
Construct the inscribed and circumscribed circles of a triangle, and prove properties of
angles for a quadrilateral inscribed in a circle.
Use coordinates to prove simple geometric theorems algebraically. For example, prove or
disprove that a figure defined by four given points in the coordinate plane is a rectangle;
prove or disprove that the point (1, 33) lies on the circle centered at the origin and
containing the point (0, 2).
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.
PCTI Mathematics: Geometry Mapping OVERVIEW
UNIT #3: Marking Period 3
TECHNOLOGY STANDARDS
Chapter 6
• http://www.mathopenref.com/similarpolygons.html (Similar polygons 6.1)
• http://www.algebra.com/algebra/homework/Triangles/Geometry-Similar-Triangles.lesson
(similar triangles 6.1)
• http://mrlarkins.com/geometry/InteractiveTextbook/Ch08/08-02/PH_Geom_ch0802_Gizmo.html (Interactive show scale factor and perimeter and area 6.1)
• http://www.mathopenref.com/similartriangles.html (interactive with notes on 6.3 & 6.4)
• http://www.mathopenref.com/similaraaa.html (6.3 AA)
• http://www.mathopenref.com/similarsas.html (6.4 SAS)
• http://www.mathopenref.com/similarsss.html (6.4 SSS)
• http://isite.lps.org/mjames2/resources/geogebra/geometry/c8_6_triangle_prop_ext.html (Using
proportionality theorems (parallel lines 6.5)
• http://isite.lps.org/mjames2/resources/geogebra/geometry/c06_5_angle_bisector_proportionalit
y_proof.html (Using proportionality theorems (angle bisector 6.5)
• http://insidemathematics.org/common-core-math-tasks/high-school/HS-G2006%20Hopewell%20Geometry.pdf
• http://www.regentsprep.org/Regents/math/geometry/GP11/indexGP11.htm
Chapter 7
• http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html (Demo Pythagorean Theorem 7.1)
• http://www.youtube.com/watch?v=CAkMUdeB06o (Demo Pythagorean Theorem 7.1)
• http://www.brainingcamp.com/resources/math/pythagorean-formula/lesson.php (Pythagorean
Theorem lesson, interaction, questions and real life applications 7.1)
• http://www.phschool.com/atschool/academy123/english/academy123_content/wl-bookdemo/ph-174s.html (Lesson on converse of Pythagorean Theorem 7.2)
• http://www.mathwarehouse.com/geometry/similar/triangles/interactive_similar_right_triangles.
html (Separates similar triangles 7.3)
• http://mrlarkins.com/geometry/InteractiveTextbook/Ch08/08-04/PH_Geom_ch0804_Gizmo.html (Separates similar triangles 7.3)
• http://www.youtube.com/watch?v=l6LUOVmix0c (Tutorial 45-45-90 7.4)
• http://www.phschool.com/atschool/academy123/english/academy123_content/wl-bookdemo/ph-112s.html (Tutorial 45-45-90 7.4)
• http://www.mathopenref.com/triangle454590.html (demo 45-45-90 7.4)
• http://www.youtube.com/watch?v=PI68l1FPRkU (Tutorial 30-60-90 7.4)
KEY VOCABULARY
Chapter 6:
• dilation
• scale factor of
dilation
• similar polygons
• scale factor of
two similar
polygons
• center of dilation
• reduction
• enlargement
Chapter 7:
• Pythagorean
triple
• trigonometric
ratio
• tangent
• sine
• cosine
• angle of
elevation
• angle of
depression
• solve a right
triangle
• inverse tangent
• inverse sine
• inverse cosine
•
•
•
•
•
•
http://www.mathopenref.com/triangle306090.html (demo 30-60-90 7.4)
http://www.mathopenref.com/trigtangent.html (demo tangent ratio 7.5)
http://www.youtube.com/watch?v=LvUsW21drOQ (tutorial tangent ration 7.5)
http://www.mathopenref.com/cosine.html (demo cosine ratio 7.6)
http://www.mathopenref.com/sine.html (demo sine ratio 7.6)
http://www.phschool.com/atschool/academy123/english/academy123_content/wl-bookdemo/ph-115s.html (tutorial all trig ratios 7.5-7.6)
• http://learni.st/users/60/boards/3457-law-of-sines-cosines-common-core-standard-9-12-g-srt10#/users/60/boards/3457-law-of-sines-cosines-common-core-standard-9-12-g-srt-10 (tutorial
law of sine and cosine)
• http://www.regentsprep.org/Regents/math/geometry/GP13/indexGP13.htm
• http://www.themathpage.com/aTrig/law-of-cosines.htm
• http://www.themathpage.com/aTrig/law-of-sines.htm
• http://www.mathwarehouse.com/trigonometry/law-of-sines-and-cosines.php
• http://illuminations.nctm.org/LessonDetail.aspx?ID=U177
• http://mathbits.com/MathBits/MathMovies/ResourceList.htm (use Star Wars, episode I)
• http://mathbits.com/MathBits/MathMovies/ResourceList.htm ( use The Englishman Who went
up a Hill but came down a Mountain)
CLUSTER WITH
CCSS codes
CCSS Description
INSTRUCTIONAL NOTES
G.C.1
G.GPE.6
G.MG.3
G.SRT.1
Similarity
G.SRT.2
G.SRT.3
G.SRT.4
Prove that all circles are similar.
Find the point on a directed line segment between two given points that partitions the
segment in a given ratio.
Apply geometric methods to solve design problems (e.g., designing an object or structure to
satisfy physical constraints or minimize cost; working with typographic grid systems based
on ratios). ★
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.
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.
Use the properties of similarity transformations to establish the AA criterion for two
triangles to be similar.
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.
G.SRT.5
G.GPE.7
G.SRT.4
G.SRT.5
Right Triangles and
Trigonometry
G.SRT.6
G.SRT.7
G.SRT.8
G.SRT.9
G.SRT.10
G.SRT.11
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g.,
using the distance formula.★
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.
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
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.
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems. ★
(+) Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line
from a vertex perpendicular to the opposite side.
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
(+) 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).
PCTI Mathematics: Geometry Mapping OVERVIEW
UNIT #4: Marking Period 4
TECHNOLOGY STANDARDS
Chapter 8:
• http://www.mathsisfun.com/geometry/interior-angles-polygons.html
• http://www.regentsprep.org/Regents/math/geometry/GP9/LParallelogram.htm
• http://www.jmap.org/htmltopics/JMAP_BY_TOPIC_POLYGONS.htm
• http://mathbits.com/MathBits/GSP/ExamAnglesPolygons.pdf (investigate angles of
polygons)
• http://insidemathematics.org/common-core-math-tasks/high-school/HS-G2005%20Quadrilaterals.pdf
• http://www.regentsprep.org/Regents/math/geometry/MultipleChoiceReviewG/Quadrilatera
ls.htm
• http://www.regentsprep.org/Regents/math/geometry/GG3/indexGG3.htm
Chapter 10
• http://insidemathematics.org/common-core-math-tasks/high-school/HS-G2007%20Circles%20in%20Triangles.pdf
• http://insidemathematics.org/common-core-math-tasks/high-school/HS-G2008%20Circle%20and%20Squares.pdf
• http://www.regentsprep.org/Regents/math/geometry/GP14/indexGP14.htm
• http://www.regentsprep.org/Regents/math/geometry/GP15/indexGP15.htm
• http://www.regentsprep.org/Regents/math/geometry/GP16/indexGP16.htm
• http://www.regentsprep.org/Regents/math/geometry/MultipleChoiceReviewG/Circles.htm
• http://www.regentsprep.org/Regents/math/geometry/GCG6/indexGCG6.htm
10.1 Use Properties of Tangents
• http://www.youtube.com/watch?v=R0lJ6WJaiW0
• http://www.youtube.com/watch?v=Zdhizxjwhqw
• http://www.youtube.com/watch?v=RnmaOwjej58
10.2 Find Arc Measures
• http://www.youtube.com/watch?v=61maaJXZT2U
• http://www.youtube.com/watch?v=oUySgjAlujU
10.3 Apply Properties of Chords
• http://www.youtube.com/watch?v=N1lh8asMIzk
• http://www.youtube.com/watch?v=W75_989AMdo
10.4 Use Inscribed Angles and Polygons
KEY VOCABULARY
Chapter 8:
• diagonal
• parallelogra
m
• rhombus
• rectangle
• square
• trapezoid
• bases of a
trapezoid
• base angles
of a
trapezoid
• legs of a
trapezoid
isosceles
trapezoid
midsegment
of a
trapezoid
• kite
Chapter 10:
• circlecenter,
radius,
diameter
• chord
• secant
• tangent
• central
angle
• minor arc
• major arc
• semicircle
• measure of
a minor arc
• measure of
a major arc
• congruent
circles
• congruent
arcs
• inscribed
angle
• intercepted
arc
• inscribed
polygon
• circumscrib
ed circle
• segments of
a chord
• secant
Chapter 11:
• circumferen
ce
• arc length
• radian
• sector of a
circle
• center of a
polygon
• radius of a
polygon
• apothem of
a polygon
• central
angle of a
regular
polygon
• probability
• geometric
pr0obability
• polyhedronface, edge,
vertex, base
• regular
polyhedron
• convex
polyhedron
• platonic
solids
• cross
section
• volume
• http://www.youtube.com/watch?v=h5S9xx1V88A
• http://www.mathplanet.com/education/geometry/circles/inscribed-angles-and-polygons
• http://prezi.com/jtp5toxmjqao/inscribed-angles-and-polygons/
10.5 Apply Other Angle Relationships in Circles
• http://www.youtube.com/watch?v=6E-wZGV7Ew8
• http://www.youtube.com/watch?v=Z8xLDVyIrko
• http://www.youtube.com/watch?v=gMbXXailkpc
10.6 Write and Graph Equations of Circles
• http://www.youtube.com/watch?v=9rLD8STilo4
• http://www.youtube.com/watch?v=J8aOmJhQtnY
• http://www.youtube.com/watch?v=unh2kbaG5Cc
Chapter 11
• http://www.learner.org/interactives/geometry/platonic.html
• http://www.regentsprep.org/Regents/math/geometry/GG2/indexGG2.htm
• http://mathbits.com/MathBits/MathMovies/ResourceListTwo.html (use Wall-E)
segment
• external
segment
• standard
equation of
a circle
• density
• solids of
revolution
• axis of
revolution
• spherecenter,
radius,
chord,
diameter
• great circle
• hemisphere
• similar
solids
• plane, axis
of symmetry
PCTI Mathematics: Geometry Mapping OVERVIEW
CLUSTER WITH
INSTRUCTIONAL NOTES
CCSS codes
G.MG.1
G.SRT.5
Quadrilaterals
G.CO.11
G.GPE.7
G.GPE.4
G.C.2
G.C.3
G.C.4
G.CO.1
Properties of Circles
G.CO.12
G.C0.13
G.GPE.1
G.C.5
CCSS Description
Use geometric shapes, their measures, and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a cylinder).*
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
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.
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles,
e.g., using the distance formula.˒
Use coordinates to prove simple geometric theorems algebraically. For example, prove or
disprove that a figure defined by four given points in the coordinate plane is a rectangle;
prove or disprove that the point (1, 33) lies on the circle centered at the origin and
containing the point (0, 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.
Construct the inscribed and circumscribed circles of a triangle, and prove properties of
angles for a quadrilateral inscribed in a circle.
(+) Construct a tangent line from a point outside a given circle to the circle.
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.
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.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
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.
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.
G.C.5
G.CO.13
G.GMD.1
G.GMD.3
Measures of Figures
and Solids
G.GMD.4
G.MG.2
G.MG.3
G.SRT.2
G.SRT.8
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.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Give an informal argument for the formulas for the circumference of a circle, area of a
circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s
principle, and informal limit arguments.
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and
identify three-dimensional objects generated by rotations of two-dimensional objects.
Apply concepts of density based on area and volume in modeling situations (e.g., persons
per square mile, BTUs per cubic foot). ★
Apply geometric methods to solve design problems (e.g., designing an object or structure
to satisfy physical constraints or minimize cost; working with typographic grid systems
based on ratios). ★
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.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.˒
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
UNIT: 1
TECHNOLOGY STANDARDS
Chapter 1
• http://www.regentsprep.org/Regents/math/geometry/GG1/in
dexGG1.htm
1.1 GEOMETRY: POINTS, LINES, PLANES
• http://www.wiziq.com/tutorial/74817-GEOMETRY
• http://www.youtube.com/watch?v=JAG1zsIt93U
• http://www.youtube.com/watch?v=GK3h7LzqsUg
1.2 SEGMENTS AND CONGRUENCE
• http://www.youtube.com/watch?v=C6Iacm9scm8
• http://www.youtube.com/watch?v=SoIDx-u3h4Q
1.3 USE MIDPOINT AND DISTANCE FORMULAS
• http://www.youtube.com/watch?v=I7LZg_oTRfo
• http://www.youtube.com/watch?v=86ueEtYRj4c
• http://www.brightstorm.com/math/precalculus/vectors-andparametric-equations/the-midpoint-and-distance-formulas-in3d/
1.4 MEASURE AND CLASSIFY ANGLES
• http://www.mathplanet.com/education/geometry/points,lines,-planes-and-angles/measure-and-classify-an-angle
1.5 DESCRIBE ANGLE PAIR RELATIONSHIPS• http://www.youtube.com/watch?v=HhphCLlonC8
• http://www.youtube.com/watch?v=yoYeUguWqnc
1.6 CLASSIFY POLYGONS
• http://freevideolectures.com/Course/3176/Introduction-toGeometry/29
• http://www.youtube.com/watch?v=4NIVYLeGjHQ
Chapter 2
• http://www.regentsprep.org/Regents/math/geometry/GP3/in
dexGP3.htm
• http://www.regentsprep.org/Regents/math/geometry/GP3b/i
ndexGP3b.htm
• http://mathbits.com/MathBits/MathMovies/ResourceList.htm
UNIT NAME: Essentials of Geometry, Reasoning and Proof, Parallel and
Perpendicular Lines
KEY VOCABULARY
Chapter 1:
• undefined terms point, line,
plane
• collinear, coplanar points
• defined terms
• line segment, endpoints
• ray, opposite rays
• intersection postulate,
axiom
• coordinate
• distance
• between
• congruent segments
• midpoint
• segment bisector
• angle- sides, vertex,
measure
• acute, right, obtuse,
straight
• congruent angles
• angle bisector
• construction
• complementary angles
• supplementary angles
• adjacent angles
• linear pair
• vertical angles
• polygon side, vertex
• convex, concave
• n-gon
• equilateral, equiangular,
regular
Chapter 2:
• conjecture
• inductive reasoning
• counterexample
• conditional
statementconverse, inverse,
contrapositive
• if-them form
hypothesis,
conclusion
• negation
• equivalent
statements
• perpendicular lines
• biconditional
statement
• deductive reasoning
• line perpendicular
to a plane
• proof
• two-column proof
• theorem
Chapter 3:
• parallel lines
• skew lines
• parallel planes
• transversal
• corresponding
angles
• alternate interior
angles
• alternate exterior
angles
• consecutive interior
angles
• paragraph proof
• slope
• slope-intercept
form
• standard form
• distance form a
point to a line
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
UNIT: 1
(use Alice in Wonderland)
2.1 Use Inductive Reasoning
• www.phschool.com/...demo/ph-346s.html
• www.brightstorm.com/.../reasoning.../inductive...
• www.virtualnerd.com/.../reasoning.../inductive/i...
2.2 Analyze Conditional Statements
• www.youtube.com/watch?v=e8atToiXkuA
• www.phschool.com/atschool/.../ph-365s.html
2.3 Deductive Reasoning
• http://www.youtube.com/watch?v=ZTfVIMPV8KY
• http://www.youtube.com/watch?v=g92lP_VdTac
2.4 Use Postulates and Diagrams
• http://www.youtube.com/watch?v=NrSaagr43t0
• http://www.youtube.com/watch?v=c9MVGlFIxog
• http://www.schooltube.com/video/2a9416b403cb4b2bbda5/
2.5 Reason Using Properties from Algebra
• http://www.youtube.com/watch?v=dZH3nNFDYdk
• http://www.youtube.com/watch?v=Ac3aFNtv9AE
2.6 Prove Statements about Segments and Angles
• http://www.youtube.com/watch?v=qp3tvVLqttk
• http://www.youtube.com/watch?v=NZ90lsy0mjE
• http://www.youtube.com/watch?v=GJh13H8-1jk
• http://www.ohschools.k12.oh.us/userfiles/225/classes/72/5p
er2-6day2sept28.pdf
2.7 Prove Angle Pair Relationships
• http://www.youtube.com/watch?v=bUwDkGr9Myk
• http://www.youtube.com/watch?v=Bhh5PbQqjCw
• http://www.youtube.com/watch?v=VTN03Q5FtUY
Chapter 3
• http://www.regentsprep.org/Regents/math/geometry/GP8/in
dexGP8.htm
• http://mathbits.com/MathBits/GSP/ParallelAngles.htm
UNIT NAME: Essentials of Geometry, Reasoning and Proof, Parallel and
Perpendicular Lines
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
#
I.
II.
III.
UNIT NAME: Essentials of Geometry, Reasoning and Proof, Parallel and
UNIT: 1
TOPICS
Essentials of Geometry
(1.1-1.6; 14 Days)
Reasoning and Proof
(2.2,2.4,2.5,2.6,2.7; 14 days)
Parallel and Perpendicular Lines
(3.1-3.4,3.6; 14 days )
Perpendicular Lines
#
STUDENT LEARNING OBJECTIVES
1
Identify points, lines and planes
2
Use segments and congruence
3
4
5
6
7
Use midpoint and distance formulas
Measure and classify angles, constructions
Construction: Bisect Segments and Angles
Describe angle relationships
Classify polygons
1
2
3
4
5
Analyze conditional statements (2.2)
Use postulates and diagrams (2.4)
Using properties from algebra (2.5)
Prove statements about segments and angles (2.6)
Prove angle pair relationships (2.7)
1
2
3
4
Identify pairs of lines and angles
Use parallel lines and transversals
Prove lines parallel
Find and use slopes of lines
5
Prove theorems about perpendicular lines
6
Construction: Parallel and Perpendicular Lines
CCSS code
G.CO.1
G.CO.1
G.GPE.7; G.MG.3
G.CO.1; G.CO.12
G.CO.12
G.CO.1; G.CO.9
G.GMD.4; G.MG.1
G.CO.9
G.CO.9; G.GMD.4
G.CO.9
G.CO.9; G.SRT.5
G.CO.9
G.CO.1; G.CO.9
G.CO.9
G.CO.9
G.GPE.5
G.CO.9; G.CO.12;
G.GPE.5
G.CO.12
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
UNIT: 1
UNIT NAME: Essentials of Geometry, Reasoning and Proof, Parallel and
Perpendicular Lines
Selected Opportunities for Connections to
Mathematical Practices
1.
2.
3.
4.
5.
6.
7.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Unit 1-Links to Open Ended Problems
Related to the Standards
G.CO.1
G.CO.6
G.CO.9
G.CO.12
G.GMD.4
G.GPE.5
G.GPE.7
G.MG.1
G.MG.3
G.SRT.5
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
CCSS Code
G.CO.1
G.CO.9
G.CO.12
G.GMD.4
G.GPE.5
G.GPE.7
G.MG.1
G.MG.3
G.SRT.5
UNIT: 1
UNIT NAME: Essentials of Geometry, Reasoning and Proof, Parallel and
Perpendicular Lines
CCSS Code DESRCIPTION
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.
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.
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.
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects
generated by rotations of two-dimensional objects.
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. (E.g. find the equation
of a line parallel or perpendicular to a given line that passes through a given point.
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso
as a cylinder).
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or
minimize cost; working with typographic grid systems based on ratios).
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
UNIT: 2
TECHNOLOGY STANDARDS
UNIT NAME:
Properties of Transformations, Congruent Triangles,
Relationships Within Triangles
Chapter 9
• http://www.mathsisfun.com/geometry/transformations.html (9.1-9.7 Transformations)
• http://www.mathsisfun.com/algebra/vectors.html (9.1 Vectors)
• http://www.mathwarehouse.com/transformations/dilations/dilations-in-math.php (9.7
Dilations)
• http://www.regentsprep.org/Regents/math/geometry/GT1a/indexGT1a.htm
• http://www.regentsprep.org/Regents/math/geometry/GT1/indexGT1.htm
• http://www.regentsprep.org/Regents/math/geometry/GT2/indexGT2.htm
• http://www.regentsprep.org/Regents/math/geometry/GT3/indexGT3.htm
• http://www.regentsprep.org/Regents/math/geometry/GT5/indexGT5.htm
• http://www.regentsprep.org/Regents/math/geometry/GT6/indexGT6.htm
• http://www.regentsprep.org/Regents/math/geometry/MultipleChoiceReviewG/Transform
ations.htm
• http://mathbits.com/MathBits/TISection/Geometry/Transformations.htm
• http://mathbits.com/MathBits/TISection/Geometry/Transformations2.htm
• http://mathbits.com/MathBits/MathMovies/ResourceList.htm (use The Matrix Revolutions)
Chapter 4
• Teacher Resources:
o http://education.ti.com/en/timathnspired/us/geometry/triangles
o http://www.mathopenref.com/constcopytriangle.html
o http://exchange.smarttech.com/search.html?q=%22congruent%20triangles%22
• Student Resources:
o http://www.mathopenref.com/congruenttriangles.html
o http://www.regentsprep.org/Regents/math/geometry/GP4/preprooftriangles.htm
o http://www.regentsprep.org/Regents/math/geometry/GP4/indexGP4.htm
Chapter 5
• http://www.regentsprep.org/Regents/math/geometry/GP10/indexGP10.htm
• http://www.regentsprep.org/Regents/math/geometry/GC3/indexGC3.htm
• http://www.regentsprep.org/Regents/math/geometry/MultipleChoiceReviewG/Constructi
ons.htm
• http://www.regentsprep.org/Regents/math/geometry/GP7/indexGP7.htm
• http://mrlarkins.com/geometry/InteractiveTextbook/Ch05/05-01/PH_Geom_ch0501_Tech.pdf
• http://mrlarkins.com/geometry/InteractiveTextbook/Ch05/05-01/PH_Geom_ch0501_Obj1.html
Chapter 9:
• image
• preimage
• isometry
• vector-initial
point, terminal
point,
horizontal
component,
vertical
component
• component
form
• matrix
• element
• dimensions
• line of
reflection
• center of
rotation
• angle of
rotation
• glide reflection
• composition of
transformatio
ns
• line symmetry
• line of
symmetry
• rotational
symmetry
• center of
symmetry
• scalar
multiplication
KEY VOCABULARY
Chapter 4:
• trianglescalene,
isosceles,
equilateral,
acute, right,
obtuse,
equiangular
• interior angles
• exterior angles
• corollary to a
theorem
• congruent
figures
• corresponding
parts
• rigid motions
• right triangle
legs,
hypotenuse
flow proof
isosceles
triangle- legs,
vertex angle,
base, base legs
• transformatio
n
• image
• congruence
transformatio
n translation,
reflection,
rotation
Chapter 5:
• midsegment of
a triangle
• coordinate
proof
• perpendicular
bisector
• equidistant
• concurrent
• point of
concurrency
• circumcenter
• incenter
• median of a
triangle
• centroid
• altitude of a
triangle
• orthocenter
• indirect proof
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
#
IV.
V.
VI.
UNIT NAME:
UNIT: 2
TOPICS
Properties of Transformations (Ch. 9)
(9.1,9.3-9.7,4.3; 13 days)
Congruent Triangles (Ch. 4)
(4.1-4.7; 14 days)
#
Properties of Transformations, Congruent Triangles,
Relationships Within Triangles
STUDENT LEARNING OBJECTIVES
1
2
3
4
Translate figures and use vectors
Perform reflections
Perform rotations
Apply compositions of transformations
5
Relate transformations and congruence (section 4.3)
6
Identify symmetry
7
Identify and perform dilations
1
Apply triangle sum properties
2
Apply congruence and triangles
3
Prove triangles congruent by SSS
4
Prove triangles congruent by SAS and HL
5
Prove triangles congruent by ASA and AAS
6
Use congruent triangles
7
Use isosceles and equilateral triangles
1
Midsegment Theorem and coordinate proof
Relationships within Triangles (Ch. 5)
(5.1-5.3,5.5-5.6; 12 days)
CCSS code
G.CO.4; G.CO.5
G.CO.2; G.CO.4
G.CO.4; G.CO.5
G.CO.2; G.CO.5
G.CO.2; G.CO.6;
G.CO.7
G.CO.3
G.CO.2; G.CO.5;
G.SRT.1;
G.SRT.2
G.CO.10
G.CO.7;
G.SRT.5
G.CO.8;
G.CO.10;
G.CO.12
G.CO.8;
G.CO.10
G.CO.8;
G.CO.10
G.CO.10;
G.CO.12;
G.CO.10
G.CO.10;
G.GPE.4
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
UNIT NAME:
UNIT: 2
2
Relationships Within Triangles
G.CO.9;
G.CO.12; G.C.3
G.CO.10; G.C.3
G.CO.10;
G.CO.12
G.CO.7;
G.CO.10
G.CO.7;
G.CO.10
Use perpendicular bisectors
3
Use angle bisectors of triangles
4
Use medians and altitudes
5
Use inequalities in a triangle
6
Inequalities in two triangles and indirect proof
Selected Opportunities for Connections to
Mathematical Practices
1.
2.
3.
4.
5.
6.
7.
Properties of Transformations, Congruent Triangles,
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Unit 2-Links to Open Ended Problems
Related to the Standards
G.CO.2
G.CO.3
G.CO.4
G.CO.5
G.CO.6
G.CO.7
G.CO.8
G.CO.9
G.CO.10
G.CO.12
G.C.3
G.GPE.4
G.SRT.1
G.SRT.2
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
CCSS Code
G.C.3
G.CO.2
G.CO.3
G.CO.4
G.CO.5
G.CO.6
G.CO.7
G.CO.8
G.CO.9
G.CO.10
G.CO.12
G.GPE.4
G.SRT.1
UNIT: 2
UNIT NAME:
Properties of Transformations, Congruent Triangles,
Relationships Within Triangles
CCSS Code DESRCIPTION
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
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).
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and
line segments.
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.
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.
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.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid
motions.
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.
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.
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.
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four
given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the
origin and containing the point (0, 2).
Verify experimentally the properties of dilations given by a center and a scale factor.
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
G.SRT.2
G.SRT.5
UNIT: 2
UNIT NAME:
Properties of Transformations, Congruent Triangles,
Relationships Within Triangles
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.
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.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
UNIT: 3
TECHNOLOGY STANDARDS
UNIT NAME:
Similarity, Right Triangles and Trigonometry
Chapter 6
• http://www.mathopenref.com/similarpolygons.html (Similar polygons 6.1)
• http://www.algebra.com/algebra/homework/Triangles/Geometry-SimilarTriangles.lesson (similar triangles 6.1)
• http://mrlarkins.com/geometry/InteractiveTextbook/Ch08/08-02/PH_Geom_ch0802_Gizmo.html (Interactive show scale factor and perimeter and area 6.1)
• http://www.mathopenref.com/similartriangles.html (interactive with notes on 6.3 &
6.4)
• http://www.mathopenref.com/similaraaa.html (6.3 AA)
• http://www.mathopenref.com/similarsas.html (6.4 SAS)
• http://www.mathopenref.com/similarsss.html (6.4 SSS)
• http://isite.lps.org/mjames2/resources/geogebra/geometry/c8_6_triangle_prop_ext.ht
ml (Using proportionality theorems (parallel lines 6.5)
• http://isite.lps.org/mjames2/resources/geogebra/geometry/c06_5_angle_bisector_pro
portionality_proof.html (Using proportionality theorems (angle bisector 6.5)
• http://insidemathematics.org/common-core-math-tasks/high-school/HS-G2006%20Hopewell%20Geometry.pdf
• http://www.regentsprep.org/Regents/math/geometry/GP11/indexGP11.htm
Chapter 7
• http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html (Demo Pythagorean
Theorem 7.1)
• http://www.youtube.com/watch?v=CAkMUdeB06o (Demo Pythagorean Theorem 7.1)
• http://www.brainingcamp.com/resources/math/pythagorean-formula/lesson.php
(Pythagorean Theorem lesson, interaction, questions and real life applications 7.1)
• http://www.phschool.com/atschool/academy123/english/academy123_content/wlbook-demo/ph-174s.html (Lesson on converse of Pythagorean Theorem 7.2)
• http://www.mathwarehouse.com/geometry/similar/triangles/interactive_similar_right
_triangles.html (Separates similar triangles 7.3)
• http://mrlarkins.com/geometry/InteractiveTextbook/Ch08/08-04/PH_Geom_ch0804_Gizmo.html (Separates similar triangles 7.3)
• http://www.youtube.com/watch?v=l6LUOVmix0c (Tutorial 45-45-90 7.4)
• http://www.phschool.com/atschool/academy123/english/academy123_content/wlbook-demo/ph-112s.html (Tutorial 45-45-90 7.4)
• http://www.mathopenref.com/triangle454590.html (demo 45-45-90 7.4)
KEY VOCABULARY
Chapter 6:
• dilation
• scale factor of
dilation
• similar polygons
• scale factor of two
similar polygons
• center of dilation
• reduction
• enlargement
Chapter 7:
•
•
•
•
•
•
•
•
•
•
•
Pythagorean triple
trigonometric ratio
tangent
sine
cosine
angle of elevation
angle of depression
solve a right triangle
inverse tangent
inverse sine
inverse cosine
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
UNIT: 3
UNIT NAME:
Similarity, Right Triangles and Trigonometry
http://www.youtube.com/watch?v=PI68l1FPRkU (Tutorial 30-60-90 7.4)
http://www.mathopenref.com/triangle306090.html (demo 30-60-90 7.4)
http://www.mathopenref.com/trigtangent.html (demo tangent ratio 7.5)
http://www.youtube.com/watch?v=LvUsW21drOQ (tutorial tangent ration 7.5)
http://www.mathopenref.com/cosine.html (demo cosine ratio 7.6)
http://www.mathopenref.com/sine.html (demo sine ratio 7.6)
http://www.phschool.com/atschool/academy123/english/academy123_content/wlbook-demo/ph-115s.html (tutorial all trig ratios 7.5-7.6)
http://learni.st/users/60/boards/3457-law-of-sines-cosines-common-core-standard-912-g-srt-10#/users/60/boards/3457-law-of-sines-cosines-common-core-standard-9-12g-srt-10 (tutorial law of sine and cosine)
http://www.regentsprep.org/Regents/math/geometry/GP13/indexGP13.htm
http://www.themathpage.com/aTrig/law-of-cosines.htm
http://www.themathpage.com/aTrig/law-of-sines.htm
http://www.mathwarehouse.com/trigonometry/law-of-sines-and-cosines.php
http://illuminations.nctm.org/LessonDetail.aspx?ID=U177
http://mathbits.com/MathBits/MathMovies/ResourceList.htm (use Star Wars, episode
I)
http://mathbits.com/MathBits/MathMovies/ResourceList.htm ( use The Englishman
Who went up a Hill but came down a Mountain)
Law of Sines and Law of Cosines
•
•
•
http://www.themathpage.com/atrig/law-of-cosines.htm
http://www.themathpage.com/atrig/law-of-sines.htm
http://library.thinkquest.org/C0121962/sincoslaws.htm
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
#
VII.
UNIT: 3
TOPICS
Similarity (Ch. 6)
(6.3-6.6; 14days)
VIII.
UNIT NAME:
#
Similarity, Right Triangles and Trigonometry
STUDENT LEARNING OBJECTIVES
1
Use similar polygons
2
Relate transformations and similarity
3
4
Prove triangles similar by AA
Prove triangles similar by SSS and SAS
5
Use proportionality theorems
1
Apply the Pythagorean Theorem
2
Use the Converse of the Pythagorean Theorem
3
Use similar right triangles
4
Special Right triangles
5
Apply the Tangent ratio
6
Apply the Sine and Cosine ratios
7
Solve right triangles
Extension: Law of Sines and Law of Cosines
(see technology links above)
Right Triangles and Trigonometry (Ch. 7)
(7.1-7.7; 20 days)
8
CCSS code
G.SRT.5
G.CO.2;
G.SRT.1;
G.SRT.2; G.C.1
G.SRT.3
G.SRT.4
G.SRT.4;
G.SRT.5;
G.GPE.6;
G.MG.3
G.SRT.4;
G.SRT.8;
G.GPE.7
G.SRT.8
G.SRT.4;
G.SRT.5
G.SRT.6
G.SRT.6;
G.SRT.8
G.SRT.6;
G.SRT.7;
G.SRT.8;
G.SRT.9
G.SRT.8
G.SRT.10;
G.SRT.11
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
UNIT: 3
UNIT NAME:
Selected Opportunities for Connections to
Mathematical Practices
1.
2.
3.
4.
5.
6.
7.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Similarity, Right Triangles and Trigonometry
Unit 3-Links to Open Ended Problems
Related to the Standards
G.C.1
G.CO.2
G.GPE.6
G.GPE.7
G.MG.3
G.SRT.1
G.SRT.2
G.SRT.3
G.SRT.4
G.SRT.5
G.SRT.6
G.SRT.7
G.SRT.8
G.SRT.9
G.SRT.10
G.SRT.11
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
CCSS Code
G.C.1
G.CO.2
G.GPE.6
G.GPE.7
G.MG.3
G.SRT.1
G.SRT.2
G.SRT.3
G.SRT.4
G.SRT.5
G.SRT.6
G.SRT.7
G.SRT.8
G.SRT.9
G.SRT.10
G.SRT.11
UNIT: 3
UNIT NAME:
Similarity, Right Triangles and Trigonometry
CCSS Code DESRCIPTION
Prove that all circles are similar.
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).
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems
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.
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.
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
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.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
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.
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
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.
Prove the Laws of Sines and Cosines and use them to solve problems.
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).
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
UNIT: 3
UNIT NAME:
Similarity, Right Triangles and Trigonometry
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
TECHNOLOGY STANDARDS
UNIT: 4
UNIT NAME:
Solids
Chapter 8:
• http://www.mathsisfun.com/geometry/interior-anglespolygons.html
• http://www.regentsprep.org/Regents/math/geometry/GP9/LParall
elogram.htm
• http://www.jmap.org/htmltopics/JMAP_BY_TOPIC_POLYGONS.ht
m
• http://mathbits.com/MathBits/GSP/ExamAnglesPolygons.pdf
(investigate angles of polygons)
• http://insidemathematics.org/common-core-math-tasks/highschool/HS-G-2005%20Quadrilaterals.pdf
• http://www.regentsprep.org/Regents/math/geometry/MultipleCh
oiceReviewG/Quadrilaterals.htm
• http://www.regentsprep.org/Regents/math/geometry/GG3/index
GG3.htm
Chapter 10
• http://insidemathematics.org/common-core-math-tasks/highschool/HS-G-2007%20Circles%20in%20Triangles.pdf
• http://insidemathematics.org/common-core-math-tasks/highschool/HS-G-2008%20Circle%20and%20Squares.pdf
• http://www.regentsprep.org/Regents/math/geometry/GP14/index
GP14.htm
• http://www.regentsprep.org/Regents/math/geometry/GP15/index
GP15.htm
• http://www.regentsprep.org/Regents/math/geometry/GP16/index
GP16.htm
• http://www.regentsprep.org/Regents/math/geometry/MultipleCh
oiceReviewG/Circles.htm
• http://www.regentsprep.org/Regents/math/geometry/GCG6/inde
xGCG6.htm
Quadrilaterals, Properties of Circles, Measures of Figures and
Chapter 8:
• diagonal
• parallelogram
• rhombus
• rectangle
• square
• trapezoid
• bases of a trapezoid
• base angles of a
trapezoid
• legs of a trapezoid
isosceles trapezoid
midsegment of a
trapezoid
• kite
KEY VOCABULARY
Chapter 10:
• circle- center,
radius, diameter
• chord
• secant
• tangent
• central angle
• minor arc
• major arc
• semicircle
• measure of a minor
arc
• measure of a major
arc
• congruent circles
• congruent arcs
• inscribed angle
• intercepted arc
• inscribed polygon
• circumscribed circle
• segments of a chord
• secant segment
• external segment
• standard equation of
a circle
Chapter 11:
• circumference
• arc length
• radian
• sector of a circle
• center of a polygon
• radius of a polygon
• apothem of a polygon
• central angle of a regular
polygon
• probability
• geometric pr0obability
• polyhedron- face, edge,
vertex, base
• regular polyhedron
• convex polyhedron
• platonic solids
• cross section
• volume
• density
• solids of revolution
• axis of revolution
• sphere- center, radius,
chord, diameter
• great circle
• hemisphere
• similar solids
• plane, axis of symmetry
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
UNIT: 4
UNIT NAME:
Solids
10.1 Use Properties of Tangents
• http://www.youtube.com/watch?v=R0lJ6WJaiW0
• http://www.youtube.com/watch?v=Zdhizxjwhqw
• http://www.youtube.com/watch?v=RnmaOwjej58
10.2 Find Arc Measures
• http://www.youtube.com/watch?v=61maaJXZT2U
• http://www.youtube.com/watch?v=oUySgjAlujU
10.3 Apply Properties of Chords
• http://www.youtube.com/watch?v=N1lh8asMIzk
• http://www.youtube.com/watch?v=W75_989AMdo
10.4 Use Inscribed Angles and Polygons
• http://www.youtube.com/watch?v=h5S9xx1V88A
• http://www.mathplanet.com/education/geometry/circles/inscribe
d-angles-and-polygons
• http://prezi.com/jtp5toxmjqao/inscribed-angles-and-polygons/
10.5 Apply Other Angle Relationships in Circles
• http://www.youtube.com/watch?v=6E-wZGV7Ew8
• http://www.youtube.com/watch?v=Z8xLDVyIrko
• http://www.youtube.com/watch?v=gMbXXailkpc
10.6 Write and Graph Equations of Circles
• http://www.youtube.com/watch?v=9rLD8STilo4
• http://www.youtube.com/watch?v=J8aOmJhQtnY
• http://www.youtube.com/watch?v=unh2kbaG5Cc
Chapter 11
• http://www.learner.org/interactives/geometry/platonic.html
• http://www.regentsprep.org/Regents/math/geometry/GG2/index
GG2.htm
• http://mathbits.com/MathBits/MathMovies/ResourceListTwo.htm
l (use Wall-E)
Quadrilaterals, Properties of Circles, Measures of Figures and
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
#
XI.
UNIT: 4
TOPICS
Quadrilaterals (Chapter 8)
(8.1-8.6; 9 days)
X.
XI.
Properties of Circles (Chapter 10)
(10.1-10.7; 12 days)
Measurements of Figures and Solids (Ch. 11)
(11.1-11.3, 1.5-11.9; 11 days)
#
UNIT NAME:
Solids
Quadrilaterals, Properties of Circles, Measures of Figures and
STUDENT LEARNING OBJECTIVES
1
Find angle measures in polygons
2
Use properties of parallelograms
3
4
5
6
Show that a quadrilateral is a parallelogram
Properties of rhombuses, rectangles and squares
Use properties of trapezoids and kites
Identify special quadrilaterals
1
2
3
4
5
6
7
8
Use properties of tangents
Find arc measures
Apply properties of chords
Use inscribed angles and polygons
Construction: Tangent lines and inscribed squares
Apply other angles relationships in circles
Find segment lengths in circles
Write and graph equations of circles
1
2
3
4
Circumference and arc length of circles (11.1)
Areas of circles and sectors (11.2)
Areas of regular polygons (11.3)
Explore solids (11.5)
5
Volume of prisms and cylinders (11.6)
6
Extension: Density
7
Volume of pyramids and cones(11.7)
CCSS code
G.MG.1
G.CO.11; G.SRT.5
G.GPE.5; G.CO.11; G.SRT.5
G.CO.11; G.SRT.5; G.GPE.7
G.SRT.5; G.GPE.4
G.CO.11
G.CO.1; G.C.2; G.C.4
G.CO.1
G.C.2; G.C.3; G.CO.12
G.C.2; G.C.3; G.C.5
G.C.4; G.CO.13
G.C.2; G.C.5
G.C.2
G.GPE.1
G.C.5; G.GMD.1
G.C.5
G.CO.13; G.SRT.8
G.GMD.4
G.GMD.1; G.GMD.3;
G.GMD.4
G.MG.2
G.GMD.1; G.GMD.3;
G.MG.3
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
UNIT: 4
8
9
10
UNIT NAME:
Solids
Quadrilaterals, Properties of Circles, Measures of Figures and
Extension: Solids of Revolutions
Surface area and volume of spheres (11.8)
Explore similar solids (11.9)
Selected Opportunities for Connections to Mathematical Practices
1.
2.
3.
4.
5.
6.
7.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
G.GMD.4
G.GMD.3
G.GMD.3; G.SRT.2
Unit 4-Links to Open Ended Problems
Related to the Standards
G.C.1
G.C.2
G.C.3
G.C.4
G.C.5
G.CO.1
G.CO.2
G.CO.3
G.CO.5
G.CO.10
G.CO.11
G.CO.13
G.GMD.1
G.GMD.3
G.GMD.4
G.GPE.1
G.GPE.4
G.GPE.5
G.GPE.7
G.MG.1
G.MG.2
G.MG.3
G.SRT.2
G.SRT.5
G.SRT.8
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
UNIT: 4
UNIT NAME:
Solids
Quadrilaterals, Properties of Circles, Measures of Figures and
CCSS Code
CCSS Code DESRCIPTION
G.C.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.
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
G.C.3
G.C.4
G.C.5
G.CO.1
G.CO.11
G.CO.12
G.CO.13
Construct a tangent line from a point outside a given circle to the circle.
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.
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.
Prove theorems about parallelograms. 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.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
PCTI MATHEMATICS DEPARTMENT
COURSE: Geometry
G.GMD.1
G.GMD.3
G.GMD.4
G.GPE.1
G.GPE.4
G.GPE.5
G.GPE.7
G.MG.1
G.MG.2
G.MG.3
G.SRT.2
G.SRT.5
G.SRT.8
UNIT: 4
UNIT NAME:
Solids
Quadrilaterals, Properties of Circles, Measures of Figures and
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and
cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects
generated by rotations of two-dimensional objects.
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.
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four
given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and
containing the point (0, 2).
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. (E.g. find the equation of a
line parallel or perpendicular to a given line that passes through a given point.
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. (E.g. find the equation of a
line parallel or perpendicular to a given line that passes through a given point.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder).
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
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.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems