Download Montclair Public Schools CCSS “Geometry High Honors” Unit MC

Montclair Public Schools
CCSS “Geometry High Honors” Unit
Subject
Geometry HH
Grade
8-10
Unit #
2
Pacing
8-10 weeks
Unit
Triangles and Quadrilaterals
Overview
Unit 2 has an intense focus on triangle relationships and theorems. Students will determine when triangle congruence can be proven.
Important relationships of inequalities for sides and angles are proven within a triangle, and between two triangles. Special relationships are
explored for isosceles triangles and right triangles. Indirect Proofs and addressing the concepts of inequalities within proofs is introduced in this
unit. Quadrilaterals are fully explored in this unit, as well, and are classified as kites, trapezoids, or parallelograms. Parallelograms are further
subdivided into rhombi, rectangles and squares. Students explore the special characteristics of each type of quadrilateral, including the sum of
interior or exterior angle measures, congruence criteria, and other relationships involving sides, diagonals, and angles.
Standard #
Writing Standards (Priority are Bold)
MC,
SLO
Student Learning Objectives
Depth of
G.CO.10
1: 20132014
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.
SC, or
AC
#
MC
1
Knowledge
Prove the following theorems about
triangles:
a) The Angle Sum Theorem
b) The Third Angle Theorem
c) The Exterior Angle Theorem
d) The Angle-Angle-Side Congruency
Theorem
e) The Side-Side-Side Inequality
Theorem
f) The Side-Angle-Side Inequality
Theorem
g) The Isosceles Triangle Theorem
h) The Triangle Midsegment Theorem
i) The medians of a triangle meet at a
point
j) The Exterior Angle Inequality
Theorem ...
k) The side opposite the greater angle
in a triangle is longer than the side
opposite the lesser angle, and the
converse of these relationships
l) The perpendicular segment from a
point to a line is the shortest distance
4
G.CO.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
MC
G.CO.8
Explain how the criteria for triangle
congruence (ASA, SAS, and SSS) follow from
the definition of congruence in terms of rigid
motions
MC
G.CO.13
Construct an equilateral triangle, a square,
and a regular hexagon inscribed in a circle
SC
G.CO.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
MC
2
3
4
5
6
from the point to the line
m) The Triangle Inequality Theorem
Define the meaning of congruent
geometric figures
Determine if two or more triangles are
congruent and explain why or why
not.
2
Demonstrate that when a certain
sequence of three component parts of
a triangle are fixed (ASA, SAS, SSS,
AAS), only one triangle can be formed,
thereby proving the criteria for
triangle congruence.
Generate formal constructions of
regular polygons inscribed in a circle
with paper folding, geometric
software or other geometric tools.
3
Prove the following theorems about
parallelograms:
a) Opposite sides are congruent
b) Opposite angles are congruent
c) The diagonals of a parallelogram
bisect each other
d) The diagonals of a rectangle are
congruent
e) The diagonals of a rhombus are
perpendicular
f) The diagonals of a rhombus bisect a
pair of opposite angles
4
Prove the following theorems proving
that a quadrilateral is a parallelogram:
a) If both pair of opposite sides of a
2: 20132014
1
3
G.SRT.5
Use congruence criteria for triangles to solve
problems and to prove relationships in
geometric figures
MC
7
8
G.GPE.4
G.GPE.7
3: 20132014
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)
Use coordinates to compute perimeters of
polygons and areas of triangles and
rectangles, e.g., using the distance formula.
MC
MC
quadrilateral are congruent
b) If both pair of opposite angles are
congruent
c) If the diagonals are congruent then
the quadrilateral is a rectangle
d) If the diagonals are perpendicular,
then the quadrilateral is a rhombus.
Use congruence criteria for triangles
to solve problems and to prove
relationships in geometric figures
3
Justify solutions to problems involving
side lengths and angle measures using
triangle congruence and similarity
criteria.
3
9
Use coordinates to prove simple
geometric theorems algebraically
2
10
Use the coordinate plane to draw
models of figures used in proofs.
2
11
Present visual models of polygons on
the coordinate plane prior to applying
the distance formula.
2
12
Use coordinates to investigate and
compute perimeters and areas of
polygons
3
Mathematical
Practice #
CCCS.MPI
CCCS.MP2
CCCS.MP3
CCCS.MP4
CCCS.MP6
Selected Opportunities for Connections to Mathematical Practices
Analyze math constraints, make conjectures, and create a pathway to solve problems.
Decontextualize the problems
Use stated assumptions, definitions, and theorems to construct arguments (proofs). Justify conclusions (via twocolumn proofs). Create plausible arguments using mathematical reasoning.
Apply knowledge to simplify a complicated or complex situation.
Make explicit use of definitions.
Big Ideas
You can prove triangles are congruent based on a specific set of given criteria. You cannot form a triangle from any given set of side lengths.
The relationship between the diagonals of a given quadrilateral can be used to further classify the polygon. Isosceles triangles have a unique
set of properties and theorems.
Essential Questions
How can two triangles be proven to be congruent? What determines the side lengths of a triangle? What are the implications when two
triangles have certain congruent parts, but not others? How does the relationship between the diagonals of a quadrilateral allow it to be
classified?
Assessments
Definitions, Theorems & Postulates Tests. Proof Tests. Concepts & Algebraic Applications Tests/Quizzes.
Key Vocabulary
Altitude, median, angle bisector, concurrency, centroid, orthocenter, circumcenter, incenter, scalene, isosceles, equilateral, triangle,
parallelogram, rhombus, rectangle, square, trapezoid, kite.
Suggested Resources (list specific chapters and or page numbers from existing text that correspond to the SLOs and Standards)
Middle School: Glencoe Geometry chapters 4 - 6
High School: Prentice Hall Geometry chapter 4, and sections, 5-1 through 5-3, 5-5, and chapter 6
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