Download GTPS Curriculum – Geometry 3 weeks Topic: 1

GTPS Curriculum – Geometry
3 weeks
Topic: 1-Points, Lines, Planes, and Angles
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G.CO.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.
Essential Questions
How will students use the definitions of the
most basic elements and terms of Geometry
to solve Geometric problems using critical
thinking and deductive reasoning?
Materials: McDougal Littel Geometry
G-CO.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
Enduring Understandings
To use correct vocabulary
Determine the distance between two
points
Determine the type of angle by its
measurement
Sketch a diagram that represents a give
Geometric problem
Find the measure of an angle
Algebraically
Find the measure of an angle with a
protractor
Name and sketch adjacent angles
Determine the locus of points that are a
solution set
Use Geometric Symbolic Notation
correctly
Apply Geometric Theorems to solve
problems
Determine the difference between equals
and congruence
1.1
1.2
1.3
1.4
1.5
A Game and Some Geometry
Points, Lines, and Planes
Segments, Rays, and Distance
Angles
Postulates and Theorems Relating Points,
Lines, and Planes
Assessments:
Formative
Teacher observation
Student sheets
Summative
Chapter tests
Quizzes
GTPS Curriculum – Geometry
3 Weeks
Topic: 2-Deductive Reasoning
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G-CO.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.
Essential Questions
Materials: McDougal Littel Geometry
G-CO.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.
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.
How are proofs used to develop conjectures
in Mathematics?
Enduring Understandings
Use correct vocabulary
Recognize the Hypothesis and Conclusion
of an if-then statement
State the converse of an if-then statement
Use a counterexample to disprove an ifthen statement
Apply if and only if statements to biconditional statements
Apply the properties of algebra and
congruence in proofs
Use the Midpoint Theorem and Angle
Bisector Theorem
Complete a two column proof
Apply the definitions of complementary and
supplementary angles in proofs and in
algebraic solutions
Apply the definitions and theorems of
vertical angles and perpendicular lines in
proofs and in algebraic solutions
2.1
2.2
2.3
2.4
2.5
2.6
If-then Statements; Converses
Properties from Algebra
Proving Theorems
Special Pairs of Angles
Perpendicular Lines
Planning a Proof
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
3 Weeks
Topic: 3-Parallel Lines and Planes
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G-SRT.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.
Essential Questions
Materials: McDougal Littel Geometry
G-SRT.5. Use congruence and similarity
criteria for triangles to solve problems and
to prove relationships in geometric figures.
How does slope influence lines’ intersection,
parallelism, and skewness?
Enduring Understandings
Use correct vocabulary when discussing
parallel lines
Distinguish the difference between
intersecting, parallel, and skew lines
Identify the angles formed by parallel lines.
Identify parallel and skew planes
Complete proofs involving parallel lines.
Use Algebra and Geometric theorems to
solve angles of parallel lines
Prove lines are parallel by Theorems,
Postulates, and Algebra.
Name a Triangle by sides and angles.
Identify the exterior and remote interior
angles of a triangle
Solve for the angles of a triangle
Use inductive reasoning to make
conclusions.
3.1
3.2
3.3
3.4
3.5
3.6
Definitions
Properties of Parallel Lines
Proving Lines Parallel
Angles of a Triangle
Angles of a Polygon
Inductive Reasoning
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
3 Weeks
Topic: 4-Congruent Triangles
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G-CO.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.
Essential Questions
How can you determine if two triangles are
congruent, or use congruent triangles to prove
segments and angles are congruent?
Materials: McDougal Littel Geometry
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.
Use correct vocabulary when discussing
congruent triangles
Determine if two objects are congruent Label
corresponding parts of congruent objects
Name a pair of congruent objects in
corresponding order
Prove two triangles are congruent by SSS, SAS,
ASA, AAS methods
Use congruent triangles in a proof
Determine if a triangle is isosceles
Use HL, LL, HA, LA in proofs involving right
triangles
Complete proofs involving overlapping
triangles Identify an altitude, median, and a
perpendicular bisector of a triangle Solve
problems involving altitudes, medians, and
bisectors
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.
Enduring Understandings
4.1 Congruent Figures
4.2 Some Ways to Prove Triangles
Congruent
4.3 Using Congruent Triangles
4.4 The Isosceles Triangle Theorems
4.5 Other Methods of Proving Triangles
Congruent
4.6 Using More than One Pair of
Congruent Triangles
4.7 Medians, Altitudes, and Perpendicular
Bisectors
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
3 Weeks
Topic: 5-Quadrilaterals
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
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.
Essential Questions
How can the properties of
parallelograms be used to determine
the name of a given four sided
polygon?
Materials: McDougal Little Geometry
Enduring Understandings
Use correct vocabulary when
discussing quadrilaterals
Apply the properties of parallelograms
to proofs
Determine the lengths of segments of
parallelograms
Determine the measure of angles in
parallelograms
Determine the length of a segment of
two sides of a triangle
Use the property of the median of a
trapezoid to determine the length of
the bases or the median
Complete proofs involving
parallelograms and trapezoids
5.1 Properties of Parallelograms
5.2 Ways to Prove that Quadrilaterals are
Parallelograms
5.3 Theorems Involving Parallel Lines
5.4 Special Parallelograms
5.5 Trapezoids
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
3 Weeks
Topic: 6- Inequalities in Geometry
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
A-REI.12. Graph the solutions to a linear
inequality in two variables as a half-plane
(excluding the boundary in the case of a
strict inequality), and graph the solution set
to a system of linear inequalities in two
variables as the intersection of the
corresponding half-planes.
Essential Questions
How do inequalities in triangles affect
the angle and the sides opposite those
angles?
Materials: McDougal Littel Geometry
G-SRT.5. Use congruence and similarity criteria
for triangles to solve problems and to prove
relationships in geometric figures.
Enduring Understandings
Use correct vocabulary when discussing
inequalities
Apply the exterior angle inequality
theorem to triangles
Determine if one segment or angle is
larger than another
Use inequalities and their properties in
proofs
Write the inverse and contrapositive of
an if-then statement
Complete an indirect proof Apply the
triangle inequality theorem to determine
the relationship between the sides of a
triangle
Apply the SAS inequality and SSS
inequality methods to determine the
relationship between two corresponding
parts of different triangles
6.1
6.2
6.3
6.4
6.5
Inequalities
Inverses and Contrapositives
Indirect Proof
Inequalities for One Triangle
Inequalities for Two Triangle
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
3 Weeks
Topic: 7- Ratio, Proportion, and Similarity
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G-SRT.1. Verify experimentally the properties of
dilations given by a center and a scale factor:
Essential Questions
Materials: McDougal Littel Geometry
G-SRT.1a. 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.
G-SRT.1b. The dilation of a line segment is longer or
shorter in the ratio given by the scale factor.
G-SRT.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.
G-SRT.3. Use the properties of similarity
transformations to establish the AA criterion for two
triangles to be similar.
G-SRT.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.
G-SRT.5. Use congruence and similarity criteria for
triangles to solve problems and to prove relationships in
geometric figures.
How can the properties of similarity
be used to determine if two polygons
are similar or to find the measures
of corresponding parts of similar
polygons?
Enduring Understandings
Use correct vocabulary when
discussing similar polygons
Set up, simplify, and solve a
proportion
Apply the properties of proportions
to solve equations
Use proportions to find parts of
similar triangles
Determine if two polygons are
similar Determine the scale factor
between two similar polygons
Determine if 2 triangles are similar
using AA similarity, SAS similarity,
or SSS similarity Use proportions
and similarity to complete proofs
Determine the lengths of segments
that have been divided
proportionally
7.1
7.2
7.3
7.4
7.5
7.6
Ratio and Proportion
Properties of Proportions
Similar Polygons
A Postulate for Similar Triangles
Theorems for Similar Triangles
Proportional Lengths
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
3 Weeks
Topic: 8-Right Triangles
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G-SRT.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.
Essential Questions
Materials: McDougal Littel Geometry
G-SRT.7. Explain and use the relationship between
the sine and cosine of complementary angles.
G-SRT.8. Use trigonometric ratios and the
Pythagorean Theorem to solve right triangles in
applied problems.
G-SRT.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.
G-SRT.10. (+) Prove the Laws of Sines and Cosines
and use them to solve problems.
G-SRT.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).
How can the properties of right
triangles and trigonometry be used to
solve mathematical and real world
problems?
Enduring Understandings
Use correct vocabulary when
discussing right triangles
Express radicals in simplest form
Apply Geometric Means to right
triangles
Use the Pythagorean theorem to solve
right triangles
Use the converse of the Pythagorean
Theorem to determine the type of
triangle (acute, right, obtuse)
Use the basic trigonometric functions
to solve for missing parts of right
triangles
Apply the Law of Sines and Law of
Cosines to solve for parts of oblique
triangles
8.1 Similarity in Right Angles
8.2 The Pythagorean Theorem
8.3 The Converse of the Pythagorean
Theorem
8.4 Special Right Triangles
8.5 The Tangent Ratio
8.6 The Sine and Cosine Ratios
8.7 Applications of Right Triangle
Trigonometry
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
3 Weeks
Topic: 9- Circles
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G-C.1. Prove that all circles are similar.
Essential Questions
Materials: McDougal Littel Geometry
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.
G-C.3. Construct the inscribed and
circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed
in a circle.
G-C.4. (+) Construct a tangent line from a point
outside a given circle to the circle.
G-C.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.
How do arcs, central angles,
segments, and circumference relate
to one another within a circle?
Enduring Understandings
Use correct vocabulary when working
with circles
Use properties of tangent lines to
find unknown lengths or angles
Use the properties of arcs to find
unknown measures of angles
Use the properties of chords to
discover lengths of segments or angle
measurements
Apply properties to complete a proof
Use properties of inscribed angles to
find the measures of arcs
Understand the relationships among
the arcs of a circle and the angles
formed by chords, secants, and
tangents
Use and understand theorems that
state relationships involving products
of parts of chords, parts of secants,
and parts of tangents and secants
9.1
9.2
9.3
9.4
9.5
9.6
9.7
Basic Terms
Tangents
Arcs and Central Angles
Arc and Chords
Inscribed Angles
Other Angles
Circles and Length of Segments
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
3 Weeks
Topic: 10- Constructions and Loci
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G-CO.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.
Essential Questions
Materials: McDougal Littel Geometry
G-CO.13. Construct an equilateral triangle, a
square, and a regular hexagon inscribed in a
circle.
How can we construct geometric figures
by knowing the properties of them?
How can loci help our understanding of
points?
Enduring Understandings
Use correct vocabulary when constructing
figures
Create congruent segments and angles
Construct an angle bisector and a
perpendicular bisector
Construct parallel lines
Construct a circle with a tangent line
Construct a circle circumscribed about a
given polygon
Construct a circle inscribed in a given
polygon Understand the properties of loci
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
What Construction Means
Perpendiculars and Parallels
Concurrent Lines
Circles
Special Segments
The Meaning of Locus
Locus Problems
Locus and Construction
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
3 Weeks
Topic: 11- Areas of Plane Figures
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G-GMD.1. 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.
Essential Questions
Materials: McDougal Little Geometry
S-CP.6. Find the conditional probability of A
given B as the fraction of B’s outcomes that
also belong to A, and interpret the answer in
terms of the model.
How can the number of sides of a polygon
be used to determine the figure’s interior
and exterior angle sums?
Which various methods can be used to find
the area of various polygons?
Enduring Understandings
Use correct vocabulary when discussing
polygons and area
Find the area of any rectangle,
parallelogram, square, rhombus, or
trapezoid
Determine the area of any triangle, right
and non-right
Apply the Area Addition Postulate to any
polygon
Use trigonometric functions to find the
missing parts needed to determine the
area of a given polygon using the Area
Addition Postulate
Determine the circumference, arc length,
area, or area of a sector of a circle
Compare the areas of polygons using scale
factors and ratios
Use area to find the theoretical probability
of an event occurring
11.1 Areas of Rectangles
11.2 Areas of Parallelograms, Triangles and
Rhombuses
11.3 Areas of Trapezoids
11.4 Areas of Regular Polygons
11.5 Circumferences and Areas of Circles
11.6 Arc Lengths and Areas of Sectors
11.7 Ratios of Areas
11.8 Geometric Probability
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
Grade 8
Topic: 12- Areas and Volumes of Solids
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G-GMD.1. 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.
Essential Questions
How can knowledge of right prisms,
right pyramids, right cylinders, right
cones, and spheres help determine the
surface area and volume of any given
object of no particular shape?
Materials: McDougal Little Geometry
G-GMD.3. Use volume formulas for cylinders,
pyramids, cones, and spheres to solve problems.
G-GMD.4. Identify the shapes of twodimensional cross-sections of three-dimensional
objects, and identify three-dimensional objects
generated by rotations of two-dimensional
objects.
G-MG.1. Use geometric shapes, their measures,
and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a
cylinder).
G-MG.2. Apply concepts of density based on
area and volume in modeling situations (e.g.,
persons per square mile, BTUs per cubic foot).
G-MG.3. 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).
Enduring Understandings
Use correct vocabulary when discussing
solids
Identify the parts of a prism, pyramid,
cylinder, sphere
Find the lateral area of a solid
Find the surface area of a solid
Determine the scale factor between two
similar solids
Find the area and volume of two similar
solids
12.1
12.2
12.3
12.4
12.5
Prisms
Pyramids
Cylinders and Cones
Spheres
Areas and Volumes of Similar Solids
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
3 Weeks
Topic: 13- Coordinate Geometry
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G-GPE.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.
Essential Questions
Materials: McDougal Little Geometry
G-GPE.2. Derive the equation of a parabola given a
focus and directrix.
Enduring Understandings
G-GPE.4. 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).
G-GPE.5. 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).
G-GPE.6. Find the point on a directed line segment
between two given points that partitions the
segment in a given ratio.
G-GPE.7. Use coordinates to compute perimeters of
polygons and areas of triangles and rectangles, e.g.,
using the distance formula.
How is coordinate algebra used when
writing geometric proofs?
Use correct vocabulary when
discussing coordinate geometry
Evaluate the distance between any
two points
Write the equation of a circle given
its center and radius
Identify the center and radius of a
circle given its equation
Determine the slope of a line
Determine if points are collinear
Determine if lines are perpendicular,
parallel, or neither from their
algebraic equations
Find the equation of lines,
perpendicular lines, parallel lines
Graph lines and vectors
Find the ordered pair and magnitude
that represents a vector
Find the sum of two vectors
Find the midpoint of any two points
Supply the coordinates needed to
complete a geometric proof
Complete a coordinate proof
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
The Distance Formula
Slope of a Line
Parallel and Perpendicular Lines
Vectors
The Midpoint Formula
Graphing Linear Equations
Writing Linear Equations
Organizing Coordinate Proofs
Coordinate Geometry Proofs
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment
GTPS Curriculum – Geometry
3 Weeks
Topic: 14- Transformations
Objectives/CPI’s/Standards
Essential Questions/Enduring
Understandings
Materials/Assessment
G-CO.3. Given a rectangle, parallelogram,
trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto
itself.
Essential Questions
Materials: McDougal Little Geometry
G-CO.4. Develop definitions of rotations,
reflections, and translations in terms of
angles, circles, perpendicular lines, parallel
lines, and line segments.
G-CO.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
G-CO.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.
Can movements of objects be explained
and modeled with mathematics using
reflections, rotations, translations, and
dilations?
Enduring Understandings
Use correct vocabulary and notation
when discussing transformations
Determine the image or pre-image of a
mapping
Determine if a mapping is isometric
Reflect a point, segment, angle, object
over a line
Determine the transformation between
two points
Complete a rotation of a point or object
about another point
Find the image of an object under a
dilation as either an expansion or
contraction
Describe the translation of an object to
its image or from an image to its preimage Find the inverse of a mapping
Locate the images of composite
mappings
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
Mapping and Functions
Reflections
Translations and Glide Reflections
Rotations
Dilations
Composites of Mappings
Inverses and the Identity
Symmetry in Plane and in Space
Web Site Resources:
Assessments:
Formative
Teacher observation
Student sheets
Summative
End of module performance
assessment
Portfolio assessment