Austin ISD Instructional Planning Guide – Mathematics
3rd Six Weeks IPG- November 11th–December 18th (25 days; 2 days for 6 weeks review/test, 4 days for Finals)
©2008 Austin ISD
Major Concept #1: Triangle Congruence
8 DAYS
Overarching
Idea
At least three corresponding parts of triangles are required to verify triangle congruence, and more than one combination of three
corresponding parts is possible to prove congruence.
Teacher
Guiding
Questions

What combinations of corresponding parts of triangles must be congruent for the triangles to be congruent?

Why do certain combinations of corresponding parts of triangles prove congruence and others do not prove congruence?

How can congruent triangles be used to verify other conjectures?
355
350
353
357
355
Geometric
Structure
Congruence and the Geometry of
Size
357
Geometric
Structure
350
Matrix
Strand
Congruence
and the
Geometry of
Size
Matrix
#
TEKS
Knowledge & Skill
The student applies
logical reasoning to
justify and prove
mathematical
statements. (G.3)
The student applies the
concept of congruence to
justify properties of
figures and solve
problems. (G.10)
The student analyzes
properties and describes
relationships in
geometric figures. (G.9)
Student Expectation
Construct and justify
statements about geometric
figures and their properties
(using flowchart proofs,
transformational proofs, or
two-column proofs). (G.3B)
Justify and apply triangle
congruence relationships
(such as using geostrips or
other manipulatives to
investigate the sufficientSSS, SAS, ASA, AAS - and
the insufficient - SSA and
AAA - conditions for proving
triangles congruent).(G.10B)
Formulate and test
conjectures about the
properties and attributes of
polygons and their
component parts based on
explorations and concrete
models. (G.9B)
TAKS
OBJ
Resource
Holt 4-4 Geometry
Lab – Explore SSS
and SAS Triangle
Congruence
Holt 4-4 Triangle
Congruence SSS and
SAS
Use deductive reasoning to
prove a statement. (G.3E)
Holt 4-5 Technology
Lab – Predict Other
Triangle Congruence
Relationships
2
days
OR
90
min.
2
days
OR
(G.10)
(G.10B)
(G.9)
(G.9B)
Holt 4-5 Triangle
Congruence ASA,
AAS, and HL
Major Concept #1: Triangle Congruence (continued)
Geometry
Time/
Pace
(G.3B)
(G.3)
Geometry
Page 1 of 4
90
min.
Teacher Tools
NOTE: Investigation Set 8 of Patty Paper Geometry includes
guided and open explorations of the triangle congruence shortcuts
using patty paper.
Vocabulary: congruence, included angle, triangle rigidity
Teacher Note: String, large paper clips, drinking straws, rulers and
protractors are needed for Lab 4-4. Many teachers have used
other materials for this type of activity. One teacher used 1.5-inch
wide strips of poster board with brads. Talk with other teachers
about the variety of materials they may have used.
Questioning Strategies:
 Does what you discovered confirm what you learned in the
Delta’s Deli Sandwich Shop problem?
 If you know two sides and one angle are congruent, are the
triangles always congruent?
 When are the triangles always congruent and when are
they not always congruent?
 If you know three sides are congruent, will the triangles
always be congruent? What if three angles are congruent?
Vocabulary: included side
Technology: Geometer’s Sketchpad or the GeoGebra website is
needed for Tech Lab 4-5.
Teacher Note: Students should be comfortable justifying
conjectures using concrete models and exploration. Encourage
students’ use of flowchart proofs as a shortcut for proving triangle
congruence. Several examples of flowchart proofs and exercises
are included as models.
The graphic organizer included in the Notes Workbook or the
Texas Know-it notebook and pictured on page 255 in the THINK
and DISCUSS box will help students organize and understand the
different triangle congruence shortcuts.
8 DAYS (Cont’d.)
8/13/07
Holt = Geometry DG = Discovering Geometry GA = Dana Center Geometry Assessments GS = Geometer’s Sketchpad MCM = Region IV Making Connections w/Measurement ETQ = Ensuring Teacher Quality Modules
MTC = Geometry Math TEKS Connections
Austin ISD Instructional Planning Guide – Mathematics
3rd Six Weeks IPG- November 11th–December 18th (25 days; 2 days for 6 weeks review/test, 4 days for Finals)
©2008 Austin ISD
Overarching
Idea
At least three corresponding parts of triangles are required to verify triangle congruence, and more than one combination of three
corresponding parts is possible to prove congruence.
Teacher
Guiding
Questions

What combinations of corresponding parts of triangles must be congruent for the triangles to be congruent?

Why do certain combinations of corresponding parts of triangles prove congruence and others do not prove congruence?

How can congruent triangles be used to verify other conjectures?
332
350
213
351
352
333
353
344
357
348
355
Geometry
Congruence
Geom
andetric
the
Congruence and the Geometry of
Geometric
Dimensionality
Structure and the Geometry of Location
Size
Struct of
Geometry
ure
Size
Matrix
Matrix
Overarching
#
Strand
Matrix
Matrix
Idea
#
Strand
348
Teacher
357
350
Guiding
Questions
352
Geometry
Major Concept #2: Quadrilaterals
11 DAYS
TEKS
Time/
TAKS
Student Expectation
Resource
Teacher Tools
OBJ
Knowledge
Pace
TEKS& Skill
TAKS
Quadrilaterals
can be categorized
according to the
similaritiesResource
and differences Time/
in their properties.
Student
Expectation
Teacher
Tools
Knowledge & Skill Use congruence transformations OBJ
Pace 4-6 Triangle Congruence: CPCTC
to make conjectures and justify
NOTE: The Alternate Opener Exploration is a proof – consider
TheWhat
student
applies the are used
(G.2B) to of
(G.2)
properties
name
andfigures
identify different types of quadrilaterals?
The
ETQ
Quadrilateral
unitstatements
explores the
interrelations
triangles
properties
geometric
having
students
cut out the
and
reasons andofcreate
a
concept of congruence
and
quadrilaterals,
using
transformations
as a tool to construct
including
(G.3B) figures represented on
proof
together
in
pairs
or
small
groups.
to
justify
properties
of
(G.3)
 How does classifying
quadrilaterals
relate to classifying other types of polygons? quadrilaterals.
This lesson
flows really well
from the previous work
a(G.3D)
coordinate
plane. (G.10A)
Vocabulary: congruent,
corresponding,
CPCTC
figures and solve
with
isosceles
triangles.
Justify and apply triangle
Teacher
Note:
The
converse
of
the
Triangle
Congruence
Use one-and
and differences
twoproblems
What (G.10)
important similarities
distinguish quadrilaterals from each other?
congruence
relationships.
Theorems can
be usedsymmetry
to reinforce student learning. For some
Vocabulary:
rotational
dimensional coordinate
ETQ Geometry Unit 3
(G.10B)
students written or flowchart proofs will be less difficult then two
systems to represent points,
Teacher Note: The bulk of this exploration could be jig sawed
Quadrilaterals
column proofs. Each bubble of the flowchart proof could be cut out
Construct
and
justify
statements
lines, line segments, rays,
between different groups of students with a sharing/discussion
and students could arrange them in the correct order supplying
about
geometric
figures
and
and figures.
(G.7A)
B11;
T11
about similarities and differences in the quadrilaterals created by
justification where needed.
their properties (using flowchart
reflected
or rotating triangles. The four big explorations are:
Use slopes and equations of
Using a flowchart proof, students can demonstrate how what is
proofs, transformational proofs,
lines to investigate geometric

Isosceles
Right
Triangle
Reflections
TXTMS Geometry
known can be used
to prove
a statement.
The student understands or two-column proofs). (G.3B)
relationships, including
Investigating
that
coordinate
systems
Questioning

Scalene
Strategies:
Right
Triangle
Reflections
Use
logical
reasoning
to prove
The student applies
parallel
lines,
perpendicular
Quadrilaterals
provide
convenient
statements
are true segments]
and find
 Scalene
What is known?
Acute/Obtuse Triangle Reflections
logical reasoning
to and
lines, and [special
2
Alternate Opener 4-6
efficient
ways
of
7
counter
examples
to
disprove
justify and prove
of triangles and other

Rotate
What are
a Triangle
you trying to prove?
days
representing
statements
that areB11;
false.T11
mathematicalgeometric
6
polygons. (G.7B)
The
TEXTEAMS
Investigating
Quadrilaterals
unitthe
explores
Holt 4-6 Triangle

How
can
you
use what you
know to identify
figures
and
uses
them
(G.3C)
statements (G.3)
days
properties
of
squares,
rectangles
and
rhombi
on
the
coordinate
Congruence:
CPCTC
intermediate
steps
in
a
proof?
OR
Quadrilateral Chart
accordingly. (G.7)
Derive and use formulas
plane.
See last page of IPG
Use
inductive
reasoning
involving
length,
slope, to
and

How does working backwards help you develop a proof?
Vocabulary: square, rectangle, rhombus (rhombi), diagonals,
Holt 4-8 Isosceles
OR
formulate
conjecture.
(G.3D)
midpoint a(such
as
90
 What
can be used
vertex,
slope,other
pointproperties/theorems/conjectures
of intersection (of diagonals)
and
Equilateral
determining the length of a
min.
toNote:
connect
what
is given tocould
whatalso
is tobe
bejig
proved?
Use deductive reasoning to
Teacher
This
investigation
sawed
between
Triangles
3 90side of a triangle given the
4-8 Isosceles
Equilateral
Trianglesor a gallery walk to
prove a statement. (G.3E)
several
groups ofand
students
with presentations
min.
coordinates of the vertices).
Vocabulary:
legs (of an of
isosceles
triangle),
vertex
angle, base,
facilitate
generalization
the similarities
and
differences
in the
The student
Develop an awareness of the
blocks
(G.7C) B11; T11
base angles
properties
of quadrilaterals. See the organizational chart on pg. 5.
understands the
structure of a mathematical
Teacher Note: This section could be taught together with the
structure of, and
system
connecting definitions,
Use congruence
CPCTC lesson
with emphasis
placed on the
unique
properties of
relationships
within, an
postulates,
logicaltoreasoning
transformations
make
The
student applies
the
Vocabulary:
parallelogram,
quadrilateral,
opposite
sides,
isosceles and
equilateral
triangles.
Because
of these
unique
axiomaticofsystem.
(G.1)to and
theoremsand
(G.1A)
conjectures
justify
concept
congruence
diagonals,
opposite
angles,
consecutive
angles,
bisect,
properties, students
may have less difficulty applying CPCTC to
properties of geometric
justify properties of
supplementary
(angles)
Holt 6-2 Properties of
isoscelesNote:
and equilateral
triangles.
Make
conjectures
about angles,
figures
including figures
figures and solve
Teacher
The sole focus
of Holt 6-2 through 6-6 is
2 days Technology: The Properties of Isosceles Triangles exploration
Parallelograms
lines,
polygons,
and
represented
oncircles,
a coordinate
problems. (G.10)
parallelogram and their properties. If you implemented the
for
from Exploring
Geometrythese
with sections
Geometer’s
Sketchpad
is a good
The student analyzes
three-dimensional
previous
investigations,
of Holt
can be used
to
plane. (G.10A) figures and
review review of properties of an isosceles triangle. The Explore More
Holt 6-3 Conditions for
geometric relationships
determine
validity
reinforce the concepts that were developed during the
Formulatethe
and
test of the
/test
Parallelograms
section includes
in order to make and
conjectures,
explorations
and connections
discussions.between isosceles and equilateral
conjectureschoosing
about thefrom a
The
analyzes
triangles
and thecan
equilateral
corollaries.
verifystudent
conjectures.
(G.2) variety
of approaches
suchofas
These
sections
be used triangle
to reinforce
previous work with
properties
and attributes
properties and describes coordinate, transformational, or
Patty Paper
Investigation
4.2 should
is a patty
paper an
parallel
linesGeometry:
and transversals.
Students
develop
polygons and their
relationships in
investigation (Open
Guided)
of the
Triangle
axiomatic.
(G.2B)
understanding
aboutorthe
properties
of Isosceles
parallelograms
to assist
component
parts based on
geometric figures. (G.9)
Conjecture
and its
with
future study
ofConverse.
quadrilaterals.
explorations and concrete
models. (G.9B)
Page 2 of 4
8/13/07
Holt = Geometry DG = Discovering Geometry GA = Dana Center Geometry Assessments GS = Geometer’s Sketchpad MCM = Region IV Making Connections w/Measurement ETQ = Ensuring Teacher Quality Modules
MTC = Geometry Math TEKS Connections
Austin ISD Instructional Planning Guide – Mathematics
3rd Six Weeks IPG- November 11th–December 18th (25 days; 2 days for 6 weeks review/test, 4 days for Finals)
©2008 Austin ISD
11 DAYS (Cont’d.)
Major Concept #2: Quadrilaterals (continued)
Overarching
Idea
Teacher
Guiding
Questions
350
353
332
213
333
Geometri
c
Structure
348
Matrix
Strand
Dimensionality and the Geometry of Location
Matrix
#
Quadrilaterals can be categorized according to the similarities and differences in their properties.

What properties are used to name and identify different types of quadrilaterals?

How does classifying quadrilaterals relate to classifying other types of polygons?

What important similarities and differences distinguish quadrilaterals from each other?
TEKS
Knowledge & Skill
(G.2)
(G.3)
The student understands
that coordinate systems
provide convenient and
efficient ways of
representing geometric
figures and uses them
accordingly. (G.7)
Student Expectation
TAKS
OBJ
Resource
Time/
Pace
(G.3B)
(G.3E)
Use one- and twodimensional coordinate
systems to represent points,
lines, line segments, rays,
and figures. (G.7A) B11; T11
Use slopes and equations of
lines to investigate geometric
relationships, including
parallel lines, perpendicular
lines, and [special segments]
of triangles and other
polygons. (G.7B) B11; T11
Derive and use formulas
involving length, slope, and
midpoint (such as
determining the length of a
side of a triangle given the
coordinates of the vertices).
(G.7C) B11; T11
Teacher Tools
Patty Paper Geometry investigation 6 – Properties of
Quadrilaterals could be included to assist students that need a
more visual representation of the theorems included in Holt
Chapter 6.
(G.2B)
Holt 6-4 Properties of
Special
Parallelograms
2 days
OR
Holt 6-5 Conditions
for Special
Parallelograms
90
min.
Holt 6-6 Technology
Lab – Explore
Isosceles Trapezoids
1 day
Holt 6-6 Properties of
Kites and Trapezoids
45
min.
UNIT or Major Test
Geometry
Geometry
Page 3 of 4
OR
2 days
OR 90
min.
Vocabulary: rectangle, rhombus, square, diagonals, quadrilateral,
parallelogram, right angle, congruent
Teacher note: These sections of Holt can be used to reinforce
students’ understanding of the properties of quadrilaterals. They
are companion lessons that apply the properties discovered
previously to proving that a quadrilateral is a particular type of
parallelogram such as a square, rectangle, or rhombus.
Chapter 4 – Quadrilaterals in the Exploring Geometry with
Geometer’s Sketchpad resource also offers a very visual
reinforcement for the study of quadrilaterals. The last section on
Summarizing Properties of Quadrilaterals also includes an
organizational chart for the properties of the different
quadrilaterals.
Vocabulary: kite, trapezoid, base of a trapezoid, leg of a trapezoid,
base angle of a trapezoid, isosceles trapezoid, midsegment of
trapezoid
Teacher note: Students may become confused that a trapezoid
has two bases. Clarify any student misconceptions surrounding
the important vocabulary in this section. Help students understand
the similarities and differences between trapezoids that are
isosceles and those that are not isosceles.
Review and major test over quadrilaterals.
8/13/07
Holt = Geometry DG = Discovering Geometry GA = Dana Center Geometry Assessments GS = Geometer’s Sketchpad MCM = Region IV Making Connections w/Measurement ETQ = Ensuring Teacher Quality Modules
MTC = Geometry Math TEKS Connections
©2008 Austin ISD
Geometry
Austin ISD Instructional Planning Guide – Mathematics
3rd Six Weeks IPG- November 11th–December 18th (25 days; 2 days for 6 weeks review/test, 4 days for Finals)
Page 4 of 4
Geometry
8/13/07
Holt = Geometry DG = Discovering Geometry GA = Dana Center Geometry Assessments GS = Geometer’s Sketchpad MCM = Region IV Making Connections w/Measurement ETQ = Ensuring Teacher Quality Modules
MTC = Geometry Math TEKS Connections