2014 - 2015
Informal Geometry
Curriculum Map
Mathematics Florida Standards
Volusia County Curriculum Maps are revised annually and updated throughout the year.
The learning goals are a work in progress and may be modified as needed.
Florida Standards
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them. (MAFS.K12.MP.1)
Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process
which sometimes requires perseverance, flexibility, and a bit of ingenuity.
2. Reason abstractly and quantitatively. (MAFS.K12.MP.2)
The concrete and the abstract can complement each other in the development of mathematical understanding:
representing a concrete situation with symbols can make the solution process more efficient, while reverting to a concrete
context can help make sense of abstract symbols.
3. Construct viable arguments and critique the reasoning of others. (MAFS.K12.MP.3)
A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and
supporting evidence.
4. Model with mathematics. (MAFS.K12.MP.4)
Many everyday problems can be solved by modeling the situation with mathematics.
5. Use appropriate tools strategically. (MAFS.K12.MP.5)
Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen
mathematical understanding.
6. Attend to precision. (MAFS.K12.MP.6)
Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical
explanations.
7. Look for and make use of structure. (MAFS.K12.MP.7)
Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea.
8. Look for and express regularity in repeated reasoning. (MAFS.K12.MP.8)
Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results
more quickly and efficiently.
Mathematics Department
Volusia County Schools
Informal Geometry Curriculum Map
May 2014
Informal Geometry: Florida Standards
The fundamental purpose of the course in Informal Geometry is to extend students’ geometric experiences from the middle grades. Students explore more
complex geometric situations and deepen their explanations of geometric relationships. Important differences exist between this Geometry course and the
historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the
introductory content for the Geometry conceptual category found in the high school CCSS. The Mathematical Practice Standards apply throughout each course
and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability
to make sense of problem situations. The critical areas, organized as follows.
Congruence, Proof, and Constructions: In previous grades, students were asked to draw triangles based on given measurements. They also have prior
experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent.
In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. Students informally prove theorems—
using a variety of formats—and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions
and explain why they work.
Similarity, Proof, and Trigonometry: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of
similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles, with particular attention to
special right triangles and the Pythagorean theorem.
Extending to Three Dimensions: Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of
circumference, area and volume formulas.
Connecting Algebra and Geometry Through Coordinates: Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a
rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and
perpendicular lines, which relates back to work done in the first course.
Circles With and Without Coordinates: In this unit students study the Cartesian coordinate system and use the distance formula to write the equation of a
circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for
solving quadratic equations, which relates back to work done in the first course, to determine intersections between lines and circles or parabolas.
Mathematics Department
Volusia County Schools
Informal Geometry Curriculum Map
May 2014
Informal Geometry: Common Core State Standards At A Glance
First Quarter
Unit 1- Geometry
Fundamentals
MAFS.912.G-GPE.2.6
MAFS.912.G-GPE.2.7
MAFS.912.G-CO.1.1
Unit 2- Geometric Properties
MAFS.912.G-GPE.2.4
MAFS.912.G-GPE.2.5
Unit 3- Properties of Polygons
and Quadrilaterals
MAFS.912.G-CO.3.11
MAFS.912.G-GPE.2.4
MAFS.912.G-GPE.2.5
Mathematics Department
Volusia County Schools
Second Quarter
Unit 4- Triangles
MAFS.912.G-CO.2.7
MAFS.912.G-CO.2.8
MAFS.912.G-SRT.2.5
Unit 5- Similarity
MAFS.912.G-SRT.1.1
MAFS.912.G-SRT.1.2
MAFS.912.G-SRT.1.3
Unit 6- Circle Concepts
MAFS.912.G-C.1.1
MAFS.912.G-C.1.2
MAFS.912.G-GPE.2.4
Third Quarter
Fourth Quarter
Unit 7- Transformational
Geometry
MAFS.912.G-CO.1.4
MAFS.912.G-CO.1.5
MAFS.912.G-CO.1.2
MAFS.912.G-CO.1.3
MAFS.912.G-CO.2.6
MAFS.912.G-SRT.1.1
Unit 9- Three-Dimensional
Measurements
MAFS.912.G-GMD.2.4
MAFS.912.G-GMD.1.3
MAFS.912.G-GMD.1.1
MAFS.912.G-MG.1.1
MAFS.912.G-MG.1.2
MAFS.912.G-MG.1.3
Unit 8- Two-Dimensional
Measurements
MAFS.912.G-MG.1.1
MAFS.912.G-MG.1.2
MAFS.912.G-MG.1.3
MAFS.912.G-GPE.2.7
Unit 10- Right Triangles and
Trigonometry
MAFS.912.G-SRT.3.6
MAFS.912.G-SRT.3.7
MAFS.912.G-SRT.3.8
Informal Geometry Curriculum Map
May 2014
The following English Language Arts CCSS should be taught throughout the course:
LAFS.910.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements or
performing tasks, attending to special cases or exceptions defined in the text.
LAFS.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in
context and topics.
LAFS.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form and translate
information expressed visually or mathematically into words.
LAFS.910.SL.1.1: Initiate and participate effectively in a range of collaborative discussions with diverse partners.
LAFS.910.SL.1.2: Integrate multiple sources of information presented in diverse media or formats evaluating the credibility and
accuracy of each source.
LAFS.910.SL.1.3: Evaluate a speaker’s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning
or exaggerated or distorted evidence.
LAFS.910.SL.2.4: Present information, findings and supporting evidence clearly, concisely, and logically such that listeners can follow
the line of reasoning.
LAFS.910.WHST.1.1: Write arguments focused on discipline-specific content.
LAFS.910.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task,
purpose, and audience.
LAFS.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research.
Mathematics Department
Volusia County Schools
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 1-Geometry Fundamentals
Essential Question(s):
How can algebra be useful when expressing geometric properties?
Standard
Learning Goals
I can:
Remarks
The students will:
MAFS.912.G-GPE.2.6
This includes the
 calculate the point(s) on a directed
Find the point on a directed line
midpoint formula.
line segment whose endpoints are
segment between two given points
Find a missing midpoint
(x1, y1) and (x2, y2) that partitions the
that partitions the segment in a given
given an endpoint and
line segment into a given ratio.
ratio.
the midpoint of the
SMP #7
segment.
-In addition to using the
formula, students may
find the midpoint
graphically using slope.
MAFS.912.G-GPE.2.7
This standard provides
 use the distance formula to compute
Use coordinates to compute
practice with the distance
segment length given two
perimeters of polygons and areas of
formula and its
coordinates
triangles and rectangles, e.g., using
connection with the
the distance formula.
Pythagorean Theorem.
SMP #1
MAFS.912.G-CO.1.1
.
 identify the undefined notions used
Know precise definitions of angle,
in geometry (point, line, plane,
circle, perpendicular line, parallel
distance) and describe their
line, and line segment, based on the
characteristics
undefined notions of point, line,
 identify angles, perpendicular lines,
distance along a line, and distance
parallel lines, rays, and line
around a circular arc.
segments.
SMP #6
 define angles, perpendicular lines,
parallel lines, rays, and line
segments precisely using the
undefined terms and “if-then” and “ifand-only-if” statements.
Mathematics Department
Volusia County Schools
Resources
Partitioning Segments:
http://learni.st/users/S33572/
boards/3128-partitioningsegments-into-a-particularratio-common-corestandard-9-12-g-gpe-6
Tasks and TI-Nspire
Lessons:
http://ccssmath.org/?page_id
=2315
First to Finish Task (toward
the bottom of this webpage)
https://commoncoregeometr
y.wikispaces.hcpss.org/Unit+
1
Midpoint resources:
http://www.shmoop.com/com
mon-core-standards/ccsshs-g-gpe-6.html
As the Crow Flies Task:
http://www.nctm.org/uploade
dFiles/Journals_and_Books/
Books/FHSM/RSMTask/Crow.pdf
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 2- Geometric Properties
Essential Question(s):
How can algebra be useful when expressing geometric properties?
Standard
Learning Goals
I can:
Remarks
The students will:
MAFS.912.G-GPE.2.4
Important formulas for
 represent the vertices of a figure in
Use coordinates to prove simple
the coordinate plane using variables.
coordinate geometry include
geometric theorems algebraically.
 connect a property of a figure to the
distance formula, slope
(For example, prove or disprove that
tool needed to verify the property.
formula, midpoint formula,
a figure defined by four given points  use coordinates and the right tool to
and definitions of
in the coordinate plane is a
prove or disprove a claim about a
quadrilaterals.
rectangle; prove or disprove that the
figure. For example:
point (1, √3) lies on the circle
a) Use slope to determine if sides
centered at the origin and containing
are parallel, intersecting, or
the point (0, 2).)
perpendicular;
SMP #3, #7
b) Use the distance formula to
determine if sides are congruent.
c) Use the midpoint formula or the
distance formula to decide if a
side has been bisected.
MAFS.912.G-GPE.2.5
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).
SMP #3, #8
Mathematics Department
Volusia County Schools

determine if lines are parallel or
perpendicular using their slopes.
Resources
http://neaportal.k12.ar.u
s/index.php/2012/02/per
pendicular-and-parallellines/
http://map.mathshell.org
/materials/download.php
?fileid=703
Apply these concepts to the
characteristics of special
quadrilaterals.
Informal Geometry Curriculum Map
May 2014
Course: Geometry
Unit 3- Properties of Quadrilaterals
Standard
The students will:
MAFS.912.G-CO.3.11
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.)
SMP #2, #3
MAFS.912.G-GPE.2.4
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).)
SMP #3, #7
Mathematics Department
Volusia County Schools
Essential Question(s):
How are quadrilaterals precisely classified?
Learning Targets
I can:
 use theorems, postulates, or definitions to
prove theorems about parallelograms,
including:
a) prove opposite sides of a
parallelogram are congruent;
b) prove opposite angles of a
parallelogram are congruent;
c) prove the diagonals of a parallelogram
bisect each other;
d) prove that rectangles are
parallelograms with congruent
diagonals.
 represent the vertices of a figure in the
coordinate plane using variables.
 connect a property of a figure to the tool
needed to verify the property.
 use coordinates and the right tool to
prove or disprove a claim about a figure.
For example:
a) Use slope to determine if sides are
parallel, intersecting, or perpendicular;
b) Use the distance formula to determine
if sides are congruent.
c) Use the midpoint formula or the
distance formula to decide if a side
has been bisected.
Remarks
The definition of a parallelogram
includes two pairs of opposite
sides parallel and congruent.
This includes rectangles,
squares, rhombi, kites, and
trapezoids.
Resources
http://www.shmoop.co
m/common-corestandards/handouts/gco_worksheet_11.pdf
http://www.shmoop.co
m/common-corestandards/handouts/gco_worksheet_11_ans.
pdf
http://ccssmath.org/?pa
ge_id=2311
Important formulas for
coordinate geometry include
distance formula, slope formula,
midpoint formula, and definitions
of quadrilaterals.
https://www.cohs.com/
editor/userUploads/file/
Meyn/321%20Student
%20Workbook.pdf
http://www.shmoop.co
m/common-corestandards/handouts/ggpe-worksheet_4.pdf
http://www.shmoop.co
m/common-corestandards/handouts/ggpeworksheet_4_ans.pdf
Informal Geometry Curriculum Map
May 2014
Course: Geometry
Unit 3- Properties of Quadrilaterals (cont)
Standard
The students will:
MAFS.912.G-GPE.2.5
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).
SMP #3, #8
Essential Question(s):
How are quadrilaterals precisely classified?
Learning Targets
I can:
 determine if lines are parallel or
perpendicular using their slopes.
Remarks
Apply these concepts to
the characteristics of
special quadrilaterals.
Resources
http://www.google.com/url?sa=t&rc
t=j&q=ggpe.4&source=web&cd=1&cad=rja
&ved=0CDIQFjAA&url=http%3A%
2F%2Flearnzillion.com%2Flesson
s%2F286-prove-whether-a-pointis-on-acircle&ei=V01kUcPTMI629gSGkY
D4Cg&usg=AFQjCNEdTlgXqxqTVyWyaIKtVmFbRE9Xg&bvm=
bv.44990110,d.eWU
http://neaportal.k12.ar.us/index.ph
p/2012/02/perpendicular-andparallel-lines/
http://map.mathshell.org/materials/
download.php?fileid=703
Mathematics Department
Volusia County Schools
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 4- Triangles
Standard
The students will:
MAFS.912.G-CO.2.7
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.
SMP #3
MAFS.912.G-CO.2.8
Explain how the criteria for
triangle congruence (ASA,
SAS, SSS and HypotenuseLeg) follow from the definition
of congruence in terms of
rigid motions.
SMP #2, #3
MAFS.912.G-SRT.2.5
Use congruence and
similarity criteria for triangles
to solve problems and to
prove relationships in
geometric figures.
SMP #1
Mathematics Department
Volusia County Schools
Essential Question(s):
In what ways can congruence be useful in the real world?
Learning Goals
I can:
Remarks
Resources
http://illuminatio
Some students may believe:
 identify corresponding sides and
ns.nctm.org/Acti
Combinations such as SSA or AAA are
corresponding angles of congruent triangles.
vityDetail.aspx?i
also
a
congruence
criterion
for
triangles.
 explain that in a pair of congruent triangles,
d=4
corresponding sides are congruent (distance is That all transformations, including
dilations, are rigid motions
preserved) and corresponding angles are
http://nlvm.usu.e
That any two figures that have the same
congruent (angle measure is preserved).
du/ennav/frames
area represent a rigid transformation.
 demonstrate that when distance is preserved
_asid_165_g_1_
(corresponding sides are congruent) and angle
t_3.html?open=i
nstructions&fro
measure is preserved (corresponding angles
m=topic_t_3.htm
are congruent) the triangles must also be
l
congruent.
Include AAS as a criterion for
 list the sufficient conditions to prove triangles
http://ccssmath.
congruence. Once you prove your
are congruent.
org/?page_id=2
triangles are congruent, the "left-over"
 map a triangle with one of the sufficient
261
conditions (e.g., SSS) onto the original triangle pieces that were not used in your method
of proof are also congruent. Remember,
and show that corresponding sides and
congruent triangles have 6 sets of
corresponding angles are congruent.
congruent pieces. We now have a
"follow-up" theorem to be used AFTER
the triangles are known to be congruent:
CPCTC.
Some students may assume that
 use triangle congruence and triangle similarity
segments, angles, and triangles are
to solve problems (e.g., indirect measure,
congruent/similar without given that
missing sides/angle measures, side splitting).
information. Remind them not to assume!
 use triangle congruence and triangle similarity
to prove relationships in geometric figures.
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 5- Similarity
Essential Question(s):
How might the features of one figure be useful when solving problems about a similar figure?
Standard
Learning Goals
I can:
Remarks
Resources
The students will:
Tasks and TI-nspire
MAFS.912.G-SRT.1.1
 define dilation.
lessons:
Verify experimentally the properties
 perform a dilation with a given center and scale
http://ccssmath.org/?page
of dilations given by a center and a
factor on a figure in the coordinate plane.
_id=2275
scale factor:
 verify that when a side passes through the center of
a) A dilation takes a line not passing
dilation, the side and its image lie on the same line.
“How Tall is the School’s
through the center of the dilation to a  verify that corresponding sides of the pre-image and
Flagpole”:
parallel line, and leaves a line
images are parallel.
http://alex.state.al.us/less
passing through the center
on_view.php?id=1669
 verify that a side length of the image is equal to the
unchanged.
scale factor multiplied by the corresponding side
b) The dilation of a line segment is
Similarity in Right
length of the pre-image.
Triangles task:
longer or shorter in the ratio given by
http://alex.state.al.us/less
the scale factor.
on_view.php?id=26341
SMP #6, #8
MAFS.912.G-SRT.1.2
Students think of
 define similarity as a composition of rigid motions
Videos, practice and
Given two figures, use the definition
similarity and
followed by dilations in which angle measure is
assessments:
of similarity in terms of similarity
congruence as
preserved and side length is proportional.
http://fabienneriesen.com/
transformations to decide if they are
separate and distinct Geometry%3A-tutorials-- identify corresponding sides and corresponding
similar; explain using similarity
categories. Remind chapter-11.php
angles of similar triangles.
transformations the meaning of
students that
 demonstrate that in a pair of similar triangles,
“Solving Geometry
similarity for triangles as the equality
corresponding angles are congruent (angle measure congruent figures
Problems: Floodlights”:
of all corresponding pairs of angles
are just similar
is preserved) and corresponding sides are
http://map.mathshell.org/
and the proportionality of all
figures with a scale
proportional.
materials/download.php?f
corresponding pairs of sides.
factor of 1:1.
 determine that two figures are similar by verifying
ileid=1257
SMP #3
that angle measure is preserved and corresponding
sides are proportional.
Mathematics Department
Volusia County Schools
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 5- Similarity (cont)
Essential Question(s):
How might the features of one figure be useful when solving problems about a similar figure?
Standard
Learning Goals
I can:
Remarks
The students will:
MAFS.912.G-SRT.1.3
Students
often
confuse the
 show and explain that when two angle
Use the properties of similarity
triangle congruence theorems
measures are known (AA), the third angle
transformations to establish the AA
measure is also known (Third Angle Theorem). with the triangle similarity
criterion for two triangles to be
theorems. It may be necessary
 conclude and explain that AA similarity is a
similar.
to go back and review SSS,
sufficient condition for two triangles to be
SMP #3
SAS, ASA and the HL theorem
similar.
to distinguish them from AA~,
SAS~ and SSS~.
Mathematics Department
Volusia County Schools
Resources
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 6- Circle Concepts
Essential Question(s):
How can the properties of circles be useful when solving geometric problems?
Standard
Learning Goals
I can:
Remarks
The students will:
MAFS.912.G-C.1.1
The definition of similarity and
 prove that all circles are similar by
Prove that all circles are similar.
showing that for a dilation centered at dilation will need to be
SMP #3
reviewed with students.
the center of a circle, the pre-image
Online applets can be helpful in
and the image have equal central
angle measures.
seeing this relationship.
MAFS.912.G-C.1.2
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.)
SMP #1, #6
MAFS.912.G-GPE.2.4
Use coordinates to prove simple
geometric theorems algebraically.
SMP #3, #7
Mathematics Department
Volusia County Schools





identify central angles, inscribed
angles, circumscribed angles,
diameters, radii, chords, and tangents.
describe the relationship between a
central angle, inscribed angle, or
circumscribed angle and the arc it
intercepts.
recognize that an inscribed angle
whose sides intersect the endpoints of
the diameter of a circle is a right angle
and that the radius of a circle is
perpendicular to the tangent where
the radius intersects the circle.
connect a property of a figure to the
tool needed to verify the property.
use coordinates and the right tool to prove
or disprove a claim about a figure.
Students may think they can
tell by inspection whether a line
intersects a circle in exactly
one point. It may be beneficial
to formally define a tangent line
as the line perpendicular to a
radius at the point where the
radius intersects the circle.
Example: prove or disprove
that the point (1, √3) lies on the
circle centered at the origin and
containing the point (0, 2); use
the distance formula to determine
if sides are congruent or to decide
if a point is inside a circle, outside
a circle, or on the circle;
Resources
-Sectors of Circles Task:
http://map.mathshell.org/
materials/lessons.php?ta
skid=441&subpage=conc
ept
-Deriving equations of
Circles:
Part 1:
http://map.mathshell.org/
materials/lessons.php?ta
skid=406&subpage=conc
ept
Part 2:
http://map.mathshell.org/
materials/lessons.php?ta
skid=425&subpage=conc
ept
-Inscribed and
Circumscribed Circles
Task:
http://map.mathshell.org/
materials/download.php?f
ileid=1194
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 7- Transformational Geometry
Essential Question(s):
In what ways can congruence be useful?
How might the features of one figure be useful when solving problems about a similar figure?
Standard
Learning Goals
I can:
Remarks
Resources
The students will:
“Translations, Reflections
MAFS.912.G-CO.1.4
The terms “mapping” and
 construct the definition of reflection,
and Rotations” task:
Develop definitions of
translation, and rotation.
“under” are used in special
http://www.shodor.org/intera
rotations, reflections, and
 construct the reflection definition by
ways when studying
ctivate/lessons/Translations
translations in terms of
connecting any point on the pre-image to its
transformations.
ReflectionsRotations/
angles, circles, perpendicular
corresponding point on the reflected image
Tessellation based Quilt
Students
sometimes
confuse
lines, parallel lines, and line
and describing the line segment’s relationship
design:
the terms “transformation”
segments.
to the line of reflection.
http://alex.state.al.us/lesson
and “translation.”
SMP #6
 construct the translation definition by
_view.php?id=29240
Remind students that that
Exploring Transformations
connecting any point on the pre-image to its
corresponding point on the translated image,
corresponding vertices have on a TI-84 graphing
calculator:
and connecting a second point on the preto be listed in order so that
http://alex.state.al.us/lesson
image to its corresponding point on the
corresponding sides and
_view.php?id=29240
translated image, and describing how the two
angles can be easily
Tessellation Applet:
segments are equal in length, point in the
http://www.shodor.org/intera
identified and that included
same direction, and are parallel.
ctivate/activities/Tessellate/
 construct the rotation definition by connecting sides or angles are
Online Transformation
apparent.
the center of rotation to any point on the preGames:
http://www.onlinemathlearnin
image and to its corresponding point on the
g.com/transformationrotated image, and describing the measure of
game.html
the angle formed and the equal measures of
Tasks and TI-nspire
the segments that formed the angle as part of
activities:
the definition.
http://ccssmath.org/?page_id
=2245
Dilations Applet:
http://nlvm.usu.edu/en/nav/fr
ames_asid_296_g_4_t_3.ht
ml
Mathematics Department
Volusia County Schools
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 7- Transformational Geometry (cont)
Essential Question(s):
In what ways can congruence be useful?
How might the features of one figure be useful when solving problems about a similar figure?
Standard
Learning Goals
I can:
Remarks
The students will:
MAFS.912.G-CO.1.5
Students may confuse rotations and
 draw a specific transformation when given a
Given a geometric figure and
reflections and be unable to differentiate
geometric figure and a rotation, reflection, or
a rotation, reflection, or
the two. Allowing them the opportunity
translation.
translation, draw the
to physically manipulate the shapes
 predict and verify the sequence of
transformed figure using, e.g.,
transformations (a composition) that will map a (such as with cut-outs or patty paper)
graph paper, tracing paper, or
can clear up misconceptions.
figure onto another.
geometry software. Specify a
sequence of transformations
that will carry a given figure
onto another.
SMP #5
MAFS.912.G-CO.1.2
Rigid transformations preserve distance
 draw transformations of reflections, rotations,
Represent transformations in
and angle measure (reflections,
translations, and combinations of these using
the plane using, e.g.,
graph paper, transparencies, and/or geometry rotations, translations, or combinations of
transparencies and geometry
those).
software.
software; describe
 determine the coordinates for the image
transformations as functions
(output) of a figure when a transformation rule
that take points in the plane
is applied to the pre-image (input).
as inputs and give other
 distinguish between transformations that are
points as outputs. Compare
rigid and those that are not.
transformations that preserve
distance and angle to those
that do not (e.g., translation
versus horizontal stretch).
SMP #6
Mathematics Department
Volusia County Schools
Resources
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 7- Transformational Geometry (cont)
Essential Question(s):
In what ways can congruence be useful?
How might the features of one figure be useful when solving problems about a similar figure?
Standard
Learning Goals
I can:
Remarks
The students will:
MAFS.912.G-CO.1.3
 describe and illustrate how a rectangle, parallelogram, This is a discussion of
Given a rectangle, parallelogram,
symmetry.
and isosceles trapezoid are mapped onto themselves
trapezoid, or regular polygon,
using transformations.
describe the rotations and
 calculate the number of lines of reflection symmetry and
reflections that carry it onto itself.
the degree of rotational symmetry of any regular
SMP #7
polygon.
MAFS.912.G-CO.2.6
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.
SMP #3

MAFS.912.G-SRT.1.1
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.
SMP #6, #8


Mathematics Department
Volusia County Schools






define rigid motions as reflections, rotations, translations
and combinations of these, all of which preserve
distance and angle measure.
define congruent figures as figures that have the same
shape and size and state that a composition of rigid
motions will map one congruent figure onto the other.
predict the composition of transformations that will map
a figure onto a congruent figure.
determine if two figures are congruent by determining if
rigid motions will turn one figure into the other.
define dilation.
perform a dilation with a given center and scale factor
on a figure in the coordinate plane.
verify that when a side passes through the center of
dilation, the side and its image lie on the same line.
verify that corresponding sides of the pre-image and
images are parallel.
Verify that a side length of the image is equal to the
scale factor multiplied by the corresponding side length
of the pre-image.
Students may believe that all
transformations, including
dilations, are rigid motions or
that any two figures that have
the same area represent a
rigid transformation. Provide
counterexamples.
Some students have prior
knowledge of dilations from
the concept of “dilated pupils”
which may lead them to
believe that dilation refers
only to objects getting larger.
Similarly, they may have
difficulty figuring out when to
multiply versus divide by the
scale factor.
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 8- Two-Dimensional Measurements
Essential Question(s):
In what ways can geometric figures be used to understand real-world situations?
Standard
Learning Goals
I can:
Remarks
Resources
The students will:
-Patchwork Task:
MAFS.912.G-GPE.2.7
Graphing the given
 use the distance formula to compute
http://map.mathshell.org/materials/d
Use coordinates to compute
coordinates of the vertices
the perimeter and area given the
ownload.php?fileid=754
perimeters of polygons and areas of
may help students visualize
coordinates of vertices of a polygon.
Square Task:
triangles and rectangles, e.g., using
the polygon in order to find
http://map.mathshell.org/materials/d
the distance formula.
the perimeter and area.
ownload.php?fileid=792
SMP #1
MAFS.912.G-MG.1.1
Use geometric shapes, their
measures, and their properties to
describe objects (e.g., modeling a tree
trunk or a human torso as a cylinder).
SMP #4



MAFS.912.G-MG.1.2
Apply concepts of density based on
area and volume in modeling
situations (e.g., persons per square
mile, BTUs per cubic foot).
SMP #1, #4




Mathematics Department
Volusia County Schools
represent real-world objects as
geometric figures.
estimate measures (circumference,
area, perimeter, volume) of real-world
objects using comparable geometric
shapes or three-dimensional figures.
apply the properties of geometric
figures to comparable real-world
objects.
decide whether it is best to calculate
or estimate the area or volume of a
geometric figure and perform the
calculation or estimation.
break composite geometric figures
into manageable pieces.
convert units of measure.
apply area and volume to situations
involving density.
Example: The spokes of a
wheel of a bicycle are equal
lengths because they
represent the radii of a circle.
Students may have issues
with estimating (rounding,
conceptual, not being exact,
etc.).
Security Camera Task:
http://map.mathshell.org/materials/d
ownload.php?fileid=798
Location, Location, Location Task:
http://illuminations.nctm.org/Lesson
Detail.aspx?id=L660
Example: Determine the
population in an area.
Students have difficulty
converting units of area and
volume due to the difference
in scale factors.
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 8- Two-Dimensional Measurements (cont)
Essential Question(s):
In what ways can geometric figures be used to understand real-world situations?
Standard
Learning Goals
I can:
Remarks
The students will:
MAFS.912.G-MG.1.3
Mathematical modeling involves
 create a visual representation of a
Apply geometric methods to
solving problems in which the
design problem.
solve design problems (e.g.,
path to the solution is not
 solve design problems using a
designing an object or structure
obvious. A challenge for
geometric model (graph, equation,
to satisfy physical constraints or
teaching modeling is finding
table, formula).
minimize cost; working with
problems that are interesting and
 interpret the results and make
typographic grid systems based
relevant to high school students
conclusions based on the geometric
on ratios).
and, at the same time, solvable
model.
SMP #1, #4
with the mathematical tools at the
students’ disposal.
Resources
Definition of pi investigation:
http://illuminations.nctm.org/
LessonDetail.aspx?id=L575
Circles and Triangles:
http://map.mathshell.org/mat
erials/lessons.php?taskid=22
2&subpage=problem
Rolling Cup Lesson:
http://map.mathshell.org/mat
erials/download.php?fileid=1
254
Evaluating Statements about
Lengths and Areas:
http://map.mathshell.org/mat
erials/download.php?fileid=6
75
Mathematics Department
Volusia County Schools
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 9- Three-Dimensional Measurements
Essential Question(s):
How can two-dimensional figures be used to understand three-dimensional objects?
Standard
Learning Goals
I can:
Remarks
The students will:
MAFS.912.G-GMD.2.4
Example: Rotating a circle produces a
 identify the shapes of the two-dimensional
Identify the shapes of twosphere, and rotating a rectangle
cross-sections of three-dimensional objects.
dimensional cross-sections  rotate a two-dimensional figure and identify
produces a cylinder.
of three-dimensional
Hands-on models of three-dimensional
the three-dimensional object that is formed.
objects, and identify threefigures will help students when they
dimensional objects
are first determining cross sections.
generated by rotations of
Rubber bands may also be stretched
two-dimensional objects.
around a solid to show a cross section.
SMP #4, #7
MAFS.912.G-GMD.1.3
Students may not understand when
 calculate the volume of a cylinder, pyramid,
Use volume formulas for
volume would be needed to solve a
cone, and sphere and use the volume
cylinders, pyramids, cones,
problem versus area, perimeter, or
formulas to solve problems.
and spheres to solve
surface area.
problems.
Students have difficulty with “B” in
SMP #4
volume/surface area formulas.
MAFS.912.G-GMD.1.1
Volumes:
The inclusion of the coefficient 1/3 in
Give an informal argument
the formulas for the volume of a
 identify the base for prisms, cylinders,
for the formulas for the
pyramid or cone and 4/3 in the formula
pyramids, and cones.
circumference of a circle,
for the volume of a sphere remains a
 calculate the area of the base for prisms,
area of a circle, volume of a
mystery for many students. In high
cylinders, pyramids, and cones.
cylinder, pyramid, and cone.  calculate the volume of a prism using the
school, students should attain a
(Use dissection arguments,
formula V = B  h and the volume of a cylinder conceptual understanding of where
Cavalieri’s principle, and
these coefficients come from. Concrete
V = πr2h.
informal limit arguments.)
demonstrations, such as pouring water
 defend the statement, “The formula for the
SMP #3, #7
from one shape into another should be
volume of a cylinder (or cone) is basically the
followed by more formal reasoning.
same as the formula for the volume of a prism
(or pyramid).”
 explain that the volume of a pyramid (or cone)
is 1/3 the volume of a prism (or cylinder) with
the same base area and height.
Mathematics Department
Volusia County Schools
Resources
Cylindrical/Conic Volume
http://www.insidemathemati
cs.org/pdfs/geometry/glasse
s/task.pdf
Rectangular Prism Fruit
Boxes:
http://map.mathshell.org/mat
erials/download.php?fileid=8
02
Prisms Fearless Frames :
http://map.mathshell.org/mat
erials/download.php?fileid=8
06
Best Size Cans Task:
http://map.mathshell.org/mat
erials/download.php?fileid=8
20
Popcorn Anyone?
http://illuminations.nctm.org/
LessonDetail.aspx?id=L797
(could extend to include
cone-shaped containers, as
well as surface area)
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 9- Three-Dimensional Measurements (cont)
Essential Question(s):
How can two-dimensional figures be used to understand three-dimensional objects?
Standard
Learning Goals
I can:
Remarks
The students will:
MAFS.912.G-MG.1.1
Example: The spokes of a wheel of a
 represent real-world objects as geometric
Use geometric shapes, their
bicycle are equal lengths because they
figures.
measures, and their
represent the radii of a circle.
 estimate measures (circumference, area,
properties to describe
perimeter, volume) of real-world objects using Students may have issues with
objects (e.g., modeling a
estimating (rounding, conceptual, not
comparable geometric shapes or threetree trunk or a human torso
being exact, etc.).
dimensional figures.
as a cylinder).
 apply the properties of geometric figures to
SMP #4
comparable real-world objects.
MAFS.912.G-MG.1.2
Apply concepts of density
based on area and volume
in modeling situations (e.g.,
persons per square mile,
BTUs per cubic foot).
SMP #1, #4




MAFS.912.G-MG.1.3

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).
SMP #1, #4
Mathematics Department
Volusia County Schools
decide whether it is best to calculate or
estimate the area or volume of a geometric
figure and perform the calculation or
estimation.
break composite geometric figures into
manageable pieces.
convert units of measure.
apply area and volume to situations involving
density.
create a visual representation of a design
problem.
solve design problems using a geometric
model (graph, equation, table, formula).
interpret the results and make conclusions
based on the geometric model.
Example: Determine the weight of
water given its density or the amount of
energy in a three-dimensional figure.
Resources
2-D figures to develop
3-D objects:
http://map.mathshell.o
rg/materials/lessons.p
hp?taskid=439&subpa
ge=concept
Evaluating Statements
about Enlargements:
http://map.mathshell.o
rg/materials/download.
php?fileid=678
Mathematical modeling involves
solving problems in which the path to
the solution is not obvious. A
challenge for teaching modeling is
finding problems that are interesting
and relevant to high school students
and, at the same time, solvable with
the mathematical tools at the students’
disposal.
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 10- Right Triangles and Trigonometry
Standard
The students will:
MAFS.912.G-SRT.3.6
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.
SMP #2, 7
MAFS.912.G-SRT.3.7
Explain and use the relationship
between the sine and cosine of
complementary angles.
SMP #2
Mathematics Department
Volusia County Schools
Essential Question(s):
How can right triangles be used to solve application problems?
Learning Goals
I can:
Remarks
Extension: Use division and the
 demonstrate that within a right triangle,
Pythagorean Theorem (a2 + b2
line segments parallel to a leg create
2
2
similar triangles by angle-angle similarity. = c ) to prove that sin A +
2
cos A = 1.
 use characteristics of similar figures to
Some students believe that
justify the trigonometric ratios.
right triangles must be oriented
 define the following trigonometric ratios
a particular way or they do not
for acute angles in a right triangle: sine,
realize that opposite and
cosine, and tangent.
adjacent sides need to be
identified with reference to a
particular acute angle in a right
triangle.
 define complementary angles.
 calculate sine and cosine ratios for
acute angles in a right triangle when
given two side lengths.
 use a diagram of a right triangle to
explain that for a pair of complementary
angles A and B, the sine of angle A is
equal to the cosine of angle B and the
cosine of angle A is equal to the sine of
angle B.
Resources
Tasks and TI-nspire
activities:
http://ccssmath.org/?page_i
d=2283
“Trig River”:
http://www.teachengineerin
g.org/view_activity.php?url=
http://www.teachengineerin
g.org/collection/cub_/activiti
es/cub_navigation/cub_navi
gation_lesson03_activity2.x
ml
Pythagorean Theorem and
Trigonometry videos,
practice, and assessments
(Chapter 9 and Chapter
12):
http://fabienneriesen.com/G
eometry.php
“Temple Geometry”:
http://map.mathshell.org/ma
terials/tasks.php?taskid=26
1&subpage=apprentice
“Proofs of the Pythagorean
Theorem”:
http://map.mathshell.org/ma
terials/lessons.php?taskid=
419&subpage=concept
Informal Geometry Curriculum Map
May 2014
Course: Informal Geometry
Unit 10- Right Triangles and Trigonometry (cont)
Standard
The students will:
MAFS.912.G-SRT.3.8
Use trigonometric ratios and the
Pythagorean Theorem to solve right
triangles in applied problems.
SMP #1, #4
Mathematics Department
Volusia County Schools
Essential Question(s):
How can right triangles be used to solve application problems?
Learning Goals
I can:
Remarks
Resources
Includes the Triangle Inequality
 use angle measures to estimate side
Theorems and Hinge Theorem
lengths and vice versa.
Some students believe that the
 solve right triangles by finding the
trigonometric ratios defined in this
measures of all sides and angles in the
cluster apply to all triangles, but they
triangles using Pythagorean Theorem
are only defined for acute angles in
and/or trigonometric ratios and their
right triangles.
inverses.
 draw right triangles that describe real
world problems and label the sides and
angles with their given measures.
 solve application problems involving
right triangles, including angle of
elevation and depression, navigation,
and surveying.
Informal Geometry Curriculum Map
May 2014