Standard

Essential Question

Bloom’s Taxonomy Activities

Vocabulary

Pacing

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.

How are points and lines related to
each other?

Create a graphic organizer involving
various geometric shapes and their definitions

Angle
Circle
Perpendicular line
Parallel line
Line segment
Arc
Point
Line

2 days

G.CO.2 Represent transformations in the plane
using, e.g., transparencies and geometry software; describe transformations
as functions that take points in the plane as inputs and give other points as
outputs. Compare transformations that preserve distance and angle to those
that do not (e.g. translation versus horizontal stretch).

How does the knowledge of
transformations assist with graphic design?

Create a poster on an approved topic
of your choice demonstrating various transformations throughout your project.

Transformations
Translations
Dilation
Reflections
Rotations
Rigid motion

2 days

G.CO.3 Given a rectangle, parallelogram,
trapezoid, or regular polygon, describe the rotations and reflections that
carry it onto itself.

How does angle measure relate to
reflections and rotations of objects?

Create models demonstrating how
rotations and reflections of various shapes operate.

 Angle measure

2 days

G.CO.4 Develop definitions of rotations,
reflections, and translations in terms of angles, circles, perpendicular
lines, parallel lines, and line segments.

What is the relationship between translations
and other forms of geometric measurement and expression?

Design a visual expression of
rotations, reflections, and translations across circles, lines and segments.

Perpendicular lines
Parallel lines

2 days

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.

How are translations interrelated?

Use multiple forms of media to create
a visual display demonstrating a series of transformations.

Glide reflection

2 days

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

What is the relationship between
congruence and rigid motion.

Predict the relationship of two given
figures based on the knowledge of how transformations relate to congruence.

Congruence
Rigid motions

2 days

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.

How do corresponding sides and angles
relate to congruence?

Visually demonstrate how corresponding
sides and angles relate to congruence in triangles.

Corresponding sides
Corresponding angles
Triangles

2 days

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

How are the triangle congruence
theorems interrelated?

Support the definitions of triangle
congruence algebraically or graphically.

ASA congruence
SAS congruence
SSS congruence

2 days

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

How are
alternate interior angle, vertical angles, and corresponding angles related?

Create
shapes similar to tangrams to prove that corresponding angles are congruent,
alternate interior angles are congruent, and vertical angles are congruent.

Corresponding
angles
 Alternate
interior angles
Vertical
angles

2 days

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.

How is knowledge
of angle measure in an triangle related to determining the total number of
degrees in a regular polygon?

Complete the
Common CoreAligned Task: Company Logo

Theorems
Isosceles
Congruent
Medians

2 days

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.

What is the
relationship between quadrilaterals, parallelograms, squares, and rectangles?

Create a
poster highlighting qualities of parallelograms. Display your poster in the
classroom or hallway.

Parallelograms
Congruent
Opposite
sides
Diagonals

2 days

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

How are various tools utilized in
constructing geometric objects?

Construct the following geometric
constructions:
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

Bisector
Segment
Perpendicular bisector

1 day

G.CO.13 Construct an equilateral triangle, a
square, and a regular hexagon inscribed in a circle

How does knowledge of angle measure
relate to the ability to draw geometric constructions?

 Construct an equilateral triangle, a square, and
a regular hexagon inscribed in a circle

Equilateral triangle
Regular hexagon
Inscribed in a circle

1 day

Standard

Essential Question

Bloom’s Taxonomy Activities

Vocabulary

Pacing

G.SRT.1
Verify experimentally the properties of dilations given by a center and a
scale factor:
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.

How does
scale factor relate to similarity in triangles?

Complete the
interactive activities in the following site and write a reflection based on
what you learned. http://bit.ly/xa7x4d

Dilations
Scale

2 days

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.

How is
knowledge of scale factor important to graphic designers?

Using a
paint program… create similar figures using the scale feature, and create dissimilar
figures by adjusting the size manually. Using a ruler, support that your
first set of figures are similar and your second set are not.

Similarity

2 days

G.SRT.3 Use
the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.

What is the
least amount of information needed to prove two triangles similar? How do you
know?

Using a
ruler and a protractor, prove AA similarity.

AA
similarity

2 days

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

How does the
Pythagorean Theorem support the case for triangle similarity?

View the
video below and create a visual proving the Pythagorean Theorem using
similarity

Proof
Triangle
Similarity

2 days

G.SRT.5 Use
congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.

How does
angle measure relate to triangle similarity?

Create a
table highlighting the six examples of similar triangles utilizing the
resource listed below.

Congruence
Similarity

2 days

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.

How are
mnemonics helpful in learning trig ratios?

Create a
worksheet of six realworld examples which use trigonometric ratios.

Sine
Cosine
Tangent
CoTangent

2 days

G.SRT.7
Explain and use the relationship between the sine and cosine of complementary
angles.

What is the
relationship between cosine and sine in relation to complementary angles?

Construct a
table demonstrating the relationship between sine and cosine of complementary
angles.

Complementary
angles

2 days

G.SRT.8 Use
trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.*

How do
trigonometric ratios assist in determining height and depth when angle and
distance to a given point are known?

Complete
the exercise listed below and create two problems demonstrating trigonometric
ratios

Trigonometric
ratios

2 days

G.SRT.9 (+)
Derive the formula A=1/2absin(C)
for the area of a triangle by drawing an auxiliary line from a vertex
perpendicular to the opposite side.

When could a
contractor use the equation A=1/2absin(C)
to find the area of a triangle?

Create a
poster showing no less than three examples of finding the area of a nonright
triangle using the formula A=1/2absin(C)

Area
Triangle
Right
triangle

1 day

G.SRT.10 (+)
Prove the Laws of Sines and Cosines and use them to solve problems.

When is it
appropriate to use Law of Sines instead of Law of Cosines?

Research
video examples of Law of Sines and Law of Cosines and create a short video
teaching others how to use Law of Sines or Law of Cosines

Law of
Sines
Law of
Cosines

1 day

G.SRT.11 (+)
Understand and apply the Law of Sines and the Law of Cosines to find unknown
measurements in right and nonright triangles (e.g. surveying problems,
resultant forces.)

What are
three realworld applications of Law of Sines and Law of Cosines?

Using the
internet, find a realworld example of Law of Sines or Law of Cosines and
evaluate which formula would be appropriate for use; justify your answers.

Law of
Sines
Law of
Cosines

1 day

Standard

Essential Question

Bloom’s Taxonomy Activities

Vocabulary

Pacing

G.C.1 Prove
that all circles are similar

Why is it
beneficial to know that all circles are similar?

Complete
the following activity and write one paragraph discussing why you believe all
circles are similar. http://illuminations.nctm.org/ActivityDetail.aspx?ID=116

Circles
Similar

1 day

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.

How is
knowledge of the properties of circles and cylinders beneficial to the
manufacturing industry?

Solve the
following problem and create similar problems for your peers to complete.
An audio CD
spins at a rate of 200 rotations per minute. If the radius of a CD is 6 cm,
how far does a point on the outer edge travel during the playing of a
57minute CD

Inscribed
angles
Radii
Chords
Central
angles
Circumscribed
angles
Tangents

1 day

G.C.3
Construct the inscribed and circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed in a circle.

Where are
inscribed circles and circumscribed used in everyday life?

Research
practical uses of inscribed and circumscribed circles pertaining to triangles
and create a visual demonstrating the results of your research.

Inscribed
angles
Circumscribed
circles

1 day

G.C.4 (+)
Construct a tangent line from a point outside a given circle to the circle.

Why would
landscape artists need to know how to create a tangent line?

Design a
garden using circles and lines, include at least two tangents in your
diagram.

Compass
Ruler

1 day

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 is area
of a sector related to the length of the arc?

Watch the
following video and create a similar problem for your peers to complete. http://www.5min.com/Video/HowtoFindtheAreaofaSectorofaCircle275614332

Similarity
Arc
Radian

2 days

Standard

Essential Question

Bloom’s Taxonomy Activities

Vocabulary

Pacing

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.

How are
radii related to the sides of an inscribed right triangle?

Watch the
following video as a guide to derive the equation of a circle using the
Pythagorean theorem: http://www.youtube.com/watch?v=pZVufpgozCw
Create a
worksheet for your peers with three sample problems

Center
Circle
Pythagorean
theorem
Distance
formula
Radius
Complete
the square

3 days

G.GPE.2
Derive the equation of a parabola given a focus *and directrix

What is the
relationship between focus and directrix to the parabola formula?

In teams,
construct a poster of various graphs of parabolas with the focus and
directrix colorcoded to match the corresponding components in each given
equation.

Parabola
Focus
Directrix

3 days

Ggpe3


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 plan 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).

How can the
use of geometry and related algebraic equations assist in proving geometric
theorems?

Support
geometric theorems algebraically and graphically.

Proof
Coordinates

3 days

G.GPE.5
Prove the slope criteria for parallel and perpendicular lines; 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).

How is slope
related to parallel and perpendicular lines?

Evaluate if
lines are parallel, intersecting, or perpendicular based on slope criteria.

Slope

3 days

G.GPE.6 Find
the point on a directed line segment between two given points that partitions
the segment in a given ratio.

How does
distance formula relate to determining ratios in a line segment?

Create a
problem on a coordinate plane in which a line segment is divided after a
certain ratio is completed; solve your problem.
For example,
1/3 of the distance between (2,3) and (3,5)

Line
segment
Ratio
Distance
formula

3 days

G.GPE.7 Use
coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.★

Why is
knowledge of geometry and algebra in relation to mathematics in the real
world?

Design
visual representations of the link between algebra and geometry in relation
to distance and area.

Distance
formula

3 days

Standard

Essential Question

Bloom’s Taxonomy Activities

Vocabulary

Pacing

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.

How is
height related to the area of a circle and the volume of a similar cylinder?

Create a
visual demonstrating the relationship between circumference, area, height,
and volume of a circle and its related cylinder

Informal
argument
Circumference
Volume

1 day

Ggmd2


G.GMD.3 Use
volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.*

In what
contexts are volume formulas necessary for math after high school?

Create a
problem for your classmates to solve in which they must use at least three
volume formulas

Volume
Cone
Cylinder
Pyramids
Sphere

1 day

G.GMD.4
Identify the shapes of twodimensional crosssections of threedimensional
objects, and identify threedimensional objects generated by rotations of
twodimensional objects.

Where is
multidimensional math most studied? What could it express about our universe?

Create a 2
sets of PowerPoint slides showing the progression of dimension from 1^{st}
to 4^{th}

Dimension
Conic
Triangle
Cone
Circle
Sphere

3 days

Standard

Essential Question

Bloom’s Taxonomy Activities

Vocabulary

Pacing

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).*

How do
artists consider geometric shapes when creating masterpieces?

Find a
picture of an object of interest to you. Use a program such as Paint or
PowerPoint to create the object using only geometric shapes

Geometric
shapes
Cylinder
Sphere
Rectangular
prism
Parabola

3 days

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).*

How does
population density in NYC assist in the saving of the world’s endangered
languages?

Research
the U.S. at night over time. Create a visual with no less than four graphics
showing the changes in population density based on images from the evening
sky

Density
Area
Volume

3 days

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).*

How is it
possible for an object o have a finite area but unlimited perimeter?

Use a
computer program to investigate fractals and discuss the apparent limitations
of the objects

Fractals

3 days

Geometry
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