# Common Core State Standards GEOMETRY

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10 Οκτ 2013 (πριν από 5 χρόνια και 5 μήνες)

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Mathematics
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Geometry

201
0

KEY ELEMENTS

CONTENT

(What Students should know)

PERFORMANCE TARGETS

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be able to do
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Common Core State Standards

GEOMETRY

Congruence

Experiment with transformations in the plane

1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined not
ions of point, line, distance
alon
g a line, and distance around a circular arc.

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

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

itself.

4.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel li
nes, and line segments.

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

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.

7. Use the definition of congruence in terms of rigid motions to show that t
wo triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.

8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of
rigid motio
ns.

Prove geometric theorems

9. Prove theorems about lines and angles.
Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior

Mathematics
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Geometry

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angles are congruent and corresponding angles are congruent; points on a p
erpendicular bisector of a line segment are exactly those equidistant
from the segment’s endpoints.

Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are
congruen
t; 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.

Theorems include: opposite sides are congruent, opposite angl
es are congruent, the diagonals of a
parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Make geometric constructions

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 s
egment; and constructing a line parallel to a given line through a point not on the line.

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

Similarity, Right Triangles, & Trigonometry

Understand similarity i
n terms of similarity transformations

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

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through
th
e center unchanged.

The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar;
explain using simil
arity
transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proporti
onality of all
corresponding pairs of sides.

Mathematics
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Geometry

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3. Use the properties of similarity transformations to establish the AA criterio
n for two triangles to be similar.

Prove theorems involving similarity

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.

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

Define trigonometric ratios and solve problems involving right triangles

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.

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

8. Use trigonometric ratios a
nd the Pythagorean Theorem to solve right triangles in applied problems.

Apply trigonometry to general triangles

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.

10. (+) Prove the Laws of Sines and Cosines and use them to solv
e problems.

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).

Circles

Understand and apply theorems about circles

1. Prove th
at all circles are similar.

Mathematics
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Geometry

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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 ci
rcle is perpendicular to the tangent where the radius intersects the circle.

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

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

Find arc lengths and areas of sectors of circles

5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and def
ine the radian measure of the
angle
as the constant of proportionality; derive the formula for the area of a sector.

Expressing Geometric Properties with Equations

Translate between the geometric description and the equation for a conic section

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.

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

3. (+) Derive the equations of ellipses and hyperbolas gi
ven the foci, using the fact that the sum or difference of distances from the foci is constant.

Use coordinates to prove simple geometric theorems algebraically

4. Use coordinates to prove simple geometric theorems algebraically.
For example, prove or disp
rove 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).

5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equ
ation of a line parallel or
perpendicular to a given

line that passes through a given point).

Mathematics
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Geometry

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

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

Geometric Measurement & Dimension

Explain volume formulas and use them to solve problems

1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyra
mid, and cone.
Use dissection
argumen
ts, Cavalieri’s principle, and informal limit arguments.

2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figu
res.

3. Use volume formulas for cylinders, pyramids, cones, and spheres to

solve problems.

Visualize relationships between two
-
dimensional and three
-
dimensional objects

4. Identify the shapes of two
-
dimensional cross
-
sections of three
-
dimensional objects, and identify three
-
dimensional objects generated by rotations of
two
-
dimensional objects.

Modeling with Geometry

Apply geometric concepts in modeling situations

1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human tor
so as a cylinder).

2. Apply concepts of density based on

area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints o
r minimize cost; working with
typogr
aphic grid systems based on ratios).

Mathematics
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Geometry

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ESSENTIALS OF GEOMETRY

Identify Points, Lines, and Planes

Use Segments and Congruence

Use Midpoint and Distance Formulas

Name and sketch geometric figures

Apply segment postulate to identify co
ngruent
segment

Students will model points, lines, and planes using
foam trays and uncooked spaghetti. They will explore
intersections of lines, and planes.

Find midpoint of segments in the coordinate plane

Find lengths of segments in the coordinate plane

FOLD A SEGMENT BISECTOR

Geometry McDougal Littell P. 15

Students will draw a line segment AB on paper, fold the
paper so that the two points fall on top of each other,
label the point M and compare the segments AM and
MB, and AB.

MODIFICATIONS

Using Geom
eter’s Sketch
-

Construct Acute, Right, Obtuse, and Straight angles

Explore Protractor Postulate

Describe Angles Pair Relationship

Classify Polygons

Use special angle relationships to find angle
m
easures.

Classify polygons

CONSTRUCT REGULAR POLYGONS

U
se the compass tool of the Geometer’s Sketchpad
to construct a circle and an inscribed square.

Using Geometer’s Sketch
-

1.

Construct various polygons; triangle, square,
pentagon, etc

2.

Measure sum of interior angles

3.

Develop a formula for sum of angles in a polygon of

Mathematics
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Geometry

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a given

number of sides

Perimeter, Circumference, and
Area

Students will be able to:

Find dimensions of a polygon

Find Perimeter

Find Circumference

Find Area

Usi
ng a Graphing Calculator students will:

1.

Construct different rectangles of area 36 square
units but of different dimensions.

2.

Plot the various dimensions that produces the
same area and make observations.

PARALLEL AND PERPENDICULAR LINES

Identify Pairs o
f Lines and Angles

Parallel Lines and Transversals

Prove Lines are Parallel

Find and Use Slopes of Lines

lines

INVESTIGATE SLOPES

Use Geometer’s Sketchpad to verify the equality of slopes
of parallel lines

Students

will be able to:

Identify angle pairs formed by three intersecting
lines

Use angles formed by parallel lines and transversals

Using
Geometer’s Sketch
-
Students will draw
two parallel lines and a transversal and explore the
relationship among angles fo
rmed.

Use angle relationships to prove that lines are
parallel

Mathematics
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Geometry

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KEY ELEMENTS

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Students will review key vocabulary and theorems

Solve algebraic problems on geometry concepts

Find and compare slopes of lines

Find equations of lines

Students will review key vocabulary and t
heorems

Write equations of parallel lines and graph them

Write equations of perpendicular lines and graph
them

Students will review key vocabulary and theorems

Apply theorems learned in section to solve
algebraic problems

CONGRUENT TRIANGLES

Apply Congru
ence and Triangles

Prove Triangles Congruent

Use two more methods to prove
congruence

Congruent triangles

Isosceles and Equilateral Triangles

Classify triangles by side and by angles

Find measures of angles algebraically

Understand congruence transformat
ions

Students will be able to:

Identify congruent figures

Use side lengths to prove triangles are congruent

Classify triangles by side and by angles

Find measures of angles algebraically

Classify triangles by side and by angles

Find measures of angles alg
ebraically

DISCOVERING ASA CONGRUENCE

Students will use tracing paper and straightedge to
investigate congruence of triangles by Angle
-
Side
-
Angle

COMPARING CONGRUENT TRIANGLES

Students will use ruler and protractor to construct two
congruent triangles an
d compare corresponding parts

Students will be able to:

Use theorems about isosceles and equilateral
triangles

Create an image congruent to a given triangle

INVESTIGATE SLIDES AND FLIPS

Students will investigate reflection and rotation on a

Mathematics
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Geometry

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coordinate ax
is.

RELATIONSHIPS WITHIN TRIANGLES

Midsegment Theorem and
Coordinate Proof

Perpendicular Bisectors

Angle Bisectors of Triangles

INVESTIGATE SEGMENTS IN TRIANGLES

Students will use Geometer’s Sketchpad to investigate
whether or not t
he midsegments of a triangle relates to
the sides of a triangles

Students will be able to
:

Use properties of midsegments and write
coordinate proofs

Use the midsegment theorem to find lengths

Place figure in a coordinate plane

Use perpendicular bisectors
to solve problems

EXPLORING PERPENDICULAR BISECTORS

Students will use tracing paper and straightedge to
investigate the relationship between the points on a
perpendicular bisector of a segment and the endpoint of
that segment

EXPLORING THE INCENTER

Stude
nts will use tracing paper, straightedge, and compass
to explore the relationship between the incenter and the
sides of a triangle

Students will be able to:

Use angle bisectors to find distance relationships

Use medians and altitudes of triangles

Use the
angle bisector theorem

Use the concurrency of angle bisectors

Medians and Altitudes

Inequalities in a Triangle

INVESTIGATING MEDIANS AND ALTITUDES

Students will use Cardboards to investigate the relationship
between segments formed by the medi
ans of a triangle

Students will be able to:

Mathematics
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Geometry

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Use medians and altitudes of triangles

Find possible side lengths of a triangle

Use the centroid of a triangle

Find centroid of a triangle

Inequalities in two Triangles and
Indirect Proof

DISCOVERING THE HING
E THEOREM

Students will use tracing paper and straightedge to
investigate triangles with two congruent sides

Students will be able to:

Use inequalities to make comparisons in two
triangles

Use the hinge theorem and its converse

Write an indirect proof

SI
MILARITY

Ratios, Proportions, and the
Geometric Mean

Proportions to solve Geometry
Problems

Similar Polygons

Prove Triangles Similar by AA

Prove Triangles Similar by SSS and
SAS

INVESTIGATING THE CROO PRODUCTS PROPERTY

Students will investigate the cross

product property by
equating the products of means and extremes

Students will be able to:

Solve problems by writing and solving proportions

Use the extended ratios and simplify ratios

Find geometric means

INVESTIGATE PROPERTIES OF PROPORTIONS

Students
will rearrange numbers to create proportions

Students will be able to:

Use proportions to solve geometry problems

Use proportions to identify similar polygons

Use properties of proportions

Find the scale of a drawing

SIMILAR POLYGONS

Mathematics
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Geometry

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Students will use G
and angles of a figure and its reduced version

Students will be able to:

Use proportions to identify similar polygons

Use the AA Similarity Postulate

DISCOVERING TRIANGLE SIMILARITY SHORTCUTS

Students will use straws

and tape to show that
corresponding sides of similar triangles are proportional

Students will be able to:

Use the SSS and SAS Similarity Theorems

Use the similarity postulate

Use indirect measurement

Use Proportionality Theorems

INVESTIGATE TRIANGLES
& CONGRUENCE

Students will use a graphing calculator to compare segment
lengths in triangles

Students will be able to:

Use proportions with a triangle or parallel lines

Find the length of a segment

Determine whether line segments are parallel

Similarity
Transformations

PERFORM SIMILARITY TRANSFORMAIONS

Students will use Geometer’s Sketchpad perform
transformations

Draw a dilation

Find a point on a dilation

RIGHT TRIANGLES AND TRIGONOMETRY

Apply the Pythagorean Theorem

PYTHAGOREAN THOEREM

Students wil
l use graph paper to explore the relationship
among sides of a right triangles

Solve problems on side lengths in right triangles

Mathematics
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Geometry

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Converse of the Pythagorean
Theorem

Similar Right Triangles

CONVERSE OF THE PYTHAGOREAN THEOREM

Students will use graphing
calculator to explore
relationship between sides and angles of a triangle

Students will be able to:

Use the converse of the Pythagorean theorem to
determine if a triangle is a right triangle

Use properties of the altitude of a right triangle

Solve problem
s on similar right triangles

Similar Right Triangles

Special Right Triangles

SIMILAR RIGHT TRIANGLES

Students will explore how geometric means are related to
the altitudes of a triangle

Students will be able to:

Use properties of the altitude of a rig
ht triangle

Use the relationships among the sides in special
right triangle

Apply the Tangent Ratio

RIGHT TRIANGLE RATIO

Students will use Geometer’s Sketchpad establish formulas
for the trigonometric ratios

Students will be able to:

Use the relationsh
ips among the sides in special
right triangle

Use the tangent ratio for indirect measurement

Sine and Cosine Ratios

APPLY SINE AND COSINE RATIOS

Students will use Geometer’s Sketchpad explore the
relationship between sides of a triangle

Students will b
e able to:

Use the sine and cosine ratios

Right Triangles

SOLVING REAL

WORLD PROBLEMS USING
TRIGONOMETRY

Students will use a calculator to find an angle measure in a
right triangle given two sides

Mathematics
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Geometry

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Students will be able to:

Use inverse tangent, sine,

and cosine ratios

Angle Measures in Polygons

INVESTIGATE ANGLE SUMS IN POLYGONS

Generator CD

Activity 8.1

Students will derive a formula for the sum of the measures
of the interior angles of a convex n
-
gon

Students will be able to:

F
ind angle measures in polygons

Find the sum of angle measures in a polygon

Find the number of sides of a polygon

Ties of Parallelograms

Show that a quadrilateral is a
parallelogram

INVESTIGATE PARALLELOGRAMS

Students will use Geometer’s Sketchpad to inv
estigate
some of the properties of parallelograms

Students will be able to:

Find angle and side measures in parallelograms

Use properties to identify parallelograms

Use the properties of a parallelogram

Find the intersection of diagonals

Properties of rh
ombuses, rectangles,
and squares

EXPLORING PROPERTIES OF RHOMBUSES

Students will explore the properties of a rhombus

Students will be able to:

Use properties of rhombuses, rectangles, and
squares

Use Properties of Trapezoids and Kites

Use properties of

Properties of Trapezoids and Kites

MIDSEGMENT OF A TRAPEZOID

Students will use Geometer’s Sketchpad explore the
properties of the midsegment of a trapezoid

Mathematics
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Geometry

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Students will be a
ble to:

Use Properties of Trapezoids and Kites

Use a coordinate plane

PROPERTIES OF TRANSFORMATIONS

Translate Figures and Use Vectors

COMPARING TRANSLATED POLYGONS

Students will use Geometer’s Sketchpad to investigate
wh
at happens to a triangle when a constant is added to its
x and y coordinates

Students will be able to:

Use a vector to translate a figure.

Translate a figure in coordinate plane

Write a rule for transformation

Properties of Matrices

INVESTIGATING MATRIX

Students will use Geometer’s Sketchpad to investigate the
effect matrix addition has on the coordinates of a triangle

Students will be able to:

Perform translations using matrix operations

Represent
a transformation using matrices

Perform reflections

REFLECTION IN THE PLANE

Students will use Geometer’s Sketchpad to explore the
relationship between the line of reflection and the segment
connecting a point and its image

Students will be able to:

Re
flect a figure in any given line

Graph reflection in horizontal and vertical lines

Use matrix multiplication to reflect polygons

Perform Rotations

Students will use Geometer’s Sketchpad to explore
igin

Mathematics
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Geometry

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Students will be able to:

Rotate a figure using the coordinate rules

Use matrices to rotate a figure

Compositions of Transformations

DOUBLE REFLECTION

Students will use a graphing calculator to reflect a figure in
tw
o lines in a plane

Students will be able to:

Perform combinations of two or more
transformations

Find the image of a glide reflection

Find image of a composition

Identify Symmetry

Identify and Perform Dilations

INVESTIGATE DILATIONS

Students will use G
dilation of a figure

Students will be able to:

Identify line and rotational symmetries of a figure

Use drawing tools and matrices to draw dilations

Identify line of symmetry

Identify rotational symmetry

PROPERTIES OF CIRC
LES

Properties of Tangents

EXPLORE TANGENT SEGMENTS

Students will use Geometer’s Sketchpad to explore how the
lengths of tangent segments are related

Students will be able to:

Use properties of a tangent to a circle

Identify special segments and line
s

Find lengths in circles in a coordinate plane

Find Arc Measures

Apply Properties of Chords

UNDERSTANDING CIRCLE VOCABULARY

Students will play a geometry vocabulary game

Mathematics
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Geometry

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Students will be able to:

Use angle measures to find arc measures

Use relati
onships of arcs and chords in a circle

Find measures of arcs

Identify congruent arcs

Use Inscribed Angles and Polygons

EXPLORE INSCRIBED ANGLES

Students will use Geometer’s Sketchpad to explore how
inscribed angles relate to central angles

Students wi
ll be able to:

Use inscribed angles of circles

Use circumscribed circles

Other Angle Relationships in Circles

Segment Lengths in Circles

Students will be able to:

Find the measures of angles inside or outside a
circle

Find segment lengths in circles

Fin
d the angle and the arc measures

Find the angle measure inside a circle

Write and Graph Equations of Circles

DETERMINING EQUATIONS OF CIRCLES

Students will use Geometer’s Sketchpad to derive the
equation of a circle

Students will be able to:

Write equa
tions of circles in the coordinate plane

MEASURING LENGTH AND AREA

Areas of Triangles and
Parallelograms

Areas of Trapezoids, Rhombuses,
and Kites

DETERMINE PRECISION AND ACCURACY

Students will use Geometer’s Sketchpad to explore the
measuring distance
s in precision

Students will be able to:

Find areas of triangles and parallelograms

Find areas of other types of quadrilaterals

Mathematics
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Geometry

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Perimeter and Area of Similar
Figures

AREA OF TRAPEZOIDS AND KITES

Students will use graph paper to explore the use of a
pa
rallelogram to find other areas

Students will be able to:

Find areas of other types of quadrilaterals

Use ratios to find areas of similar figures

Find the area of a quadrilateral

Find an area in the coordinate plane

Circumference and Arc Length

EXPLO
RE CIRCUMFERENCE

Students will explore the ratio of circumference to
diameter and establish a formula for finding the
circumference of a circle when given diameter

Students will be able to:

Find arc lengths and other measures

Use the formula for circumfer
ence

Use arc length to find measures

Areas of Circles and Sectors

AREAS OF CIRCLES AND SECTORS

Students will explore the area of circles and sectors

Students will be able to:

Find areas of circles and sectors of circles

Use the formula for area of a c
ircle

Find the area of sectors

Areas of Regular Polygons

FINDING THE AREA OF REGULAR POLYGONS

Students will establish an equation for the area of a regular
polygon

Students will be able to:

Find areas of regular polygons inscribed in circles

Find angle
s measures in a regular polygon

Mathematics
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Geometry

201
0

KEY ELEMENTS

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(What Students should know)

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Find the perimeter and area of a regular polygon

Geometric Probability

INVESTIGATE GEOMETRIC PROBABILITY

Students explore how theoretical and experimental
probabilities compare

Students will be able to:

Use lengths and
areas to find geometric
probabilities

SURFACE AREA AND VOLUME OF SOLIDS

Explore Solids

INVESTIGATE SOLIDS

Students will investigate what solids can be made using
congruent regular polygons

Students will be able to:

Identify Solids

Identify and name
polyhedra

Use euler’s theorem with platonic solids

Surface Area of Prism and Cylinders

Surface Area of Pyramids and Cones

INVESTIGATE SURFACE AREA

Students will explore how you can find the surface area of
a polyhedron

Students will be able to:

Find t
he surface areas of prisms and cylinders

Find the surface areas of pyramids and cones

Volume of Prisms and Cylinders

SURFACE AREAS OF PYRAMIDS AND CONES

Students will use Geometer’s Sketchpad to prove triangles
are congruent by Side
-
Side
-
Angle

Student
s will be able to:

Find the surface areas of pyramids and cones

Find the volume of prisms and cylinders

Find the area of a lateral face of a pyramid

Mathematics
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Geometry

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Volume of Pyramids and Cones

VOLUME OF PRISMS AND CYLINDERS

Students will derive a formula for finding
the volume of
Prisms and Cylinders

Students will be able to:

Find the volume of prisms and cylinders

Find the volume of pyramids and cones

Use volume of prism

Volume of Pyramids and Cones

Surface Area and Volume of Spheres

INVESTIGATE TRIANGLES & CONGR
UENCE

Students will explore the surface area of a pyramid

Students will be able to:

Find the volume of pyramids and cones

Find the surface

area and volume of spheres

Find the volume of a solid

Use trigonometry to find the volume of a cone

Explore Simila
r Solids

Surface Area and Volume of Spheres

Students will play a game on surface area and volume of
spheres

Students will be able to:

Find the surface

area and volume of spheres

Use properties of similar solids