Download Geometry CURRICULUM GUIDE AND INSTRUCTIONAL ALIGNMENT

TRENTON PUBLIC SCHOOLS
Department of Curriculum and Instruction
108 NORTH CLINTON AVENUE
TRENTON, NEW JERSEY 08609
Secondary Schools
Geometry
CURRICULUM GUIDE AND INSTRUCTIONAL ALIGNMENT
August 2013-Revised June 2004
1 *M.C. = Major Content
*S.C. = Supporting Content
*A.C. = Additional Content
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Geometry Units at Glance
(From NJDOE Model Curriculum - each unit is designed to take approximately 30 days.)
Overview
Moving towards formal mathematical arguments, the standards presented in this high school geometry course are meant to formalize and
extend middle grades geometric experiences. Transformations are presented early in the year to assist with the building of conceptual
understandings of the geometric concepts.
In unit 1, triangle congruence conditions are established using analysis of rigid motion and formal constructions. Various formats will be
used to prove theorems about angles, lines, triangles and other polygons. The work in unit 2 will build on the students understanding of
dilations and proportional reasoning to develop a formal understanding of similarity.
The standards included in unit 3 extend the notion of similarity to right triangles and the understanding of right triangle trigonometry. In
developing the Laws of Sines and Cosines, the students are expected to find missing measures of triangles in general, not just right
triangles.
Work in unit 4 will focus on circles and using the rectangular coordinate system to verify geometric properties and to solve geometric
problems. Concepts of similarity will be used to establish the relationship among segments on chords, secants and tangents as well as to
prove basic theorems about circles.
The standards in unit 5 will extend previous understandings of two- dimensional objects in order to explain, visualize, and apply geometric
concepts to three-dimensional objects. Informal explanations of circumference, area and volume formulas will be analyzed.
Unit 1: Congruence, Proof, and Constructions
Big idea
Experiment with
transformations in
the plane.
Pre-req
standard
Prerequisite description
CCSS
Description
Solve real-world and mathematical problems
by graphing points in all four quadrants of the
Know precise definitions of angle, circle, perpendicular
6.NS.8 coordinate plane. Include the use of
line, parallel line, and line segment, based on the
G.CO.1
Gr.6 Unit2 coordinates and absolute value to find
undefined notions of point, line, distance along a line, and
distances between points with the same first
SLO 4
distance around a circular arc.
coordinate or the same second
coordinate.
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
8.G.1
Gr.8 Unit1
SLO 1
8.G.3
Gr.8 Unit1
SLO 3
8.G.2
Understand
Congruence
interms of rigid
motions.
Gr.8 Unit1
SLO 2
8.G.4
Gr.8 Unit1
SLO 3
Utilize the properties of rotation, reflection,
and translation to model and relate pre-images
of lines, line segments, and angles to their
resultant image through physical
representations and/or Geometry software.
Verify experimentally the properties of
rotations, reflections, and translations.
a. Lines are taken to lines, and line segments
to line segments of the same length.
b. Angles are taken to angles of the same
measure.
c. Parallel lines are taken to parallel
Describe the effect of dilations, translations,
rotations, and reflections on two-dimensional
figures using coordinates. Use the coordinate
plane to locate pre-images of two-dimensional
figures and determine the coordinates of a
resultant image after applying dilations,
rotations, reflections, and translations.
Apply an effective sequence of rotations,
reflections, and transitions to prove that two
dimensional figures are congruent.
Apply an effective sequence of transformations
to determine similar figures in which
corresponding angles are congruent and
corresponding sides are proportional. Write
similarity statements based on such
G.CO.2
Represent transformations in the plane using, e.g.,
transparencies and geometrysoftware; 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).
G.CO.3
Given a rectangle, parallelogram, trapezoid, or regular
polygon, describe the rotations and reflections that carry
it onto itself.
G.CO4
G.CO.5
G.CO.6
G.CO.7
Develop definitions of rotations, reflections, and
translations in terms of angles, circles, perpendicular
lines, parallel lines, and line segments.
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.
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.
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.
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
transformations. Understand that a twodimensional figure is similar to another if the
second can be obtained from the first by a
sequence of rotations, reflections, translations,
and dilations; given two similar twodimensional figures, describe a sequence that
exhibits the similarity between them.
7.G.5
Gr.7 Unit5
SLO5
Prove geometric
theorems.
Make Geometric
Constructions
8.G.5
Gr.8 Unit1
SLO 6 & 7
7.G.2
Gr.7 Unit3
SLO7
Use facts about supplementary,
complementary, vertical, and adjacent angles
in a multi-step problem to write and solve
simple equations for an unknown angle in a
figure.
Justify facts about angles created when parallel
lines are cut by a transversal. Use informal
arguments to establish facts about the angle
sum and exterior angle of triangles, about the
angles created when
parallel lines are cut by a transversal, and the
angle-angle criterion for similarity of triangles.
For example, arrange three copies of the same
triangle so that the sum of the three angles
appears to form a line, and give an argument in
terms of transversals why this is so.
Draw (freehand, with ruler and protractor, and
with technology) geometric shapes with given
conditions. Focus on constructing triangles
from three measures of angles or sides,
noticing when the conditions determine a
unique triangle, more than one triangle, or no
triangle.
G.CO.8
Explain how the criteria for triangle congruence (ASA,
SAS, and SSS) follow fromthe definition of congruence in
terms of rigid motions.
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.
G.CO.10
Prove theorems about triangles. Theorems include:
measures of interior angles of atriangle sum to 180°;
base angles of isosceles triangles are congruent; the
segment joining midpoints of two sides of a triangle is
parallel to the third side and half the length; the medians
of a triangle meet at a point.
G.CO.11
Prove theorems about parallelograms. Theorems include:
opposite sides are congruent, opposite angles are
congruent, the diagonals of a parallelogram bisect each
other, and conversely, rectangles are parallelograms with
congruent diagonals.
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.
G.CO.13
Construct an equilateral triangle, a square, and a regular
hexagon inscribed in a circle.
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Unit 2: Similarity and Proof.
Big idea
Pre-req
standard
8.G.3
Gr.8 Unit1
SLO 4
Understand
similarity in terms
of similarity
transformations
8.G.4
Gr.8 Unit1
SLO 5
Prove geometric
theorems.
Understand and
apply theorems
about circles
8.G.5
Gr.8Unit1
SLO 6&7
Prerequisite description
Recognize dilation as a reduction or an
enlargement of a figure and determine the
scale factor. Describe the effect of dilations...
on two-dimensional figures using coordinates.
Use the coordinate plane to locate pre-images
of two-dimensional figures and determine the
coordinates of a resultant image after applying
dilations, ...
Apply an effective sequence of transformations
to determine similar figures in which
corresponding angles are congruent and
corresponding sides are proportional. Write
similarity statements based on such
transformations. Understand that a twodimensional figure is similar to another if the
second can be obtained from the first by a
sequence of ...dilations; given two similar twodimensional figures, describe a sequence that
exhibits the similarity between them.
Use informal arguments to establish facts
about the angle sum and exterior angle of
triangles, about the angles created when
parallel lines are cut by a transversal, and the
angle-angle criterion for similarity of triangles.
For example, arrange three
copies of the same triangle so that the sum of
the three angles appears to form a line, and
give an argument in terms of transversals why
this is so.
N/A
CCSS
CCSS Description
Verify experimentally the properties of dilations given by
a center and a scale factor.
G.SRT.1 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.
The dilation of a line segment is longer or shorter in the
ratio given by the scale factor.
G.SRT.2
Given two figures, use the definition of similarity in terms
of similarity transformations to decide if they are similar;
explain using similarity transformations the meaning of
similarity for triangles as the equality of all corresponding
pairs of angles and the proportionality of all
corresponding pairs of sides.
Use the properties of similarity transformations to
G.SRT.3 establish the AA criterion for two triangles to be similar.
G.CO.10
G.SRT.4
G.C.1
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.
Prove that all circles are similar
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Unit 3: Trigonometry
Big idea
Use coordinates to
prove simple
geometric
theorems
Pre-req
standard
6.NS.8
Gr.6 Unit2
SLO 4
8.G.6
Prove theorems
involving similarity.
Gr.8 Unit4
SLO5
N/A
7.G.1
Define
trigonometric
ratios and solve
problems involving
right triangles.
Apply trigonometry
to general
triangles.
Prerequisite description
Solve real-world and mathematical problems
by graphing points in all four quadrants of the
coordinate plane. Include the use of
coordinates and absolute value to find
distances between points with the same first
coordinate or the same second
coordinate.
Explain a proof of the Pythagorean Theorem
and its converse.
N/A
Solve problems involving scale drawings of
geometric figures, including computing actual
lengths and areas from a scale
drawing and reproducing a scale drawing at a
different scale.
CCSS
G.GPE.6
G.SRT.4
G.SRT.5
Description
Find the point on a directed line segment between two
given points that partitions the segment in a given ratio.
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.
Use congruence and similarity criteria for triangles to
solve problems and to prove relationships in geometric
figures.
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.
N/A
G.SRT.7
Explain and use the relationship between the sine and
cosine of complementary angles.
Gr.8 Unit4
SLO6
Apply the Pythagorean Theorem to determine
unknown side lengths in right triangles in realworld and mathematical problems in two or
three dimensions.
G.SRT.8
Use trigonometric ratios and the Pythagorean Theorem
to solve right triangles in applied problems.★
N/A
N/A
G.SRT. 9
N/A
N/A
G.SRT.10
N/A
N/A
G.SRT.11
Gr.7 Unit3
SLO6
N/A
8.G.7
(+) Derive the formula A = 1/2 ab sin(C) for the area of a
triangle by drawing anauxiliary line from a vertex
perpendicular to the opposite side.
(+) Prove the Laws of Sines and Cosines and use them
to solve problems.
(+) 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).
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Unit 4: Circles and Expressing Geometric Properties through Equations
Big idea
Understand and
apply theorems
about circles
Find arc lengths
and areas of
sectors of circles.
Pre-req
standard
Description
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.
Construct the inscribed and circumscribed circles of a
triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
N/A
G.C.2
N/A
N/A
G.C.3
N/A
N/A
G.C.4
(+) Construct a tangent line from a point outside a given
circle to the circle.
G.C.5
Derive using similarity the fact that the length of the arc
intercepted by an angle is proportional to the radius, and
define the radian measure of the angle as the constant of
proportionality; derive the formula for the area of a sector.
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
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.
7.G.4
Gr.7 Unit5
SLO5
Gr.8 Unit4
SLO7
Alg 1
A.REI.4
Unit 3
Prove geometric
theorems.
CCSS
N/A
8.G.8
Translate between
the geometric
description and the
equation of a conic
section.
Prerequisite description
N/A
Know the formulas for the area and
circumference of a circle and use them to solve
problems; give an informal derivation of the
relationship between the circumference and
area of a circle.
Apply the Pythagorean Theorem to find the
distance between two points in a coordinate
system.
Solve quadratic equations in one variable.
a. Use the method of completing the square to
transform any quadratic equation in x into an
equation of the form (x – p)2 = q that has the
same solutions. Derive the quadratic formula
from this form.
N/A
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
8.G.8
Gr.8 Unit4
SLO7
Use coordinates to
prove simple
geometric
theorems
algebraically.
8.EE.5,6
Gr.8 Unit3
SLO1&2
8.G.8
Gr.8 Unit4
SLO7
Apply the Pythagorean Theorem to find the
distance between two points in a coordinate
system.
Graph proportional relationships, interpreting
the unit rate as the slope of a graph. Compare
two different proportional relationships
represented in different ways. Use similar
triangles to explain why the slope m is the
same between any two distinct points on a
non-vertical line in the coordinate plane; derive
the equation y = mx for a line through the
origin and the equation y = mx +b for a line
intercepting the vertical axis at b.
Apply the Pythagorean Theorem to find the
distance between two points in a coordinate
system.
G.GPE.4
Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a
figure defined by four given points in the coordinate plane
is a rectangle; prove or disprove that the point (1, √3)
lies on the circle centered at the origin and containing the
point (0, 2).
G.GPE.5
Prove the slope criteria for parallel and perpendicular
lines and use them to solve geometric problems. (e.g.
find the equation of a line parallel or perpendicular to a
given line that passes through a given point.
G.GPE.7
Use coordinates to compute perimeters of polygons and
areas of triangles and rectangles, e.g., using the distance
formula.★
CCSS
Description
Unit 5: Extending to Three Dimensions
Big idea
Pre-req
standard
7.G.4
Explain volume
formulas and use
them to solve
problems.
Gr.7 Unit5
SLO5
8.G.9
Gr.8 Unit5
SLO 6
Prerequisite description
Know the formulas for the area and
circumference of a circle and use them to solve
problems;
give an informal derivation of the relationship
between the circumference and area of a
circle.
Know and apply the appropriate formula for
the volume of a cone, a cylinder, or a sphere to
solve real-world and mathematical problems.
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.
G.GMD. 3
Use volume formulas for cylinders, pyramids, cones, and
spheres to solve problems.★
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Visualize the
relation between
two-dimensional
and threedimensional
objects.
Apply geometric
concepts in
modeling
situations.
7.G.3
Gr.7 Unit5
SLO6
7.G.6
Gr.7 Unit5
SLO3
Describe, using drawings or written
descriptions, the 2-dimensional figures that
result when 3-dimemsional figures (right
rectangular prisms and pyramids) are sliced
from multiple angles given both concrete
models and a written description of the 3dimensional figure.
Solve real-world and mathematical problems
involving area, volume and surface area of twoand three-dimensional objects composed of
triangles, quadrilaterals, polygons, cubes, and
right prisms.
G.GMD. 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.
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 acylinder). ★
G.MG. 2
Apply concepts of density based on area and volume in
modeling situations (e.g., persons per square mile, BTUs
per cubic foot). ★
G.MG. 3
Apply geometric methods to solve design problems (e.g.,
designing an object orstructure to satisfy physical
constraints or minimize cost; working with typographic
grid systems based on ratios). ★
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
UNIT 1: Congruence, Proof, and Construction
http://www.state.nj.us/education/modelcurriculum/math/Geometryu1.shtml
Grade level: HS
District-Approved Text: Geometry Common Core, Pearson 2012
5 weeks of instruction, 2 weeks of review, enrichment and assessment
Stage 1 – Desired Results
Enduring Understandings/Goals:
Undefined geometric notions form the basis of defining interrelated geometric terms.
Rigid transformations provide a means of understanding congruence, a useful concept in modeling various geometric
situations.
Non-rigid transformations enable us to understand similarity.
Geometric definitions as well as geometric postulates/axioms allow mathematicians to prove geometric theorems.
We can use theorems about angles formed by parallel lines cut by a transversal in order to develop theorems about
triangles and parallelograms.
Essential Questions:
How can we change a figure’s position without changing its size or shape? How can we change a figure’s size without
changing its shape?
How can we represent transformations in the coordinate plane?
Why do congruence transformations, based upon “rigid motions”, preserve distance in figures?
How can “non-rigid” transformations, which preserve angle but not distance, help us understand similarity?
How can we systematically use our understanding of geometric definitions and postulates (axioms) in order to develop
geometric theorems?
Mathematical Practices:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
7 *M.C. = Major Content
*S.C. = Supporting Content
*A.C. = Additional Content
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Standard:
(Common Core)
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. S.C.*
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). S.C.*
Suggested Instructional Strategies
Student Learning
Objectives (SLO’s)
SLO1: Use the undefined
Explain the undefined terms.
notion of a point, line, distance
Have students write their own
along a line and distance
around a circular arc to
understanding of a given term.
develop definitions for angles,
Give students formal and informal
circles,
parallel
lines,
perpendicular lines, and line
definitions of each term and compare
segments.
them.
SLO2: Apply the definitions of
Develop precise definitions through
angles, circles, parallel lines,
perpendicular lines, and line
use of examples and non-examples
segments to describe rotations,
(Frayer model).
reflections, and translations.
Discuss the importance of having
precise definitions.
SLO3: Develop and perform
rigid transformations that
include reflections, rotations,
and
translations
using
geometric software, graph
paper, tracing paper, and
geometric tools, and compare
them
to
non-rigid
transformations.
8 *M.C. = Major Content
A function has one output for every
input, which can be a point in the
plane; hence, transformations act as
functions upon a set of points.
Transform < L. transformare, “to
change the shape or form of.”
Rigid <L. rigere, “be stiff”
Rigid transformation: change position
not size or shape.
Suggested
Resources
Pearson Geometry:
1-2: Points, Lines & Planes
1-3: Measuring Segments
1-4: Measuring Angles
1-6: Basic Constructions
3-1: Lines and Angles
10-6: Circles and Arcs
Unit 1 assessment (2012):
#1 Define using undefined
www.geogebra.org
http://www.engageny.org/mathematics
Geometry, module 1
Pearson Geometry:
9-1: Translations
9-22: Reflections
9-33: Rotations
9-6: Dilations
Translate <L. transferre “to carry
across”
Unit 1 assessment (2012):
#5 Reflection
#7 Compare/contrast rigid
and non-rigid
transformations
Reflect <L. reflectere “to bend back”
www.geogebra.org
Rotate <L. rotare “to turn around”
Compare/ contrast rigid & non-rigid
transformations in Escher drawings.
*S.C. = Supporting Content
*A.C. = Additional Content
http://www.engageny.org/mathematics
Geometry, module 1
www.etymonline.com
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
G.CO.3 Given a
rectangle,
parallelogram,
trapezoid, or regular
polygon, describe the
rotations and
reflections that carry
it onto itself. S.C.*
SLO3: Develop and perform
rigid transformations that
include reflections, rotations,
and translations using
geometric software, graph
paper, tracing paper, and
geometric tools, and compare
them to non-rigid
transformations.
G.CO.4
Develop
definitions
of
rotations, reflections,
and translations in
terms
of
angles,
circles, perpendicular
lines, parallel lines,
and line segments.
S.C.*
SLO2: Apply the definitions
of angles, circles, parallel
lines, perpendicular lines, and
line segments to describe
rotations, reflections, and
translations.
SLO3: Develop and perform
rigid transformations that
include reflections, rotations,
and translations using
geometric software, graph
paper, tracing paper, and
geometric tools, and compare
them to non-rigid
transformations.
SLO3: Develop and perform
rigid transformations that
include reflections, rotations,
and translations using
geometric software, graph
paper, tracing paper, and
geometric tools, and compare
them to non-rigid
transformations.
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
carries a given figure
onto another. S.C.*
9 *M.C. = Major Content
Provide sets of polygons for students
to manipulate.
Use mirrors or a reflective device to
help students see lines of symmetry.
Remind students of etymological
meaning of rotation and reflections.
Connect to prior knowledge of mirror
image (reflection), turning (rotation),
sliding (translation)
Draw rotations, reflections, and
translations.
Use geometry software to model
rotations, reflections, and
translations.
Pearson Geometry:
9-3 Concept Byte:
symmetry p. 568-9
http://www.engageny.org/mathematics
Geometry, module 1
www.geogebra.org
Unit 1 assessment (2012):
#5 Reflection
#6 Translation, rotation
Pearson Geometry:
9-1, 9-2, 9-3
9-2 Concept Byte: p. 553
Paper folding/reflections
Unit 1 assessment (2012):
#2 Translation
#3 Rotation
#4 Reflection
http://www.engageny.org/mathematics
Geometry, module 1
www.geogebra.org
Have students use a variety of tools
to explore and perform simple, multistep, and composite rotations,
reflections, and translations.
Given a transformation, work
backwards to discover the sequence
that led to that transformation.
*S.C. = Supporting Content
Pearson Geometry:
9-4 Composition of
isometries
Unit 1 assessment (2012):
#5 Reflection
#6 Translation & rotation
http://www.cut-the-knot.org/
http://www.engageny.org/mathematics
Geometry, module 1
*A.C. = Additional Content
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
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. M.C.*
SLO4: Use rigid
transformations to determine,
explain and prove congruence
of geometric figures.
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.
M.C.*
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.
M.C.*
SLO4: Use rigid
transformations to determine,
explain and prove congruence
of geometric figures.
Use graph paper, tracing paper,
physical models and geometry
software to verify predictions
regarding rigid motion and
congruence.
Use frieze patterns and Escher art to
explore congruency in
transformations.
Pearson Geometry:
9-5 Congruence
transformations (rigid
motions)
9-4 Composition of
isometries
Unit 1 assessment (2012):
#10 quadrilateral
congruence transformation
http://www.cut-the-knot.org/
http://www.engageny.org/mathematics
Geometry, module 1
Match pairs of cardboard congruent
triangles and justify congruence.
Measure angles and side lengths of
triangles resulting from rigid
transformations using a variety of
technology and paper based methods
(e.g., patty paper).
Pearson Geometry:
9-5 Congruence
transformations (rigid
motions)
Unit 1 assessment (2012):
#8 triangle congruence
transformation: reflection
http://www.cut-the-knot.org/
http://www.engageny.org/mathematics
Geometry, module 1
SLO4: Use rigid
transformations to determine,
explain and prove congruence
of geometric figures.
10 *M.C. = Major Content
Explore minimum conditions to show
triangles are congruent using
technology, reflective devices,
spaghetti, or grid paper.
Establish triangle congruence criteria
using properties of rigid motion.
*S.C. = Supporting Content
*A.C. = Additional Content
Pearson Geometry:
9-5 Congruence/rigid
transformations
Unit 1 assessment (2012):
#9 triangle congruence
transformation (criteria:
SSS, SAS, ASA)
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
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.
M.C.*
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. (also
appear in unit 2)
SLO5: Create proofs of
theorems involving lines,
angles, triangles, and
parallelograms.*
SLO5: Create proofs of
theorems involving lines,
angles, triangles, and
parallelograms.*
Prove triangle angle sum
theorem
Other triangle theorems
also appear in unit 2
Connect to properties of lines and
angles, triangles and parallelograms.
Let the students explain and defend
conjectures
Encourage multiple ways of writing
proofs (narratives, paragraphs, flow
charts/diagrams)
Make sure students learn the
meanings of key terms:
o Vertical angles
o Parallel
o Transversal
o Alternate interior angles
o Corresponding angles
o Perpendicular
o Bisect “cut in two”
o Equidistant “same distance”
Connect to properties of lines and
angles, triangles and parallelograms.
Students explain/defend conjectures
Encourage multiple ways of writing
proofs (narratives, paragraphs, flow
charts/diagrams)
Prove triangle sum theorem now
(other theorems appear in unit 2)
Review corresponding angles
theorem
Review angle sum postulate
Review straight angle =180 degrees
Review substitution property
M.C.*
11 *M.C. = Major Content
*S.C. = Supporting Content
*A.C. = Additional Content
Pearson Geometry:
2-6 Proving angles
congruent
3-2 Properties of parallel
lines
5-2 Perpendicular and
angle bisectors
Unit 1 assessment (2012):
#13 prove vertical angles
congruent
www.geogebra.org
http://www.engageny.org/mathematics
Geometry, module 1
Pearson Geometry:
3-5 Parallel lines and
triangles
4-5 Isosceles and
equilateral triangles
5-1 Midsegments of
triangles(appears in unit 2)
5-4 Medians and altitudes
(appears in unit 2)
Unit 1 assessment (2012):
#12 use alt. interior angles
to prove triangle angle
sum theorem
http://www.engageny.org/mathematics
Geometry, module 1
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
G.CO.11 Prove
theorems about
parallelograms.
Theorems include:
opposite sides are
congruent, opposite
angles are
congruent, and the
diagonals of a
parallelogram bisect
each other and
conversely, rectangles
are parallelograms
with congruent
diagonals. (Content
included in unit 2 & 4
repeated to assess
fluency.)
M.C.*
G.CO.12 Make formal
geom. 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,
incl. the perpendicular
bisector of a line
segment; constructing
a line parallel to a
given line through a
point not on the line.
*SC
SLO5: Create proofs of
theorems involving lines,
angles, triangles, and
parallelograms.*
(Content included in units 2
and 4, repeated to assess
fluency.)
SLO6: Generate formal
constructions with paper
folding, geometric software
and geometric tools to
include, but not limited to,
the construction of regular
polygons inscribed in a circle.
12 *M.C. = Major Content
Connect to properties of lines and
angles, triangles and parallelograms.
Students should understand
definition of a parallelogram as two
pairs of opposite parallel sides.
Students should understand that the
sum of interior angles of a
quadrilateral = 360 degrees, a
special case of polygon angle sum
theorem.
Let the students explain and defend
conjectures
Encourage multiple ways of writing
proofs (narratives, paragraphs, flow
charts/diagrams)
Have students explore how to make
a variety of constructions using
different tools. Ask students to justify
how they know their method results
in the desired construction.
Discuss the underlying principles that
different tools rely on to produce
desired constructions (e.g., compass:
circles, mira: reflections).
Pearson Geometry:
6-2 Properties of
parallelograms
6-3 Proving that a
quadrilateral is a
parallelogram
6-1 The polygon angle
sum theorem
Unit 1 assessment (2012):
#11 Prove that a
quadrilateral with both
pairs of congruent
opposite angles is a
parallelogram
http://www.engageny.org/mathematics
Geometry, module 1
Pearson Geometry:
1-6 Basic constructions
3-6 Constructing parallel
and perpendicular lines
Geometric software:
http://www.geogebra.org
Unit 1 assessment (2012):
#14 construct
perpendicular bisector
using paper folding
#15 construct angle using
compass and straight edge
http://www.engageny.org/mathematics
Geometry, module 1
*S.C. = Supporting Content
*A.C. = Additional Content
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
G.CO.13 Construct an
equilateral triangle, a
square, and a regular
hexagon inscribed in a
circle. S.C.*
SLO6: Generate formal
constructions with paper
folding, geometric software
and geometric tools to
include, but not limited to,
the construction of regular
polygons inscribed in a circle.
Allow students to explore possible
methods for constructing equilateral
triangles, squares, and hexagons,
and methods for constructing each
inscribed in a circle.
Pearson Geometry:
10-3 Areas of regular
polygons
Unit 1 assessment (2012):
#16 construct regular
hexagon in a circle
http://www.engageny.org/mathematics
Geometry, module 1
Suggested Performance Tasks:
Exemplars
Extended projects
Math Webquests
Writing in Math/Journal
13 *M.C. = Major Content
Stage 2 – Assessment Evidence
Other Evidence:
Classwork
Exit Slips
Homework
Individual and group tests
Open-ended questions
Portfolio
Quizzes
*S.C. = Supporting Content
*A.C. = Additional Content
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Stage 3 – Learning Plan
Lesson Plan Template
with suggested pacing for required 80 minute math block
Lesson
Objective
Using 3-part,
studentfriendly
language.
Ex. With 80%
proficiency, I
will solve 10
addition word
problems.
Instructional Components
Opening/Do
Now
Homework
Review
10-15 minutes
5-10
minutes
Do Now could
include:
Spiral review
of prerequisite
skills for
today’s lesson,
Pretest skills
to see where
students are
regarding
today’s
objective, or
Contain a
writing in
math type of
May choose to Whole group mini-lesson
review a few
with a check for
specific
understanding afterwards.
problems from
previous
nights
homework to
review for
understanding.
Students may
also have a
few they
struggled with
and need reteaching.
14 *M.C. = Major Content
Mini Lesson(s)
I DO/ WE DO
15-20 minutes (each)
*S.C. = Supporting Content
Independent/Partner/Group
Work
YOU DO
20-30 minutes
Lesson activity including at least
one check for understanding.
*A.C. = Additional Content
Summary and
Exit Slip
10 minutes
As a class,
teacher should
facilitate a
summary of
today’s targeted
objective then
provide an exit
question (last
check for
understanding)
that allows
students to
individually prove
their
understanding of
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
the objective.
prompt/
question for
students to
explain their
thinking, etc.
15 *M.C. = Major Content
*S.C. = Supporting Content
*A.C. = Additional Content
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
UNIT 2: Similarity and Proof
http://www.state.nj.us/education/modelcurriculum/math/Geometryu2.shtml
Grade level: HS
District-Approved Text: Geometry Common Core, Pearson 2012
5 weeks of instruction, 2 weeks of review, enrichment and assessment
Stage 1 – Desired Results
Enduring Understandings/Goals:
We can understand similar figures by using dilations (non-rigid transformations), which provide the same shape in a
different size.
Angle relationships of similar figures are constant, but not distance relationships.
Dilations, similarity and the properties of similar triangles give rise to the trigonometric ratios, which we can apply to
many real world situations.
Essential Questions:
How can we use our understanding of the features of one figure in order to solve problems relating to a similar figure?
Mathematical Practices:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
16 *M.C. = Major Content
*S.C. = Supporting Content
*A.C. = Additional Content
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Standard:
(Common Core)
G.C.1 Prove that all circles are
similar. A.C.*
G.SRT.1 Verify experimentally
the properties of dilations
given by a center and a scale
factor. M.C.*
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. M.C.*
b. The dilation of a line
segment is longer or shorter in
the ratio given by the scale
factor. M.C.*
G.SRT.2 Given two figures,
use the definition of similarity
17 *M.C. = Major Content
Student Learning
Objectives (SLO’s)
SLO1: Generate proofs
that demonstrate that
all circles are similar
SLO2: Verify and use
the properties of
dilations, use the
definition of similarity to
determine whether
figures are similar, and
establish the AA
criterion using
similarity.
SLO3: Verify and use
the properties of
*S.C. = Supporting Content
Suggested Instructional
Strategies
Use foldables to prove that
all circles are similar
Circles are dilations of one
another: same shape,
different size.
Scale factor proportional to
radius or diameter.
Concentric circles same
center
Explore dilations visually
and physically
Illustrate two-dimensional
dilations using scale
drawings and photocopies
Use graph paper and rulers
to obtain the image of a
given figure under dilations
having specified centers and
scale factors
Preserve angle but not
distance (length).
Use the theorem that the
angle sum of a triangle is
*A.C. = Additional Content
Suggested
Resources
Pearson Geometry:
10-6 Circles and arcs
Unit2 assessment version A:
#1 Circles are similar: plot
diameter vs. circumference.
#2 Circles are similar proof
http://www.geogebra.org
Pearson Geometry:
9-6: Dilations
Unit2 assessment version A:
# 3 dilation properties: line
through center of dilation
says same, but line not
through center parallel.
#4 dilation properties with
center and scale factor:
preserve angle not length;
sides proportional
#5 dilated line segment
proportional to scale factor.
http://www.geogebra.org
Pearson Geometry:
9-7 Similarity
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
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.
M.C.*
dilations, use the
definition of similarity to
determine whether
figures are similar, and
establish the AA
criterion using
similarity.
G.SRT.3 Use the properties of
similarity transformations to
establish the AA criterion for
two triangles to be similar.
M.C.*
SLO4: Verify and use
the properties of
dilations, use the
definition of similarity to
determine whether
figures are similar, and
establish the AA
criterion using
similarity.
G.SRT.4 Prove theorems
about triangles. Theorems
include: a line parallel to
18 *M.C. = Major Content
SLO5: Prove theorems
about triangles and use
triangle congruence and
*S.C. = Supporting Content
180 degrees; verify that the
AA criterion is equivalent to
the AAA criterion
Given two triangles for
which AA holds, use rigid
motions to map a vertex of
one triangle onto the
corresponding vertex of the
other. Then show that
dilation will complete the
mapping of one triangle
onto the other
Use the theorem that the
angle sum of a triangle is
180 degrees; verify that the
AA criterion is equivalent to
the AAA criterion
Given two triangles for
which AA holds, use rigid
motions to map a vertex of
one triangle onto the
corresponding vertex of the
other. Then show that
dilation will complete the
mapping of one triangle
onto the other
Review triangle congruence
criteria and similarity criteria
Review the angle sum
*A.C. = Additional Content
transformations
Unit2 assessment version A:
#6 similar pentagons:
dilated and reflected image
#7 determine whether 2
parallelograms are similar
by using transformations
#8 dilation properties to
show similar triangles have
equal
angles and proportional
sides:
http://www.geogebra.org
Pearson Geometry:
7-3 Proving Triangles
Similar
9-7 Similarity
transformations
Unit2 assessment version A:
#9 Why AA criterion is
sufficient to prove triangle
similarity.
#10 ASA congruence test of
a triangle’s dilated image
proves the triangles similar.
Pearson Geometry:
7-5 Proportions in Triangles
8-1 Pythagorean Theorem
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
one side of a triangle
divides the other two
proportionally, and
conversely; the Pythagorean
Theorem proved using triangle
similarity. M.C.*
G.CO.10 Prove theorems
about triangles. Theorems
include: measures of interior
angles of triangle sum 180
degrees, 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. M.C.*
19 *M.C. = Major Content
similarity to solve
problems and prove
relationships in
geometric figures.
SLO5: Prove theorems
about triangles and use
triangle congruence and
similarity to solve
problems and prove
relationships in
geometric figures.
Prove isosceles triangle
theorem
Prove the segment
joining midpoints of two
sides of a triangle is
parallel to third side
*S.C. = Supporting Content
theorem for triangles, the
alternate interior angle
theorem and its converse,
and properties of
parallelograms
Use cardboard cutouts to
illustrate that the altitude to
the hypotenuse divides a
right triangle into two
triangles that are similar to
the original triangle. Then,
use AA to prove this
theorem
Establish the Pythagorean
relationship among right
triangle’s sides to prove the
Pythagorean Theorem.
Compare/contrast triangle
congruence criteria and
similarity criteria
Review corresponding parts
of congruent triangles are
congruent (CPCTC) theorem
Review the reflexive
property (applied to same
angle)
Review definition of
midpoint
Review dilation properties:
o Lines passing
through center point
are proportional with
*A.C. = Additional Content
and its converse
Unit2 assessment version A:
#11 prove line parallel to
one side of a triangle
divides the other 2 sides
proportionally.
#13 Prove that line dividing
2 sides of a triangle
proportionally is parallel to
the other side
http://blossoms.mit.edu
search Pythagorean
theorem (video)
Pearson Geometry:
4-5 Isosceles and
equilateral triangles
5-1 Midsegments of
triangles
5-4 Medians and altitudes
Unit2 assessment version A:
#12 Prove the isosceles
triangle theorem.
#13 Prove that the segment
joining midpoints of two
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
and half the length
scale factor
o Lines not on center
point of dilation
parallel to each other
Prove the medians of a
triangle meet at a point
sides of a triangle is parallel
to the third side…
Stage 2 – Assessment Evidence
Other Evidence:
Classwork
Exit Slips
Homework
Individual and group tests
Open-ended questions
Portfolio
Quizzes
Suggested Performance Tasks:
Exemplars
Extended projects
Math Webquests
Writing in Math/Journal
Stage 3 – Learning Plan
Lesson Plan Template
with suggested pacing for required 80 minute math block
Lesson
Objective
Opening/Do
Now
Homework
Review
10-15 minutes
5-10
minutes
20 *M.C. = Major Content
Instructional Components
Mini Lesson(s)
I DO/ WE DO
15-20 minutes (each)
*S.C. = Supporting Content
Independent/Partner/Group
Work
YOU DO
20-30 minutes
*A.C. = Additional Content
Summary and
Exit Slip
10 minutes
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Using 3-part,
studentfriendly
language.
Ex. With 80%
proficiency, I
will solve 10
addition word
problems.
Do Now could
include:
Spiral review
of prerequisite
skills for
today’s lesson,
Pretest skills
to see where
students are
regarding
today’s
objective, or
Contain a
writing in
math type of
prompt/questi
on for
students to
explain their
thinking, etc.
May choose to Whole group mini-lesson
review a few
with a check for
understanding afterwards.
specific
problems from
previous
nights
homework to
review for
understanding.
Students may
also have a
few they
struggled with
and need reteaching.
21 *M.C. = Major Content
*S.C. = Supporting Content
Lesson activity including at least
one check for understanding.
*A.C. = Additional Content
As a class,
teacher should
facilitate a
summary of
today’s targeted
objective then
provide an exit
question (last
check for
understanding)
that allows
students to
individually prove
their
understanding of
the objective.
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
UNIT 3: Trigonometry
http://www.state.nj.us/education/modelcurriculum/math/Geometryu3.shtml
Grade level: HS
District-Approved Text: Geometry Common Core, Pearson 2012
5 weeks of instruction, 2 weeks of review, enrichment and assessment
Stage 1 – Desired Results
Enduring Understandings/Goals:
Dilations, similarity and the properties of similar triangles allow for the application of trigonometric ratios to real world
situations.
Essential Questions:
How might the features of one figure be useful when solving problems about a similar figure?
Mathematical Practices:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Standard:
(Common Core)
G.GPE.6 Find the point on a
directed line segment between
two given points that partitions
the segment in a given ratio.
M.C.*
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. M.C.*
22 *M.C. = Major Content
Student Learning
Objectives(SLOs)
SLO 1: Find the
point on a directed
line segment
between two given
points that
partitions the
segment in a given
ratio.
SLO 2: Prove
theorems about
triangles and use
triangle congruence
and similarity to
solve problems and
prove relationships
in geometric
figures.
*S.C. = Supporting Content
Suggested Instructional
Strategies
Give students two points and
use the distance formula to find
the coordinates of the point
halfway between them.
Generalize this for two arbitrary
points to derive the midpoint
formula.
Use linear interpolation to
generalize the midpoint formula
and find the point that
partitions a line segment in any
specified ratio.
Use similar triangles from rise/
run of partition and end points
Review triangle congruence
criteria and similarity criteria
Review the angle sum theorem
for triangles, the alternate
interior angle theorem and its
converse, and properties of
parallelograms
Use cardboard cutouts to
illustrate that the altitude to the
*A.C. = Additional Content
Suggested
Resources
Pearson Geometry:
1-3 Measuring Segments
1-7 Midpoint and distance in
the coordinate plane
Houghton-Mifflin: 8-3
Partitioning a Segment
www.schmoop.com type in
G.GPE.6; for a formula
Unit 3 assessment version A:
#1 - 4 Directed line segment
Pearson Geometry:
7-5: Proportions in triangles
8-1: The Pythagorean
Theorem and its converse
Unit 3 assessment version A:
#5 Prove the Pythagorean
theorem
#6 Apply the Pythagorean
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
G.SRT.5 Use congruence and
similarity criteria for triangles to
solve problems and to prove
relationships in geometric
figures. M.C.*
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. M.C.*
23 *M.C. = Major Content
SLO 3: Use
congruence and
similarity criteria for
triangles to solve
problems and to
prove relationships
in geometric
figures.
SLO 4: Derive the
definitions for
trigonometric ratios
using similarity of
right triangles.
*S.C. = Supporting Content
hypotenuse divides a right
triangle into two triangles that
are similar to the original
triangle. Then, use AA to
prove this theorem
Establish the Pythagorean
relationship among the sides of
a right triangle to obtain an
algebraic proof of the
Pythagorean Theorem
Review triangle congruence
criteria and similarity criteria
Review the angle sum theorem
for triangles, the alternate
interior angle theorem and its
converse, and properties of
parallelograms
Use cardboard cutouts to
illustrate that the altitude to the
hypotenuse divides a right
triangle into two triangles that
are similar to the original
triangle.
Review vocabulary associated
with right triangles(Opposite
and adjacent sides, legs,
hypotenuse and
complementary angles)
Make cutouts or drawings of
right triangles and ask students
to measure side lengths and
compute side ratios.
*A.C. = Additional Content
theorem
#7 Apply theorem that a line
parallel to one side of a
triangle divides the other 2
sides proportionally.
#8 Apply midpoint formula,
slope, equation of a line,
medians intersect at a
common point
Pearson Geometry:
4-2 Triangle congruence by
SSS and SAS
4-3 Triangle congruence by
ASA and AAS
4-4 Using corresponding parts
of congruent triangles
4-6 Congruence in rt triangles
4-7 Congruence in overlapping
triangles
Unit 3 assessment version A:
#9 Proof using ASA & CPCTC
#10 Triangle similarity; AA
#11 Apply triangle similarity
Pearson Geometry:
P. 506 Concept Byte, Exploring
trigonometric ratios
8-3 Trigonometry
Unit 3 assessment version A:
#12 define trig ratio - sine
#13 define trig ratio - tangent
#14 Define 3 trig ratios
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
G.SRT.7 Explain and use the
relationship between the sine
and cosine of complementary
angles. M.C.*
G.SRT.8 Use trigonometric
ratios and the Pythagorean
Theorem to solve right triangles
in applied problems.★ M.C.*
24 *M.C. = Major Content
SLO 5: Develop
and apply the
definitions of
trigonometric ratios
for the acute angles
of a right triangle,
the relationship
between the sine
and cosine of
complementary
angles, and the
Pythagorean
Theorem on right
triangles in applied
problems. ★
Investigate sines and cosines of
complimentary angles, and
guide students to discover that
they are equal to one another.
Point out to students that the
“co: in cosine refers to the
“sine of the compliment”.
SLO 6: Develop
and apply the
definitions of
trigonometric ratios
for the acute angles
of a right triangle,
the relationship
between the sine
and cosine of
complementary
angles, and the
Pythagorean
Have students make their own
diagrams showing a right
triangle with labels showing
trigonometric ratios
Use the Pythagorean Theorem
to obtain exact trigonometric
ratios for 30 degrees, 45
degrees and 60 degree angles.
Use cooperative learning in
small groups for discovery
activities and outdoor
measurement projects
*S.C. = Supporting Content
*A.C. = Additional Content
Pearson Geometry:
8-3 Trigonometry p. 512 p#37
Houghton Mifflin: 6-2 The Sine
and Cosine Ratios or Holt
McDougal Explorations in Core
Math, Geometry: 8-2
Trigonometry
Unit 3 assessment version A:
#15 Sine of an angle = cosine
of its complement
#16 Sine of an angle = cosine
of its complement
#17 Sine of an angle = cosine
of its complement
#18 Sine of an angle = cosine
of its complement
Pearson Geometry:
8-3 Trigonometry
8-1 The Pythagorean Theorem
and its converse
8-2 Special Right Triangles
Concept byte p. 515
Unit 3 assessment version A:
#19 Applied trig: sine
#20 Pythagorean Theorem
#21 Inverse of trig functions:
find the angle, given the sides
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
G.SRT.9 (+) Derive the formula
A = 1/2 ab sin(C) for the area of
a triangle by drawing an
auxiliary line from a vertex
perpendicular to the opposite
side.
Theorem on right
triangles in applied
problems. ★
SLO 7: Derive and
use the formula for
the area of an
oblique triangle
(A = 1/2 ab sin
(C)).
Draw an auxiliary line from a
vertex perpendicular to the
opposite side to illustrate the
formula
A = 1/2 ab sin(C) for the area
of a triangle*
*Note that C is the included angle
between sides a and b.
G.SRT.10 (+) Prove the Laws
of Sines and Cosines and use
them to solve problems.
G.SRT.11 (+) Understand and
apply the Law of Sines and the
Law of Cosines to find unknown
measurements in right and nonright triangles (e.g., surveying
problems, resultant forces).
25 *M.C. = Major Content
SLO 8: Prove and
apply the Laws of
Sines and Cosines
to solve both right
and oblique
triangles.
Use real world problems that
involve solving a triangle by
using the Pythagorean
Theorem, right triangle
trigonometry and/or the Law of
Sines and the Law of Cosines
SLO 8: Prove and
apply the Laws of
Sines and Cosines
to solve both right
and
oblique
triangles.
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).
*S.C. = Supporting Content
*A.C. = Additional Content
Pearson Geometry:
10-5 Trigonometry and area
8-5 Law of Sines
Unit 3 assessment version A:
#22 Use definition of Sine and
formula for triangle area to
derive SAS formula for area.
#23 Apply area formula of
oblique triangle given SAS
#24 Apply area formula of
triangle given SAS to calculate
approximate area of land mass
Pearson Geometry:
8-5 Law of Sines
8-6 Law of Cosines
Unit 3 assessment version A:
#27 Derive Law of Sines
#28 Use the Law of Cosines to
solve for the largest angle;
note that largest angle
opposite the largest side of the
triangle.
Pearson Geometry:
8-5 Law of Sines
8-6 Law of Cosines
Unit 3 assessment version A:
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
#25 given 1 side and 3 angles,
apply either Law of Cosines of
Law of Sines to find a side
#26 Given 2 sides and 1
angle, apply Law of Sines to
find an angle
Suggested Performance Tasks:
Exemplars
Extended projects
Math Webquests
Writing in Math/Journal
26 *M.C. = Major Content
Stage 2 – Assessment Evidence
Other Evidence:
Classwork
Exit Slips
Homework
Individual and group tests
Open-ended questions
Portfolio
Quizzes
*S.C. = Supporting Content
*A.C. = Additional Content
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Stage 3 – Learning Plan
Lesson Plan Template with suggested pacing for required 80 minute math block
Lesson
Objective
Using 3-part,
studentfriendly
language.
Ex. With 80%
proficiency, I
will solve 10
addition word
problems.
Instructional Components
Opening/Do
Now
Homework
Review
10-15 minutes
5-10
minutes
Do Now could
include:
Spiral review
of prerequisite
skills for
today’s lesson,
Pretest skills
to see where
students are
regarding
today’s
objective, or
Contain a
writing in
math type of
prompt/questi
on for
students to
explain their
thinking, etc.
May choose to Whole group mini-lesson
review a few
with a check for
specific
understanding afterwards.
problems from
previous
nights
homework to
review for
understanding.
Students may
also have a
few they
struggled with
and need reteaching.
27 *M.C. = Major Content
Mini Lesson(s)
I DO/ WE DO
15-20 minutes (each)
*S.C. = Supporting Content
Independent/Partner/Group
Work
YOU DO
20-30 minutes
Lesson activity including at least
one check for understanding.
*A.C. = Additional Content
Summary and
Exit Slip
10 minutes
As a class,
teacher should
facilitate a
summary of
today’s targeted
objective then
provide an exit
question (last
check for
understanding)
that allows
students to
individually prove
their
understanding of
the objective.
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
UNIT 4: Circles and Expressing Geometric Properties through Equations
http://www.state.nj.us/education/modelcurriculum/math/Geometry4.shtml
Grade level: HS
District-Approved Text: Geometry Common Core, Pearson 2012
5 weeks of instruction, 2 weeks of review, enrichment and assessment
Stage 1 – Desired Results
Enduring Understandings/Goals:
The properties of polygons, lines and angles can be used to understand circles; the properties of circles can be used to
solve problems involving polygons, lines and angles.
Algebra can be used to efficiently and effectively describe and apply geometric properties.
Essential Questions:
How can the properties of circles, polygons, lines and angles be useful when solving geometric problems?
How can algebra be useful when expressing geometric properties?
Mathematical Practices:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Standard:
(Common Core)
Student Learning
Objectives (SLO’s)
27 *M.C. = Major Content
Suggested Instructional
Strategies
*S.C. = Supporting Content
*A.C. = Additional Content
Suggested
Resources
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
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. A.C.*
G.C.4 (+) Construct
a tangent line from a
point outside a given
circle to the circle.
A.C.
SLO 1: 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.
Use properties of congruent
triangles and perpendicular
lines to prove theorems about
diameters, radii, chords and
tangent lines.
SLO 2: Prove the properties
of angles for a quadrilateral
inscribed in a circle and
construct inscribed and
circumscribed circles of a
triangle, and a tangent line to
a circle from a point outside a
circle, using geometric tools
and geometric software.
Constructing tangent to a circle from
a point outside a circle is a useful
application of the result that an angle
inscribed in a semi-circle is a right
angle.
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
SLO 3: Use similarity to show
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.
Calculate lengths of arcs that
are simple fractional parts of a
circle and do this for circles of
various radii so that students
discover a proportionality
relationship.
Compute areas of sectors by
first considering them as
fractional parts of a circle.
Derive formulas that relate
Pearson Geometry:
10-6: Circles and arcs
Concept Byte p. 658-659
Concept Byte p. 770
12-3: Inscribed Angles
Unit 4 assessment version A:
#1 Circumscribed angle
#2 Inscribed angle
#3 Chords, central angle
#4 Tangent-radius theorem
Pearson Geometry:
12-3: Inscribed angles
Unit 4 assessment version A:
#7 Construct a tangent line
to a circle.
Pearson Geometry:
10-6 Circles and arcs
10-7 Areas of circles and
sectors
Unit 4 assessment version A:
#8 Radian measure as
constant of proportionality
#9 Arc length proportional to
radius
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
for the area of a
sector. A.C.
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. A.C.
SLO 4: Derive the formula
for the area of a circular
sector, the equation of a
circle (given the center and
radius using the Pythagorean
Theorem), and the equation
of a parabola (given the focus
and the directrix).
G.GPE.5 Prove the
slope criteria for
parallel and
perpendicular lines
and use them to
solve geometric
problems. (E.g. find
the equation of a line
parallel or
perpendicular to a
given line that passes
through a given
point. M.C.
SLO 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.)
degrees and radians.
Introduce arc measures that
are equal to the measures of
the intercepted central angles
in degrees or radians.
Emphasize appropriate use of
terms such as angle, arc,
radian, degree and sector.
Review the definition of a
circle as a set of points whose
distance from a fixed point is
constant.
Review the algebraic method
of completing the square and
demonstrate it geometrically.
Use the Pythagorean theorem
to derive the distance formula
Given two points, find the
equation of a circle passing
through one of the points and
having the other as its center.
Discuss the role of algebra in
providing a precise means of
representing a visual image.
Use right triangle similarity
and the definition of slope to
show perpendicular lines have
negative reciprocal slopes.
Relate systems of linear
equations to graphs having no
solution (parallel lines) or
infinitely many solutions (same
line different equation).
www.regentsprep.org
www.mathisfun.com
Pearson Geometry:
12-5 Circles in the
Coordinate Plane
Unit 4 assessment version A:
#10 Derive the equation for
the center of a circle using
the Pythagorean Theorem
#11 Relate distance formula
to the equation of a circle
#12 Use completing the
square to rewrite equation of
a circle in standard form.
Pearson Geometry:
3-8 Slopes of parallel and
perpendicular lines
Unit 4 assessment version A:
#13 Prove that the slopes of
perpendicular lines are
negative reciprocals of each
other.
#14 Rewrite line equations
to determine whether they
are parallel or perpendicular.
#15 Equation of a parallel
line passing through a point.
30
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
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.
M.C.
G.GPE.4 Use
coordinates to prove
simple geometric
theorems
algebraically. For
example, prove or
disprove that a figure
defined by four given
points in the
coordinate plane is a
rectangle; prove or
disprove that the
point (1, √3) lies on
the circle centered at
the origin and
containing the point
SLO 6: Construct formal
proofs using theorems,
postulates, and definitions
involving parallelograms. *
Encourage students to justify
their ideas.
Build visual and empirical
foundation for higher levels of
geometric thought.
Sketch, draw and construct
figures and relationships
between geometric objects.
*G.C0.11: (Parallelograms) Theorems include: opposite
sides are congruent, opposite
angles are congruent, and
the diagonals of a
parallelogram bisect each
other and conversely,
rectangles are parallelograms
with congruent diagonals.
(Content included in unit 2,
repeated to assess fluency.)
SLO 7: Use the coordinate
system to generate simple
geometric proofs algebraically
and to compute perimeters
and areas of geometric
figures using the distance
formula. ★
29 *M.C. = Major Content
31
Explore properties of geometric
figures plotted on a coordinate
axes system using graphing
technology and dynamic software.
Generalize coordinates of
geometric figures using variables
for one or more of the vertices.
Derive the equation for a line
through two points using similar
right triangles.
*S.C. = Supporting Content
*A.C. = Additional Content
Pearson Geometry:
6-2 Properties of
parallelograms
6-3 Prove that a quadrilateral
is a parallelogram
6-4 Properties of rhombuses,
rectangles and squares
6-5 Conditions for
rhombuses, rectangles and
squares
Unit 4 assessment version A:
#16 Prove opposite angles of
parallelogram are congruent.
#17 Prove opposite sides of
parallelogram are congruent
#18 Prove that
parallelogram with congruent
diagonals is a rectangle
Pearson Geometry:
6-9 Proofs using coordinate
geometry
Unit 4 assessment version A:
#19 Calculate lengths and
slopes of quadrilateral in
coordinate plane and then
classify the quadrilateral as
parallelogram, rhombus,
rectangle or square.
#20 Find 2 vertices of a
parallelogram, given 2
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
(0, 2). M.C.
G.GPE.7 Use
coordinates to
compute perimeters
of polygons and
areas of triangles and
rectangles, e.g.,
using the distance
formula.★ M.C.
SLO 8: Use the coordinate
system to generate simple
geometric proofs algebraically
and to compute perimeters
and areas of geometric
figures using the distance
formula. ★
32 *M.C. = Major Content
Graph polygons using coordinates.
Determine side lengths and
perimeters of polygons. Calculate
areas of triangles and rectangles.
Given a triangle, use slopes to
verify that the base and height are
perpendicular. Find the area.
Explore perimeter and area of a
variety of polygons, including
convex, concave, and irregularly
shaped polygons.
*S.C. = Supporting Content
*A.C. = Additional Content
vertices and diagonal
intersection point.
Pearson Geometry:
6-7 Polygons in the
coordinate plane
10-1 Areas of parallelograms
and triangles
Unit 3
assessment version A:
#21 Apply the distance
formula to calculate the
dimensions of a house’s base
from 4 points plotted on the
coordinate grid.
#22 Calculate the perimeter
and area of various pieces of
a tangram
#23 Apply distance formula
to calculate triangle
perimeter, then utilize a
formula to calculate triangle
area.
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Lesson
Objective
Using 3part,
studentfriendly
language.
Ex. With
80%
proficiency,
I will solve
10 addition
word
problems.
Opening/Do
Now
Homework
Review
5-10
minutes
5-10
minutes
Do Now could
include:
Spiral review
of
prerequisite
skills for
today’s
lesson,
Pretest skills
to see where
students are
regarding
today’s
objective, or
Contain
writing in
math type of
prompt/questi
on for
students to
explain their
thinking, etc.
May choose to
review a few
specific
problems from
previous
night’s
homework to
review for
understanding.
Students may
also have a
few they
struggled with
and need reteaching.
33 *M.C. = Major Content
Instructional Components
Mini Lesson (s)
I DO/ WE DO
15-20 minutes
Whole group
mini-lesson with
a check for
understanding
afterwards.
*S.C. = Supporting Content
Independent/Partner/Group
Work
YOU DO
20-30 minutes
Lesson activity including at least
one check for understanding.
Math centers should be
implemented during this time.
Suggestions:
Technology
Problem-based/Skill-based
Task
Vocabulary Work
Writing in Math
Art/Music Connections
*A.C. = Additional Content
Summary
and
Exit Slip
10 minutes
As a class,
teacher should
facilitate a
summary of
today’s
targeted
objective then
provide an exit
question (last
check for
understanding)
that allows
students to
individually
prove their
understanding
of the
objective.
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
UNIT 5: Extending to Three Dimensions
http://www.state.nj.us/education/modelcurriculum/math/Geometry5.shtml
Grade level: HS
District-Approved Text: Geometry Common Core, Pearson 2012
5 weeks of instruction, 2 weeks of review, enrichment and assessment
Stage 1 – Desired Results
Enduring Understandings/Goals:
Two-dimensional figures can be “stacked” to create three-dimensional objects and generate volume formulas.
Essential Questions:
How can two-dimensional figures be used to understand three-dimensional objects?
Mathematical Practices:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Standard:
Suggested Instructional Strategies
Student Learning
Suggested
Objectives
Resources
Pearson
Geometry:
G.GMD.1 Give an
SLO 1: Develop informal
Revisit formulas for a
11-4 Volumes of prisms and
informal argument
arguments to justify
Circumference
for the formulas for
cylinders;
formulas for the
Us alternative ways to derive the
the circumference of
Concept Byte p.659
circumference of a circle,
formula for the area of the circle
a circle, area of a
Unit 5 assessment version A:
circle, volume of a
area of a circle, volume of a
Introduce Cavalierri’s principle
cylinder, pyramid,
#1 Plot points in 3-D
cylinder, pyramid, and cone
using concrete models, such as a
and cone. Use
coordinate system, apply
(use dissection arguments,
deck of cards.
dissection arguments,
distance formula, compare
Cavalieri’s principle, and
Use Cavalierri’s principle with
Cavalieri’s principle,
volume of cube & pyramid.
informal limit arguments).
cross sections of cylinders,
and informal limit
#2 Compare volume of cone
arguments. A.C.
with volume of pyramid then
pyramids and cones to justify
apply Cavalieri’s principle.
34 *M.C. = Major Content
*S.C. = Supporting Content
*A.C. = Additional Content
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
their volume formulas
G.GMD.3 Use
volume formulas for
cylinders, pyramid,
cones, and spheres
to solve problems.★
A.C.
G.GMD.4 Identify
the shapes of twodimensional crosssections of threedimensional objects,
and identify threedimensional objects
generated by
rotations of twodimensional objects.
A.C.
SLO 2: Solve problems
using volume formulas for
cylinders, pyramids, cones,
and spheres.★
SLO 3: Identify the shape
of a two-dimensional crosssection of a threedimensional figure and
identify three-dimensional
objects created by the
rotation of two-dimensional
objects.
35 *M.C. = Major Content
For pyramids and cones, use
containers to explain the factor 1/3.
Use Geoblocks to model
Find the capacity of a pipeline,
compare the amount of food in cans
of various shapes; compare
capacities of cylindrical, conical and
spherical storage tanks
Use a combination of concrete
models and formal reasoning to
develop conceptual understanding of
the volume formula
Review vocabulary for names of
solids(right prism, cylinder, cone,
sphere)
Slice various solids to illustrate
their cross sections
Cut a half-inch slit in the end of a
drinking straw, and insert a
cardboard cutout shape. Rotate
the straw and observe the threedimensional solid of revolution
generated by the –dimensional
*S.C. = Supporting Content
*A.C. = Additional Content
#3 Apply dissection
argument: compare pyramid
volumes with triangular &
polygonal bases.
Pearson Geometry:
11-4 Volumes of prisms and
cylinders
Concept Byte p. 725
11-5 Volumes of pyramids
and cones
11-6 Surface areas and
volumes of spheres
Unit 5 assessment version A:
4. Composite volumes:
cylinder + hemisphere;
calculate radius
5. Calculate volume of
pyramid
6. Calculate cone’s height,
after calculating the equal
volume of a sphere.
Pearson Geometry:
11-1 Space figures and cross
sections; p. 691 #31-33
Unit 5 assessment version A:
#7 Cross-section of plane
perpendicular to base of
cone intersects the vertex.
#8 Rotate a circle around its
diameter to get a 3dimensional figure
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
cutout.
Make three-dimensional models
out of modeling clay and slice
through them with a plastic knife
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).
★ M.C.
SLO 4: Use geometric
shapes, their measures, and
their properties to describe
objects (e.g., modeling a
tree trunk or a human torso
as a cylinder). ★
Show students different pictures and
ask them to name various simple
shapes they can use in order to
approximate the images in the
picture. Ask the students to explain
their reasoning.
#9 Rotate a shape to get a
3-dimensional figure
#10 Shaded region rotated
about y-axis to form a solid
Pearson Geometry:
p. 508, p. 511 #29 & #36
11-2 Surface areas of prisms
and cylinders
11-3 Surface areas of
pyramids and cones
11-4 Volumes of prisms and
cylinders
11-5 Volumes of pyramids
and cones
11-6 Surface areas and
volumes of spheres
11-7 Areas and volumes of
similar solids
http://map.mathshell.org
Unit 5 assessment version A:
#11 Model shape of
evergreen with a
combination of simple 3dimensional shapes (i.e.
cylinder, cone, etc.)
#12 Model shape of a trunk
with a combination of 3-D
shapes.
#13 Model alternative cross-
36 *M.C. = Major Content
*S.C. = Supporting Content
*A.C. = Additional Content
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
sectional shapes in order to
meet a criterion.
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). ★ M.C.
G.MG.3 Apply
geometric methods
to solve design
problems (e.g.,
designing an object
SLO 5: Use density
concepts in modeling
situations based on area
and volume. (e.g., persons
per square mile, BTUs per
cubic foot). ★
SLO 6: Solve design
problems using geometric
methods. (e.g., designing
an object or structure to
satisfy physical constraints
37 *M.C. = Major Content
Each group of students is tasked
with filling a box up with a
number of balls (Styrofoam,
tennis balls, marbles). Students
calculate the volumes of each box
and weigh each group of balls in
order to calculate density.
Archimedes principle “Eureka,
I found it” story.
Ask students to measure the
room in meters and feet and then
calculate the population density of
the class in students per square
foot, students per square inch and
students per square meter. Have
students describe a way to
calculate the population density of
the school.
Use ratio and proportion as well
as composite area in relation to
designing thumbnail images on a
website.
Break students into groups and
*S.C. = Supporting Content
*A.C. = Additional Content
?Pearson Geometry:
11-7 Areas and volumes of
similar solids
Type G.MG.2 into schmoop:
http://www.shmoop.com
Holt McDougal Explorations
in Core Math Geometry:
p. 432 and p. 462
Unit 5 assessment version A:
#14 Model the amount of
energy required to cool
rooms of various volumes
#15 Model the population
density of a rectangular
town.
#16 Use density of paint and
volume of paint in order to
calculate mass of paint.
Pearson Geometry:
p. 164-166
Type in G.MG.3 into
www.schmoop.com
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
or structure to satisfy
physical constraints
or minimize cost;
working with
typographic grid
systems based on
ratios). ★ M.C.
or minimize cost; working
with typographic grid
systems based on ratios). ★
have them calculate different
areas by shaping the same length
of string into rectangles of
different dimensions. Allow the
students to discover which
dimensions optimize the area
enclosed inside the string
perimeter.
Holt McDougal Explorations
in Core Math Geometry:
p.309-312, p.428, p.464
Unit 5 assessment version A:
#17 Find scaled size of 3
paintings in a website
#18 Given perimeter, find
the dimensions that
maximize area of a rectangle
#19 Design a hemisphere on
a rectangular base.
#20 Carve pyramid out of
wooden block.
Lesson Plan Template
with suggested pacing for required 80 minute math block
Lesson
Objective
Opening/Do
Now
Homework
Review
5-10 minutes
5-10
minutes
38 *M.C. = Major Content
Instructional Components
Mini Lesson (s)
I DO/ WE DO
15-20 minutes
*S.C. = Supporting Content
Independent/Partner/Group
Work
YOU DO
20-30 minutes
*A.C. = Additional Content
Summary and
Exit Slip
10 minutes
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013
Using 3-part,
studentfriendly
language.
Ex. With 80%
proficiency, I
will solve 10
addition word
problems.
Do Now could
include:
Spiral review
of prerequisite
skills for
today’s lesson,
Pretest skills
to see where
students are
regarding
today’s
objective, or
Contain
writing in
math type of
prompt/questi
on for
students to
explain their
thinking, etc.
39 *M.C. = Major Content
May choose to Whole group mini-lesson
review a few
with a check for
understanding afterwards.
specific
problems from
previous
night’s
homework to
review for
understanding.
Students may
also have a
few they
struggled with
and need reteaching.
*S.C. = Supporting Content
Lesson activity including at least
one check for understanding.
Math centers should be
implemented during this time.
Suggestions:
Technology
Problem-based/Skill-based
Task
Vocabulary Work
Writing in Math
Art/Music Connections
*A.C. = Additional Content
As a class,
teacher should
facilitate a
summary of
today’s targeted
objective then
provide an exit
question (last
check for
understanding)
that allows
students to
individually prove
their
understanding of
the objective.
★ = Modeling Standard
(Identified by PARCC Model Content Frameworks). Trenton BOE Approved August 26, 2013