Download Student Learning Targets

Standards
Last Review: April 4, 2012
G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of a point, line,
distance along a line and distance around a circular arc.
Knowledge Learning Target(s)
DART Statements
1. Describe the undefined terms: point, line, and
1. I can describe a point, line, plane, and distance along a line in a plane.
distance along a line in a plane.
2. Define perpendicular lines, parallel lines, line
segments, and angles.
2. I can define perpendicular lines, parallel lines, line segments, and angles.
3. Define circle and the distance around a circular arc.
3. I can define the circumference of a circle as the distance around a circle.
Reasoning Learning Target(s)
DART Statements
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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)
Knowledge Learning Target(s)
1. Describe the different types of transformations
including translations, reflections, rotations and
dilations.
2. Describe transformations as functions that take
DART Statements
1. I can describe the different types of transformations including translations,
reflections, rotations and dilations.
2. I can describe transformations as functions.
points in the coordinate plane as inputs and
give other points as outputs
Reasoning Learning Target(s)
1. Represent transformations in the plane using,
e.g., transparencies and geometry software.
2. Write functions to represent transformations.
3. Compare transformations that preserve
distance and angle to those that do not (e.g.,
translation versus horizontal stretch)
DART Statements
1. I can represent transformations in the plane using physical models.
2. I can write a function to represent a transformation.
3. I can compare transformations that preserve congruency and those that do not.
From Appendix A: Build on student experience with
rigid motions from earlier grades. Point out the basis
of rigid motions in geometric concepts, e.g,
translations move points a specific distance along a
line parallel to a specified line; rotations move objects
along a circular arc with a specified center through a
specified angle.
Performance Skill Learning Target(s)
DART Statements
Standards
Product Learning Target(s)
Last Review: April 4, 2012
DART Statements
Standards
Last Review: April 4, 2012
G.CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Knowledge Learning Target(s)
DART Statements
1. Given a rectangle, parallelogram, trapezoid, or
1. I can describe the rotations and/or reflections that carry a 2-D figure onto itself.
regular polygon, describe the rotations and/or
reflections that carry it onto itself.
2. I can explain how transformations follow rigid motions using geometric concepts of
lines and circles.
From Appendix A: Build on student experience with
rigid motions from earlier grades. Point out the basis
of rigid motions in geometric concepts, e.g,
translations move points a specific distance along a
line parallel to a specified line; rotations move objects
along a circular arc with a specified center through a
specified angle.
Reasoning Learning Target(s)
DART Statements
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.CO.4 Develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.
Knowledge Learning Target(s)
DART Statements
1. Recall definitions of angles, circles, perpendicular
1. I can define angles, circles, perpendicular and parallel lines, and line segments.
and parallel lines and line segments.
Reasoning Learning Target(s)
1. Develop definitions of rotations, reflections and
translations in terms of angles, circles,
perpendicular lines, parallel lines and line
segments.
DART Statements
1. I can develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
2. I can explain how transformations follow rigid motions using geometric concepts of
lines and circles.
From Appendix A: Build on student experience with rigid
motions from earlier grades. Point out the basis of rigid
motions in geometric concepts, e.g., translations move
points a specific distance along a line parallel to a specified
line; rotations move objects along a circular arc with a
specified center through a specified angle.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.CO.5 Given a geometric figure and a rotation, reflection or translation, draw the transformed figure using, e.g. graph paper, tracing paper
or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Knowledge Learning Target(s)
1. Given a geometric figure and a rotation,
reflection or translation, draw the transformed
figure using, e.g. graph paper, tracing paper or
geometry software.
DART Statements
1. I can draw an image of a figure given a transformation.
Reasoning Learning Target(s)
1. Draw a transformed figure and specify the
sequence of transformations that were used to
carry the given figure onto the other.
DART Statements
1. I can determine the sequence of transformations used to create a transformed
figure.
From Appendix A: Build on student experience with rigid
motions from earlier grades. Point out the basis of rigid
motions in geometric concepts, e.g., translations move
points a specific distance along a line parallel to a specified
line; rotations move objects along a circular arc with a
specified center through a specified angle.
2. I can explain how transformations follow rigid motions using geometric concepts of
lines and circles.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
DART Statements
1. Use geometric descriptions of rigid motions to
1. I can use geometric descriptions of rigid motions to transform figures.
transform figures.
2. Predict the effect of a given rigid motion on a given
figure.
2. I can predict the effect of a given rigid motion on a given figure.
3. Define congruence in terms of rigid motions
3. I can define congruence in terms of rigid motions.
(i.e. two figures are congruent if there exists a
rigid motion, or composition of rigid motions,
that can take one figure to the second).
Reasoning Learning Target(s)
1. Decide if two figures are congruent in terms of rigid
motions (it is not necessary to find the precise
transformation(s) that took one figure to a second,
only to understand that such a transformation or
composition exists).
DART Statements
1. I can determine if two figures are congruent.
2. I understand that rigid motions are at the foundation of the definition of
congruence.
From Appendix A: Rigid motions are at the foundation of
the definition of congruence. Students reason from the
basic properties of rigid motions (that they preserve
distance and angle), which are assumed without proof.
Rigid motions and their assumed properties can be used to
establish the usual triangle congruence criteria, which can
then be used to prove other theorems.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
1. Identify corresponding angles and sides of two
triangles.
2. Identify corresponding pairs of angles and sides of
congruent triangles after rigid motions.
Reasoning Learning Target(s)
1. Use the definition of congruence in terms of rigid
motions to show that two triangles are congruent if
corresponding pairs of sides and corresponding
pairs of angles are congruent.
2. Use the definition of congruence in terms of rigid
motions to show that if the corresponding pairs of
sides and corresponding pairs of angles of two
triangles are congruent then the two triangles are
congruent.
3. Justify congruency of two triangles using
transformations.
From Appendix A: Rigid motions are at the foundation of
the definition of congruence. Students reason from the
basic properties of rigid motions (that they preserve
distance and angle), which are assumed without proof.
Rigid motions and their assumed properties can be used to
establish the usual triangle congruence criteria, which can
then be used to prove other theorems.
Performance Skill Learning Target(s)
DART Statements
1. I can identify corresponding angles and sides of two triangles.
2. I can identify corresponding pairs of angles and sides of congruent triangles after
rigid motions.
DART Statements
1. I can use the definition of congruence to show that two triangles are congruent.
2. I can use the definition of congruence to show that if corresponding parts are
congruent, then triangles are congruent.
3. I can justify congruency of two triangles using transformations.
4. I understand that rigid motions are at the foundation of the definition of
congruence.
DART Statements
Standards
Product Learning Target(s)
Last Review: April 4, 2012
DART Statements
Standards
Last Review: April 4, 2012
G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS) follow from the definition of congruence in terms of rigid motions.
Knowledge Learning Target(s)
DART Statements
1. Informally use rigid motions to take angles to
1. I can use rigid motions to create an image of angles and segments.
angles and segments to segments (from 8th grade).
2. Formally use dynamic geometry software or
2. I can construct a congruent angle and segment using a compass and straightedge.
straightedge and compass to take angles to
angles and segments to segments.
Reasoning Learning Target(s)
1. Explain how the criteria for triangle congruence
(ASA, SAS, SSS) follows from the definition of
congruence in terms of rigid motions (i.e. if two
angles and the included side of one triangle are
transformed by the same rigid motion(s) then the
triangle image will be congruent to the original
triangle).
DART Statements
1. I can explain the criteria for triangle congruence in terms of rigid motion.
From Appendix A: Rigid motions are at the foundation of
the definition of congruence. Students reason from the
basic properties of rigid motions (that they preserve
distance and angle), which are assumed without proof.
Rigid motions and their assumed properties can be used to
establish the usual triangle congruence criteria, which can
then be used to prove other theorems.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
1. Identify and use properties of;
o
o
o
o
o
o
o
Vertical angles
Parallel lines with transversals
All angle relationships
Corresponding angles
Alternate interior angles
Perpendicular bisector
Equidistant from endpoint
Reasoning Learning Target(s)
1. Prove vertical angles are congruent.
DART Statements
1. I can identify and use properties of:
o
o
o
o
o
o
o
Vertical angles
Parallel lines with transversals
All angle relationships
Corresponding angles
Alternate interior angles
Perpendicular bisector
Equidistant from enpoint
DART Statements
1. I can prove vertical angles are congruent.
2. Prove corresponding angles are congruent when
two parallel lines are cut by a transversal and
converse.
2. I can prove corresponding angles are congruent when two parallel lines are cut by a
transversal and converse.
3. Prove alternate interior angles are congruent when
two parallel lines are cut by a transversal and
converse.
3. I can prove alternate interior angles are congruent when two parallel lines are cut
by a transversal and converse.
4. Prove points are on a perpendicular bisector of a
line segment are exactly equidistant from the
segments endpoint.
From Appendix A: Encourage multiple ways of writing
proofs, such as in narrative paragraphs, using flow
diagrams, in two-column format, and using diagrams
without words. Students should be encouraged to focus on
the validity of the underlying reasoning while exploring a
variety of formats for expressing that reasoning.
4. I can prove points are on a perpendicular bisector of a line segment are exactly
equidistant from the segments endpoint.
Standards
Last Review: April 4, 2012
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
1. Identify the hypothesis and conclusion of a
theorem.
DART Statements
1. I can identify the hypothesis and conclusion of a theorem.
Reasoning Learning Target(s)
1. Design an argument to prove theorems about triangles.
DART Statements
1. I can design an argument to prove theorems about triangles.
2. Analyze components of the theorem.
2. I can analyze components of the theorem.
3. Prove theorems about triangles.
3. I can prove theorems about triangles.
From Appendix A: Encourage multiple ways of writing
proofs, such as in narrative paragraphs, using flow
diagrams, in two-column format, and using diagrams
without words. Students should be encouraged to focus on
the validity of the underlying reasoning while exploring a
variety of formats for expressing that reasoning.
Implementations of G.CO.10 may be extended to include
concurrence of perpendicular bisectors and angle bisectors
as preparation for G.C.3 in Unit 5.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
1. Classify types of quadrilaterals.
2. Explain theorems for parallelograms and relate
to figure.
Reasoning Learning Target(s)
1. Use the principle that corresponding parts of
congruent triangles are congruent to solve
problems.
2. Use properties of special quadrilaterals in a
proof.
DART Statements
1. I can classify types of quadrilaterals.
2. I can explain theorems for parallelograms and relate to a figure.
DART Statements
1. I can use the principle that corresponding parts of congruent triangles are
congruent to solve problems.
2. I can use properties of special quadrilaterals in a proof.
From Appendix A: Encourage multiple ways of writing
proofs, such as in narrative paragraphs, using flow
diagrams, in two-column format, and using diagrams
without words. Students should be encouraged to focus on
the validity of the underlying reasoning while exploring a
variety of formats for expressing that reasoning.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
1. Explain the construction of geometric figures
using a variety of tools and methods.
DART Statements
1. I can explain the construction of geometric figures using a variety of tools
and methods.
Reasoning Learning Target(s)
DART Statements
1. Apply the definitions, properties and theorems
about line segments, rays and angles to
support geometric constructions.
1. I can apply the definitions, properties and theorems about line segments,
rays and angles to support geometric constructions.
2. Apply properties and theorems about parallel
2. I can apply properties and theorems about parallel and perpendicular lines
and perpendicular lines to support
constructions.
Performance Skill Learning Target(s)
1. Perform geometric constructions including:
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, using a variety
of tools and methods (compass and
straightedge, string, reflective devices, paper
folding, dynamic geometric software, etc.).
From Appendix A: Build on prior student experience with
simple constructions. Emphasize the ability to formalize
and explain how these constructions result in the desired
objects.
to support constructions.
DART Statements
1. I can perform geometric constructions.
Standards
Last Review: April 4, 2012
Some of these constructions are closely related to previous
standards and can be introduced in conjunction with them.
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.CO.13 Construct an equilateral triangle, a square and a regular hexagon inscribed in a circle.
Knowledge Learning Target(s)
Note: Underpinning performance, reasoning, and
knowledge targets, if applicable, are addressed in G.CO.12
DART Statements
From Appendix A: Build on prior student experience with
simple constructions. Emphasize the ability to formalize
and explain how these constructions result in the desired
objects.
Some of these constructions are closely related to
previous standards and can be introduced in
conjunction with them.
Reasoning Learning Target(s)
DART Statements
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
1. Construct an equilateral triangle, a square and a
regular hexagon inscribed in a circle.
DART Statements
1. I can construct an equilateral triangle, a square and a regular hexagon inscribed in a
circle.
Standards
Last Review: April 4, 2012
G.SRT.1a Verify experimentally the properties of dilations given by a center and a scale factor.
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center
unchanged.
Knowledge Learning Target(s)
1. Define image, pre-image, scale factor, center, and
similar figures as they relate to transformations.
DART Statements
1. I can define image, pre-image, scale factor, center, and similar figures as they relate
to transformations.
2. I can identify a dilation stating its scale factor and center
2. Identify a dilation stating its scale factor and center
Reasoning Learning Target(s)
1. Verify experimentally that a dilated image is similar
to its pre-image by showing congruent
corresponding angles and proportional sides.
DART Statements
1. I can verify experimentally that a dilated image is similar to its pre-image by
showing congruent corresponding angles and proportional sides.
2. Verify experimentally that a dilation takes a line not
passing through the center of the dilation to a
parallel line by showing the lines are parallel.
2. I can verify experimentally that a dilation takes a line not passing through the
center of the dilation to a parallel line by showing the lines are parallel.
3. Verify experimentally that dilation leaves a line
passing through the center of the dilation
unchanged by showing that it is the same line.
3. I can verify experimentally that dilation leaves a line passing through the center of
the dilation unchanged by showing that it is the same line.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.SRT.1b Verify experimentally the properties of dilations given by a center and a scale factor.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
Knowledge Learning Target(s)
1. Define image, pre-image, scale factor, center, and
similar figures as they relate to transformations.
DART Statements
1. I can explain that the scale factor represents how many times longer or shorter a
dilated line segment is than its pre-image
2. Identify a dilation stating its scale factor and center
3. Explain that the scale factor represents how
many times longer or shorter a dilated line
segment is than its pre-image.
Reasoning Learning Target(s)
1. Verify experimentally that the dilation of a line
segment is longer or shorter in the ratio given by
the scale factor.
DART Statements
1. I can verify experimentally that the dilation of a line segment is longer or shorter in
the ratio given by the scale factor.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
1. By using similarity transformations, explain
that triangles are similar if all pairs of
corresponding angles are congruent and all
corresponding pairs of sides are proportional.
Reasoning Learning Target(s)
1. Given two figures, decide if they are similar by
using the definition of similarity in terms of
similarity transformations.
DART Statements
1. I can explain that all triangles are similar if all pairs of corresponding angles are
congruent and all corresponding pairs of sides are proportional.
DART Statements
1. Given two figures I can decide if they are similar.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Knowledge Learning Target(s)
1. Recall the properties of similarity
transformations.
Reasoning Learning Target(s)
1. Establish the AA criterion for similarity of triangles
by extending the properties of similarity
transformations to the general case of any two
similar triangles.
DART Statements
1. I can recall the properties of similarity transformations.
DART Statements
1. I can establish the AA criterion for similarity of triangles by extending the properties
of similarity transformations to the general case of any two similar triangles.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
1. Recall postulates, theorems, and definitions to
prove theorems about triangles.
DART Statements
1. I can recall postulates, theorems, and definitions to prove theorems about triangles
Reasoning Learning Target(s)
1. Prove theorems involving similarity 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.)
DART Statements
1. I can prove theorems involving similarity about triangles.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Knowledge Learning Target(s)
1. Recall congruence and similarity criteria for
triangles.
Reasoning Learning Target(s)
1. Use congruency and similarity theorems for
triangles to solve problems.
2. Use congruency and similarity theorems for
triangles to prove relationships in geometric
figures.
DART Statements
1. I can recall congruence and similarity criteria for triangles.
DART Statements
1. I can use congruency and similarity theorems for triangles to solve problems.
2. I can use congruency and similarity theorems for triangles to prove relationships in
geometric figures.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
1. Names the sides of right triangles as related to an
acute angle.
2. Recognize that if two right triangles have a pair
of acute, congruent angles that the triangles
are similar.
Reasoning Learning Target(s)
1. Compare common ratios for similar right triangles
and develop a relationship between the ratio and
the acute angle leading to the trigonometry ratios.
DART Statements
1. I can name the sides of right triangles as related to an acute angle.
2. I can recognize that if two right triangles have a pair of acute, congruent angles that
the triangles are similar.
DART Statements
1. I can compare common ratios for similar right triangles and develop a relationship
between the ratio and the acute angle leading to the trigonometry ratios.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
Knowledge Learning Target(s)
1. Use the relationship between the sine and
cosine of complementary angles.
Reasoning Learning Target(s)
2. Explain how the sine and cosine of complementary
angles are related to each other.
DART Statements
1. I can use the relationship between the sine and cosine of complementary angles.
DART Statements
1. I can explain how the sine and cosine of complementary angles are related to each
other.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. *(Modeling standard)
Knowledge Learning Target(s)
1. Recognize which methods could be used to solve
right triangles in applied problems.
2. Solve for an unknown angle or side of a right
triangle using sine, cosine, and tangent.
Reasoning Learning Target(s)
1. Apply right triangle trigonometric ratios and the
Pythagorean Theorem to solve right triangles in
applied problems.
DART Statements
1. I can recognize which methods could be used to solve right triangles in applied
problems.
2. I can solve for an unknown angle or side of a right triangle using sine, cosine, and
tangent.
DART Statements
1. I can apply right triangle trigonometric ratios and the Pythagorean Theorem to
solve right triangles in applied problems.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
DART Statements
1. Recall right triangle trigonometry to solve
1. I can recall right triangle trigonometry to solve mathematical problems.
mathematical problems.
Reasoning Learning Target(s)
1. 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.
DART Statements
1. I can 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.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.SRT.10(+) Prove the Laws of Sines and Cosines and use them to solve problems.
Knowledge Learning Target(s)
1. Use the Laws of Sines and Cosines this to find
missing angles or side length measurements.
Reasoning Learning Target(s)
DART Statements
1. I can use the Laws of Sines and Cosines to find missing angles or side length
measurements.
DART Statements
1. Prove the Law of Sines
1. I can prove the Law of Sines
2. Prove the Law of Cosines
2. I can prove the Law of Cosines
3. Recognize when the Law of Sines or Law of
Cosines can be applied to a problem.
3. I can recognize when the Law of Sines or Law of Cosines can be applied to a
problem.
From Appendix A: With respect to the general case of
Laws of Sines and Cosines, the definition of sine and
cosine must be extended to obtuse angles.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.SRT.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).
Knowledge Learning Target(s)
1. Determine from given measurements in right
and non-right triangles whether it is
appropriate to use the Law of Sines or Cosines.
Reasoning Learning Target(s)
1. 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).
DART Statements
1. I can determine from given measurements in right and non-right triangles
whether it is appropriate to use the Law of Sines or Cosines.
DART Statements
1. I can 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).
From Appendix A: With respect to the general case of
the Laws of Sines and Cosines, the definition of sine
and cosine must be extended to obtuse angles.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
G.C.1 Prove that all circles are similar.
Knowledge Learning Target(s)
1. Recognize when figures are similar. (Two
figures are similar if one is the image of the
other under a transformation from the plane
into itself that multiplies all distances by the
same positive scale factor, k. That is to say,
one figure is a dilation of the other. )
Reasoning Learning Target(s)
1. Compare the ratio of the circumference of a circle
to the diameter of the circle.
Last Review: April 4, 2012
DART Statements
1. I can recognize when figures are similar.
DART Statements
1. I can compare the ratio of the circumference of a circle to the diameter of the circle.
2. Discuss, develop and justify this ratio for several
circles.
2. I can discuss, develop and justify this ratio for several circles.
3. Determine that this ratio is constant for all circles.
3. I can determine that this ratio is constant for all circles.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
DART Statements
1. Identify inscribed angles, radii, chords, central
1. I can identify inscribed angles, radii, chords, central angles, circumscribed angles,
angles, circumscribed angles, diameter, tangent.
diameter, tangent.
2. Recognize that inscribed angles on a diameter are
right angles.
3. Recognize that radius of a circle is perpendicular to
the radius at the point of tangency.
Reasoning Learning Target(s)
1. Examine the relationship between central,
inscribed and circumscribed angles by applying
theorems about their measures.
2. I can recognize that inscribed angles on a diameter are right angles.
3. I can recognize that a tangent of a circle is perpendicular to the radius at the point
of tangency.
DART Statements
1. I can examine the relationship between central, inscribed and circumscribed angles
by applying theorems about their measures.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Knowledge Learning Target(s)
DART Statements
1. Define inscribed and circumscribed circles of a
1. I can define inscribed and circumscribed circles of a triangle.
triangle.
2. Recall midpoint and bisector definitions.
2. I can recall midpoint and bisector definitions.
3. Define a point of concurrency.
3. I can define a point of concurrency.
Reasoning Learning Target(s)
1. Prove properties of angles for a quadrilateral
inscribed in a circle.
DART Statements
1. I can prove properties of angles for a quadrilateral inscribed in a circle.
Performance Skill Learning Target(s)
1. Construct inscribed circles of a triangle
DART Statements
1. Construct inscribed circles of a triangle
2. Construct circumscribed circles of a triangle.
Product Learning Target(s)
2. Construct circumscribed circles of a triangle.
DART Statements
Standards
Last Review: April 4, 2012
G.C. 4 (+) Construct a tangent line from a point outside a given circle to the circle.
Knowledge Learning Target(s)
DART Statements
1. Recall vocabulary:
1. I can recall the definition of Tangent, Radius, Perpendicular bisector, and Midpoint
Tangent
Radius
Perpendicular bisector
Midpoint
2. Identify the center of the circle
Reasoning Learning Target(s)
1. Synthesize theorems that apply to circles and
tangents, such as:
2. I can identify the center of the circle
DART Statements
1. I can synthesize the following theorems:
2. Tangents drawn from a common external point are congruent.
2. Tangents drawn from a common external point are
congruent.
3. A radius is perpendicular to a tangent at the point of tangency.
3. A radius is perpendicular to a tangent at the point
of tangency.
Performance Skill Learning Target(s)
1. Construct the perpendicular bisector of the line
segment between the center C to the outside point
P.
DART Statements
1. I can construct the perpendicular bisector of the line segment between the center C
to the outside point P.
2. Construct arcs on circle C from the midpoint Q,
having length of CQ.
2. I can construct arcs on circle C from the midpoint Q, having length of CQ.
3. Construct the tangent line.
3. I can construct the tangent line.
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
DART Statements
1. Recall how to find the area and circumference of a
1. I can recall how to find the area and circumference of a circle.
circle.
2. Explain that 1° = ∏/180 radians
2. I can explain that 1° = ∏/180 radians
3. Recall from G.C.1, that all circles are similar.
3. I can recall that all circles are similar.
4. I can determine the constant of proportionality (scale factor).
4. Determine the constant of proportionality
(scale factor).
Reasoning Learning Target(s)
Justify the radii of any two circles (r1 and r2) and the
1.
arc lengths (s1 and s2) determined by congruent
central angles are proportional, such that r1 /s1 =
r2/s2
1.
DART Statements
I can justify the radii of any two circles (r1 and r2) and the arc lengths (s1 and s2)
determined by congruent central angles are proportional, such that r1 /s1 = r2/s2
2. Verify that the constant of a proportion is the same
as the radian measure, Θ, of the given central
angle.
2. Verify that the constant of a proportion is the same as the radian measure, Θ, of the
given central angle.
3. Conclude s = r Θ
3. I can conclude s = r Θ
From Appendix A: Emphasize the similarity of all circles.
Note that by similarity of sectors with the same central
angle, arc lengths are proportional to the radius. Use this
as a basis for introducing radian as a unit of measure. It is
not intended that it be applied to the development of
circular trigonometry in this course.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
DART Statements
1. Define a circle.
1. I can define a circle.
2. Use Pythagorean Theorem.
2. I can use the Pythagorean Theorem.
3. Complete the square of a quadratic equation.
3. I can complete the square of a quadratic equation.
Reasoning Learning Target(s)
DART Statements
Derive equation of a circle using the Pythagorean
Theorem – given coordinates of the center and length
of the radius.
1. I can derive the equation of a circle using the Pythagorean Theorem – given
coordinates of the center and length of the radius.
Determine the center and radius by completing the
square.
2. I can determine the center and radius by completing the square.
From Appendix A: Emphasize the similarity of all
circles. Note that by similarity of sectors with the
same central angle, arc lengths are proportional to the
radius. Use this as a basis for introducing radian as a
unit of measure. It is not intended that it be applied to
the development of circular trigonometry in this
course.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.GPE.2 Derive the equation of a parabola given a focus and directrix.
Knowledge Learning Target(s)
1. Define a parabola including the relationship of
the focus and the equation of the directrix to
the parabolic shape.
DART Statements
1. I can define a parabola using focus and directrix.
From Appendix A: The directrix should be parallel to a
coordinate axis.
Reasoning Learning Target(s)
1. Derive the equation of parabola given the
focus and directrix.
DART Statements
1. I can derive the equation of parabola given the focus and directrix.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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).
Knowledge Learning Target(s)
1. Recall previous understandings of coordinate
geometry (including, but not limited to:
distance, midpoint and slope formula, equation
of a line, definitions of parallel and
perpendicular lines, etc.)
Reasoning Learning Target(s)
1. Use coordinates to prove simple geometric
theorems algebraically.
2. 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).
DART Statements
1. I can recall previous understandings of coordinate geometry (including, but
not limited to: distance, midpoint and slope formula, equation of a line,
definitions of parallel and perpendicular lines, etc.)
DART Statements
1. I can use coordinates to prove simple geometric theorems algebraically.
2. 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).
From Appendix A: Include simple proofs involving
circles.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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).
Knowledge Learning Target(s)
1. Recall previous understandings of coordinate
geometry (including, but not limited to:
distance, midpoint and slope formula, equation
of a line, definitions of parallel and
perpendicular lines, etc.)
Reasoning Learning Target(s)
1. Use coordinates to prove simple geometric
theorems algebraically.
2. 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).
DART Statements
1. I can recall previous understandings of coordinate geometry (including, but
not limited to: distance, midpoint and slope formula, equation of a line,
definitions of parallel and perpendicular lines, etc.)
DART Statements
1. I can use coordinates to prove simple geometric theorems algebraically.
2. 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).
From Appendix A: This unit has a close connection
with the next unit. For example, a curriculum might
merge G.GPE.1 and the unit 5 treatment of G.GPE.4
with the standards in this unit. Reasoning with
triangles in this unit is limited to right triangles; e.g.,
derive the equation of a line through 2 points using
similar right triangles.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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).
Knowledge Learning Target(s)
DART Statements
1. Recognize that slopes of parallel lines are equal.
1. I can recognize that slopes of parallel lines are equal.
2. Recognize that slopes of perpendicular lines are
opposite reciprocals (i.e, the slopes of
perpendicular lines have a product of -1)
2. I can recognize that slopes of perpendicular lines are opposite reciprocals (i.e, the
slopes of perpendicular lines have a product of -1)
3. Find the equation of a line parallel to a given line
that passes through a given point.
3. I can find the equation of a line parallel to a given line that passes through a given
point.
4. Find the equation of a line perpendicular to a given
line that passes through a given point.
Reasoning Learning Target(s)
1. Prove the slope criteria for parallel and
perpendicular lines and use them to solve
geometric problems.
DART Statements
1. I can prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems.
From Appendix A: Relate work on parallel lines in G.GPE.5
to work on A.REI.5 in High School Algebra 1 involving
systems of equations having no solution or infinitely many
solutions.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Knowledge Learning Target(s)
DART Statements
1. Recall the definition of ratio.
1. I can recall the definition of ratio.
2. Recall previous understandings of coordinate
geometry.
2. I can recall previous understandings of coordinate geometry.
Reasoning Learning Target(s)
1. Given a line segment (including those with
positive and negative slopes) and a ratio, find
the point on the segment that partitions the
segment into the given ratio.
DART Statements
1. Given a line segment and a ratio, I can find the point on the segment that
partitions the segment into the given ratio.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.GPE.7 Use coordinates to compute perimeters of polygons and area of triangles and rectangles, e.g., using the distance formula.*(Modeling
standard)
Knowledge Learning Target(s)
DART Statements
1. Use the coordinates of the vertices of a polygon to
1. I can use coordinates of the vertices of a polygon to find the perimeter.
find the necessary dimensions for finding the
perimeter (i.e., the distance between vertices).
2. I can use the coordinates of the vertices of a triangle to find its area.
2. Use the coordinates of the vertices of a triangle to
find the necessary dimensions (base, height) for
finding the area (i.e., the distance between vertices
by counting, distance formula, Pythagorean
Theorem, etc.).
3. Use the coordinates of the vertices of a
rectangle to find the necessary dimensions
(base, height) for finding the area (i.e., the
distance between vertices by counting,
distance formula).
Reasoning Learning Target(s)
1. Formulate a model of figures in contextual
problems to compute area and/or perimeter.
3. I can use the coordinates of the vertices of a rectangle to find its area.
DART Statements
1. I can formulate a model of a figure to compute area and /or perimeter.
From Appendix A: G.GPE.7 provides practice with the
distance formula and its connection with the Pythagorean
theorem.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
1. Recognize cross-sections of solids as twodimensional shapes
2. Recognize formulas for area and circumference of a
circle and volume of a cylinder, pyramid, and cone
DART Statements
1. I can recognize cross-sections of solids as two-dimensional shapes
2. I can recognize formulas for area and circumference of a circle and volume of a
cylinder, pyramid, and cone
3. Use the techniques of dissection and limit
arguments.
3. I can use the techniques of dissection and limit arguments.
4. Recognize Cavalieri’s principle.
4. I can recognize Cavalieri’s principle.
Reasoning Learning Target(s)
1. Decompose volume formulas into area formulas
using cross-sections.
DART Statements
1. I can decompose volume formulas into area formulas using cross-sections.
2. Apply dissection and limit arguments (e.g.
Archimedes’ inscription and circumscription of
polygons about a circle and) as a component of the
informal argument for the formulas for the
circumference and area of a circle.
2. I can apply dissection and limit arguments (e.g. Archimedes’ inscription and
circumscription of polygons about a circle and) as a component of the informal
argument for the formulas for the circumference and area of a circle.
3. Apply Cavalieri’s Principle as a component of the
informal argument for the formulas for the volume
of a cylinder, pyramid, and cone.
3. I can apply Cavalieri’s Principle as a component of the informal argument for the
formulas for the volume of a cylinder, pyramid, and cone.
From Appendix A: Informal arguments for area and volume
formulas can make use of the way in which area and
volume scale under similarity transformations: when one
figure in the plane results from another by applying a
similarity transformation with scale factor K, its area is K2
times the area of the first. Similarly, volumes of solid
figures scale by K3 under a similarity transformations with
scale factor K.
Standards
Last Review: April 4, 2012
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*(Modeling standard)
Knowledge Learning Target(s)
1. Utilize the appropriate formula for volume
depending on the figure.
Reasoning Learning Target(s)
2. Use volume formulas for cylinders, pyramids,
cones, and spheres to solve contextual
problems.
DART Statements
1. I can utilize the appropriate formula for volume depending on the figure.
DART Statements
1. I can use volume formulas for cylinders, pyramids, cones, and spheres to
solve contextual problems.
From Appendix A: Informal arguments for area and
volume formulas can make use of the way in which
area and volume scale under similarity
transformations: when one figure in the plane results
from another by applying a similarity transformation
with scale factor K, its area is K2 times the area of the
first. Similarly, volumes of solid figures scale by K3
under a similarity transformations with scale factor K.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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.
Knowledge Learning Target(s)
1. Use strategies to help visualize relationships
between two-dimensional and three
dimensional objects
Reasoning Learning Target(s)
DART Statements
1. I can use strategies to help visualize relationships between two-dimensional
and three dimensional objects
DART Statements
1. Relate the shapes of two-dimensional crosssections to their three-dimensional objects
1. I can relate the shapes of two-dimensional cross-sections to their threedimensional objects
2. Discover three-dimensional objects generated
by rotations of two-dimensional objects.
2. I can discover three-dimensional objects generated by rotations of twodimensional objects.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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).* (Modeling standard)
Knowledge Learning Target(s)
1. Use measures and properties of geometric
shapes to describe real world objects.
Reasoning Learning Target(s)
1. Given a real world object, classify the object as a
known geometric shape – use this to solve
problems in context.
DART Statements
1. I can use measures and properties of geometric shapes to describe real world
objects.
DART Statements
1. Given a real world object, I can classify the object as a known geometric shape – use
this to solve problems in context.
From Appendix A: Focus on situations well modeled by
trigonometric ratios for acute angles.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
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).*(Modeling standard)
Knowledge Learning Target(s)
1. Define density.
Reasoning Learning Target(s)
1. Apply concepts of density based on area and
volume to model real-life situations (e.g., persons
per square mile, BTUs per cubic foot).
DART Statements
1. I can define density.
DART Statements
1. I can apply concepts of density based on area and volume to model real-life
situations (e.g., persons per square mile, BTUs per cubic foot).
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or
minimize cost; working with typographic grid systems based on ratios).* (Modeling standard)
Knowledge Learning Target(s)
1. Describe a typographical grid system.
DART Statements
1. I can describe a typographical grid system.
Reasoning Learning Target(s)
1. Apply geometric methods to solve design problems
DART Statements
1. I can apply geometric methods to solve design problems
(e.g., designing an object or structure to satisfy
physical constraints or minimize cost; working
with typographic grid systems based on ratios).
From Appendix A: Focus on situations well modeled by
trigonometric ratios for acute angles.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as
unions, intersections, or complements of other events (“or”, “and”, “not”).
Knowledge Learning Target(s)
1. Define unions, intersections and complements
of events.
Reasoning Learning Target(s)
1. Describe events as subsets of a sample space
(the set of outcomes) using characteristics (or
categories) of the outcomes, or as unions,
intersections, or complements of other events
(“or”, “and”, “not”).
DART Statements
1. I can define unions, intersections and complements of events.
DART Statements
1. I can describe events as subsets of a sample space (the set of outcomes)
using characteristics (or categories) of the outcomes, or as unions,
intersections, or complements of other events (“or”, “and”, “not”).
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
S.CP 2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their
probabilities, and use this characterization to determine if they are independent.
Knowledge Learning Target(s)
1. Categorize events as independent or not using
the characterization that two events A and B
are independent when the probability of A and
B occurring together is the product of their
probabilities.
DART Statements
1. I can categorize events as independent or not.
From Appendix A: Build on work from 2-way tables
from Algebra 1 Unit 3 (S.ID.5) to develop
understanding of conditional probability and
independence.
Reasoning Learning Target(s)
DART Statements
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
S.CP 3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the
conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the
probability of B.
Knowledge Learning Target(s)
DART Statements
1. Know the conditional probability of A given B
as P(A and B)/P(B)
1. I know the conditional probability of A given B as P(A and B)/P(B)
2. Interpret independence of A and B as saying
2. I can interpret independence of A and B in terms of conditional proability.
that the conditional probability of A given B is
the same as the probability of A, and the
conditional probability of B given A is the same
as the probability of B.
Reasoning Learning Target(s)
DART Statements
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
S.CP. 4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the twoway table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a
random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly
selected student from your school will favor science given that the student is in 10 th grade. Do the same for other subjects and compare the results.
Knowledge Learning Target(s)
DART Statements
1. Use the two-way table as a sample space to
decide if events are independent and to
approximate conditional probabilities.
Reasoning Learning Target(s)
1. Interpret two-way frequency tables of data
when two categories are associated with each
object being classified. (For example, collect
data from a random sample of students in your
school on their favorite subject among math,
science, and English. Estimate the probability
that a randomly selected student from your
school will favor science given that the student
is in 10th grade. Do the same for other subjects
and compare the results.)
1. I can use the two-way table as a sample space to decide if events are
independent and to approximate conditional probabilities.
DART Statements
1. I can interpret two-way frequency tables of data when two categories are
associated with each object being classified.
From Appendix A: Build on work with two-way tables
from Algebra 1 Unit 3 (S.ID.5) to develop
understanding of conditional probability and
independence.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Knowledge Learning Target(s)
1. Recognize the concepts of conditional
probability and independence in everyday
language and everyday situations.
Reasoning Learning Target(s)
1. Explain the concepts of conditional probability and
independence in everyday language and everyday
situations. (For example, compare the chance of
having lung cancer if you are a smoker with the
chance of being a smoker if you have lung cancer.)
DART Statements
1. I can recognize the concepts of conditional probability and independence in
everyday language and everyday situations.
DART Statements
1. I can explain the concepts of conditional probability and independence in
everyday language and everyday situations
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A and interpret the answer in terms
of the model.
Knowledge Learning Target(s)
1. Find the conditional probability of A given B as
the fraction of B’s outcomes that also belong to
A.
Reasoning Learning Target(s)
1. Interpret the answer in terms of the model.
DART Statements
1. I can find the conditional probability of A given B as the fraction of B’s
outcomes that also belong to A.
DART Statements
1. I can interpret the answer in terms of the model.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
S.CP.7 Apply the Additional Rule, P(A or B) = P(A) + P(B) – P(A and B) and interpret the answer in terms of the model.
Knowledge Learning Target(s)
1. Use the Additional Rule, P(A or B) = P(A) + P(B)
– P(A and B)
Reasoning Learning Target(s)
1. Interpret the answer in terms of the model.
DART Statements
1. I can use the Additional Rule, P(A or B) = P(A) + P(B) – P(A and B)
DART Statements
1. I can interpret the answer in terms of the model.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the
answer in terms of the model.
Knowledge Learning Target(s)
1. Use the multiplication rule with correct
notation.
Reasoning Learning Target(s)
DART Statements
1. I can use the multiplication rule with correct notation.
DART Statements
1. Apply the general Multiplication Rule in a
uniform probability model P(A and B) =
P(A)P(B|A) = P(B)P(A|B).
1. I can apply the general Multiplication Rule in a uniform probability model
P(A and B) = P(A)P(B|A) = P(B)P(A|B).
2. Interpret the answer in terms of the model.
2. I can interpret the answer in terms of the model.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
Knowledge Learning Target(s)
1. Identify situations that are permutations and
those that are combinations.
Reasoning Learning Target(s)
1. Use permutations and combinations to
compute probabilities of compound events and
solve problems.
DART Statements
1. I can identify situations that are permutations and those that are
combinations.
DART Statements
1. I can use permutations and combinations to compute probabilities of
compound events and solve problems.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
S.MD.6: (+) Use probabilities to make fair decisions (e.g. drawing by lots, using a random number generator.)
This unit sets the stage for work in Algebra II, where the ideas of statistical inference are introduced. Evaluating the risks associated with
conclusions drawn from sample data (i.e. incomplete information) requires an understanding of probability concepts.
Knowledge Learning Target(s)
1. Compute Theoretical and Experimental
Probabilities.
Reasoning Learning Target(s)
1. Use probabilities to make fair decisions (e.g.
drawing by lots, using a random number
generator.)
DART Statements
1. I can compute Theoretical and Experimental Probabilities.
DART Statements
1. I can use probabilities to make fair decisions (e.g. drawing by lots, using a
random number generator.)
From Appendix A: This unit sets the stage for work in
Algebra II, where the ideas of statistical inference are
introduced. Evaluating the risks associated with
conclusions drawn from sample data (i.e. incomplete
information) requires an understanding of probability
concepts.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the
end of a game.)
Knowledge Learning Target(s)
1. Recall prior understandings of probability.
Reasoning Learning Target(s)
1. Analyze decisions and strategies using
probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the
end of a game.)
DART Statements
1. I can recall prior understandings of probability.
DART Statements
1. I can analyze decisions and strategies using probability concepts.
From Appendix A: This unit sets the stage for work in
Algebra II, where the ideas of statistical inference are
introduced. Evaluating the risks associated with
conclusions drawn from sample data (i.e. incomplete
information) requires an understanding of probability
concepts.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
8.G.1abc 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 lines.
Knowledge Learning Target(s)
1. Define & identify rotations, reflections, and
translations.
2. Identify corresponding sides & corresponding
angles.
3. Understand prime notation to describe an image
after a translation, reflection, or rotation.
4. Identify center of rotation.
5. Identify direction and degree of rotation.
6. Identify line of reflection.
Reasoning Learning Target(s)
1. Use physical models, transparencies, or geometry
software to verify the properties of rotations,
reflections, and translations (ie. Lines are taken to
lines and line segments to line segments of the
same length, angles are taken to angles of the same
measure, & parallel lines are taken to parallel lines.)
Performance Skill Learning Target(s)
Product Learning Target(s)
DART Statements
1. I can define & identify rotations, reflections, and translations.
2. I can identify corresponding sides & corresponding angles.
3. I can use prime notation to describe an image after a translation, reflection, or
rotation.
4. I can identify center of rotation.
5. I can identify direction and degree of rotation.
6. I can identify line of reflection.
DART Statements
1. I can explain how transformations follow rigid motions using geometric concepts of
lines and circles.
DART Statements
DART Statements
Standards
Last Review: April 4, 2012
8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of
rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
Knowledge Learning Target(s)
DART Statements
1. Define congruency.
1. I can define congruency.
2. Identify symbols for congruency.
2. I can identify symbols for congruency.
Reasoning Learning Target(s)
DART Statements
1. Apply the concept of congruency to write
congruent statements.
1. I can apply the concept of congruency to write congruent statements.
2. Reason that a 2-D figure is congruent to
another if the second can be obtained by a
sequence of rotations, reflections, translation.
2. I can explain that a 2-D figure is congruent to another if the second can be obtained
by a series of congruence-preserving transformations.
3. Describe the sequence of rotations, reflections,
translations that exhibits the congruence
between 2-D figures using words.
3. I can describe the sequence of rotations, reflections, translations that
exhibits the congruence between 2-D figures using words.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Knowledge Learning Target(s)
1. Define dilations as a reduction or enlargement
of a figure.
2. Identify scale factor of the dilation.
Reasoning Learning Target(s)
1. Describe the effects of dilations, translations,
rotations, & reflections on 2-D figures using
coordinates.
DART Statements
1. I can define dilations as a reduction or enlargement of a figure.
2. I can identify the scale factor of the dilation.
DART Statements
1. I can describe the effects of dilations on 2-D figures using coordinates.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
8.G.4 Understand that a two-dimensional 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 two-dimensional figures, describe a sequence that exhibits the
similarity between them.
Knowledge Learning Target(s)
1. Define similar figures as corresponding angles are
congruent and corresponding sides are
proportional.
DART Statements
1. I can define similar figures.
2. I can recognize the symbol for similarity.
2. Recognize symbol for similar.
Reasoning Learning Target(s)
1. Apply the concept of similarity to write similarity
statements.
DART Statements
1. I can write a similarity statement.
2. Reason that a 2-D figure is similar to another if the
second can be obtained by a sequence of rotations,
reflections, translation, or dilation.
2. I can reason that a 2-D figure is similar to another if the second can be obtained by
a sequence of rotations, reflections, translation, or dilation.
3. Describe the sequence of rotations, reflections,
translations, or dilations that exhibits the similarity
between 2-D figures using words and/or symbols.
Performance Skill Learning Target(s)
3. I can describe the sequence of rotations, reflections, translations, or dilations that
exhibits the similarity between 2-D figures using words and/or symbols.
Product Learning Target(s)
DART Statements
DART Statements
Standards
Last Review: April 4, 2012
8.G.5 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 three angles appear to form a line, and give an argument in terms of transversals why this is so.
Knowledge Learning Target(s)
DART Statements
1. Define similar triangles
1. I can define similar triangles.
2. Define and identify transversals
2. I can define and identify transversals.
3. Identify angles created when parallel line is cut
3. I can identify angles created when parallel lines are cut by a transversal.
by transversal (alternate interior, alternate
exterior, corresponding, vertical, adjacent, etc.)
Reasoning Learning Target(s)
1. Justify that the sum of interior angles equals
180. (For example, arrange three copies of the
same triangle so that the three angles appear
to form a line.)
DART Statements
1. I can justify that the sum of interior angles equals 180.
2. Justify that the exterior angle of a triangle is
equal to the sum of the two remote interior
angles.
2. I can justify that the exterior angle of a triangle is equal to the sum of the
two remote interior angles.
3. Use Angle-Angle Criterion to prove similarity
among triangles. (Give an argument in terms of
transversals why this is so.)
3. I can use Angle-Angle Criterion to prove similarity among triangles.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
8.G.6 Explain a proof of the Pythagorean Theorem and it’s converse.
Knowledge Learning Target(s)
DART Statements
1. Define key vocabulary: square root,
Pythagorean Theorem, right triangle, legs a &
b, hypotenuse, sides, right angle, converse,
base, height, proof.
1. I can define key vocabulary: square root, Pythagorean Theorem, right
2. Be able to identify the legs and hypotenuse of
a right triangle.
2. I can be able to identify the legs and hypotenuse of a right triangle.
3. Explain a proof of the Pythagorean Theorem.
3. I can explain a proof of the Pythagorean Theorem.
4. Explain a proof of the converse of the
4. I can explain a proof of the converse of the Pythagorean Theorem.
triangle, legs a & b, hypotenuse, sides, right angle, converse, base, height,
proof.
Pythagorean Theorem.
Reasoning Learning Target(s)
DART Statements
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
8.G.7 Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in
two and three dimensions.
Knowledge Learning Target(s)
1. Recall the Pythagorean theorem and its
converse.
Reasoning Learning Target(s)
DART Statements
1. I can recall the Pythagorean theorem and its converse.
DART Statements
1. Solve basic mathematical Pythagorean
theorem problems and its converse to find
missing lengths of sides of triangles in two and
three-dimensions.
1. I can solve basic mathematical Pythagorean theorem problems and its
converse to find missing lengths of sides of triangles in two and threedimensions.
2. Apply Pythagorean theorem in solving real-
2. I can apply Pythagorean theorem in solving real-world problems dealing with
world problems dealing with two and threedimensional shapes.
two and three-dimensional shapes.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Knowledge Learning Target(s)
1. Recall the Pythagorean Theorem and its
converse.
DART Statements
2. I can recall the Pythagorean Theorem and its converse.
Reasoning Learning Target(s)
DART Statements
1. Determine how to create a right triangle from
two points on a coordinate graph.
1. I can determine how to create a right triangle from two points on a
coordinate graph.
2. Use the Pythagorean Theorem to solve for the
2. I can use the Pythagorean Theorem to solve for the distance between the
distance between the two points.
two points.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements
Standards
Last Review: April 4, 2012
8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Knowledge Learning Target(s)
1. Identify and define vocabulary:
cone, cylinder, sphere, radius,
diameter, circumference, area,
volume, pi, base, height
2. Know formulas for volume of cones, cylinders,
DART Statements
1. I can Identify and define:
cone, cylinder, sphere, radius,
diameter, circumference, area,
volume, pi, base, height
2. I know the formulas for volume of cones, cylinders, and spheres
and spheres
Reasoning Learning Target(s)
1. Compare the volume of cones, cylinders, and
spheres.
2. Determine and apply appropriate volume formulas
in order to solve mathematical and real-world
problems for the given shape.
3. Given the volume of a cone, cylinder, or sphere,
find the radii, height, or approximate for π.
DART Statements
1. I can compare the volume of cones, cylinders, and spheres.
2. I can determine and apply appropriate volume formulas in order to solve
mathematical and real-world problems for the given shape.
3. Given the volume of a cone, cylinder, or sphere, I can find the radii, height, or
approximate for π.
Performance Skill Learning Target(s)
DART Statements
Product Learning Target(s)
DART Statements