GEOMETRY 2015-2016
S-1 Unit 1 - Foundations of Geometry (3 weeks)
Drag
1.1
To identify, name and draw points, lines, and planes.
G.CO.A.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.
Drag
1.2
To identify, name and draw line segments, rays, and angles.
G.CO.A.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.
1.3
To construct congruent segments and midpoints.
G.CO.D.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.
1.4
To understand and use the distance formulas as they relate to line segments on a number line and in a
coordinate plane.
8.G.B.8 - Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
G.CO.A.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.
G.GPE.B.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).
1.5
To understand and use the midpoint formulas as they relate to line segments on a number line and in a
coordinate plane.
G.CO.A.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.
G.GPE.B.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.B.6 - Find the point on a directed line segment between two given points that partitions the segment in a
given ratio.
1.6
To construct congruent angles and angle bisectors.
G.CO.D.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.
1.7
To identify, name and draw pairs of angles, including: complementary angles, supplementary angles,
adjacent angles, vertical angles, and linear pairs.
G.CO.C.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.
1.8
To understand and use properties of angles to find missing angle measures.
G.CO.C.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.
S-1 Unit 2 - Transformations in the Plane (3 weeks)
2.1
To identify whether a figure has line symmetry or rotational symmetry.
G.CO.A.3 - Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections
that carry it onto itself.
2.2
To know and identify the different types of transformations, e.g., reflections, rotations, translations,
dilations, etc.
G.CO.A.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).
G.CO.A.4 - Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular
lines, parallel lines, and line segments.
2.3
To know the difference between transformations that are isometries (rigid) and those that are not
isometries.
G.CO.A.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).
2.4
To know that isometries preserve distance and angle measure and be able to create the image given a
transformation.
G.CO.A.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).
2.5
To describe transformations as functions that take the original points (called the pre-image or input) and
give other points (called the image or output).
G.CO.A.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).
G.CO.A.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.
2.6
To understand and perform isometric transformations.
G.CO.A.4 - Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular
lines, parallel lines, and line segments.
G.CO.A.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.
2.7
To perform composite transformations on a given figure.
G.CO.A.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.
G.CO.B.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.
2.8
To prove figures are congruent using rigid motion(s).
G.CO.B.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.
G.CO.B.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.
S-1 Unit 3 - Reasoning and Proof (1 week)
3.1
To determine the difference between inductive and deductive reasoning.
G.CO.C.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.
3.2
To identify, write and analyze conditional statements and their converse.
G.CO.C.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.
3.3
To reason with properties of Algebra.
G.CO.C.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.
3.4
To solve problems using segment and angle addition.
G.CO.A.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.
G.CO.D.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.
S-1 Unit 4 - Parallel and Perpendicular Lines (3 weeks)
4.1
To identify, name and draw parallel and perpendicular lines.
G.CO.A.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.
G.CO.C.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.
4.2
To identify pairs of angles formed by a transversal.
G.CO.C.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.
4.3
To use transformations to prove properties about angles formed by parallel lines cut by a transversal.
G.CO.C.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.
4.4
To prove lines are parallel using angles formed by a transversal.
G.CO.C.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.
4.5
To create equations of lines parallel or perpendicular to a given line.
G.GPE.B.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).
4.6
To perform constructions involving parallel and perpendicular lines.
G.CO.D.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.
S-1 Unit 5 - Triangles and Triangle Congruence (4 weeks)
5.1
To identify and classify triangles.
G.CO.C.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.
5.2
To prove theorems about triangles, such as the triangle sum theorem, exterior angle theorem, isosceles
triangle theorems, midsegment theorem, and triangle proportionality theorem.
G.CO.C.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.
G.CO.D.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.SRT.B.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.
5.3
To use inequalities to describe the relationships among side lengths and angle measures in a triangle.
G.CO.C.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.
G.SRT.B.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
5.4
To use coordinate geometry to prove statements about triangles.
G.GPE.B.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).
5.5
To develop and write a congruency statement for two triangles, matching corresponding angles and
sides.
G.SRT.B.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
5.6
To apply the criteria of triangle congruence (ASA, SAS, SSS, AAS, and HL) to prove triangle congruency.
G.SRT.B.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
5.7
To prove properties of the perpendicular bisectors, medians, altitudes and angle bisectors of a triangle.
G.CO.C.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.C.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.
G.CO.D.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.C.A.3 - Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
5.8
To construct the inscribed and circumscribed circles of a triangle.
G.C.A.3 - Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
S-1 Unit 6 - Quadrilaterals (3 weeks)
6.1
To classify polygons based on their sides and angles.
G.CO.C.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.
6.2
To find and use the measures of interior and exterior angles of regular and irregular polygons.
G.CO.C.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.
6.3
To prove theorems about parallelograms, such as opposite sides are congruent and parallel, opposite
angles are congruent, consecutive angles are supplementary, and diagonals bisect each other.
G.CO.C.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.
6.4
To use criteria to prove that a quadrilateral is a parallelogram.
G.CO.C.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.
6.5
To know and use the properties for special parallelograms, such as rectangle, rhombus, and square.
G.CO.C.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.SRT.B.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
6.6
To use criteria to prove that a parallelogram is a rectangle, rhombus, or square.
G.CO.C.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.SRT.B.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
6.7
To know and use properties for kites and trapezoids.
G.SRT.B.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
6.8
To use coordinate geometry to prove statements about quadrilaterals.
G.GPE.B.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.B.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).
6.9
To construct regular polygons (triangle, square, and hexagon) inscribed in a circle.
G.CO.D.13 - Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
S-2 Unit 7 - Similarity (3 weeks)
7.1
To perform a dilation given a center and scale factor.
G.CO.A.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).
G.SRT.A.1a - Verify experimentally the properties of dilations given by a center and a scale factor: a dilation takes
a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center
unchanged.
G.SRT.A.1b - Verify experimentally the properties of dilations given by a center and a scale factor: the dilation of
a line segment is longer or shorter in the ratio given by the scale factor.
7.2
To understand that dilations preserve corresponding angle measures and create proportional
corresponding side lengths between a pre-image and its dilated image.
G.SRT.A.1a - Verify experimentally the properties of dilations given by a center and a scale factor: a dilation takes
a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center
unchanged.
G.SRT.A.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.
7.3
To define similarity using dilations, i.e., corresponding angles are congruent and corresponding sides are
proportional.
G.SRT.A.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.
7.4
To prove triangles are similar using Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS).
G.SRT.A.3 - Use the properties of similarity transformations to establish the AA criterion for two triangles to be
similar.
G.SRT.B.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
7.5
To use properties of similar triangles to solve problems.
G.SRT.A.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.
G.SRT.B.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
7.6
To find the point on a directed line segment that partitions the given segment in a given ratio.
G.CO.D.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.GPE.B.6 - Find the point on a directed line segment between two given points that partitions the segment in a
given ratio.
S-2 Unit 8 - Right Triangles and Trigonometry (4 weeks)
8.1
To apply similarity relationships, i.e. Geometric Mean and Pythagorean Theorems, in right triangles to
solve problems.
G.SRT.B.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.
G.SRT.B.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
8.2
To derive special right triangles and use the side length relationships to solve problems.
G.SRT.C.8 - Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
*(Modeling Standard)
8.3
To understand that similarity in right triangles leads to proportional relationships, which produces the
trigonometric ratios for the acute angles in a right triangle.
G.SRT.C.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.
8.4
To define sine and cosine as cofunctions.
G.SRT.C.7 - Explain and use the relationship between the sine and cosine of complementary angles.
8.5
To use the trigonometric functions to find unknown sides of a right triangle.
G.SRT.C.8 - Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
*(Modeling Standard)
8.6
To use the trigonometric inverse functions to find unknown angles of a right triangle.
G.SRT.C.8 - Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
*(Modeling Standard)
8.7
To solve applications of right triangles using trigonometric ratios and the Pythagorean Theorem.
G.SRT.C.8 - Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
*(Modeling Standard)
S-2 Unit 9 - Probability (3 weeks)
9.1
To understand set theory (set, subset, sample space, event, complement, union, intersection) using
models such as two-way tables, Venn diagrams, tree diagrams, etc.
S.CP.A.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”).
*(Modeling Standard)
S.CP.A.4 - Construct and interpret two-way frequency tables of data when two categories are associated with
each object being classified. Use the two-way 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 tenth grade. Do the same for other subjects
and compare the results. *(Modeling Standard)
9.2
To calculate probabilities of subsets of a sample space, including unions, intersections, and
complements.
7.SP.C.7 - Develop a probability model and use it to find probabilities of events. Compare probabilities from a
model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
Specifically:
9.3
To use two-way tables to determine probabilities.
S.CP.A.4 - Construct and interpret two-way frequency tables of data when two categories are associated with
each object being classified. Use the two-way 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 tenth grade. Do the same for other subjects
and compare the results. *(Modeling Standard)
9.4
To apply the General Addition Rule for probability and interpret the answers in terms of the model.
S.CP.B.7 - Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the
model. *(Modeling Standard)
9.5
To understand that for two events to be independent, the product of their probabilities is equal to the
probability of their intersection.
S.CP.A.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. *(Modeling
Standard)
S.CP.A.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. *(Modeling Standard)
S.CP.A.4 - Construct and interpret two-way frequency tables of data when two categories are associated with
each object being classified. Use the two-way 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 tenth grade. Do the same for other subjects
and compare the results. *(Modeling Standard)
S.CP.A.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. *(Modeling Standard)
9.6
To understand, calculate, and interpret the conditional probability of A given B as P(A|B) = P(A and
B)/P(B) based on real-world situations.
S.CP.A.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. *(Modeling Standard)
9.7
To determine whether two events are independent using conditional probability in context of real-world
situations.
S.CP.A.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. *(Modeling Standard)
S.CP.B.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. *(Modeling Standard)
S-2 Unit 10 - Circles (4 weeks)
10.1
To prove all circles are similar using transformations.
G.C.A.1 - Prove that all circles are similar.
10.2
To determine the measures of central angles and inscribed angles of a circle, including angles of a
quadrilateral inscribed in a circle.
G.C.A.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.
G.C.A.3 - Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
10.3
To find the measure of angles formed by tangents, secants, chords, and radii of a circle.
G.C.A.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.
10.4
To find segment lengths formed in a circle by radii, chords, secants, and tangents.
G.C.A.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.
10.5
To develop and use the formulas for circle area and circumference.
G.GMD.A.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.
10.6
To define and understand the radian measure of an angle; include conversions from radian to degrees
and vice versa.
G.C.A.1 - Prove that all circles are similar.
G.C.B.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.
10.7
To find arc length and sector area of a circle.
G.C.B.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.GMD.A.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.
10.8
To derive the standard form of the equation of a circle using Pythagorean Theorem, given center (h, k)
and radius, r.
G.GPE.A.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.GPE.B.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).
10.9
To maneuver between the parts of a circle (radius and center), the equation of a circle, and its graph.
G.GPE.A.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.GPE.B.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).
10.10
To derive the equation of a parabola given its focus and directrix.
G.GPE.A.2 - Derive the equation of a parabola given a focus and directrix.
G.GPE.B.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).
10.11
To maneuver between the parts of a parabola (focus and directrix), the equation of a parabola, and its
graph.
G.GPE.A.2 - Derive the equation of a parabola given a focus and directrix.
S-2 Unit 11 - Surface Area and Volume (3 weeks)
11.1
To find the perimeter and area of plane figures, including the use of coordinates.
G.GPE.B.7 - Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g.,
using the distance formula. *(Modeling Standard)
G.MG.A.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)
11.2
To know and apply the formula for the surface area of prisms and cylinders.
7.G.B.6 - Solve real-world and mathematical problems involving area, volume and surface area of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
G.MG.A.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)
8.G.C.9 - Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world
and mathematical problems.
11.3
To know and apply the formula for the surface area of pyramids and cones.
7.G.B.6 - Solve real-world and mathematical problems involving area, volume and surface area of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
G.MG.A.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)
8.G.C.9 - Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world
and mathematical problems.
11.4
To know and apply the formula for the volume of prisms and cylinders.
G.GMD.A.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.A.3 - Use volume formulas for cylinders, pyramids, cones, and spheres, and the surface area of a sphere
to solve problems. *(Modeling Standard)
G.MG.A.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)
11.5
To know and apply the formula for the volume of pyramids and cones.
G.GMD.A.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.A.3 - Use volume formulas for cylinders, pyramids, cones, and spheres, and the surface area of a sphere
to solve problems. *(Modeling Standard)
G.MG.A.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)
11.6
To know and apply the formulas for the surface area and volume of spheres.
G.GMD.A.3 - Use volume formulas for cylinders, pyramids, cones, and spheres, and the surface area of a sphere
to solve problems. *(Modeling Standard)
G.MG.A.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)
11.7
To identify the shapes of two-dimensional cross-sections of three-dimensional objects.
G.GMD.B.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.
11.8
To identify the three-dimensional objects generated by rotating two-dimensional shapes about a line.
G.GMD.B.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.
11.9
To understand Cavalieri’s Principle for volume of solids.
G.GMD.A.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.A.2 - Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere
and other solid figures.
11.10
To apply the concepts of density (density = mass/volume) based on area and volume in modeling
situations.
G.MG.A.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)
G.MG.A.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)
11.11
To solve real-world problems in which restrictions are imposed, i.e., maximizing volume, limited
dimensions, minimizing cost, etc.
G.MG.A.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)
G.MG.A.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)