Theme/Essential Questions/Concept/Unit Name

Theme/Essential Questions/Concept/Unit Name

Pacing StandardI Can Statement/ Essential Question

Evidence of Understanding including Assessments

Suggested Resources and Instructional Strategies Key Terms

(All but ELA) Anchor Standard Extra Column

Theme/Essential Questions/Concept/Unit Name: Transformations andCongruence

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.

How do you draw a segment and measure its length?

• Pre Readiness • Your turn problems • Assessement Readiness • Tier 1-3 Worksheet • Personal Math Trainer • Teacher created assignments on hmh player app • Practice AIR Problems

• Houghton Mifflin Harcourt Textbook •HMH Player App •Assessment rediness book • Math on the spot videos • Check answer option •View example option •Explicit instruction •Peer tutoring •Cooperative learning

•point •line •plane •line segment •endpoints •ray •coplanar •parallel •collinear •postulate •midpoint •segment bisector •angle •vertex •side •degrees •angle bisector •transformation •preimage •image •rigid motion •conjecture •inductive reasoning •deductive theorem •counterexample •conditional statement •linear pair MP.5 Using Tools 1.1

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.

How is measuring an angle similar to and different from measuring a line segment?



•point •line •plane •line segment •endpoints •ray •coplanar •parallel •collinear •postulate •midpoint •segment bisector •angle •vertex •side •degrees •angle bisector •transformation •preimage •image •rigid motion •conjecture •inductive reasoning •deductive theorem •counterexample •conditional statement •linear pair MP.5 Using Tools 1.2

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

How can you describe transformations in the coordinate plane using algebraic representations and using words?



•point •line •plane •line segment •endpoints •ray •coplanar •parallel •collinear •postulate •midpoint •segment bisector •angle •vertex •side •degrees •angle bisector •transformation •preimage •image •rigid motion •conjecture •inductive reasoning •deductive theorem •counterexample •conditional statement •linear pair MP.6 Precision 1.3

G.CO.9 Prove theorems about lines and angles.

How do you go about proving a statement?



•point •line •plane •line segment •endpoints •ray •coplanar •parallel •collinear •postulate •midpoint •segment bisector •angle •vertex •side •degrees •angle bisector •transformation •preimage •image •rigid motion •conjecture •inductive reasoning •deductive theorem •counterexample •conditional statement •linear pair MP.8 Patterns 1.4

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

How do you draw the image of a figure under a translation?



•vector •initial point •terminal point •translation •perpendicular lines •perpendicular bisector •reflection •rotation •center of rotation •angle of rotation •symmetry •line symmetry •line of symmetry •rotational symmetry •angle of rotational symmetry MP.7 Using Structure 2.1


How do you draw the image of a figure under a reflection?



•vector •initial point •terminal point •translation •perpendicular lines •perpendicular bisector •reflection •rotation •center of rotation •angle of rotation •symmetry •line symmetry •line of symmetry •rotational symmetry •angle of rotational symmetry MP.5 Using Tools 2.2


How do you draw the image of a figure under a rotation?




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

How do you determine whether a figure has line symmetry or rotational symmetry?



•vector •initial point •terminal point •translation •perpendicular lines •perpendicular bisector •reflection •rotation •center of rotation •angle of rotation •symmetry •line symmetry •line of symmetry •rotational symmetry •angle of rotational symmetry MP.7 Using Structure 2.4

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.

What happens when you apply more than one transformation to a figure?




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.

How can you determine whether two figures are congruent?



•vector •initial point •terminal point •translation •perpendicular lines •perpendicular bisector •reflection •rotation •center of rotation •angle of rotation •symmetry •line symmetry •line of symmetry •rotational symmetry •angle of rotational symmetry MP.3 Logic 3.2

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.

What can you conclude about two figures that are congruent?



•vector •initial point •terminal point •translation •perpendicular lines •perpendicular bisector •reflection •rotation •center of rotation •angle of rotation •symmetry •line symmetry •line of symmetry •rotational symmetry •angle of rotational symmetry MP.2 Reasoning 3.3

Pacing Standard I Can Statement




Theme/Essential Questions/Concept/Unit Name: Lines, Angles, and Triangles

G-CO.C.9 Prove theorems about lines and angles.

How can you find the measures of angles formed by intersecting lines?



•vertical angles •complementary angles •supplementary angles •transversal •indirect proof MP.3 Logic 4.1


How can you prove and use theorems about angles formed by transversals that intersect parallel lines?





How can you prove that two lines are parallel?





What are the key ideas about perpendicular bisectors of a segment?



•vertical angles •complementary angles •supplementary angles •transversal •indirect proof MP.5 Using Tools 4.4

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).

How can you find the equation of a line that is parallel or perpendicular to a given line?



•vertical angles •complementary angles •supplementary angles •transversal •indirect proof MP.2 Reasoning 4.5

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.

How can you show that two triangles are congruent?



•vertical angles •complementary angles •supplementary angles •transversal •indirect proof MP.2 Reasoning 5.1

G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS) follow from the definition of congruence in terms of rigid motions.

What does the ASA Triangle Congruence Theorem tell you about triangles?



•vertical angles •complementary angles •supplementary angles •transversal •indirect proof MP.4 Modeling 5.2


What does the SAS Triangle Congruence Theorem tell you about triangles?





What does the SSS Triangle Congruence Theorem tell you about triangles?



•vertical angles •complementary angles •supplementary angles •transversal •indirect proof MP.7 Using Structure 5.4

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.).

How can you be sure that the result of a construction is valid?


• Houghton Mifflin Harcourt Textbook •HMH Player App •Assessment rediness book • Math on the spot videos • Check answer option •View example option •Explicit instruction •Peer tutoring •Cooperative learning •hypotenuse •legs MP.3 Logic 6.1

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

What does the AAS Triangle Congruence Theorem tell you about two triangles?


• Houghton Mifflin Harcourt Textbook •HMH Player App •Assessment rediness book • Math on the spot videos • Check answer option •View example option •Explicit instruction •Peer tutoring •Cooperative learning •hypotenuse •legs MP.3 Logic 6.2


What does the HL Triangle Congruence Theorem tell you about two triangles?


• Houghton Mifflin Harcourt Textbook •HMH Player App •Assessment rediness book • Math on the spot videos • Check answer option •View example option •Explicit instruction •Peer tutoring •Cooperative learning •hypotenuse •legs MP.7 Using Structure 6.3

G-CO.C.10 Prove theorems about triangles.

What can you say about the interior and exterior angles of a triangle and other polygons?



•interior angle •auxiliary line •exterior angle •remote interior angle •isosceles triangle •legs •vertex angle •base •base angles •equilateral triangle •equiangular triangle MP.8 Patterns 7.1


What are the special relationships among angles and sides in isosceles and equilateral triangles?



•interior angle •auxiliary line •exterior angle •remote interior angle •isosceles triangle •legs •vertex angle •base •base angles •equilateral triangle •equiangular triangle MP.3 Logic 7.2


How can you use inequalities to describe the relationships among side lengths and angle measures in a triangle?



•interior angle •auxiliary line •exterior angle •remote interior angle •isosceles triangle •legs •vertex angle •base •base angles •equilateral triangle •equiangular triangle MP.5 Using Tools 7.3

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

How can you use perpendicular bisectors to find the point that is equidistant from all the vertices of a triangle?



•altitude of a triangle •centroid of a triangle •circumcenter of a triangle •circumscribed circle •concurrent •distance from a point to a line •equidistant •incenter of a triangle •inscribed circle •median of a triangle •midsegment of a triangle •orthocenter of a triangle •point of concurrency MP.6 Precision 8.1


How can you use angle bisectors to find the point that is equidistant from all the sides of a triangle?



•altitude of a triangle •centroid of a triangle •circumcenter of a triangle •circumscribed circle •concurrent •distance from a point to a line •equidistant •incenter of a triangle •inscribed circle •median of a triangle •midsegment of a triangle •orthocenter of a triangle •point of concurrency MP.3 Logic 8.2


How can you find the balancing point, or center of gravity, of a triangle?



•altitude of a triangle •centroid of a triangle •circumcenter of a triangle •circumscribed circle •concurrent •distance from a point to a line •equidistant •incenter of a triangle •inscribed circle •median of a triangle •midsegment of a triangle •orthocenter of a triangle •point of concurrency MP.5 Using Tools 8.3


How are the segments that join the midpoints of a triangle’s sides related to the triangle’s sides?



•altitude of a triangle •centroid of a triangle •circumcenter of a triangle •circumscribed circle •concurrent •distance from a point to a line •equidistant •incenter of a triangle •inscribed circle •median of a triangle •midsegment of a triangle •orthocenter of a triangle •point of concurrency MP.2 Reasoning 8.4





Theme/Essential Questions/Concept/Unit Name: Quadrilaterals and CoordinateProof

G-CO.C.11 Prove theorems about parallelograms.

What can you conclude about the sides, angles, and diagonals of a parallelogram?



•diagonal •isosceles trapezoid •kite •midsegment of a trapezoid •parallelogram •quadrilateral •rectangle •rhombus •square •trapezoid MP.3 Logic 9.1


What criteria can you use to prove that a quadrilateral is a parallelogram?



•diagonal •isosceles trapezoid •kite •midsegment of a trapezoid •parallelogram •quadrilateral •rectangle •rhombus •square •trapezoid MP.2 Reasoning 9.2


What are the properties of rectangles, rhombuses, and squares?



•diagonal •isosceles trapezoid •kite •midsegment of a trapezoid •parallelogram •quadrilateral •rectangle •rhombus •square •trapezoid MP.6 Precision 9.3


How can you use given conditions to show that a quadrilateral is a rectangle, rhombus, or square?



•diagonal •isosceles trapezoid •kite •midsegment of a trapezoid •parallelogram •quadrilateral •rectangle •rhombus •square •trapezoid MP.7 Using Structure 9.4


What are the properties of kites and trapezoids?



•diagonal •isosceles trapezoid •kite •midsegment of a trapezoid •parallelogram •quadrilateral •rectangle •rhombus •square •trapezoid MP.6 Precision 9.5


How can you use slope to solve problems involving parallel lines?



•coordinate proof •composite figure MP.7 Using Structure 10.1


How can you use slope to solve problems involving perpendicular lines?



•coordinate proof •composite figure MP.7 Using Structure 10.2

G-GPE.B.4 Use coordinates to prove simple geometric theorems.

How do you write a coordinate proof?



•coordinate proof •composite figure MP.6 Precision 10.3


How can you use slope and the distance formula in coordinate proofs?



•coordinate proof •composite figure MP.6 Precision 10.4

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

How do you find the perimeter and area of polygons in the coordinate plane?



•coordinate proof •composite figure MP.1 Problem Solving 10.5





Theme/Essential Questions/Concept/Unit Name: Similarity

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

How does a dilation transform a figure?



•center of dilation •dilation •scale factor •Side-Side-Side Similarity •Side-Angle-Side Similarity •similar •similarity transformation MP.5 Using Tools 11.1

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

How can similarity transformations be used to show two figures are similar?



•center of dilation •dilation •scale factor •Side-Side-Side Similarity •Side-Angle-Side Similarity •similar •similarity transformation MP.6 Precision 11.2

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.

If two figures are similar, what can you determine about measures of corresponding angles and lengths?



•center of dilation •dilation •scale factor •Side-Side-Side Similarity •Side-Angle-Side Similarity •similar •similarity transformation MP.4 Modeling 11.3

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

How can you show that two triangles are similar?



•center of dilation •dilation •scale factor •Side-Side-Side Similarity •Side-Angle-Side Similarity •similar •similarity transformation MP.5 Using Tools 11.4

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.

When a line parallel to one side of a triangle intersects the other two sides, how does it divide those sides?



•indirect measurement •geometric mean MP.5 Using Tools 12.1

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

How do you find the point on a directed line segment that partitions the given segment in a given ratio?





How can you use similar triangles to solve problems?




G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

How does the altitude to the hypotenuse of a right triangle help you use similar right triangles to solve problems?



•indirect measurement •geometric mean MP.8 Patterns 12.4





Theme/Essential Questions/Concept/Unit Name: Trigonometry

G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

How do you find the tangent ratio for an acute angle?



•adjacent •opposite •tangent •tan -1 A •inverse tangent of ∠A •trigonometric ratio •sine •cosine •inverse trigonometric ratios •Pythagorean triple MP.4 Modeling 13.1

G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

How can you use the sine and cosine ratios, and their inverses, in calculations involving right triangles?



•adjacent •opposite •tangent •tan -1 A •inverse tangent of ∠A •trigonometric ratio •sine •cosine •inverse trigonometric ratios •Pythagorean triple MP.4 Modeling 13.2

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

What do you know about the side lengths and the trigonometric ratios in special right triangles?



•adjacent •opposite •tangent •tan -1 A •inverse tangent of ∠A •trigonometric ratio •sine •cosine •inverse trigonometric ratios •Pythagorean triple MP.2 Reasoning 13.3

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

How can you solve a right triangle?



•adjacent •opposite •tangent •tan -1 A •inverse tangent of ∠A •trigonometric ratio •sine •cosine •inverse trigonometric ratios •Pythagorean triple MP.2 Reasoning 13.4

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

How can you use trigonometric ratios to find side lengths and angle measures of non-right triangles?



•cosine •sine •trigonometric ratio MP.3 Logic 14.1

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

How can you use the Law of Cosines to find measures of any triangle?



•cosine •sine •trigonometric ratio MP.3 Logic 14.2





Theme/Essential Questions/Concept/Unit Name: Properties of Circles

G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

How can you determine the measures of central and inscribed angles of a circle?



•chord •central angle •inscribed angle •arc •minor arc •major arc •semicircle •adjacent arcs •Inscribed Angle Theorem •Inscribed Quadrilateral Theorem •tangent •point of tangency •Tangent-Radius Theorem •Chord-Chord Product Theorem •secant •secant segment •external secant segment •Secant-Secant Theorem •tangent segment •Intersecting Chords Angle Measure Theorem •Tangent-Secant Interior Angle Measure Theorem •Tangent-Secant •Exterior Angle Measure Theorem MP.2 Reasoning 15.1


What can you conclude about the angles of a quadrilateral inscribed in a circle?



•chord •central angle •inscribed angle •arc •minor arc •major arc •semicircle •adjacent arcs •Inscribed Angle Theorem •Inscribed Quadrilateral Theorem •tangent •point of tangency •Tangent-Radius Theorem •Chord-Chord Product Theorem •secant •secant segment •external secant segment •Secant-Secant Theorem •tangent segment •Intersecting Chords Angle Measure Theorem •Tangent-Secant Interior Angle Measure Theorem •Tangent-Secant •Exterior Angle Measure Theorem MP.4 Modeling 15.2


What are the key theorems about tangents to a circle?





What are the relationships between the segments in circles?





What are the relationships between angles formed by lines that intersect a circle?



•chord •central angle •inscribed angle •arc •minor arc •major arc •semicircle •adjacent arcs •Inscribed Angle Theorem •Inscribed Quadrilateral Theorem •tangent •point of tangency •Tangent-Radius Theorem •Chord-Chord Product Theorem •secant •secant segment •external secant segment •Secant-Secant Theorem •tangent segment •Intersecting Chords Angle Measure Theorem •Tangent-Secant Interior Angle Measure Theorem •Tangent-Secant •Exterior Angle Measure Theorem MP.3 Logic 15.5

G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments

How can you justify and use the formulas for the circumference and area of a circle?



•circle •parabola •focus •directrix MP.8 Patterns 16.1

G.C.1 Prove that all circles are similar.

How do you find the length of an arc?



•circle •parabola •focus •directrix MP.4 Modeling 16.2

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

How do you find the area of a sector of a circle?



•circle •parabola •focus •directrix MP.7 Using Structure 16.3

G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How can you write the equation of a circle if you know its radius and the coordinates of its center?



•circle •parabola •focus •directrix MP.6 Precision 17.1

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

How do you write the equation of a parabola that opens up or down given its focus and directrix? If the parabola



•circle •parabola •focus •directrix MP.3 Logic 17.2





Theme/Essential Questions/Concept/Unit Name: Measurement and Modeling inTwo and Three Dimensions


How do the formulas for the volume of a prism and a cylinder relate to area formulas that you already know?


• Houghton Mifflin Harcourt Textbook •HMH Player App •Assessment rediness book • Math on the spot videos • Check answer option •View example option •Explicit instruction •Peer tutoring •Cooperative learning MP.4 Modeling 18.1


How do you find the volume of a pyramid?


• Houghton Mifflin Harcourt Textbook •HMH Player App •Assessment rediness book • Math on the spot videos • Check answer option •View example option •Explicit instruction •Peer tutoring •Cooperative learning MP.2 Reasoning 18.2


How do you calculate the volumes of composite figures that include cones?



G.GMD.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

How can you use the formula for the volume of a sphere to calculate the volumes of composite figures?



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.

What tools can you use to visualize solid figures accurately?



G.MG.1 Use geometric shapes, their measures, and their properties to describe objects

How can you find the surface area of a prism or a cylinder?




How is the formula for the lateral area of a regular pyramid similar to the formula for the lateral area of a right cone?




How can you use the formula for the surface area of a sphere to calculate the surface areas of composite figures?



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

How does multiplying one or more of the dimensions of a figure affect its attributes?




How can you model real-world situations involving density?




How can you model situations to meet real-world constraints?







Theme/Essential Questions/Concept/Unit Name: Probability

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”).

How are sets and their relationships used to calculate probabilities?


• Houghton Mifflin Harcourt Textbook •HMH Player App •Assessment rediness book • Math on the spot videos • Check answer option •View example option •Explicit instruction •Peer tutoring •Cooperative learning MP.6 Precision 21.1

S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

When are permutations useful in calculating probability?


• Houghton Mifflin Harcourt Textbook •HMH Player App •Assessment rediness book • Math on the spot videos • Check answer option •View example option •Explicit instruction •Peer tutoring •Cooperative learning MP.7 Using Structure 21.2

S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

What is the difference between a permutation and a combination?



S.CP. 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.

How are probabilities affected when events are mutually exclusive or overlapping?




How do you calculate a conditional probability?



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.

What does it mean for two events to be independent?



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.

How do you find the probability of dependent events?



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.

How can you use probability to help you make fair decisions?




How can conditional probability help you make real-world decisions?



Documents

Theme/Essential Questions/Concept/Unit Name