Geometry Mathematics: 2012 -13 to 2017 18 Hoover City Schools

Geometry Mathematics: 2012-13 to 2017-18 Hoover City Schools

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Jump to… Scope and Sequence Map Units of Study Correlation of Standards Special Notes

Scope and Sequence Map

Conceptual Categories, Domains, Content Clusters, & Standard Numbers Units of Study (expected proficiency)

1st nwks 2nd nwks 3rd nwks 4th nwks

GEOMETRY (G)

Congruence (CO) Experiment with transformations in the plane: 1, 2, 3, 4, 5 Understand congruence in terms of rigid motions: 6, 7, 8 Prove geometric theorems: 9, 10, 11 Make geometric constructions: 12, 13

1 , 2 , 3 4 , 5 , 6 7 10

Similarities, Right Triangles, and Trigonometry (SRT) Understand similarity in terms of similarity transformations: 14, 15, 16 Prove theorems involving similarity: 17, 18 Define trigonometric ratios and solve problems involving right triangles: 19, 20, 21* Apply trigonometry to general triangles: 22(+), 23(+)

4 7 , 8 , 9

Circles (C) Understand and apply theorems about circles: 24, 25, 26, 27(+) Find arc lengths and areas of sectors of circles: 28

5 10 , 11

Expressing Geometric Properties With Equations (GPE) Translate between the geometric description and the equation for a conic section: 29 Use coordinates to prove simple geometric theorems algebraically: 30, 31, 32, 33*, 34(AL)

1 , 3 6 10

Geometric Measurement and Dimension (GMD) Explain volume formulas and use them to solve problems: 35, 36*, 37(AL) Visualize relationships between two-dimensional and three-dimensional objects: 38

10 , 12

Modeling With Geometry (MG) Apply geometric concepts in modeling situations: 39*, 40*, 41*

12

STATISTICS AND PROBABILITY (S)

Using Probability to Make Decisions (MD) Use probability to evaluate outcomes of decisions: 42, 43

10

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Units of Study

Unit 1- Introduction to Geometry Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Know precise definitions of angle, circle, and line segment based on the undefined notions of point, line, and distance along a line.

1 G-CO1 Recall, identify and use definitions appropriately.

Recall, identify and use definitions appropriately.


Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle.

12 G-CO12

Create constructions with accuracy.



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

32 G-GPE6 Find a point on a segment with accuracy.

Find a point on a segment with accuracy.

Find a point on a segment with accuracy.

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

33 G-GPE7

Compute areas and perimeters of simple figures given the formulas.

Compute areas and perimeters of simple figures without the formulas.

Compute areas and perimeters of more complex figures without the formulas.

Instructional Recommendations / Resources: All courses

Know definitions and notations for point, line, plane, collinear/noncollinear, line segment, angle, ray, circle, segment length, and bisector.

Apply Distance and Midpoint formula.

Know types of angles and use a protractor to measure.

Define angle pair relationships (complementary, supplementary, vertical, adjacent, linear pair, etc…).

Constructions: Copy a segment, copy an angle, bisect a segment, trisect a segment, bisect an angle.

Calculate perimeter of polygons on a coordinate plane.

Calculate area of triangles and rectangles on a coordinate plane.

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Unit 2- Reasoning and Proof Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusions

QC C-1b

Apply inductive and deductive reasoning.



Identify and write conditional and biconditional statements along with the converse, inverse, and contrapositive of a conditional statement. Use these statements to form conclusions

QC C-1c

Define conditional statements and their inverse, converse, contrapositive.



Prove theorems about lines and angles. Theorems include vertical angles are congruent; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

9 G-CO9

Complete guided proofs by filling in missing reasons…

Complete guided proofs by filling in missing reason as well as fill in missing steps.

Complete guided proofs by filling in missing reason as well as fill in missing step and generate entire proofs.

Instructional Recommendations / Resources: All courses

Recognize patterns and apply inductive/deductive reasoning.


Use algebraic properties to justify steps in solving equations.

Segment and angle theorems/postulates to know and apply: Reflexive, Symmetric and Transitive, Vertical Angles theorem, Linear Pair postulate, Segment and Angle Addition postulates, Right Angle Congruence theorem.

Proofs Formats to use: two-column, flow chart, paragraph and indirect. Regular and PreAP

Additional Theorems: Congruent Complements and Congruent Supplements theorem

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Unit 3- Parallel and Perpendicular Lines Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Know precise definitions perpendicular and parallel lines.




Prove theorems about lines and angles. Theorems include; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

9 G-CO9

Complete guided proofs by filling in missing reasons.

Complete guided proofs by filling in missing reason as well as fill in missing steps.

Complete guided proofs by filling in missing reason as well as fill in missing steps and generate entire proofs.


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Unit 3- Parallel and Perpendicular Lines Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include; 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.

12 G-

CO12




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).

31 G-GPE5

Given slope-intercept form, determine slope and find equations of parallel and perpendicular lines.

Given slope-intercept form, determine slope and find equations of parallel and perpendicular lines. Also, be able to determine slope and find equations of parallel and perpendicular lines given other forms of equations.

Given slope-intercept form, determine slope and find equations of parallel and perpendicular lines. Also, be able to determine slope and find equations of parallel and perpendicular lines given other forms of equations.

Instructional Recommendations / Resources: All courses:

Know and apply definitions of parallel and perpendicular lines.

Recognize and apply angle relationships formed by parallel lines cut by a transversal (corresponding, same-side interior, same-side exterior, alternate interior, and alternate exterior)

Complete proofs using relationships with parallel lines.

Constructions: Perpendicular lines, perpendicular bisector, and a line parallel to a given line.

Determine slope of parallel and perpendicular lines.

Given the slope and y-intercept, find the equation of parallel and perpendicular lines. Regular and PreAP:

Include the converse of each angle relationship.

Find the equation of parallel and perpendicular lines given a variety of information.

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Unit 4- Triangle Congruence Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Identify and classify triangles by their sides and angles. QC

D -2a

Classify different types of triangles.



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. 7 G-CO7

Know the definition of congruence and apply it when naming corresponding parts and writing congruence statements as well as solving problems.



Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions.

8 G-CO8

Given simple diagrams or information, students will determine and/or prove triangle congruence.

Given more difficult diagrams or information, students will determine and/or prove triangle congruence.

Show and/or prove triangle congruence using more abstract information.

Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180⁰, base angles of isosceles triangles are congruent.

10 G-

CO10

Participate in activities used to prove theorems.

Complete guided proofs by filling in missing reasons as well as fill in missing steps.

Complete guided proofs by filling in missing reasons as well as fill in missing steps and generate entire proofs.

Use congruence criteria for triangles to solve problems and to prove relationships in geometric figures. 18 G-SRT5

Solve problems and prove relationships using simple diagrams or information.

Solve problems and prove relationships given more difficult diagrams or information.

Solve problems and prove relationships using more abstract information.

Instructional Recommendations / Resources: All Classes:


Define congruence in triangles and name congruent corresponding parts.

Write congruency statements for triangles.

Solve problems using congruent triangles.

Use SSS, SAS, ASA, AAS and HL to show triangle congruence.

Use CPCTC.

Apply triangle sum and isosceles triangle theorem (converse).

Use manipulatives or software to prove the triangle sum theorem and base angles theorem. Regular and PreAP

Apply exterior angle theorem.

Complete formal proofs of triangle congruence. PreAP

Generate formal proofs of triangle congruence.


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Unit 5- Relationships within Triangles Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Prove theorems about triangles. Theorems include segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and the medians of a triangle meet at a point.

10 G-

CO10



Participate in activities used to prove theorems. Solve problems using triangle relationship theorems.

Construct the inscribed and circumscribed circles of a triangle.

26 G-C3 Create constructions with accuracy.




Identify angle bisectors, perpendicular bisectors, medians, altitudes and midsegments in a triangle.

Investigate points of concurrency in triangles and apply their properties.

Use manipulatives or software to prove the midsegment theorem.

Know criteria for possible side lengths of a triangle.

Explore inequalities in one triangle. (order the sides/angles of a triangle)

Construct the inscribed and circumscribed circles of a triangle. PreAP

Explore inequalities in one and two triangles. (Hinge Theorem)

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Unit 6- Polygons / Quadrilaterals Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Identify and classify regular and nonregular polygons based on the number of sides, angle measure, and the side lengths.

QC D-2h

Identify and classify polygons up to 12-sided.



Identify and classify quadrilaterals using their properties.

QC D-2g

Know and use properties of special parallelograms.



Apply the Angle Sum Theorem for triangles and polygons to find interior and exterior angle measures given the number of sides, to find the number of sides given angle measures, and to solve real world problems.

QC D-2i

Know and apply the properties of polygons to determine measures of missing angles and sides.




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Unit 6- Polygons / Quadrilaterals Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced 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.

11 G-CO11

Participate in activities used to prove theorems and solve problems.

Participate in activities used to prove theorems and apply properties of quadrilaterals to solve problems.

Participate in activities used to prove theorems and apply properties of quadrilaterals to solve problems.

Use coordinates to prove simple geometric theorems algebraically.

30 G-GPE4

Given coordinates, prove characteristics of parallelograms algebraically.

Given coordinates, prove characteristics of parallelograms and other quadrilaterals algebraically.

Given coordinates, prove characteristics of parallelograms and other quadrilaterals algebraically and write coordinate proofs.

Instructional Recommendations / Resources All Classes:


Apply Interior/Exterior Angle Theorems of polygons.

Identify and classify parallelograms, rectangles, rhombi, squares, kites, trapezoids and isosceles trapezoids

Know and use properties of special parallelograms in a proof.

Know and apply the properties of polygons to determine measures of missing angles and sides. PreAP

Apply the properties of all quadrilaterals.

Given coordinates of vertices, determine type of quadrilateral.

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Unit 7 – Transformations Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Determine points or lines of symmetry and apply the properties of symmetry to figures.

QC E-1a

Apply properties of line and rotational symmetry.



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

2 G-CO2

Identify the basic transformations in a plane.

Identify the basic transformations in a plane and determine if a transformation is an isometry as well as equate pre-image and image as input and output of a

Identify the basic transformations in a plane and determine if a transformation is an isometry as well as equate pre-image and image as input and output of a


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Standards COS # CCSS # Basic Proficient Advanced horizontal stretch). relationship. relationship and apply

additional methods of identifying transformations using technology tools.

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

3 G-CO3

Perform basic transformations on plane figures.

Perform basic transformations on plane figures and combine transformations that result in image points that coincide with corresponding pre-image points.

Perform basic transformations on plane figures and combine transformations that result in image points that coincide with corresponding pre-image point and apply multiple methods that have the same results.

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

4 G-CO4

Recognize the differences in the transformations of rotation, reflection and translation.

Recognize the differences in the transformations of rotation, reflection and translation and describe these transformations using the definition of each as their guide in terms of angles, circles, lines, parallel lines and line segments.

Recognize the differences in the transformations of rotation, reflection and translation and describe these transformations using the definition of each as their guide in terms of angles, circles, lines, parallel lines and line segments and use technology that demonstrates how transformations occur clearly showing how the definitions are applied.

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. 5 G-CO5

Can develop a rough sketch of the result of each transformation requested.

Can develop a rough sketch of the result of each transformation requested and perform these transformations using traditional or technology tools and identify the transformations required on a given pre-image that results in a given image.

Can develop a rough sketch of the result of each transformation requested and perform these transformations using traditional or technology tools and identify the transformations required on a given pre-image that results in a given image and


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Standards COS # CCSS # Basic Proficient Advanced use technology that demonstrates how transformations occur clearly demonstrating the sequence in a step by step fashion.

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.

6 G-CO6

Identify if a transformation is an Isometry.

Identify if a transformation is an Isometry and, using the definitions of each transformation, conclude corresponding parts between pre-image and image are congruent.

Identify if a transformation is an Isometry and, using the definitions of each transformation, conclude corresponding parts between pre-image and image are congruent show, using technology tools, the congruent objects formed using the definitions of each transformation.

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. 14 G-SRT1

Identify a dilation as an enlargement or reduction based on observation and the scale factor.

Identify a dilation as an enlargement or reduction based on observation and the scale factor and identify corresponding parts as parallel and, using the scale factor, find the lengths of corresponding parts of pre-image and image.

Identify a dilation as an enlargement or reduction based on observation and the scale factor and identify corresponding parts as parallel and, using the scale factor, find the lengths of corresponding parts of pre-image and image and identify the relationships between the triangles formed by the images and the lines connecting the corresponding image and pre-image points to the center of dilation.



Recognize different types of transformations including reflections, rotations, translations and dilations.

Perform each type of transformation in the coordinate plane.


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Unit 8- Similarity Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced 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.

15 G-SRT2

Know the definition of similarity and apply it when naming corresponding parts and writing proportionality statements as well as solving problems.



Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. (Quality Core includes SAS, SSS)

16 G-SRT3 Use various tools to show similarity by AA, SSS, SAS.

Use various tools to show similarity by AA, SSS, SAS.

Use various tools to show similarity by AA, SSS, SAS.

Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely. 17 G-SRT4


Participate in activities used to prove theorems. Solve more difficult problems relating to theorems.

Participate in activities used to prove theorems. Solve more difficult problems relating to theorems. Include real-world applications.

Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 18 G-SRT5

Solve problems and prove relationships using simple diagrams or information.

Solve problems and prove relationships given more difficult diagrams or information.

Solve problems and prove relationships using more abstract information.

Solve problems involving the relationships formed when the altitude to the hypotenuse of a right triangle is drawn.

QC

D-2d

Solve problems with altitude to hypotenuse of right triangles using geometric mean.




Define similarity in triangles and name corresponding parts.

Write similarity statements for triangles.

Solve problems using similar triangles.

Use AA, SAS, SSS criteria to show triangle similarity.

Solve proportions.

Find geometric mean between two numbers.

Solve problems with altitude to hypotenuse of right triangles using geometric mean. Pre-AP

Solve proportions that involve quadratics.


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Unit 9- Right Triangles and Trigonometry Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Prove theorems about triangles. Prove the Pythagorean Theorem using triangle similarity.

17 G-SRT4

Participate in activities used to prove theorems. Can state the Pythagorean Theorem and use it to find lengths of the sides of a right triangle.

Can state the Pythagorean Theorem and use it to find lengths of the sides of a right triangle and prove the Pythagorean Theorem.

Can state the Pythagorean Theorem and use it to find lengths of the sides of a right triangle and prove the Pythagorean Theorem use multiple ways to prove the Pythagorean theorem.

Apply properties of 45-45-90 and 30-60-90 triangles to determine the lengths of sides of triangles.

QC H-1a

Apply 45-45-90 / 30-60-90 relationships.



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.

19 G-SRT6

Know the trig ratios and how their definitions relate to similarity.



Explain and use the relationship between the sine and cosine of complementary angles. 20 G-SRT7

Know the relationship between sine and cosine of complementary angles.



Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* 21 G-SRT8

Set up trig ratios in right triangles to determine missing sides.

Set up trig ratios in right triangles to determine missing sides and angles.

Set up trig ratios in right triangles to determine missing sides and angles.

(+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. 22

G-SRT10

*Not in Geometry Principles.

*Not in Geometry. Prove the law of sines and cosines and use them to solve triangles.

(+) 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).

23

G-

SRT11

*Not in Geometry Principles.

*Not in Geometry. Apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.


Use manipulatives or software to prove the Pythagorean Theorem.

Solve problems using the Pythagorean Theorem and its converse, including Pythagorean triples.


Determine trigonometric ratios of acute angles.

Know the relationship between the sine and cosine of complementary angles.

Determine missing sides and angles of right triangles applying trigonometric ratios.

Solve trigonometric problems using angle of elevation and angle of depression.


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Unit 9- Right Triangles and Trigonometry Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Pre-AP

Derive inductively and inductively the formulas for 45-45-90 / 30-60-90.

Define additional trigonometric ratios.

Apply the Law of Sines and the Law of Cosines in right and non-right triangles.

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Unit 10- Geometric Measurement in Two Dimensions Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Know precise definitions of circle and distance around a circular arc.




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

G-CO13

Construct with accuracy. Construct with accuracy. Construct with accuracy.

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.

28 G-C5

Derive facts and formulas as a guided classroom activity and use to solve problems. Define radian measure.

Derive facts and formulas with some guidance and use to solve problems. Define radian measure.

Derive facts and formulas with minimal guidance and use to solve problems. Define radian measure.

Find perimeter and area of common plane figures including triangles, quadrilaterals, irregular figures

QC F-1a

Given the formulas, find area and perimeter.



Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics.

34


Recall the formulas and find area and perimeter.

Recall and derive formulas and find area and perimeter.

Give an informal argument for the formulas for the circumference of a circle; area of a circle.

35 G-

GMD1

Give informal arguments for formulas through guided classroom activities, and use the formulas to solve problems.

Give informal arguments for formulas through classroom activities, and use the formulas to solve problems.

Give informal arguments for formulas through classroom activities, and use the formulas to solve problems.

(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 42 S-MD6

*Not in Geometry Basic. Students use probabilities to make fair decision in various situations.

Students use probabilities to make fair decision in various situations and justify their results.


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Unit 10- Geometric Measurement in Two Dimensions Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

43 S-MD7 *Not in Geometry Basic. Students use probability to make decisions in various real-life situations.

Students use probability to make decisions in various real-life situations and explain their reasoning and make further predications based on probabilities.


Identify characteristics and types of two-dimensional figures.

Constructions: Equilateral triangle, square and regular hexagon inscribed in a circle.

Calculate area and perimeter of triangles, quadrilaterals, irregular figures, circles, sectors, inscribed/circumscribed polygons.

Calculate missing length given area.

Apply relationships between perimeter and area of similar figures.

Calculate circumference and arc length.

Use proportionality to define radian.

Use area to solve problems involving geometric probability. Regular

Calculate the area and perimeter of regular hexagons. PreAP

Calculate the area and perimeter of regular polygons requiring trigonometry.

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Unit 11- Circles Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Prove that all circles are similar.

24 G-C1 Participate in activities used to verify all circles are similar.

Verify that all circles are similar.

Verify that all circles are similar.

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.

25 G-C2

Given circle theorems, find missing segment lengths, angles and arcs.

Apply circle theorems to find missing segment lengths, angles and arcs.

Apply and prove circle theorems to find missing segment lengths, angles and arcs.


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Unit 11- Circles Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Prove properties of angles for a quadrilateral inscribed in a circle.

26 G-C3

Participate in activities used to prove properties of angles for a quadrilateral inscribed in a circle.

Prove and apply properties of angles for a quadrilateral inscribed in a circle.

Prove and apply properties of angles for a quadrilateral inscribed in a circle and discover a pattern if the number of sides of the inscribed polygon increases.

(+) Construct a tangent line from a point outside a given circle to the circle. 27 G-C4

*Not in Geometry Basic. Construct a tangent line from a point outside a given circle to the circle.

Construct a tangent line from a point outside a given circle to the circle.

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. 29 G-GPE1

Participate in activities used to derive the equation of a circle. Complete the square by filling in missing information in order to find the center and radius of a circle.

Participate in activities used to derive the equation of a circle. Use simple equations to complete the square and derive the equation of a circle of given center and radius.

Use more complex equations to complete the square and derive the equation of a circle of given center and radius.


Identify and define line segments and angle measures associated with circles.

Apply angle/arc relationships inside and outside the circle including angles of a quadrilateral inscribed in a circle.

Apply segment relationships inside and outside the circles.

From given information, write the equation of a circle and/or given the equation determine information and graph circles.

Locate, describe and draw a locus in a plane or space.

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Unit 12- Geometric Measurement in Three Dimensions Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Define lateral area and calculate surface area of prism, cylinders, cones, pyramids and spheres

QC

F-2a/c

Calculate the surface area and volume of cylinders, cones, spheres.



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

35 G-

GMD1

Give informal arguments for formulas through guided classroom activities.

Give informal arguments for formulas through classroom activities.

Give informal arguments for formulas through classroom activities.


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Unit 12- Geometric Measurement in Three Dimensions Indicators of Proficiency

Standards COS # CCSS # Basic Proficient Advanced Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* 36

G-GMD3

Given the formulas, solve problems.

Recall and use formulas to solve problems.

Recall and use formulas to solve more complex problems.

Determine the relationship between surface areas of similar figures and volumes of similar figures.

37

Determine the relationships with guided activities, and solve problems using the relationships.



Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. 38

G-GMD4

Identify the cross-sectional shapes and three-dimensional objects.

Identify the cross-sectional shapes and three-dimensional objects.

Identify the cross-sectional shapes and three-dimensional objects and calculate areas of cross-sections and volumes of three dimensional object generated by rotating two-dimensional objects.

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

39 G-MG1 Use real-life examples as models of geometric shapes.

Use real-life examples as models of geometric shapes.

Use real-life examples as models of geometric shapes.

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).*

40 G-MG2 Use real-life examples to model density and solve problems.

Use real-life examples to model density and solve problems.

Use real-life examples to model density and solve problems.

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).*

41 G-MG3

Apply geometric methods to solve basic design problems.

Apply geometric methods to solve more complex design problems.

Apply geometric methods to solve challenging design problems.


Identify characteristics and types of 3-dimensional figures including cross sections.


Calculate the surface area and volume of pyramids and prisms with rectangular or triangular bases.

Model 3-dimensional figures.

Apply the relationship between surface area and volume of similar solids. Regular

Calculate the surface area and volume of pyramids and prisms with regular hexagonal bases. Pre-AP

Calculate the surface area and volume of pyramids and prisms with bases which are regular polygons.


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Correlation of Standards Standards Key AL COS # CCSS # HCS Unit #

GEOMETRY: Congruence 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 G-CO1 1, 3 , 10

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 G-CO2 7

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

3 G-CO3 7

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

4 G-CO4 7

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.

5 G-CO5 7

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.

6 G-CO6 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.

7 G-CO7 4

Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions.

8 G-CO8 4

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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

9 G-CO9 2, 3

Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180⁰, 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, and the medians of a triangle meet at a point.

10 G-CO10 4 , 5 ,

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.

11 G-CO11 6

Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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.

12 G-CO12 1, 3


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Standards Key AL COS # CCSS # HCS Unit #

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 13 G-CO13 10

GEOMETRY: Similarity, Right Triangles, and Trigonometry Verify experimentally the properties of dilations given by a center and a scale factor.

c) 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.

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

14 G-SRT1 7

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.

15 G-SRT2 8

Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar.

16 G-SRT3 8

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

17 G-SRT4 8, 9

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 18 G-SRT5 4 , 8 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.

19 G-SRT6 9

Explain and use the relationship between the sine and cosine of complementary angles. 20 G-SRT7 9 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* 21 G-SRT8 9 (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. 22 G-SRT10 9 (+) 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).

23 G-SRT11 9

GEOMETRY: Circles Prove that all circles are similar. 24 G-C1 11 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.

25 G-C2 11

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

26 G-C3 5, 11

(+) Construct a tangent line from a point outside a given circle to the circle. 27 G-C4 11 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.

28 G-C5 10

GEOMETRY: Expressing Geometric Properties With Equations 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.

29 G-GPE1 11

Use coordinates to prove simple geometric theorems algebraically. 30 G-GPE4 6


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Standards Key AL COS # CCSS # HCS Unit #

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).

31 G-GPE5 3

Find the point on a direct line segment between two given points that partitions the segment in a given ratio. 32 G-GPE6 1 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.*

33 G-GPE7 1

Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics.

34

10

GEOMETRY: Geometric Measurement and Dimension Given an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

35 G-GMD1 10 , 12

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* 36 G-GMD3 12 Determine the relationship between surface areas of similar figures and volumes of similar figures. 37

12

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

38 G-GMD4 12

GEOMETRY: Modeling With Geometry Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*

39 G-MG1 12

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).*

40 G-MG2 12

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).*

41 G-MG3 12

STATISTICS AND PROBABILITY: Using Probability to Make Decisions Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 42 S-MD6 10 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

43 S-MD7 10

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Special Notes This curriculum document was written to apply to basic, regular and advanced levels of the Geometry course. Specific notes are provided in the notes section of each unit specifying particular topics which might require more or less emphasis for the various levels of the course. The notation QC refers to content included from the ACT Quality Core Standards. The letter denotes the theme with the number being the concept and the lower case letter representing a specific skill. This local curriculum document was developed from the 2010 Alabama Course of Study for Mathematics which was itself based on the newly adopted Common Core State Standards for Mathematics. State COS standards are keyed to CCSS (i.e. Common Core) standards using the lettering and number system employed by the CCSS so that instructional resources which are subsequently designed to support the CCSS can be easily matched back to lessons based on state and local requirements. The symbol (+) is used to designate standards based on mathematics content that students should learn in order to take advanced courses such as calculus. An asterisk (*) is used to designate standards where modeling with mathematics should be stressed because modeling is best interpreted through a relevant context rather than as an isolated topic. The state map of Alabama or the designation (AL) is used to denote additional standards required by the Alabama Board of Education which extend beyond the minimum content defined by the CCSS. The Standards for Mathematical Practice describe the varieties of expertise that mathematics educators at all levels should seek to develop in their students. These standards were developed with input from the National Council of Teachers of Mathematics and the National Research Council, and math teachers should reinforce these process skills when designing daily instructional lessons for students at all grade levels in the Hoover school system:

1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

According to the Alabama Quality Teaching Standards (AQTS), teachers of all grade levels and subjects are required to model and reinforce literacy skills for all students. The Alabama Course of Study for Mathematics defines specific college and career readiness “anchor standards” for grades 6-12 in the areas of reading and writing. Specific grade-appropriate criteria can be found in the state course of study document, but the general anchor standards are defined below:

Reading


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Key Ideas and Details 1. Read closely to determine what the text says explicitly and to make logical inferences from it; cite specific textual

evidence when writing or speaking to support conclusions drawn from the text. 2. Determine central ideas or themes of a text and analyze their development; summarize the key supporting details and

ideas. 3. Analyze how and why individuals, events, or ideas develop and interact over the course of a text.

Craft and Structure 4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative

meanings, and analyze how specific word choices shape meaning or tone. 5. Analyze the structure of texts, including how specific sentences, paragraphs, and larger portions of the text (e.g. a

section, chapter, scene, or stanza) relate to each other and the whole. 6. Assess how point of view or purpose shapes the content and style of a text.

Integration of Knowledge and Ideas 7. Integrate and evaluate content presented in diverse formats and media, including visually and quantitatively, as well as

in words. 8. Delineate and evaluate the argument and specific claims in a text, including the validity of the reasoning as well as the

relevance and sufficiency of the evidence. 9. Analyze how two or more texts address similar themes or topics in order to build knowledge or to compare the

approaches the authors take. Range of Reading and Level of Text Complexity

10. Read and comprehend complex literary and informational texts independently and proficiently. Writing

Text Types and Purposes 1. Write arguments to support claims in an analysis of substantive topics or texts using valid reasoning and relevant and

sufficient evidence. 2. Write informative / explanatory texts to examine and convey complex ideas and information clearly and accurate

through the effective selection, organization, and analysis of content. 3. Write narratives to develop real or imagined experiences or events using effective technique, well-chosen details, and

well-structured event sequences. Production and Distribution of Writing

4. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.

5. Develop and strengthen writing as needed by planning, revising, editing, rewriting, or trying a new approach. 6. Use technology, including the Internet, to produce and publish writing and to interact and collaborate with others.


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Research to Build and Present Knowledge 7. Conduct short as well as more sustained research projects based on focused questions, demonstrating understanding

of the subject under investigation. 8. Gather relevant information from multiple print and digital sources, assess the credibility and accuracy of each source,

and integrate the information while avoiding plagiarism. 9. Draw evidence form literary or informational texts to support analysis, reflection, and research.

Range of Writing 10. Write routinely over extended time frames (time for research, reflection, and revision) and short time frames (a single

sitting or a day or two) for a range of tasks, purposes, and audiences. Back to top

Geometry Mathematics: 2012 -13 to 2017 18 Hoover City Schools

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