IXL Skill Plan | California Common Core Content Standards: Geometry

California Common Core Content Standards: Geometry

Skills available for California Geometry standards

Domain

Geometry
Statistics and Probability

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G	Geometry
G-CO Congruence
Experiment with transformations in the plane
Experiment with transformations in the plane

G-CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.		1. Angle vocabulary
G-CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
1. Angle vocabulary
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).		Translations 1. Translations: graph the image 2. Translations: find the coordinates 3. Translations: write the rule Reflections 4. Reflections: graph the image 5. Reflections: find the coordinates Rotations 6. Rotations: graph the image 7. Rotations: find the coordinates Congruence transformations 8. Classify congruence transformations 9. Reflections and rotations: write the rule Dilations 10. Dilations: graph the image 11. Dilations: find the coordinates 12. Dilations: find the scale factor
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).
Translations 1. Translations: graph the image 2. Translations: find the coordinates 3. Translations: write the rule Reflections 4. Reflections: graph the image 5. Reflections: find the coordinates Rotations 6. Rotations: graph the image 7. Rotations: find the coordinates Congruence transformations 8. Classify congruence transformations 9. Reflections and rotations: write the rule Dilations 10. Dilations: graph the image 11. Dilations: find the coordinates 12. Dilations: find the scale factor
G-CO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.		1. Transformations that carry a polygon onto itself
G-CO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
1. Transformations that carry a polygon onto itself
G-CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.		1. Rotate polygons about a point
G-CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
1. Rotate polygons about a point
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.		Graph an image 1. Translations: graph the image 2. Reflections: graph the image 3. Rotations: graph the image Sequences of transformations 4. Sequences of congruence transformations: graph the image 5. Sequences of congruence transformations: choose the sequence 6. Sequences of congruence transformations: find the rules
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.
Graph an image 1. Translations: graph the image 2. Reflections: graph the image 3. Rotations: graph the image Sequences of transformations 4. Sequences of congruence transformations: graph the image 5. Sequences of congruence transformations: choose the sequence 6. Sequences of congruence transformations: find the rules
Checkpoint opportunity		1. Checkpoint: Definitions of geometric objects 2. Checkpoint: Congruence transformations 3. Checkpoint: Transformations in the plane
Checkpoint opportunity
1. Checkpoint: Definitions of geometric objects 2. Checkpoint: Congruence transformations 3. Checkpoint: Transformations in the plane

Understand congruence in terms of rigid motions
Understand congruence in terms of rigid motions

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.		1. Sequences of congruence transformations: graph the image 2. Sequences of congruence transformations: choose the sequence 3. Transformations that carry a polygon onto itself 4. Congruence transformations: mixed review 5. Determine if two figures are congruent: justify your answer
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.
1. Sequences of congruence transformations: graph the image 2. Sequences of congruence transformations: choose the sequence 3. Transformations that carry a polygon onto itself 4. Congruence transformations: mixed review 5. Determine if two figures are congruent: justify your answer
G-CO.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.		1. Prove triangles are congruent using rigid motions
G-CO.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
1. Prove triangles are congruent using rigid motions
G-CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G-CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Checkpoint opportunity		1. Checkpoint: Rigid motion and congruence
Checkpoint opportunity
1. Checkpoint: Rigid motion and congruence

Prove geometric theorems
Prove geometric theorems

G-CO.9: Prove theorems about lines and angles.		Angles 1. Angle bisectors 2. Proofs involving angles Parallel lines 3. Transversals of parallel lines: solve for x 4. Proofs involving parallel lines I 5. Proofs involving parallel lines II Perpendicular bisectors 6. Perpendicular Bisector Theorem
G-CO.9: Prove theorems about lines and angles.
Angles 1. Angle bisectors 2. Proofs involving angles Parallel lines 3. Transversals of parallel lines: solve for x 4. Proofs involving parallel lines I 5. Proofs involving parallel lines II Perpendicular bisectors 6. Perpendicular Bisector Theorem
G-CO.10: Prove theorems about triangles.		Isosceles triangles 1. Proofs involving isosceles triangles Angles in triangles 2. Proofs involving triangles I Segments in triangles 3. Proofs involving triangles II Centroids or orthocenters of triangles 4. Construct the centroid or orthocenter of a triangle 5. Find the centroid of a triangle
G-CO.10: Prove theorems about triangles.
Isosceles triangles 1. Proofs involving isosceles triangles Angles in triangles 2. Proofs involving triangles I Segments in triangles 3. Proofs involving triangles II Centroids or orthocenters of triangles 4. Construct the centroid or orthocenter of a triangle 5. Find the centroid of a triangle
G-CO.11: Prove theorems about parallelograms.		1. Properties of parallelograms 2. Proving a quadrilateral is a parallelogram 3. Properties of rhombuses 4. Properties of squares and rectangles
G-CO.11: Prove theorems about parallelograms.
1. Properties of parallelograms 2. Proving a quadrilateral is a parallelogram 3. Properties of rhombuses 4. Properties of squares and rectangles
Checkpoint opportunity		1. Checkpoint: Line and angle theorems 2. Checkpoint: Triangle theorems 3. Checkpoint: Parallelogram theorems
Checkpoint opportunity
1. Checkpoint: Line and angle theorems 2. Checkpoint: Triangle theorems 3. Checkpoint: Parallelogram theorems

Make geometric constructions
Make geometric constructions

G-CO.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).		Angles 1. Construct an angle bisector 2. Construct a congruent angle Points, lines, and segments 3. Construct a congruent segment 4. Construct the midpoint or perpendicular bisector of a segment 5. Construct a perpendicular line 6. Construct parallel lines Regular polygons 7. Construct an equilateral triangle or regular hexagon 8. Construct a square Triangles 9. Construct the circumcenter or incenter of a triangle 10. Construct the centroid or orthocenter of a triangle Circles 11. Construct a tangent line to a circle 12. Construct the inscribed or circumscribed circle of a triangle
G-CO.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Angles 1. Construct an angle bisector 2. Construct a congruent angle Points, lines, and segments 3. Construct a congruent segment 4. Construct the midpoint or perpendicular bisector of a segment 5. Construct a perpendicular line 6. Construct parallel lines Regular polygons 7. Construct an equilateral triangle or regular hexagon 8. Construct a square Triangles 9. Construct the circumcenter or incenter of a triangle 10. Construct the centroid or orthocenter of a triangle Circles 11. Construct a tangent line to a circle 12. Construct the inscribed or circumscribed circle of a triangle
G-CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.		1. Construct an equilateral triangle inscribed in a circle 2. Construct a square inscribed in a circle 3. Construct a regular hexagon inscribed in a circle
G-CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
1. Construct an equilateral triangle inscribed in a circle 2. Construct a square inscribed in a circle 3. Construct a regular hexagon inscribed in a circle
Checkpoint opportunity		1. Checkpoint: Geometric constructions
Checkpoint opportunity
1. Checkpoint: Geometric constructions
G-SRT Similarity, Right Triangles, and Trigonometry
Understand similarity in terms of similarity transformations
Understand similarity in terms of similarity transformations

G-SRT.1: Verify experimentally the properties of dilations given by a center and a scale factor:
G-SRT.1: Verify experimentally the properties of dilations given by a center and a scale factor:

G-SRT.1.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.		1. Dilations: graph the image 2. Dilations and parallel lines
G-SRT.1.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.
1. Dilations: graph the image 2. Dilations and parallel lines
G-SRT.1.b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.		1. Dilations: find length, perimeter, and area 2. Dilations: find the scale factor 3. Dilations: find the scale factor and center of the dilation
G-SRT.1.b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
1. Dilations: find length, perimeter, and area 2. Dilations: find the scale factor 3. Dilations: find the scale factor and center of the dilation
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.		1. Identify similar figures 2. Side lengths and angle measures in similar figures 3. Similar triangles and similarity transformations 4. Determine if two figures are similar: justify your answer
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.
1. Identify similar figures 2. Side lengths and angle measures in similar figures 3. Similar triangles and similarity transformations 4. Determine if two figures are similar: justify your answer
G-SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.		1. Similarity rules for triangles
G-SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
1. Similarity rules for triangles
Checkpoint opportunity		1. Checkpoint: Dilations 2. Checkpoint: Similarity transformations
Checkpoint opportunity
1. Checkpoint: Dilations 2. Checkpoint: Similarity transformations

Prove theorems involving similarity
Prove theorems involving similarity

G-SRT.4: Prove theorems about triangles.		1. Proofs involving similarity in right triangles 2. Proofs involving triangles II 3. Prove the Pythagorean theorem
G-SRT.4: Prove theorems about triangles.
1. Proofs involving similarity in right triangles 2. Proofs involving triangles II 3. Prove the Pythagorean theorem
G-SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.		Congruent triangles 1. SSS and SAS Theorems 2. ASA and AAS Theorems 3. SSS, SAS, ASA, and AAS Theorems 4. SSS Theorem in the coordinate plane Proofs involving congruent triangles 5. Proving triangles congruent by SSS and SAS 6. Proving triangles congruent by ASA and AAS 7. Proving triangles congruent by SSS, SAS, ASA, and AAS 8. Proofs involving corresponding parts of congruent triangles 9. Proofs involving triangles and quadrilaterals Isosceles, equilateral, and right triangles 10. Congruency in isosceles and equilateral triangles 11. Proofs involving isosceles triangles 12. Hypotenuse-Leg Theorem Similar triangles 13. Similar triangles and indirect measurement 14. Similarity rules for triangles 15. Triangle Proportionality Theorem 16. Similarity and altitudes in right triangles Proofs involving similar triangles 17. Prove similarity statements 18. Prove proportions or angle congruences using similarity
G-SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Congruent triangles 1. SSS and SAS Theorems 2. ASA and AAS Theorems 3. SSS, SAS, ASA, and AAS Theorems 4. SSS Theorem in the coordinate plane Proofs involving congruent triangles 5. Proving triangles congruent by SSS and SAS 6. Proving triangles congruent by ASA and AAS 7. Proving triangles congruent by SSS, SAS, ASA, and AAS 8. Proofs involving corresponding parts of congruent triangles 9. Proofs involving triangles and quadrilaterals Isosceles, equilateral, and right triangles 10. Congruency in isosceles and equilateral triangles 11. Proofs involving isosceles triangles 12. Hypotenuse-Leg Theorem Similar triangles 13. Similar triangles and indirect measurement 14. Similarity rules for triangles 15. Triangle Proportionality Theorem 16. Similarity and altitudes in right triangles Proofs involving similar triangles 17. Prove similarity statements 18. Prove proportions or angle congruences using similarity
Checkpoint opportunity		1. Checkpoint: Triangle similarity and congruence
Checkpoint opportunity
1. Checkpoint: Triangle similarity and congruence

Define trigonometric ratios and solve problems involving right triangles
Define trigonometric ratios and solve problems involving right triangles

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.		1. Trigonometric ratios: sin, cos, and tan 2. Trigonometric ratios with radicals: sin, cos, and tan 3. Trigonometric ratios: csc, sec, and cot 4. Trigonometric ratios in similar right triangles
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.
1. Trigonometric ratios: sin, cos, and tan 2. Trigonometric ratios with radicals: sin, cos, and tan 3. Trigonometric ratios: csc, sec, and cot 4. Trigonometric ratios in similar right triangles
G-SRT.7: Explain and use the relationship between the sine and cosine of complementary angles.		1. Sine and cosine of complementary angles
G-SRT.7: Explain and use the relationship between the sine and cosine of complementary angles.
1. Sine and cosine of complementary angles
G-SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.		Pythagorean Theorem 1. Pythagorean theorem 2. Converse of the Pythagorean theorem 3. Pythagorean Inequality Theorems Trigonometric ratios 4. Trigonometric ratios: find a side length 5. Trigonometric ratios: find an angle measure Solve a right triangle 6. Solve a right triangle
G-SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Pythagorean Theorem 1. Pythagorean theorem 2. Converse of the Pythagorean theorem 3. Pythagorean Inequality Theorems Trigonometric ratios 4. Trigonometric ratios: find a side length 5. Trigonometric ratios: find an angle measure Solve a right triangle 6. Solve a right triangle
G-SRT.8.1: Derive and use the trigonometric ratios for special right triangles (30°,60°,90°and 45°,45°,90°).		1. Special right triangles 2. Find trigonometric functions of special angles
G-SRT.8.1: Derive and use the trigonometric ratios for special right triangles (30°,60°,90°and 45°,45°,90°).
1. Special right triangles 2. Find trigonometric functions of special angles
Checkpoint opportunity		1. Checkpoint: Right triangle trigonometry
Checkpoint opportunity
1. Checkpoint: Right triangle trigonometry

Apply trigonometry to general triangles
Apply trigonometry to general triangles

G-SRT.9: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.		1. Area of a triangle: sine formula
G-SRT.9: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
1. Area of a triangle: sine formula
G-SRT.10: Prove the Laws of Sines and Cosines and use them to solve problems.		1. Law of Sines 2. Law of Cosines 3. Area of a triangle: Law of Sines
G-SRT.10: Prove the Laws of Sines and Cosines and use them to solve problems.
1. Law of Sines 2. Law of Cosines 3. Area of a triangle: Law of Sines
G-SRT.11: Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).		1. Solve a triangle
G-SRT.11: Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
1. Solve a triangle
Checkpoint opportunity		1. Checkpoint: Laws of Sines and Cosines
Checkpoint opportunity
1. Checkpoint: Laws of Sines and Cosines
G-C Circles
Understand and apply theorems about circles
Understand and apply theorems about circles

G-C.1: Prove that all circles are similar.		1. Similarity of circles
G-C.1: Prove that all circles are similar.
1. Similarity of circles
G-C.2: Identify and describe relationships among inscribed angles, radii, and chords.		Central and inscribed angles 1. Central angles and arc measures 2. Inscribed angles 3. Angles in inscribed right triangles Chords, secants, and tangents 4. Arcs and chords 5. Tangent lines 6. Angles formed by chords, secants, and tangents 7. Segments formed by chords, secants, and tangents
G-C.2: Identify and describe relationships among inscribed angles, radii, and chords.
Central and inscribed angles 1. Central angles and arc measures 2. Inscribed angles 3. Angles in inscribed right triangles Chords, secants, and tangents 4. Arcs and chords 5. Tangent lines 6. Angles formed by chords, secants, and tangents 7. Segments formed by chords, secants, and tangents
G-C.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.		Constructions 1. Construct the circumcenter or incenter of a triangle 2. Construct the inscribed or circumscribed circle of a triangle Properties of angles 3. Angles in inscribed quadrilaterals I 4. Angles in inscribed quadrilaterals II
G-C.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Constructions 1. Construct the circumcenter or incenter of a triangle 2. Construct the inscribed or circumscribed circle of a triangle Properties of angles 3. Angles in inscribed quadrilaterals I 4. Angles in inscribed quadrilaterals II
G-C.4: Construct a tangent line from a point outside a given circle to the circle.		1. Construct a tangent line to a circle
G-C.4: Construct a tangent line from a point outside a given circle to the circle.
1. Construct a tangent line to a circle
Checkpoint opportunity		1. Checkpoint: Prove circles are similar 2. Checkpoint: Angles and lines in circles 3. Checkpoint: Inscribed and circumscribed circles
Checkpoint opportunity
1. Checkpoint: Prove circles are similar 2. Checkpoint: Angles and lines in circles 3. Checkpoint: Inscribed and circumscribed circles

Find arc lengths and areas of sectors of circles
Find arc lengths and areas of sectors of circles

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. Convert between degrees and radians.		1. Arc length 2. Area of sectors
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. Convert between degrees and radians.
1. Arc length 2. Area of sectors
Checkpoint opportunity		1. Checkpoint: Arc length and area of sectors
Checkpoint opportunity
1. Checkpoint: Arc length and area of sectors
G-GPE Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic section
Translate between the geometric description and the equation for a conic section

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.		Complete the square 1. Convert equations of circles from general to standard form 2. Find properties of circles from equations in general form Write equations and graph circles 3. Derive equations of circles using the Pythagorean theorem 4. Write equations of circles in standard form from graphs 5. Write equations of circles in standard form using properties 6. Graph circles from equations in standard form 7. Graph circles from equations in general form
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.
Complete the square 1. Convert equations of circles from general to standard form 2. Find properties of circles from equations in general form Write equations and graph circles 3. Derive equations of circles using the Pythagorean theorem 4. Write equations of circles in standard form from graphs 5. Write equations of circles in standard form using properties 6. Graph circles from equations in standard form 7. Graph circles from equations in general form
G-GPE.2: Derive the equation of a parabola given a focus and directrix.		Find properties 1. Find the focus or directrix of a parabola 2. Find properties of a parabola from equations in general form Write equations 3. Write equations of parabolas in vertex form from graphs 4. Write equations of parabolas in vertex form using the focus and directrix 5. Write equations of parabolas in vertex form using properties Graph parabolas 6. Graph parabolas
G-GPE.2: Derive the equation of a parabola given a focus and directrix.
Find properties 1. Find the focus or directrix of a parabola 2. Find properties of a parabola from equations in general form Write equations 3. Write equations of parabolas in vertex form from graphs 4. Write equations of parabolas in vertex form using the focus and directrix 5. Write equations of parabolas in vertex form using properties Graph parabolas 6. Graph parabolas
G-GPE.3: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.		1. Write equations of ellipses in standard form using properties 2. Write equations of hyperbolas in standard form using properties
G-GPE.3: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
1. Write equations of ellipses in standard form using properties 2. Write equations of hyperbolas in standard form using properties
G-GPE.3.1: Given a quadratic equation of the form ax² + by² + cx + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola, and graph the equation.		1. Graph a quadratic function 2. Complete the square 3. Graph parabolas 4. Convert equations of circles from general to standard form 5. Graph circles 6. Convert equations of ellipses from general to standard form 7. Convert equations of hyperbolas from general to standard form
G-GPE.3.1: Given a quadratic equation of the form ax² + by² + cx + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola, and graph the equation.
1. Graph a quadratic function 2. Complete the square 3. Graph parabolas 4. Convert equations of circles from general to standard form 5. Graph circles 6. Convert equations of ellipses from general to standard form 7. Convert equations of hyperbolas from general to standard form
Checkpoint opportunity		1. Checkpoint: Equations of circles 2. Checkpoint: Equations of parabolas
Checkpoint opportunity
1. Checkpoint: Equations of circles 2. Checkpoint: Equations of parabolas

Use coordinates to prove simple geometric theorems algebraically
Use coordinates to prove simple geometric theorems algebraically

G-GPE.4: Use coordinates to prove simple geometric theorems algebraically.		1. Classify shapes on the coordinate plane: justify your answer 2. Determine if a point lies on a circle
G-GPE.4: Use coordinates to prove simple geometric theorems algebraically.
1. Classify shapes on the coordinate plane: justify your answer 2. Determine if a point lies on a circle
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).		1. Slopes of parallel and perpendicular lines 2. Equations of parallel and perpendicular lines 3. Find the distance between a point and a line 4. Find the distance between two parallel lines
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).
1. Slopes of parallel and perpendicular lines 2. Equations of parallel and perpendicular lines 3. Find the distance between a point and a line 4. Find the distance between two parallel lines
G-GPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.		1. Midpoints 2. Midpoint formula: find the midpoint 3. Partition a line segment in a given ratio 4. Midpoint formula: find the endpoint
G-GPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
1. Midpoints 2. Midpoint formula: find the midpoint 3. Partition a line segment in a given ratio 4. Midpoint formula: find the endpoint
G-GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.		1. Distance formula 2. Area and perimeter in the coordinate plane I 3. Area and perimeter in the coordinate plane II
G-GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
1. Distance formula 2. Area and perimeter in the coordinate plane I 3. Area and perimeter in the coordinate plane II
Checkpoint opportunity		1. Checkpoint: Partition a line segment 2. Checkpoint: Coordinate proofs 3. Checkpoint: Parallel and perpendicular lines 4. Checkpoint: Area and perimeter in the coordinate plane
Checkpoint opportunity
1. Checkpoint: Partition a line segment 2. Checkpoint: Coordinate proofs 3. Checkpoint: Parallel and perpendicular lines 4. Checkpoint: Area and perimeter in the coordinate plane
G-GMD Geometric Measurement and Dimension
Explain volume formulas and use them to solve problems
Explain volume formulas and use them to solve problems

G-GMD.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
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.

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

G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.		1. Volume of prisms and cylinders 2. Volume of pyramids and cones 3. Volume of spheres
G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
1. Volume of prisms and cylinders 2. Volume of pyramids and cones 3. Volume of spheres
Checkpoint opportunity		1. Checkpoint: Volume
Checkpoint opportunity
1. Checkpoint: Volume

Visualize relationships between two-dimensional and three-dimensional objects
Visualize relationships between two-dimensional and three-dimensional objects

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.		1. Cross sections of three-dimensional figures 2. Solids of revolution
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.
1. Cross sections of three-dimensional figures 2. Solids of revolution
G-GMD.5: Know that the effect of a scale factor k greater than zero on length, area, and volume is to multiply each by k, k², and k³, respectively; determine length, area and volume measures using scale factors.		1. Scale drawings: word problems 2. Dilations: find the scale factor 3. Ratios in similar figures 4. Side lengths and angle measures in similar figures 5. Perimeters of similar figures 6. Areas of similar figures 7. Area and perimeter of similar figures 8. Surface area and volume of similar solids
G-GMD.5: Know that the effect of a scale factor k greater than zero on length, area, and volume is to multiply each by k, k², and k³, respectively; determine length, area and volume measures using scale factors.
1. Scale drawings: word problems 2. Dilations: find the scale factor 3. Ratios in similar figures 4. Side lengths and angle measures in similar figures 5. Perimeters of similar figures 6. Areas of similar figures 7. Area and perimeter of similar figures 8. Surface area and volume of similar solids
G-GMD.6: Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems.		1. Angle-side relationships in triangles 2. Triangle Inequality Theorem
G-GMD.6: Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems.
1. Angle-side relationships in triangles 2. Triangle Inequality Theorem
Checkpoint opportunity		1. Checkpoint: Cross sections and solids of revolution
Checkpoint opportunity
1. Checkpoint: Cross sections and solids of revolution
G-MG Modeling with Geometry
Apply geometric concepts in modeling situations
Apply geometric concepts in modeling situations

G-MG.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).		1. Pythagorean theorem
G-MG.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
1. Pythagorean theorem
G-MG.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).		1. Calculate density, mass, and volume
G-MG.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
1. Calculate density, mass, and volume
G-MG.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
G-MG.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Checkpoint opportunity		1. Checkpoint: Geometric modeling and design 2. Checkpoint: Density
Checkpoint opportunity
1. Checkpoint: Geometric modeling and design 2. Checkpoint: Density

S	Statistics and Probability
S-CP Conditional Probability and the Rules of Probability
Understand independence and conditional probability and use them to interpret data
Understand independence and conditional probability and use them to interpret data

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").		1. Theoretical and experimental probability 2. Outcomes of compound events
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").
1. Theoretical and experimental probability 2. Outcomes of compound events
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.		1. Identify independent and dependent events 2. Probability of independent and dependent events 3. Identify independent events
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.
1. Identify independent and dependent events 2. Probability of independent and dependent events 3. Identify independent events
S-CP.3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.		1. Find conditional probabilities 2. Independence and conditional probability
S-CP.3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
1. Find conditional probabilities 2. Independence and conditional probability
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.		1. Find probabilities using two-way frequency tables 2. Find conditional probabilities using two-way frequency tables
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.
1. Find probabilities using two-way frequency tables 2. Find conditional probabilities using two-way frequency tables
S-CP.5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
S-CP.5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.


Use the rules of probability to compute probabilities of compound events in a uniform probability model
Use the rules of probability to compute probabilities of compound events in a uniform probability model

S-CP.6: Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.		1. Find conditional probabilities using two-way frequency tables 2. Geometric probability
S-CP.6: Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
1. Find conditional probabilities using two-way frequency tables 2. Geometric probability
S-CP.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.		1. Find probabilities using the addition rule
S-CP.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.
1. Find probabilities using the addition rule
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.		1. Find conditional probabilities
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.
1. Find conditional probabilities
S-CP.9: Use permutations and combinations to compute probabilities of compound events and solve problems.		1. Counting principle 2. Permutations 3. Permutation and combination notation 4. Find probabilities using combinations and permutations
S-CP.9: Use permutations and combinations to compute probabilities of compound events and solve problems.
1. Counting principle 2. Permutations 3. Permutation and combination notation 4. Find probabilities using combinations and permutations
Checkpoint opportunity		1. Checkpoint: Understand independence and conditional probability 2. Checkpoint: Probabilities of compound events
Checkpoint opportunity
1. Checkpoint: Understand independence and conditional probability 2. Checkpoint: Probabilities of compound events
S-MD Using Probability to Make Decisions
Use probability to evaluate outcomes of decisions
Use probability to evaluate outcomes of decisions

S-MD.6: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
S-MD.6: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

S-MD.7: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
S-MD.7: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).