geometry c pacing guide - lorain city school...

Lorain City School District Scope, Sequence and Pacing Guides

Ohio’s New Learning Standards: Mathematics-‐-‐Geometry

Common Formative Assessments will be implemented daily.

Mathematics: Geometry – 1st Quarter

Days: 5 Essential Question: In what ways can the problem be solved, and why should one method be chosen over another?

Domain Cluster Ohio’s New Learning Standards

Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning

A. 9-‐12. REI. 1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Clear Learning Targets Content Standard Vocabulary • I can apply order of operations and inverse operations to solve equations. \

• I can construct an argument (proof) to justify my solution process. Inverse operation Division property Addition property Distributive property Subtraction property Substitution property Multiplication property

Resources [adopted and supplemental] Assessment & Notes Holt Geometry (2007) Chapter 2: Section 5 Holt Geometry Resource Materials

Holt Geometry Resource Materials This standard provides an algebra review as well as introduction to proof.

Days: 25 Essential Question: In what ways can congruence be helpful?


Congruence Experiment with transformations in the plane

G. 9-‐12. 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.

Prove geometric theorems

G. 9-‐12. CO. 9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.




Clear Learning Targets Content Standard Vocabulary • I can identify the undefined notions used in geometry (point, line, plane, distance) and describe their characteristics.

• I can identify angles, circles, perpendicular lines, parallel lines, rays, and line segments. • I can define angles, circles, perpendicular lines, parallel lines, rays, and line segments precisely using the undefined terms and "if-‐then" and "if-‐and-‐only-‐if" statements.

• I can distinguish between inductive and deductive reasoning.

• I can identify and compose conditional statements, converses, inverses, and contrapositives.

• I can identify and compose a biconditional statement.

• I can identify and use the properties of congruence and equality (reflexive, symmetric, transitive) in my proofs.

• I can order statements based on the Law of Syllogism when constructing my proof.

• I can correctly interpret geometric diagrams by identifying what can and cannot be assumed.

• I can use theorems. postulates, or definitions to prove theorems about lines and angles including: Vertical angles are congruent; When a transversal crosses parallel lines alternate interior and alternate exterior angles are congruent, Corresponding angles are congruent, and Same-‐side interior (or consecutive interior) angles are supplementary; Points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Undefined terms Converse Point Inverse Line Contrapositive Plane Theorem Distance Postulate Ray Definition Angle Linear pair Vertex Vertical angles Line Segment Complementary angles Endpoint Supplementary angles Circle Adjacent Arc Consecutive Collinear Alternate interior angles Coplanar Alternate exterior angles Equidistant Corresponding angles Intersect Same-‐side interior angles Perpendicular (Consecutive interior angles) Parallel Corresponding angles Right Angle Perpendicular bisector Inductive Reasoning Midpoint Deductive Reasoning Law of Syllogism Conditional Statement Segment Addition Postulate Biconditional Statement Angle Addition postulate Counterexample

Resources [adopted and supplemental] Assessment & Notes Holt Geometry (2007) Chapter 1: Sections 1-‐4 Chapter 11: Page 746 Chapter 2: Sections 1-‐4, 6-‐7 Chapter 3: Sections 1-‐4 Holt Geometry Resource Materials OGT Packet “Angles & More” ACT Packet “Reasoning” ACT Packet “Angles & More” ACT Packet “Proof—Lines & Angles”

Holt Geometry Resource Materials The topics involving the various types of angles have been OGT items. Honors classes could include the Chapter 2 Extension: Introduction to Symbolic Logic (truth tables).




Days: 8 Essential Question: How can algebraic equations be used to model, analyze, and solve mathematical situations?


Geometry Understand and apply the Pythagorean Theorem

8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Interpreting Functions Analyze functions using different representations

A.9-‐12.IF.7 Graph functions expressed symbolically and show key features of the graph. a) Graph linear functions and show intercepts.

Clear Learning Targets Content Standard Vocabulary • I can connect any two points on a coordinate grid to a third point so that the three points form a right triangle, and use that triangle along with the Pythagorean Theorem to find the distance between the original two points.

• I can identify the point slope form of a linear function as y – y1 = m(x –x1) and use it to graph lines stating a point on the line and its slope.

• I can identify the slope-‐intercept form of a linear function as y = mx + b and use it to graph lines stating the slope and y-‐intercept of those lines.

Midpoint Distance Slope Parallel lines Pythagorean Theorem Perpendicular lines Positive slope Coincident lines Negative slope Slope-‐Intercept Form Zero slope Point-‐Slope Form Undefined slope

Resources [adopted and supplemental] Assessment & Notes Holt Geometry (2007)

Chapter 1: Section 6 Chapter 3: Sections 5-‐6

Holt Geometry Resource Materials


This serves as an algebra review (also eighth grade review) for assorted problems in various units using coordinate geometry to solve problems or create proofs.

Days: 5 Essential Question: How does performing constructions relate to the characteristics of various geometric objects?


Congruence Make geometric constructions

G. 9-‐12. CO. 12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G. 9-‐12. CO. 13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.




Clear Learning Targets Content Standard Vocabulary • I can identify and use the tools (string, reflective devices such as Miras®, tracing paper, and/or geometric software) needed in formal constructions to: Copy a segment; Copy an angle; Bisect a segment; Bisect an angle; Construct perpendicular lines and bisectors; and Construct a line parallel to a given line through a point not on the line.

• I can informally perform the constructions listed above using string, reflective devices, paper folding, and/or dynamic geometric software.

• I can define inscribed polygons (the vertices of the figure must be points on the circle).

• I can explain and perform the steps to construct an equilateral triangle, square, and a regular hexagon inscribed in a circle.

Segment Equilateral triangle Angle Hexagon Perpendicular lines Formal construction Perpendicular bisector Informal construction Parallel lines Compass Bisect Straightedge Circle Mira® Inscribed polygon Patty paper


Pages 14, 16, 22, 23, 27, 79, 163, 170-‐171, 172, 177, 179, 220, & 380

Holt Geometry Resource Materials ACT Packet “Constructions”


End 1st Quarter (9 Weeks)




Mathematics: Geometry – 2nd Quarter

Days: 13 Essential Question: How are transformations related to congruence?


Congruence Experiment with transformations in the plane

G. 9-‐12. 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). G. 9-‐12. CO. 4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G. 9-‐12. 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. G. 9-‐12. CO. 3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G. 9-‐12. 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.

Clear Learning Targets Content Standard Vocabulary • I can draw transformations of reflections, rotations, translations, and combinations of these using graph paper, transparencies, and/or geometry software.

• I can determine the coordinates for the image (output) of a figure when a transformation rule is applied to the preimage (input).

• I can distinguish between transformations that are rigid (preserve distance and angle measure -‐ reflections, rotations, translations, or combinations of these) and those that are not (dilations or rigid motions followed by dilations).

• I can construct the reflection definition by showing that the line of reflection is the perpendicular bisector of any segment connecting a preimage point to its corresponding image point.

• I can construct the translation definition by showing that the line segments connecting preimage points with the corresponding image points are all equal in length, and are parallel (or pointing in the same direction).

transformation reflection rotation dilation image preimage rigid motion congruent input output coordinates distance




• I can construct the rotation definition by showing that all segments drawn from the center of rotation to a preimage point and to its corresponding image point are equal in length.

• I can describe that the measure of the angles formed from the center of rotation to any pair of preimage and corresponding image points is the same.

• I can draw a specific transformation when given a geometric figure and a reflection, rotation, or translation.

• I can predict and verify the sequence of transformations (a composition) that will map a figure onto another. • I can describe and illustrate how a rectangle, parallelogram, or isosceles trapezoid is mapped onto itself using transformations.

• I can calculate the number of lines of reflectional symmetry and the degree of rotational symmetry of any regular polygon.

• I can define rigid motions as transformations which preserve distance and angle measure: reflections, rotations, translations, and combinations of these.

• I can define congruent figures as those that have the same shape and size and state that a composition of rigid motions will map one congruent figure onto the other.

•I can predict the composition of transformations that will map a figure onto a congruent figure.

• I can determine if two figures are congruent by determining if rigid motions will turn one figure into the other.

angle measure translation perpendicular bisector line segment parallel lines center of rotation figure map composition rectangle parallelogram trapezoid isosceles trapezoid regular polygon rotational symmetry reflectional symmetry point symmetry

Resources [adopted and supplemental] Assessment & Notes Holt Geometry (2007) Chapter 1: Section 7 Chapter 12: Sections 1-‐5, 7 Chapter 6: Sections 1, 2, 4, 6 Holt Geometry Resource Materials

Quadrilateral Book Activity ACT Packet “Transformations” OGT Packet “Transformations & Tessellations”

Holt Geometry Resource Materials Transformations have been a topic included on the OGT. Students in eighth grade worked with highly similar standards in regard to transformations and their relation to congruence. Students may need a review of the properties of the special quadrilaterals that they learned in seventh grade.




Days: 30 Essential Question: How does applying and proving congruence provide a basis for modeling situations geometrically?


Congruence Understand congruence in terms of rigid motions.

G. 9-‐12. 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. G. 9-‐12. CO. 8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Prove geometric theorems

G. 9-‐12. CO. 10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G. 9-‐12. CO. 11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Clear Learning Targets Content Standard Vocabulary

• I can identify corresponding sides and corresponding angles of congruent triangles.

• I can explain that if in a pair of congruent triangles, corresponding sides are congruent (distance is preserved) and corresponding angles are congruent (angle measure is preserved).

• I can demonstrate that when distance is preserved (corresponding sides are congruent) and angle measure is preserved (corresponding angles are congruent), then the triangles must also be congruent.I can define rigid motions as reflections, rotations, translations, and combinations of these, all of which preserve distance and angle measure.

• I can list the sufficient conditions to prove triangles are congruent.

• I can define rigid motions as reflections, rotations, translations, and combinations of these, all of which preserve distance and angle measure.

• I can map a triangle with one of the sufficient conditions (e.g., SSS) on to the original triangle and show that corresponding sides and corresponding angles are congruent. • I can order statements based on the Law of Syllogism when constructing my proof. • I can correctly interpret geometric diagrams (what can and cannot be assumed). • I can use theorems, postulates, or definitions to prove theorems about triangles, including: Measures of interior angles of a triangle sum to 180 degrees;

Rigid motions SSS Reflection SAS Translation ASA Rotation AAS Distance HL Congruence Quadrilateral Map Parallelogram Composition Rectangle Triangle Rhombus Equilateral triangle Square Isosceles triangle Diagonal Corresponding sides Corresponding angles Angle measure Adjacent Consecutive




Base angles of isosceles triangles are congruent; The segment joining the 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. • I can use theorems, postulates, or definitions to prove theorems about parallelogram, including: Prove opposite sides of a parallelogram are congruent; Prove opposite angles of a parallelogram are congruent; Prove the diagonals of a parallelogram bisect each other; and Prove that rectangles are parallelograms with congruent diagonals.

Law of Syllogism Coordinate proof Midpoint Midpoint formula Midsegment Median Centroid

Resources [adopted and supplemental] Assessment & Notes Holt Geometry (2007) Chapter 4: Sections 3-‐8 Chapter 12: Sections 1-‐3 Chapter 5: Sections 3-‐4 Chapter 6: Sections 2-‐5 Holt Geometry Resource Materials ACT Packet “Proofs with Triangles & Quadrilaterals” SmartBoard Activity: Kooshball for SSS, etc.

Holt Geometry Resource Materials The characteristics (not necessarily proofs) of triangles and parallelograms are topics that have been included on the OGT.

End 2nd Quarter (9 Weeks)




Mathematics: Geometry – 3rd Quarter

Days: 10 Essential Question: How might the features of one figure be useful when solving problems about a similar figure?


Similarity, Right Triangles, and Trigonometry

Understand similarity in terms of similarity transformations.

G. 9-‐12. SRT.1 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. G. 9-‐12. 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. G. 9-‐12. SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity

G. 9-‐12. 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. G. 9-‐12. SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Clear Learning Targets Content Standard Vocabulary • I can define dilation.

• I can perform a dilation with a specific center and scale factor on a figure in the coordinate plane.

• I can verify that when a side passes through the center of dilation, the side and its image lie on the same line.

• I can verify that the corresponding sides of the preimage and image are parallel.

• I can verify that a side length of the image is equal to the scale factor multiplied by the side length of the preimage.

• I can define similarity as a composition of congruence transformations followed by dilations in which angle measure is preserved and side length is proportional.

• I can identify corresponding sides and angles of similar triangles.

• I can determine that two figures are similar by verifying that angle measure is congruent and corresponding sides are proportional.

Dilation Segment Center Ratio Scale factor Similarity Preimage Composition Image Intersect Slope Distance Parallel Corresponding sides Corresponding angles Rigid motion Angle measure




• I can show and explain that when two angle measures are known (AA), the third angle measure is also known (Third Angle Theorem).

• I can conclude and explain that AA similarity is a sufficient condition for two triangles to be similar.

• I can use theorems, postulates, or definitions to prove theorems about triangles including: A line parallel to one side of a triangle divides the other two proportionally; If a line divides two sides of a triangle proportionally, then it is parallel to the third side; The Pythagorean Theorem proved using triangle similarity.

• I can use triangle congruence and triangle similarity to solve problems (e.g., indirect measure, missing sides, angle measures, side splitting).

• I can use triangle congruence and triangle similarity to prove relationships in geometric figures.

Side length Proportional Similarity transformation Proof Segment addition Pythagorean Theorem Congruence Triangle congruence Triangle similarity Geometric mean Mean proportional


Chapter 12: Section 7 Chapter 7: Sections 2-‐6 Chapter 8: Section 1 Holt Geometry Resource Materials ACT Packet: “Similarity” OGT Packet: “Trigonometry & Similarity”

Holt Geometry Resource Materials Section 7.1 can be used as a review of ratio and proportion if needed. Students worked extensively with ratio, proportion and similarity in both grades seven and eight. Ratio, proportions, and similarity are topics that have been included on the OGT. (Time has been allotted in Quarter 3 for the OGT.)

Days: 12 Essential Question: How are the trigonometric ratios used to solve real-‐world problems?


Similarity, Right Triangles, and Trigonometry

Define trigonometric ratios and solve problems involving right triangles

G. 9-‐12. 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. G. 9-‐12. SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G. 9-‐12. SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.




Clear Learning Targets Content Standard Vocabulary • I can demonstrate that within a right triangle, line segments parallel to a leg create similar triangles by AA similarity.

• I can use the characteristics of similar triangles to justify that the trigonometric ratios are dependent on the angles, not on the size of the triangle.

• I can define the sine (sin), cosine (cos), and tangent (tan) ratios in a right triangle.

• I can define complementary angles.

• I can calculate sine and cosine ratios for acute angles in a right triangle when given two side lengths.

• I can explain (using diagrams) that for complementary angles A and B, the sine of angle A is equal to the cosine of angle B, and the cosine of angle A is equal to the sine of angle B.

• I can use side lengths to estimate angles measures (e.g., the angle opposite of a 10 cm side will be larger than the angle across from a 9 cm side).

• I can use angle measures to estimate side lengths (e.g., the side opposite from a 33o angle will be shorter than the side across from a 57o angle).

• I can use the Pythagorean Theorem to find an unknown length of a right triangle.

• I can identify common Pythagorean triples.

• I can use special triangles to find exact trigonometric values for angles of 30, 45 or 60 degrees.

• I can use sine, cosine, tangent, and their inverses to solve for the unknown side lengths and angle measures of a

right triangle.

• I can solve right triangles by finding measures of all sides and angles in the triangle.

• I can draw right triangles that describe real world problems and label the sides and angles with their given measures.

Right triangle Acute angle Complementary angles Leg Hypotenuse Ratio Proportion Constant AA similarity Corresponding sides Pythagorean Theorem Pythagorean triple Special triangle Trigonometry Trigonometric ratio Sine Ratio Cosine Ratio Tangent Ratio Inverse trigonometric ratio Angle of elevation Angle of depression

Resources [adopted and supplemental] Assessment & Notes Holt Geometry (2007) Chapter 5: Sections 7-‐8 Chapter 8: Sections 1-‐4 Holt Geometry Resource Materials

ACT Packet: “Right Triangles & Trigonometry” OGT Packet: “Trigonometry & Similarity”

Holt Geometry Resource Materials Honors students could include using the Laws of Sines and Cosines (G. 9-‐12. SRT.10-‐11) and using ½ absinC to find area of triangles (G. 9-‐12. SRT.9). The OGT has included only the definition of the trigonometric functions (usually tangent). (Time has been allotted in Quarter 3 for the OGT.)




Days: 8 Essential Question: How can two-‐dimensional figures be used to understand three-‐dimensional figures?


Geometric Measure and Dimension

Explain volume formulas and use them to solve problems

G. 9-‐12. 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 G. 9-‐12. GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

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

G. 9-‐12. 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.

Modeling with Geometry

Apply geometric concepts in modeling situations

G. 9-‐12. 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). ★ G. 9-‐12. MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). ★ G. 9-‐12. 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).�

Clear Learning Targets Content Standard Vocabulary • I can define п (pi) as the ratio of a circle’s circumference to its diameter, and I can use algebra to demonstrate that the formula for a circle's circumference must be: C = пd.

• I can inscribe a regular polygon in a circle and break it into many congruent triangles in order to find its area and explain how to use this dissection method to generate and use the formula: A = ½ apothem • perimeter.

• I can identify the base(s) for prisms, cylinders, pyramids, and cones and calculate their areas.

• I can calculate the volume of a prism using the formula V= Bh and the volume of a cylinder with V = пr2h.

• I can defend statements like, ”The formula for the volume of a cylinder is basically the same as that of a prism” and “The volume of a cone is basically the same as that of a pyramid.”

• I can explain that the volume of a pyramid is 1/3 the volume of the prism of the same base & height and that the volume of a cone is 1/3 the volume of a cylinder with the same base & height. • I can calculate the volume of a cylinder, pyramid, cone, and sphere; and use the appropriate volume formulas to solve problems.

Pi Circle Circumference Diameter Radius Dissection Equivalent Ratio Area Regular Polygon Perimeter Side




• I can identify the shapes of two-‐dimensional cross-‐sections of three-‐dimensional objects (e.g., the cross-‐section of a sphere is a circle).

• I can rotate a two-‐dimensional figure and identify the three-‐dimensional object that is formed (e.g., rotating a circle produces a sphere and rotating a rectangle produces a cylinder).

• I can represent real-‐world objects as geometric figures.

• I can estimate measures of real-‐world objects by using comparable geometric shapes.

• I can estimate measures (circumference, area, perimeter, volume) of real-‐world objects using comparable geometric shapes or three-‐dimensional figures.

• I can apply properties of geometric figures to real-‐world objects (e.g., the spokes of a wheel of a bicycle are equal lengths since they represent the radii of a circle).

• I can decide whether it is best to calculate or estimate the area or volume of a geometric figure and perform the calculation or estimation.

• I can break composite geometric figures into manageable pieces.

• I can use scale factors when solving area and volume problems of similar figures.

• I can convert units of measure (e.g., convert square feet to square miles).

• I can apply area and volume to situations involving density (e.g., determine the population in an area, the weight of water given its density, or the amount of energy in a three-‐dimensional figure).

• I can create a visual representation of a design problem.

• I can solve design problems using a geometric model (graph, table, equation, or formula).

• I can interpret the results and make conclusions based on the geometric model.

Apothem Base Prism Cylinder Pyramid Cone Sphere Volume Substitute Altitude Height Slant Height Cross-‐section Rotate Cavalieri’s Principle Cross-‐sectional area Composite figures Scale factor Unit of measure Convert Density Geometric model


Chapter 9: Sections 1-‐5 Chapter 10: Sections 1, 3 & 6-‐8 Holt Geometry Resource Materials

ACT Packet: “Area & Volume” OGT Packet: “Polygons & Polyhedra”


Area and volume are topics that have been covered on the OGT. (Time has been allotted in Quarter 3 for the OGT.)

� denotes modeling

End 3rd Quarter (9 Weeks)




Mathematics: Geometry – 4th Quarter

Days: 18 Essential Question: How can the properties of circles, polygons, lines, and angles be useful when solving geometric problems?


Circles Understand and apply theorems about circles.

G. 9-‐12. C.1 Prove that all circles are similar. G. 9-‐12. 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. G. 9-‐12. C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Find arc lengths and areas of sectors of circles.

G. 9-‐12. 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.

Clear Learning Targets Content Standard Vocabulary • I can prove that all circles are similar by showing that for a dilation centered at the center of a circle, the pre-‐image and the image have equal central angle measures.

• I can identify central angles, inscribed angles, circumscribed angles, diameters, radii, chords, and tangents.

• I can describe the relationships between intercepted arcs and central, inscribed, and circumscribed angles.

• I can recognize that an inscribed angle whose sides intersect the endpoints of the diameter of a circle is a right angle.

• I can prove that opposite angles in an inscribed quadrilateral are supplementary.

• I can recognize that the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

• I can define the terms inscribed, circumscribed, angle bisector, and perpendicular bisector.

•I can construct the inscribed circle whose center is the point of intersection of the angle bisectors (the incenter).

• I can construct the circumscribed circle whose center is the point of intersection of the perpendicular bisectors of each side of the triangle (the circumcenter).

• I can apply the Arc Addition Postulate to solve for missing arc measures.

Circle Diameter Similar figures Radius Rigid motion Chord Dilation Arc Preimage Image Central angle Inscribed angle Circumscribed angle Intercepted arc Tangent Right angle Perpendicular Angle bisector Perpendicular bisector Arc Addition Postulate




• I can show that all intercepted arcs of inscribed regular polygons are congruent.

• I can show that any regular polygon circumscribed about a circle is tangent to that circle at the midpoint of each side.

• I can define similarity as rigid motions with dilations, which preserve angle measures and makes lengths proportional.

• I can use similarity to calculate the length of an arc.

• I can define radian measure of an angle as the ratio of an arc length to its radius and calculate a radian measure when given an arc length and its radius.

• I can use proportionality to convert from degrees to radians and vice-‐versa.

• I can calculate the area of a circle.

• I can define a sector and calculate the area of a sector using proportionality or the ratio of the intercepted arc measure and 360o multiplied by the area of the circle.

• I can calculate geometric probability (e.g., the chance of a dart hitting a particular region of a circle or regular polygon) that is based on area of sectors, or on lengths of arcs.

Incenter Circumcenter Quadrilateral Opposite angles Supplementary Proportional Constant of proportionality Length of arc Radian Area Sector Probability Geometric probability

Resources [adopted and supplemental] Assessment & Notes Holt Geometry (2007) Chapter 12: Section 7 Chapter 11: Sections 1-‐6 Chapter 9: Section 6


ACT Packet: “Angles & Segments of Circles” ACT Packet: “Circles: Area, Sectors & Probability” Khan Academy—Trigonometry: Radians

Holt Geometry Resource Materials Honors students could include G. 9-‐12. C.4 which involves constructing a tangent line from a point outside the circle.

(Time has been allotted for exam review in Quarter 4.)




Days: 15 Essential Question: How can algebra be useful when expressing geometric properties?


Expressing Geometric Properties with Equations

Use coordinates to prove simple geometric theorems algebraically

G. 9-‐12. GPE.4 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). G. 9-‐12. GPE.5 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). G. 9-‐12. GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G. 9-‐12. GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Translate between the geometric description and the equation for a conic section

G. 9-‐12. 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.

Clear Learning Targets Content Standard Vocabulary • I can represent the vertices of a figure in the coordinate plane using variables.

• I can connect a property of a figure to the tool needed to verify that property.

• I can use coordinates and the right tool to prove or disprove a claim about a figure: Use slope to determine if sides are parallel, intersecting, or perpendicular; Use the distance formula to determine if sides are congruent or to decide if a point is inside a circle, outside a circle, or on the circle; Use the midpoint formula or the distance formula to decide if a side has been bisected. • I can draw a line on a coordinate plane and translate that line to produce its image, and I can explain that these lines are parallel since translations preserve angle measure.

• I can determine the slope of the original line and its image after translation and show that they have the same slope using both specific examples and general coordinates (x, y).

• I can state that parallel lines have the same slope.

• I can determine if lines are parallel using their slopes.

Side length Vertex Quadrants Slope Distance Midpoint Bisect Intersecting Slope Parallel Perpendicular Product Line Linear equation




• I can write an equation for a line that is parallel to a given line that passes through a given point.

• I can draw a line on a coordinate plane and rotate that line 90o to produce a perpendicular image.

• I can determine the slopes of the original line and its image after a 90o rotation and show that they have opposite reciprocal slopes using both specific examples and general coordinates (x, y).

• I can state that perpendicular lines have the opposite reciprocal slopes.

• I can determine if lines are perpendicular using their slopes.

• I can use the coordinates of the vertices of a polygon graphed in the coordinate plane and the distance formula to compute the perimeter.

• I can use the coordinates of the vertices of triangles and rectangles graphed in the coordinate plane to compute area.

• I can write an equation for a line that is perpendicular to a given line that passes through a given point.

• I can calculate the point(s) on a directed line segment whose endpoints are (x1, y1) and (x2, y2) that partitions the line segment into a given ratio, r1 to r2 using the formula: x = r2·x1 + r1·x2 and y = r2·y1 + r1·y2 r1 + r2 r1 + r2

• I can draw a right triangle with a horizontal leg, a vertical leg, and the radius of a circle as its hypotenuse.

• I can identify the center and radius of a circle given its equation.

• I can use the distance formula or Pythagorean Theorem, the coordinates of a circle’s center, and the circle’s radius to write the equation of a circle.

• I can convert an equation of a circle in general (quadratic) form to standard form by completing the square.

Slope-‐intercept form Point-‐slope form Coordinate plane Coordinates Distance formula Pythagorean Theorem Perimeter Polygon Area Triangle Rectangle Directed line segment Endpoint Ratio Leg Hypotenuse Complete the square Perfect square trinomial

Resources [adopted and supplemental] Assessment & Notes Holt Geometry (2007) Chapter 3: Sections 5-‐6 Chapter 9: Section 4 Chapter 11: Section 7 Pages 267-‐8, 393, 400, 410, 420-‐1, 434, 617 Holt Geometry Resource Materials

McDougal Littell Algebra 2 (2004) Chapter 10: Section 3

OGT Packet “Coordinate Geometry” ACT Packet “Coordinate Geometry”

Holt Geometry Resource Materials Honors students could include the topic of finding the distance between parallel lines. (Time has been allotted for exam review in Quarter 4.)

End 4th Quarter (9 Weeks)




SUGGESTED GEOMETRY PROJECT #1

At the teacher’s discretion, allow students to work in teams or as individuals. Students will take pictures (or find pictures in magazines) of various geometric shapes, etc. in our world and mount them on a poster. Highlight the geometric shape, etc in the picture, and place a label below the picture stating the shape, etc. Students should have unique pictures—the same picture should not be shared by individuals or teams.

Include the following shapes, etc:

Point Line Plane Ray

Parallel Lines Perpendicular Lines Skew Lines Acute Angle

Obtuse Angle Right Angle Alternate Interior Angles Vertical Angles

Same-‐Side Interior Angles Equilateral Triangle Isosceles Triangle Scalene Triangle

Parallelogram Rectangle Rhombus Trapezoid

Kite Square Ellipse Parabola

Circle Pentagon Hexagon Octagon

Teachers can add or delete shapes as desired. Extra points could be given for additional shapes, etc.





At the teacher’s discretion, allow students to work in teams or as individuals. Use a computer to make a template for the 1” pattern blocks, the GeoFix shapes, etc. Each shape will have sides of one inch unless otherwise specified. Each shape is listed below. On another paper, show each shape and its dimensions (width & height). You will need to use trigonometry to determine some of these dimensions. That work must be shown as well.

Shapes to be included in the template: Sample Template (not drawn to scale)

1) equilateral triangle 2) square 3) rhombus (angles of 60o & 120o) 4) rhombus (angles of 30o & 150o) 5) isosceles trapezoid (angles of 60o & 120o) 6) regular hexagon 7) regular pentagon 8) regular octagon 9) regular nonagon 10) regular decagon 12) regular dodecagon 13) rectangle (sides of 1” & 2”) 14) large equilateral triangle (sides of 2”) 15) isosceles triangle (base 1”, legs 2”) 16) 45o-‐45o-‐90o triangle (legs of 1”) 17) 30o-‐60o-‐90o triangle (legs of 1” & 3")

Etc.

1

c15

2

c13

6

c7

c

5

c





At the teacher’s discretion, allow students to work in teams or as individuals. Make a poster clearly showing and explaining a proof of the Pythagorean Theorem other than the one shown below. There are many different proofs—as soon as students decide on a proof, they should sign up with the instructor. The same proof may not be used too often (specify) in any class. Be prepared to present your proof to the class.


At the teacher’s discretion, allow students to work in teams or as individuals. For each of the six triangles given, construct the centroid, circumcenter, orthocenter, and incenter. Be sure to use a straightedge! [The midpoint of each side is shown, but not all segments will go through the midpoint.] You can use the grid to find the first three, but a compass is needed to find the incenter. Recall that perpendicular lines have opposite reciprocal slopes. Be sure to notice the location of each point of concurrency. Draw the circumscribed and inscribed circles where appropriate. Show or calculate the coordinates of the centroid, circumcenter, and orthocenter. Write a summary describing the location of each of the four points of concurrency (on the triangle, in the interior, or in the exterior, etc.) for the different cases that you have found. (The needed worksheets are included at the end of this document).

a2

c2 b2





At the teacher’s discretion, allow students to work in teams or as individuals. Have students select (or be assigned) one or more of the special quadrilaterals. Students must draw the quadrilateral on the grid (provided—see pages at the end of this document), calculate lengths and slopes of sides and diagonals, and list all characteristics. The quadrilateral must not contain any horizontal/vertical sides or diagonals. Try to avoid slopes of ±1. Your quad should not be the same as any other student’s quad! This is a Project—neatness & use of color are expected. Points will be earned based on accuracy, completeness, difficulty & presentation. The teacher should develop a specific rubric.

CENTROID CIRCUMCENTER

INCENTER ORTHOCENTER

ISOSCELES RIGHT TRIANGLE #4




SCALENE RIGHT TRIANGLE #4



ORTHOCENTER INCENTER

ISOSCELES ACUTE TRIANGLE #4


CENTROID CIRCUMCENTER NTER

ORTHOCENTER INCENTER

ISOSCELES OBTUSE TRIANGLE #4




SCALENE ACUTE TRIANGLE #4



INCENTER

ORTHOCENTER

SCALENE OBTUSE TRIANGLE #4

CH

AR

AC

TE

RIS

TIC

S

CALCULATIONS (Length & Slope of each

Side & Diagonal; Midpoint of each Diagonal)

geometry c pacing guide - lorain city school...

Documents