MASSAPEQUA PUBLIC SCHOOLS

Geometry

Summer 2015

COMMITTEE MEMBERS Noreen Reinle

BOARD OF EDUCATION Jane Ryan – President

Maryanne Fisher – Vice President Gary Baldinger – Secretary Timothy Taylor – Trustee Joseph LaBella – Trustee

ADMINISTRATION

Lucille F. Iconis, Superintendent Alan C. Adcock, Deputy Superintendent

Thomas Fasano, Ed.D., Assistant to the Superintendent for Curriculum & Instruction Robert Schilling, Executive Director Assessment, Student Data and Technology Services

Diana Haanraadts, Asst. to the Superintendent for Instructional Support & General Administration Dina Maggiacomo, Executive Director for Human Resources & General Administration Jean Castelli, Executive Director of Special Education and Student Support Services

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Table of Contents Course Descriptions/Rationale.......................................................................................3 Key Words for Curriki......................................................................................................3 Common Core State Standards, Learning Standards, Key Ideas and Performance Indicators...........................................................................4 Unit 1: Angles in Geometry............................................................................................8 Unit 2: Constructions and Transformations....................................................................11 Unit 3: Triangle Proofs...................................................................................................14 Unit 4: Introduction to Similarity.....................................................................................20 Unit 5: Similar Triangles.................................................................................................22 Unit 6: Trigonometry.......................................................................................................25 Unit 7: Quadrilateral Proofs............................................................................................27 Unit 8: Circles.................................................................................................................29 Unit 9: Circle Proofs.......................................................................................................32 Unit 10: Volume and Area...............................................................................................34 Unit 11: Coordinate Geometry.........................................................................................39 Unit 12: Coordinate Proofs..............................................................................................42

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Course Description/Rationale The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school CCSS. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

Key Words for Curriki Geometry Constructions Transformations Proofs Congruence Similarity Trigonometry Coordinates

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Common Core State Standards, Learning Standards, Key Ideas & Performance Indicators For Content Area

Module 1: Congruence, Proof, and Constructions

Experiment with transformations in the plane: G­CO.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. G­CO.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). G­CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G­CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G­CO.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. Understand congruence in terms of rigid motions: G­CO.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. G­CO.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. G­CO.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: G­CO.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. G­CO.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

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triangle meet at a point. G­CO.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. Make geometric constructions: G­CO.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.

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

Module 2: Similarity, Proof, and Trigonometry

Understand similarity in terms of similarity transformations: G­SRT.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.

G­SRT.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.

G­SRT.3 Use the properties of similarity transformations to establish the AA criterion for two

triangles to be similar. Prove theorems involving similarity: G­SRT.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.

G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove

relationships in geometric figures. Define trigonometric ratios and solve problems involving right triangles: G­SRT.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. G­SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G­SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied

problems.

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Apply geometric concepts in modeling situations: G­MG.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). G­MG.2 Apply concepts of density based on area and volume in modeling situations (e.g.,

persons per square mile, BTUs per cubic foot). G­MG.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).

Module 3: Extending to Three Dimensions

Explain volume formulas and use them to solve problems: G­GMD.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.

G­GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Visualize relationships between two­dimensional and three­dimensional objects: G­GMD.4 Identify the shapes of two­dimensional cross­sections of three­dimensional objects,

and identify three­dimensional objects generated by rotations of two­dimensional objects.

Apply geometric concepts in modeling situations: G­MG.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).

Module 4: Connecting Algebra and Geometry through Coordinates

Use coordinates to prove simple geometric theorems algebraically: G­GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove

or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

G­GPE.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).

G­GPE.6 Find the point on a directed line segment between two given points that partitions the

segment in a given ratio. G­GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and

rectangles, e.g., using the distance formula.

Module 5: Circles with

Understand and apply theorems about circles: G­C.1 Prove that all circles are similar.

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and without Coordinates

G­C.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.

G­C.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: G­C.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.

Translate between the geometric description and the equation for a conic section: G­GPE.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.

Use coordinates to prove simple geometric theorems algebraically: G­GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Apply geometric concepts in modeling situations: G­MG.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).

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Unit 1: Angles in Geometry (12 Days) Instructions Days: 8 Days Review Days: 2 Days Assessments: 2 Days Technology: Graphing Calculator, Chromebook, Smartboard Day 1 – Review of Algebra

Solving Quadratic Equations Solving Fractional Equations Solving equations with multiply binomials

A­REI.14 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. Day 2 – Angles

Supplementary, complementary Vertical angles Angles at a point (360)

G­CO.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. G­CO.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.

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Day 3 – Parallel Lines

Alternate interior, corresponding Interior angles on same side of transversal Auxiliary lines

G­CO.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. G­CO.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 Day 4 – Angles of a Triangle

Sum is 180 Isosceles Equilateral Include parallel lines

G­CO.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 Day 5 – Exterior Angle Theorem G-CO.10 Theorems include but are not limited to the listed theorems. Example: an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the triangle. Day 6 – Mixed Practice

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Day 7 – Quiz Day 8 – Proofs involving angles of triangles

Angles Vertical angles Exterior angle theorem

G­CO.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. G­CO.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. Day 9 – Proofs involving parallel lines

Auxiliary lines G­CO.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. Day 10 – Proofs involving more complex parallel lines

See Module 1, Lesson 18 G­CO.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. G­CO.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.

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Day 11 – Review Day 12 – Test

Unit 2: Constructions and Transformations (12 Days) Instructions Days: 9 Days Review Days: 1 Day Assessments: 2 Days Technology: Graphing Calculator, Chromebook, Smartboard Day 1 – Constructions #1

Copy segment Equilateral triangle

Circles with same radii Copy angle Bisect angle

G­CO.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. G­CO.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. G­CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle Day 2 – Constructions #2

Perpendicular Bisector (midpoint) Divide segments into fourths circumcenter

Perpendicular Lines Point on/off line

Parallel Lines Using Perpendicular point off line

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G­CO.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. Day 3 – Mixed Construction Practice G­CO.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. G­CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle Day 4 – Construct Square, Hexagon

Orthocenter (altitude) 9 point circle

G­CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle Day 5 – Constructions on drawings G­CO.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. G­CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle Day 6 – Quiz Day 7 – Rotations

Center of Rotation

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Find angle of rotation – how??? Perform a rotation about a point

Example from module 1, page 136 G­CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Day 8 – Reflections

Construct line of reflection Use perpendicular bisector

Reflect image over line of reflection preserves distance

show AB = A'B' (p 136 in Module 1) G­CO.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). G­CO.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. Day 9 – Symmetry

Rotational Symmetry Exterior angle of regular polygons

With reflections and rotations G­CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Day 10 – Translations

Vectors (direction) Given vector, translate figure

See page 136 G­CO.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

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outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Day 11 – Review Day 12 – Test

Unit 3: Triangle Proofs (16 Days) Instructions Days: 12 Days Review Days: 2 Days Assessments: 2 Days Technology: Graphing Calculator, Chromebook, Smartboard Day 1 – Correspondence with Transformations

Sequence of rigid motions Module 1, Lesson 19

Correspondence vs Congruent Triangle Congruence See Module 1, Lessons 19, 20, 21

G­CO.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. G­CO.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. Day 2 – Developing Axioms

SAS, ASA, SSS, AAS Fill in chart in module

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

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Day 3 – Choosing an Axiom SAS, ASA, SSS, AAS

Develop that AAA, SSA do not work G­CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions Day 4 – Introduction to Triangle Proofs

Reflexive Property Vertical Angles Bisector (angle and segment), Midpoint All proofs should include what rigid motion maps the triangles

G­CO.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. G­CO.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. G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity. Day 5 – Triangle Proofs #2

Perpendicular Lines Parallel Lines Supplementary All proofs should include what rigid motion maps the triangles

G­CO.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. G­CO.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.

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G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity.

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Day 6 – HL Proofs All proofs should include what rigid motion maps the triangles

G­CO.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. G­CO.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. G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity. Day 7 – CPCTC

Correspondence with Rigid Motion All proofs should include what rigid motion maps the triangles

G­CO.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. G­CO.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. G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity.

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Day 8 – Triangle Proofs Median Altitude All proofs should include what rigid motion maps the triangles

G­CO.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. G­CO.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. G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity. Day 9 – Mixed Practice Day 10 – Quiz Day 11 – Overlapping Triangles G­CO.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. G­CO.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. G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity.

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Day 12 and 13 – Addition/Subtraction Proofs Addition/Subtraction Property Substitution property Transitive Property Division property

G­CO.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. G­CO.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. G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity. Day 14 – Isosceles Triangles

An isosceles triangle has at least two congruent sides/angles Base angles of an isosceles triangle are congruent If two angles (sides) of a triangle are congruent, then the sides (angles) opposite are congruent

G­CO.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. G­CO.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. G­CO.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. G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity.

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Day 15 – Review Day 16 – Test

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Unit 4: Introduction to Similarity (8 Days) Instructions Days: 6 Days Review Days: 1 Day Assessments: 1 Day Technology: Graphing Calculator, Chromebook, Smartboard Day 1 – Construct Similar Triangles

Use given scale factor Scale factor can be integer or fraction

Construct midsegments Perimeter, area, sides, angle relationship

G­SRT.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. Day 2 – Scale Drawings ­ Dilations

Ratio method Module 2, Lesson 2 (page 33 #4) Include angle measures, write sides in ratios

Dilating figures from different points produce congruent figures Dilate

Regular polygons Circles Find scale factor

Day 3 – Similarity Transformation using Constructions

Describe rigid motion (include compositions) State scale factor (r < 1 or r >1)

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

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Day 4 – Develop Similarity Axioms SAS, SSS, AA Include 3 triangles Proofs with AA Include rigid motion

G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity. Day 5 – CSSTP

Include rigid motion G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity. Day 6 – Product Proofs

Include rigid motion G­SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity. Day 7 – Review Day 8 – Test

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Unit 5: Similar Triangles (13 Days) Instructions Days: 9 Days Review Days: 2 Days Assessments: 2 Days Technology: Graphing Calculator, Chromebook, Smartboard Day 1 – Radicals

Simplifying Add/Sub/Mult/Divide

8.EE.A.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. Day 2 – Pythagorean Theorem

Prove Pythagorean Theorem 8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real­world and mathematical problems in two and three dimensions. Day 3 – Similar Triangles

Similar figures Triangle Side Splitter Theorem (parallel lines) Determine if triangles are similar

G­SRT.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.

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Day 4 – Similar Triangles – Word Problems Triangle Side Splitter Theorem (parallel lines) Families of Parallel Lines

Module 2, Lesson 19 G­SRT.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. Day 5 – Midsegments and Centroid

Midsegment of a triangle Centroid

Module 1, Lesson 30 G­SRT.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. Day 6 – Mixed Practice Day 7 – Quiz Day 8 – Angle Bisector Theorem G­SRT.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. Day 9 – Right Triangle Proportions #1

Altitude G­SRT.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. Theorems include but are not limited to the listed theorems.

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Example: the length of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the lengths of the two segments of the hypotenuse Day 10 – Right Triangle Proportions #2

Leg G­SRT.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. Theorems include but are not limited to the listed theorems. Example: the length of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the lengths of the two segments of the hypotenuse Day 11 – Right Triangle Proportions #3

Altitude vs Leg G­SRT.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. Theorems include but are not limited to the listed theorems. Example: the length of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the lengths of the two segments of the hypotenuse Day 12 – Review Day 13 – Test

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Unit 6: Trigonometry (7 Days) Instructions Days: 5 Days Review Days: 1 Day Assessments: 1 Days Technology: Graphing Calculator, Chromebook, Smartboard Day 1 – Trig Ratios

Use radicals

G­SRT.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

Day 2 – Trig Ratios

Sine and Cosine are complementary See example from Fall 2014 Sampler

G­SRT.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

G­SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. Day 3 – Solve for Missing Side

Numerator and Denominator G­SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Day 4 –Solve for Missing Angle/Word Problems

Angle of elevation/depression G­SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

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Day 5 – Angle of elevation/depression Off the ground

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

Day 6 – Review

Day 7 – Test

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Unit 7: Quadrilateral Proofs (9 Days) Instructions Days: 7 Days Review Days: 1 Day Assessments: 1 Day Technology: Graphing Calculator, Chromebook, Smartboard Day 1 – Properties of Quadrilaterals #1

Algebra G­CO.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. Day 2 – Properties of Quadrilaterals #2

Algebra G­CO.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. Day 3 – Given Parallelogram and Rectangle Proofs G­CO.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. Day 4 – Given Square and Rhombus Proofs G­CO.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.

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Day 5 – Prove a Quadrilateral is a Parallelogram/Rectangle G­CO.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. Day 6 ­ Prove a Quadrilateral is a Rhombus/Square G­CO.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. Day 7 – Trapezoid Proofs G­CO.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. Day 8 – Review Day 9 – Test

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Unit 8: Circles (17 Days) Instructions Days: 13 Days Review Days: 2 Days Assessments: 2 Days Technology: Graphing Calculator, Chromebook, Smartboard

Day 1 – Circles Center and radius Graphing Write equation of circle given

Center and radius Center 2 endpoints

G­GPE.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. Day 2 and 3 – Equation of Circles by completing the square

Include fractions G­GPE.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. Day 4 – Inscribed and Central Angles

Diameter and Radius Major and Minor Arcs Thale

Angle inscribed in semicircle

G­C.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.

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Day 5 – Inscribed and Central Angles #2

Use Pythagorean Theorem Isosceles Triangles

See p 15 #2 More difficult problems

G­C.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. Day 6 – Tangents and Secants

Prove through constructions Given angles and arcs

G­C.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. Day 7 – Chords

Angles formed by 2 chords Parallel and Perpendicular Chords

G­C.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. Day 8 – Circles and Tangents

Equations of tangents (extension standard)? Number of tangent lines

G­C.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. Day 9 – Mixed Angle practice

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Day 10 – Quiz Day 11 – Segments formed by Secants and Tangents G­C.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. Day 12 – Segments formed by Chords

Lesson 2 G­C.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. Day 13 – Construct a rectangle in a circle G­C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Day 14 – Arc Length and Area of Sector

Radians G­C.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 Day 15 – Shaded Region G­C.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 Day 16 – Review Day 17 – Test

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Unit 9: Circle Proofs (7 Days) Instructions Days: 5 Days Review Days: 1 Day Assessments: 1 Day Technology: Graphing Calculator, Chromebook, Smartboard

Day 1 – Circle Proofs Proofs Like Unit 1

G­C.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. G­C.3 Construct the inscribed and circumscribed circles of a triangle, and prove 37 properties of angles for a quadrilateral inscribed in a circle. Day 2 – Circle Proofs #2

Similar Triangles G­C.1 Prove that all circles are similar. Day 3 – Circle Proofs #3

Congruent Triangles G­C.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. G­C.3 Construct the inscribed and circumscribed circles of a triangle, and prove 37 properties of angles for a quadrilateral inscribed in a circle.

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Day 4 – Cyclic Quadratics Day 5 – Ptolemy Day 6 – Review Day 7 – Test

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Unit 10: Volume and Area (19 Days) Instructions Days: 15 Days Review Days: 2 Days Assessments: 2 Days Technology: Graphing Calculator, Chromebook, Smartboard

Day 1 – Basic Area/Conversions Review of basic area formulas Conversions

Inches, feet, cm, m, etc

7.G.B.6 Solve real­world and mathematical problems involving area, volume and surface area of two­ and three­dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. 5.MD.A.1 Convert among different­sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi­step, real world problems. Day 2 – Estimates of Polygonal Regions

Circles, ovals, parabolas Day 3 – Properties of Area

Regions with area Include overlapping Use union and intersection

7.G.B.6 Solve real­world and mathematical problems involving area, volume and surface area of two­ and three­dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

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Day 4 – Scaling with area

Scale area Make Chart

Day 5 – Develop Circle Formulas

Area/Circumference G­GMD.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. Day 6 – Prisms, Cylinders, Pyramids, and Cones

Cross sections Lateral edge, faces Oblique, right See page 137 (Lesson 9)

G­GMD.4 Identify the shapes of two­dimensional cross­sections of three­dimensional objects, and identify three­dimensional objects generated by rotations of two­dimensional objects. Day 7 – Area with Cone and Pyramid

Area of cross section with scaling

G­GMD.4 Identify the shapes of two­dimensional cross­sections of three­dimensional objects, and identify three­dimensional objects generated by rotations of two­dimensional objects.

Day 8 – Prism Volume, Surface area, lateral area

G­MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). G­GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

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G­MG.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). Day 9 – Cylinder

Volume

G­MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). G­GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G­MG.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). Day 10 ­ Word Problems

Lateral Area Volume

Day 11 – Mixed Practice Day 12 – Quiz Day 13 – Volume of Pyramid

Lesson 11 G­MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

G­GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G­MG.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). Day 14 – Volume of Cone

Lesson 11

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G­MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

G­GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G­MG.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). Day 15 – Sphere

Volume and surface area Hemispheres

G­MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). G­GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G­MG.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). Day 16 – Overlapping Volume

Overlapping with Prisms Use intersection and union symbols Density formula Water tower

G­MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). G­GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G­MG.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). Day 17 – Overlapping Volume#2 and Cavalieri’s Principle

Lesson 13 G­GMD.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

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limit arguments G­GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G­MG.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). Day 18 – Review Day 19 – Test

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Unit 11: Coordinate Geometry (16 days) Instructions Days: 13 Days Review Days: 1 Day Assessments: 2 Days Technology: Graphing Calculator, Chromebook, Smartboard Day 1 – Distance Formula

Derive Distance from point to line

Day 2 – Slope Formula

Given slope, missing coordinate Day 3 – Midpoint Formula

Derive midpoint formula Finding midpoint Given midpoint

G­GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio Day 4 – Midpoint and Partitions

¼ of way, 1:3, etc directed segment

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

Day 5 – Equation of a Line

Write equation of line, given Slope and point 2 points

8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance­time graph to a distance­time equation to determine which of two moving objects has greater speed.

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8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non­vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Day 6 – Perpendicular Lines (Normal Segment)

Module 4, Lessons 5 and 6 Perpendicular, use Pythagorean Theorem (and converse) Perpendicular with one coordinate at (0, 0)

x1y1 + x2y2 = 0 Translate

Day 7 – Parallel and Perpendicular Lines

Using Slope Parallel to axis

G­GPE.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). Day 8 – Dilating Lines and Mixed Practice G­SRT.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. Day 9 – Quiz Day 10 – Inequalities and Boundaries

Equation of lines that bound region Triangles, rectangles, trapezoids

REI.12 Graph the solutions to a linear inequality in two variables as a half­plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half­planes

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Day 11 – Perimeter and Area in Plane #1

Perimeter and Area Include translations Decomposition method

Triangles and rectangles

G­GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Day 12 – Perimeter and Area in Plane #2

Shoe Lace Method G­GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Day 13 and 14 – Area of Region bounded by Inequalities

Boundaries Length of boundaries Include rotations (lesson 4)

G­GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Day 15 – Review Day 16 – Test

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Unit 12: Coordinate Proofs (9 days) Instructions Days: 7 Days Review Days: 1 Day Assessments: 1 Day Technology: Graphing Calculator, Chromebook, Smartboard

Day 1 – Triangle Coordinate Proofs

Isosceles, Right, Isosceles Right, Equilateral

G­GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Day 2 – Quadrilateral Coordinate Proofs

Parallelogram, Rectangle, Rhombus, and Square

G­GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Day 3 – Quadrilateral Coordinate Proofs “Not”

G­GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Day 4 – Variable Coordinate Proofs G­GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

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Day 5 – Trapezoid Coordinate Proofs G­GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Day 6 – Proving Properties on Coordinate Plane #1

Prove Medians concurrent and 1/3 from vertex Prove properties of quadrilaterals

Diagonals bisect Area of triangle ¼ area of triangle

G­GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Day 7 – Proving Properties on Coordinate Plane #2

Name point on diagonal Equation of perpendicular bisector

G­GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Day 8 – Review Day 9 – Test

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