Somerville Schools 2018 CURRICULUM MAP WITH SCOPE AND SEQUENCE. Course: Geometry CP Subject Area: Mathematics Grade Level: 9-10

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1 Somerville Schools 2018 CURRICULUM MAP WITH SCOPE AND SEQUENCE Course: Geometry CP Subject Area: Mathematics Grade Level: 9-10 Enduring Understandings Unit 1: Tools of Geometry (16 Days) The undefined terms (points, lines and planes) help lay the foundation for many geometric concepts. Geometry is a mathematical system built on accepting facts, basic terms, and definitions. Special angle pairs are a foundational geometric topic that help us prove many geometric theorems. All conclusions in geometry must have clear and logical mathematical explanations. You can use number operations to find and compare the lengths of segments and measures of angles. Special angle pairs can help you identify geometric relationships and use angle pairs to find angle measures. You can use formulas to find the midpoint and length of any segment in the coordinate plane. Perimeter and area are two different ways of measuring geometric figures. You can use special geometric tools to make a figure that is congruent to an original figure without measuring. Unit 2: Reasoning, Proof and Lines (22 Days) You can describe some mathematical relationships using a variety of if-then statements. Algebraic properties of equality are used in geometry and will help with problem solving and justification. The use of given information, definitions, theorems, properties and postulates can be used as justification for a proof. Cutting parallel lines with a transversal creates special angle pairs that can be used to solve various problems. Comparing the slopes of two lines allows us to identify if the lines are parallel, perpendicular or neither. Special angle pairs formed by parallel lines and a transversal are congruent, supplementary, or both. Certain angle pairs will allow you to determine whether two lines are parallel. The relationships of two lines to a third line can be used to determine two lines are parallel or perpendicular to each other. Unit 3: Congruent Triangles (22 Days) You can determine whether two figures are congruent by comparing their corresponding parts. You can prove that two triangles are congruent without have to show all corresponding parts are congruent. (SSS, SAS, ASA, AAS, HL) If you know two triangles are congruent, then you know that every pair of corresponding parts is also congruent. The angles and sides of isosceles and equilateral triangles have special relationships. You may use corresponding parts of congruent triangles when proving one triangle congruent to another. You must use logical reasoning to develop an argument to prove something to be true.

2 Essential Questions What are the underlying principles of Geometry? How can you describe the attributes of a segment or angle? How is the visualization of points, lines and angles essential to the study of the real world? How are special angle pairs useful in the real world? The sum of the angle measures of a triangle is always the same. A straightedge and a compass can be used to construct parallel and perpendicular lines. You can graph a line and write its equation when you know certain facts about the line such as the slope and a point or two points on the line. How can you make a conjecture and prove that it is true? How do we logically prove theorems and arguments? How do you prove that two lines are parallel? What is the sum of the measures of the angles of a triangle? How do you write an equation of a line in the coordinate plane? How do you know if two objects are congruent? How do you identify corresponding parts of congruent triangles? How do you show that two triangles are congruent? How can you tell whether a triangle is isosceles or equilateral? How can you solve problems involving measurements within triangles? What are properties of isosceles triangles and how can they be used to solve problems? Content Knowledge Unit vocabulary including: a. collinear b. coplanar c. congruent d. midpoint e. segment bisector f. angle f. vertex g. angle bisector h. perpendicular i. polygon k. perimeter l. Circumference Unit vocabulary including: a. inductive reasoning b. conjecture c. counterexample d. negation e. if-then statement f. hypothesis g. conclusion h. converse i. inverse j. postulate k. proof Unit Vocabulary: a. Equiangular triangle b. equilateral triangle c. isosceles triangle d. scalene triangle e. congruent f. congruent polygons g. corresponding parts h. included angle i. included side j. base angle The sum of the interior angles in any triangle is 180 degrees.

3 How to identify and model points, lines, and planes. Identify intersecting lines and planes. How to measure segments and calculate using measures. How to find the distance between two points. How to find the midpoint of a segment. How to measure and classify angles. How to identify and use congruent angles and the bisector of an angle. How to identify and use special pairs of angles. l. Theorem m. parallel lines n. skew lines o. parallel planes p. transversal q. interior angles r. exterior angles s. corresponding angle t. slope u. rate of change v. slope-intercept form How to find counterexamples using logic and deductive reasoning. How to connect definitions, postulates, and logical reasoning. How to use logical reasoning to prove statements are true. How to analyze statements in ifthen form. How to write the converse, inverse, and contrapositive of ifthen statements. How to use deductive reasoning to prove a statement. How to construct and justify statements about geometric figures, using basic postulates. How to use algebra to write twocolumn proofs and use the properties of equality to write geometric proofs. How to write proofs involving segment addition and segment congruence. The sum of the exterior angles of any polygon is 360 degrees. How to prove triangles congruent using the triangle congruence theorems. How to apply the isosceles triangle theorem to find the measures of angles and legs of isosceles and equilateral triangles.

4 Major Skills Applying the undefined terms (point, line, and plane) to fundamental geometric concepts. Comparing lengths of segments and measures of angles. Identifying special angle pairs and using their relationships to find angle measures. Justifying an argument using reasoning in algebra and geometry. Solving for the midpoint of a segment. Using the midpoint formula to algebraically calculate the How to write proofs involving supplementary and complementary angles. How to use deductive reasoning to prove a statement. How to identify relationships between lines and planes. How to name angle pairs formed by parallel lines and transversals. How to recognize angle pair relationships when parallel lines are cut by a transversal. How to use theorems to determine the relationships between special angle pairs. How to use algebra to find measures of angles. How to use slope and other key characteristics to write equations of lines. How to identify parallel and perpendicular lines using slope. Identifying and using basic postulates about points, lines, and planes. Use algebraic reasoning to write two column proofs. Applying properties of equality and congruence to write two column proofs. Applying the segment and angle addition postulates to write two column proofs. Identifying special angle pairs to justify reasoning when writing two column proofs. Identifying and classifying triangles based on angle measures and side lengths. Applying the triangle sum theorem and exterior angle theorem to find the measures of missing angles. Naming corresponding parts of congruent triangles and identifying relationships between congruent triangles. Proving triangles congruent using the definition of congruence.

5 coordinates of a midpoint in the coordinate plane. Calculating the distance between two points in the coordinate plane. Applying the distance formula to algebraically calculate distance. Calculating the perimeter, circumference, and area of basic two dimensional shapes. Make basic constructions using a compass and straightedge. Identifying the relationship between lines and planes. Identifying angles formed by two lines and a transversal. Using properties of parallel lines to determine congruent and supplementary angles. Proving theorems involving parallel lines. Proving lines to be parallel using angle pair relationships. Using properties of parallel lines to find angle measures. Determining whether two lines are parallel using special angle pairs. Relating slope to equations of parallel and perpendicular lines. Graphing and writing equations of lines provided specific characteristics. Using properties of isosceles and equilateral triangles. Performance Based Assessments PERFORMANCE TASK(S): Algebra Benchmark: Benchmark to assess skill level in algebra including solving linear equations, simplifying radicals, exponent properties, factoring, and solving quadratic equations Pre-Assessment for SGO: Test given to assess student understanding of Geometry Concepts addressed throughout the entire year, used as a benchmark for post assessment. 1.2 Points, Lines, Planes edulastic assignments Quiz: Formative assessment allowing students to self-assess their PERFORMANCE TASK(S): 2.5 Proofs in Algebra Edulastic Assignment: Formative assessment allowing students to self-check their understanding and practice identifying properties of equality and congruence 2.6 Proofs in Geometry Edulastic Assignment 3.1 Lines and Angles Quizziz: Student centered activity where students identify angle pairs involving parallel lines being cut by a transversal Equations of Lines Kahoot: Formative assessment where students can self-assess their understanding of lines in PERFORMANCE TASK(S): 4.6 Congruent Triangles Edulastic Assignment: Formative assessment of student understanding of the 5 triangle congruence theorems 4.5 Isosceles and Equilateral Triangle Quizziz: Formative assessment of the isosceles triangles theorem and all of its corollaries 4.7 Congruence in Overlapping Triangles Project: OTHER EVIDENCE: 4.1 Congruent Figures Desmos Investigation: 4.2 SAS and SSS Congruence Investigation:

6 Digital Platforms Sources: Instructional Materials understanding of Special Angle Pairs and Coordinate Geometry OTHER EVIDENCE: Around the Room activity: Students demonstrate understanding of the basics of Geometry and new terminology 1.5 Exploring Angle Pairs QR Code Activity: 1.6 Constructions Partner Activity: Always, Sometimes, and Never Coloring Activity: Periodic group-work throughout the unit Kuda Desmos Chromebooks Pens Pencils Paper Anchor Charts SMARTboard Teacher Laptop Modifications the coordinate plane and determine if lines are parallel or perpendicular. OTHER EVIDENCE: 2.2 Conditional Statements Activity: 2.5 Introduction to Proof Writing in Algebra Activity: 2.6 Proving Angles Congruent Around the Room Activity: 3.2 Properties of Parallel Lines Investigation: Stations Activity 3.5 Triangle Sum Theorem and Exterior Angles Theorem Proof Activity 3.6 Constructions Activity 3.7 Equations of Lines Investigation: Periodic group-work throughout the unit Kuda Desmos Chromebooks Pens Pencils Paper Anchor Charts SMARTboard Teacher Laptop Modifications 4.3 ASA and AAS Congruence Investigation: 4.4 CPCTC Around the Room Activity: Periodic group-work throughout the unit Kuda Desmos Chromebooks Pens Pencils Paper Anchor Charts SMARTboard Teacher Laptop Modifications SEL Competencies SEL Competencies SEL Competencies NJSLS Standards Priority Standards: G.CO.A.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 Priority Standards: G.CO.A.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. Priority Standards: G.CO.B.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of

7 line, and distance around a circular arc. G.CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). G.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. G.GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G.GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G.MG.A.1 Use geometric shapes, their measures, and their properties to describe objects. G.MG.A.3 Apply geometric methods to solve design problems. N.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Support Standards: NJSLSA.R4. Interpret words and phrases as they are used in a text, including determining technical, G.CO.C.9 Prove theorems about lines and angles. G.CO.C.10 Prove theorems about triangles. G.CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). G.CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle G.MG.A.3 Apply geometric methods to solve design problems. G.GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. N.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Support Standards: NJSLSA.R4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. NJSLSA.R7. Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words. sides and corresponding pairs of angles are congruent. G.CO.B.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. C. Prove geometric theorems G.CO.C.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.SRT.B.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. Support Standards: NJSLSA.R4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. NJSLSA.R7. Integrate and evaluate content presented in

8 connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. NJSLSA.R7. Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words D.4 Research and understand the positive and negative impact of one s digital footprint D.5 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address personal, social, lifelong learning, and career needs F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs C.1 Review career goals and determine steps necessary for attainment C.3 Identify transferable career skills and design alternate career plans CRP Standards: CRP1. Act as a responsible and contributing citizen and employee. CRP2. Apply appropriate academic and technical skills. CRP4. Communicate clearly and effectively and with reason. CRP11. Use technology to enhance productivity. CRP12. Work productively in teams while using cultural global competence D.4 Research and understand the positive and negative impact of one s digital footprint D.5 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address personal, social, lifelong learning, and career needs F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs C.1 Review career goals and determine steps necessary for attainment C.3 Identify transferable career skills and design alternate career plans CRP Standards: CRP1. Act as a responsible and contributing citizen and employee. CRP2. Apply appropriate academic and technical skills. CRP4. Communicate clearly and effectively and with reason. CRP11. Use technology to enhance productivity. CRP12. Work productively in teams while using cultural global competence. diverse media and formats, including visually and quantitatively, as well as in words D.4 Research and understand the positive and negative impact of one s digital footprint D.5 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address personal, social, lifelong learning, and career needs F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs C.1 Review career goals and determine steps necessary for attainment C.3 Identify transferable career skills and design alternate career plans CRP Standards: CRP1. Act as a responsible and contributing citizen and employee. CRP2. Apply appropriate academic and technical skills. CRP4. Communicate clearly and effectively and with reason. CRP11. Use technology to enhance productivity. CRP12. Work productively in teams while using cultural global competence.

9 Course: Geometry CP Subject Area: Mathematics Grade Level: 9-10 Enduring Understandings Unit 4: Relationships Within Triangles (12 Days) There are two special relationships between the midsegment of a triangle and the third side of that triangle. There is a special relationship between the points on the perpendicular bisector or a segment and the endpoint of the segment. There is a special relationship between points on an angle bisector and the perpendicular distance to the rays that for the angle. In all triangles, certain sets of lines are always concurrent which provide us with a universal rule regarding each point of concurrency. The three medians of any triangle are always concurrent. The angles and sides of a triangle have special relationships that involve inequalities. In triangles that have two pairs of congruent sides, there is a Unit 5: Similarity (17 Days) You can write a ratio to compare two quantities. You can use ratios and proportions to determine whether two polygons are similar and to find missing side lengths of similar figures. You can show two triangles are similar when you know the relationship between only two or three parts of the triangles. When you draw the altitude to the hypotenuse of a right triangle, you form three similar right triangles. When two or more parallel lines intersect other lines, proportional segments are formed. Unit 6: Transformations (15 Days) You can change the position of a geometric figure so that the angle measures and the distance between any two points of a figure stay the same. When you reflect a figure across a line, each point of the figure maps to another point the same distance from the line but on the other side. (Orientation is reversed) Rotations preserve distance, angle measures, and orientation of figures. You can express all isometries as compositions of reflections. You can use compositions of rigid motions to understand congruence. You can use a scale factor to make a figure that is similar to the original figure. You can use compositions of rigid motions to understand properties of similarity.

10 relationship between the included angles and the third pair of sides. Essential Questions How do you use coordinate geometry to find relationships within triangles? How do you solve problems that involve measurements of triangles? What information do we get from knowing that a segment is a median, altitude or angle bisector of a triangle? What inequalities apply to the relationships of angles and sides in triangles? What does it mean for figures to be similar? How do you use proportions to find side lengths in similar polygons? How do you show two triangles are similar? How do you identify corresponding parts of similar triangles? How are similar figures useful in finding indirect measurement? How can you change a figure s position without changing its size and shape? How can you change a figures size without changing its shape? What are transformations and how can they be used in real life? How can congruence and similarity be related to transformations? How can you represent a transformation in the coordinate plane? Content Knowledge Unit vocabulary including: A. Perpendicular bisector B. Angle bisector C. Median D. Altitude E. Point of concurrency F. Circumcenter G. Incenter H. Centroid I. Orthocenter J. Midsegment How to identify each of the four points of concurrency. The relationship between an exterior angle of a triangle compares to any interior angle of that same triangle. The relationship between the sides and angles of any triangle. How two angles are related if we know the measures of the sides opposite those angles. How two sides are related if we know the measures of the angles opposite those sides. How to apply the hinge theorem when knowing two sides of Unit Vocabulary Including: A. Ratio B. Proportion C. Cross Product D. Scale Factor E. Similar F. Similar Polygons G. Indirect Measurement H. Geometric Mean How to solve a proportion. How to identify similar triangles. How to use proportions to find missing side lengths between similar figures. How to use similar figures to find a measurement indirectly. Unit Vocabulary Including: A. Transformation B. Orientation C. Pre-Image D. Image E. Rigid Motion F. Translation G. Reflection H. Glide Reflection I. Rotation J. Angle of Rotation K. Isometry L. Dilation M. Line of Reflection N. Center of Rotation O. Center of Dilation P. Composition of Transformations Q. Symmetry How to translate a figure in the coordinate plane. How to determine if a transformation preserves orientation or distance.

11 multiple triangles are congruent and the measures of the included angles. The relationship between the midsegment and the third side of any triangle. Major Skills Using perpendicular and angle bisectors to find the distance to sides and angles of a triangle. Using medians and altitudes to calculate distance. Calculating the length of a midsegment or third side of a triangle by using the midsegment formula. Identifying possible side lengths using the triangle inequalities. Applying the hinge theorem to make comparisons between two triangles. Writing ratios and solving proportions. Using proportions to find missing side lengths of similar polygons. Identifying and applying properties of similar figures. Using the Angle Angle Postulate and similar triangle theorems. Using similarity to find measures indirectly. Find and using the relationships of similar right triangles. Applying the side splitter theorem when parallel lines are being intersected. Applying the triangle angle bisector theorem. How to reflect a figure over a line of reflection and identify corresponding points. How to dilate figures using scale factor. How to determine the scale factor when given an image and its pre-image. Identifying if transformed figures are isometries. Performing congruent transformations in the coordinate plane. Performing similar transformations in the coordinate plane. Performing rotations in the coordinate plane about the origin and about a point other than the origin. Identifying rotated figures and determining the angle of rotation. Identifying the center of dilation and scale factor between a preimage and image. Exploring the effects of compositions of transformations and how certain compositions of transformations can be achieved by only one transformation. Identifying line and rotational symmetries of two dimensional figures.

12 Identifying plane and axis symmetries of three dimensional figures. Performance Based Assessments Quizzes Unit Test Periodic Formative Assessments Teacher Observation Anecdotal Notes Assessments: Mid-segments and Points of Concurrency Quiz Unit 4 Test Unit 4 Problem of the Week Modifications PERFORMANCE TASK(S): 7.1 Ratios and Proportions Kahoot: Formative assessment where students solve proportions by applying skills from algebra 7.3 Proving Triangles Similar Edulastic Assignment: 7.4 Similarity in Right Triangles Investigation: OTHER EVIDENCE: 7.2 Similar Polygons Desmos Investigation: 7.3 Proving Triangles Similar Geogebra Investigation: 7.5 Proportions in Triangles Investigation: PERFORMANCE TASK(S): 9.2 Reflections Investigation: Congruent Transformations Edulastic Assignment: Similar Transformations Edulastic Assignment: Compositions of Transformations Project: OTHER EVIDENCE: 9.1 Translations Investigation: 9.3 Rotations Investigation 9.6 Dilations Desmos investigation: Digital Platforms Sources: Instructional Materials Kuda Desmos Chromebooks Pens Pencils Paper Anchor Charts SMARTboard Teacher Laptop Modifications Kuda Desmos Chromebooks Pens Pencils Paper Anchor Charts SMARTboard Teacher Laptop Modifications Kuda Desmos Chromebooks Pens Pencils Paper Anchor Charts SMARTboard Teacher Laptop Modifications SEL Competencies SEL Competencies SEL Competencies

13 NJSLS Standards G.CO.C.9. Prove theorems about lines and angles. G.CO.C.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.CO.D.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). G.SRT.B.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.C.A. 3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Support Standards: NJSLSA.R4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. NJSLSA.R7. Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words. Priority Standards: G.SRT.A.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.SRT.A.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.B.4. Prove theorems about triangles. G.SRT.B.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.GPE.B.4. Use coordinates to prove simple geometric theorems algebraically. Support Standards: NJSLSA.R4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. NJSLSA.R7. Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words. Priority Standards: G.CO.A.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. G.CO.A.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.A.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.A.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.CO.B.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

14 D.4 Research and understand the positive and negative impact of one s digital footprint D.5 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address personal, social, lifelong learning, and career needs F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs C.1 Review career goals and determine steps necessary for attainment C.3 Identify transferable career skills and design alternate career plans CRP Standards: CRP1. Act as a responsible and contributing citizen and employee. CRP2. Apply appropriate academic and technical skills. CRP4. Communicate clearly and effectively and with reason. CRP11. Use technology to enhance productivity. CRP12. Work productively in teams while using cultural global competence D.4 Research and understand the positive and negative impact of one s digital footprint D.5 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address personal, social, lifelong learning, and career needs F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs C.1 Review career goals and determine steps necessary for attainment C.3 Identify transferable career skills and design alternate career plans CRP Standards: CRP1. Act as a responsible and contributing citizen and employee. CRP2. Apply appropriate academic and technical skills. CRP4. Communicate clearly and effectively and with reason. CRP11. Use technology to enhance productivity. CRP12. Work productively in teams while using cultural global competence. motions to decide if they are congruent. G.SRT.A.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.SRT.A.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. 3. Use the properties of similarity Support Standards: NJSLSA.R4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. NJSLSA.R7. Integrate and evaluate content presented in diverse media and formats, including visually and

15 quantitatively, as well as in words D.4 Research and understand the positive and negative impact of one s digital footprint D.5 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address personal, social, lifelong learning, and career needs F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs C.1 Review career goals and determine steps necessary for attainment C.3 Identify transferable career skills and design alternate career plans CRP Standards: CRP1. Act as a responsible and contributing citizen and employee. CRP2. Apply appropriate academic and technical skills. CRP4. Communicate clearly and effectively and with reason. CRP11. Use technology to enhance productivity. CRP12. Work productively in teams while using cultural global competence.

16 Course: Geometry CP Subject Area: Mathematics Grade Level: 9-10 Enduring Understandings Unit 7: Right Triangles and Trigonometry (16 days) If you know the lengths of any two sides of a right triangle, you can find the length of the third side using the Pythagorean Theorem. Certain right triangles have properties that allow you to know the lengths of each of the three sides when only given one side of the triangle. If you know certain combinations of side lengths and angle measures of a right triangle, you can use ratios to find the other side lengths and angle measures. If you take the sine of an angle measure, the cosine of that angles compliment will be equivalent. You can use the angles of elevation and depression as the acute angles of right triangles formed by a horizontal distance and vertical height. If you know the measures of two angles and the length of a side (AAS or ASA), or the length of two sides and the non-included angle (SSA) then you can determine the missing parts of any triangle. Unit 8: Quadrilaterals (12 Days) The sum of the interior angles measures of a polygon depends on the number of sides the polygon has. Parallelograms have special properties regarding their sides, angles, and diagonals. You can determine if a quadrilateral is a parallelogram if its sides, angles, or diagonals have certain properties. Special parallelograms have all of the basic properties of parallelograms but have unique properties that help to identify them. Using properties of special quadrilaterals can help in determining missing side lengths and angle measures. You can determine if a parallelogram is a rhombus or a rectangle based on the properties of its diagonals. The angles, sides, and diagonals of trapezoids and kites have certain properties. You can classify figures in the coordinate plane by determining slopes, distances, and midpoints. Unit 9: Circles (16 days) A radius of a circle and the tangent that intersects the endpoint of the radius on a circle have a special relationship. You can find the arc length of a circle by comparing the circumference to an angle in the circle. You can use information about congruent parts of a circle or congruent circles, to find information about other parts of the circle(s). Angles formed by intersecting lines have a special relationship to the arcs they are intersecting. Angles formed by intersecting lines have a special relationship to the related arcs when the lines intersect inside or outside the circle. You can write the equation of a circle if you know its center and radius. The information in the equation of a circle allows you to identify points inside, outside, or on a circle and graph the circle in the coordinate plane.

17 If you know the measures of two sides lengths and the measure of the included angle or all three sides lengths of a triangle then you can find the measures of the missing sides and angles. Essential Questions How do you find a side length or angle measure in a right triangle? What is the significance of right triangle relationships? How do trigonometric ratios relate to similar triangles? How are sine and cosine related? How can you classify triangles by their angles when only given their side lengths? How can you use special right triangles to find missing side lengths of triangles? How can you use trigonometric ratios to find the measure of an object indirectly? To prove relationships of quadrilaterals true for a general case, use variables as coordinates to demonstrate properties. How can you find the sum of the measures of the angles in a polygon? How are triangles the building blocks to quadrilaterals? How can you classify quadrilaterals? What is the major difference between parallelograms, kites, and trapezoids? How can you use coordinate geometry to prove general relationships between quadrilaterals? How can you prove the relationship between angles and arcs in a circle? How can circles be studied algebraically and geometrically? How can properties of circles be used to find the measurements of unknown angles, arcs, and segments of circles? How do you find the equation of a circle in the coordinate plane? How can you prove relationships between angles and arcs in a circle? Content Knowledge Unit Vocabulary Including: A. Pythagorean Triple B. Trigonometry C. Trigonometric Ratio D. Sine E. Cosine F. Tangent G. Inverse Trigonometry H. Angle of Elevation I. Angle of Depression J. Law of Sines K. Law of Cosines How and when to apply the Pythagorean Theorem. Unit Vocabulary Including: A. Diagonal B. Parallelogram C. Rectangle D. Rhombus E. Square F. Trapezoid G. Isosceles Trapezoid H. Leg of Trapezoid I. Bases J. Base Angles K. Kite L. Midsegment Unit Vocabulary Including: A. Circle B. Center C. Radius D. Diameter E. Chord F. Secant G. Tangent H. Point of Tangency I. Arc J. Major Arc K. Minor Arc L. Semi-Circle M. Circumference

18 How to classify triangles using the Converse of the Pythagorean Theorem. How to apply the and triangle ratios to find missing sides of special right triangles. How to apply the trigonometric ratios to find missing side lengths of right triangles. How to use inverse trigonometry to find missing angles in a right triangle. How to identify which method can be used to solve a triangle based on the given information. Major Skills Using the Pythagorean Theorem and its converse to find missing side lengths of a triangle or classify a triangle. Applying the and ratios to find missing sides of special right triangles. Identifying and applying the sine, cosine, and tangent ratios to find missing side lengths and angles of right triangles. Using angles of elevation and depression to find measurements indirectly. Applying the law of sines and law of cosines to solve non-right triangles. (H) Properties of the following Quadrilaterals: A. Parallelogram B. Rectangle C. Rhombus D. Square E. Trapezoid F. Isosceles Trapezoid G. Kite How to find the sum of the interior angles of a polygon. How to determine if a quadrilateral is a parallelogram How to distinguish between a parallelogram, rhombus, rectangle, and square. How to apply properties of quadrilaterals to find missing angles and side lengths. Determining the sum of the interior angles of a polygon. Determining the measure of each individual interior angle of a regular polygon. Classifying quadrilaterals based on their properties. Verifying a quadrilateral is a parallelogram. Determining the most precise name of a quadrilateral based on properties being displayed. Applying properties of trapezoids and kites to find missing side lengths and angle measures. Using trigonometry and properties of special N. Segment O. Central Angle P. Inscribed Angle Q. Pi R. Standard form of the equation of a circle How the radius and a chord are related. The relationship formed when a tangent line meets the radius on the circumference of a circle. The Arc-Chord-Central angle relationship How to apply the inscribed angles theorem. The relationships between angles formed by secants, tangents, and chords. How to write and interpret the equation of a circle in standard form. Identifying parts of circles. Solving problems involving the circumference of a circle. Identifying and measuring central and inscribed angles. Recognizing the relationship between arcs and chords. Finding the measures of angles of inscribed polygons. Using properties of tangents to solve problems involving circumscribed polygons. Finding the measures of angles formed by lines intersecting on, inside, or outside a circle.

19 quadrilaterals to find missing side lengths and angle measures. Applying the midsegment formula to find the length of a midsegment or base of a trapezoid. Finding the measures of segments that intersect inside or outside a circle. Writing the equation of a circle in standard form. Graphing a circle in the coordinate plane when given the equation in standard form. Using the equation of a circle to identify if a given point is inside, outside, or on the circle. Performance Based Assessments PERFORMANCE TASK(S): 8.1 Pythagorean Theorem Activity: 8.1 Pythagorean Theorem and It s Converse Edulastic Assignment: 8.3 Trigonometry Kahoot 8.4 Angles of Elevation Activity: Scavenger Hunt: OTHER EVIDENCE: 8.2 Special Right Triangles Investigation: 8.3 Trigonometry Investigation: 8.5 Law of Sines Investigation: PERFORMANCE TASK(S): Properties of Parallelograms Kahoot: Properties of Special Parallelograms Edulastic Assignment: Coordinate Geometry Activity: OTHER EVIDENCE: 6.1 Angles in Polygons Investigation: Student centered activity 6.2 Special Quadrilaterals Activity: Student centered activity 6.4 Special Parallelograms Investigation: 6.6 Trapezoids and Kites Investigation: Periodic group-work throughout the unit PERFORMANCE TASK(S): 12.1 Tangent Lines Edulastic: Student centered activity Angles in Circles Quizziz: 12.5 Equations of a Circle Scavenger Hunt: Students will self-assess their understanding of the equation of a circle and how to convert from standard form to center-radius form OTHER EVIDENCE: 12.1 Tangent Lines Investigation: Students will investigate properties of tangent lines to circles on Desmos 12.2 Arcs and Chords Investigation: Students will investigate properties of arcs and chords using Geogebra 12.3 Inscribed Angles Investigation: 12.4 Angle Measures and Segment Length Investigation 12.5 Equation of a Circle Desmos Activity: Digital Platforms Kuda Desmos Kuda Desmos Kuda Desmos

20 Sources: Instructional Materials Chromebooks Pens Chromebooks Pens Chromebooks Pens Pencils Pencils Pencils Paper Paper Paper Anchor Charts Anchor Charts Anchor Charts SMARTboard SMARTboard SMARTboard Teacher Laptop Teacher Laptop Teacher Laptop Modifications Modifications Modifications SEL Competencies SEL Competencies SEL Competencies NJSLS Standards Priority Standards: G.SRT.B.4. Prove theorems about triangles. G.SRT.C.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.SRT.C.7. Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.C.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.SRT.D.10. Prove the Laws of Sines and Cosines and use them to solve problems. (H) G.SRT.D.11. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles. (H) Priority Standards: G.CO.11 Prove theorems about parallelograms. G.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. G.MG.1 Use geometric shapes, their measures, and their properties to describe objects G.MG.3 Apply geometric methods to solve design problems. G.SRT.B.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Support Standards: Write out standards completely NJSLSA.R4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. Priority Standards: G.C.A.1 Prove that all circles are similar. G.C.A.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.C.A.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.C.A.4. (+) Construct a tangent line from a point outside a given circle to the circle. G.C.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is

21 G.MG.A.1 Use geometric shapes, their measures, and their properties to describe objects. G.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. Support Standards: NJSLSA.R4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. NJSLSA.R7. Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words D.4 Research and understand the positive and negative impact of one s digital footprint D.5 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address personal, social, lifelong learning, and career needs F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs C.1 Review career goals and determine steps necessary for attainment. NJSLSA.R7. Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words D.4 Research and understand the positive and negative impact of one s digital footprint D.5 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address personal, social, lifelong learning, and career needs F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs C.1 Review career goals and determine steps necessary for attainment C.3 Identify transferable career skills and design alternate career plans CRP Standards: CRP1. Act as a responsible and contributing citizen and employee. CRP2. Apply appropriate academic and technical skills. CRP4. Communicate clearly and effectively and with reason. CRP11. Use technology to enhance productivity. CRP12. Work productively in teams while using cultural global competence. 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. G.CO.A.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. G.GPE.A.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. G.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. Support Standards: Write out standards completely NJSLSA.R4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. NJSLSA.R7. Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words.

22 C.3 Identify transferable career skills and design alternate career plans CRP Standards: CRP1. Act as a responsible and contributing citizen and employee. CRP2. Apply appropriate academic and technical skills. CRP4. Communicate clearly and effectively and with reason. CRP11. Use technology to enhance productivity. CRP12. Work productively in teams while using cultural global competence D.4 Research and understand the positive and negative impact of one s digital footprint D.5 Analyze the capabilities and limitations of current and emerging technology resources and assess their potential to address personal, social, lifelong learning, and career needs F.1 Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal and or social needs C.1 Review career goals and determine steps necessary for attainment C.3 Identify transferable career skills and design alternate career plans Course: Geometry CP Subject Area: Mathematics Grade Level: 9-10 CRP Standards: CRP1. Act as a responsible and contributing citizen and employee. CRP2. Apply appropriate academic and technical skills. CRP4. Communicate clearly and effectively and with reason. CRP11. Use technology to enhance productivity. CRP12. Work productively in teams while using cultural global competence. Enduring Understandings Unit 10: Areas and Volume (32 Days) Students will understand that You can find the area of a parallelogram or a triangle when Unit 11: Unit 12:

23 you know the length of its base and its height. You can find the area of a trapezoid when you know its height and the lengths of the bases. You can use trigonometry to find missing side lengths of parallelograms, triangles, or trapezoids which will help in finding the area. The area of a regular polygon is related to the distance from the center to a side. Regular polygons can be viewed as composite shapes comprised of isosceles triangles to find the area of the entire polygon. You can use ratios to compare the perimeters and areas of similar figures. You can find the length of an arc of a circle by comparing the circumference of the circle to the central angle formed by the two radii. You can find the area of a sector of a circle by comparing the area of the circle to the central angle formed by the two radii. You can use geometric models to solve certain types of probability problems. You can analyze a three dimensional figure by using the relationship among its vertices, edges, and faces. To find the surface area of a three dimensional figure, find the sum