**Geometry Scope and Sequence**

**General Information**

- Suggested Pacing is 1 lesson per day.
- Each lesson has an exit ticket that may be used as a formative assessment.
**As of now, Engagenyonly has Module 1 completed, the rest of the modules will be added once they are complete. The modules listed are suggested topics and lessons. If the remaining modules are unavailable when needed, please refer to the Topics/Standards below and use materials from 2013-2014.**

Suggestions

- Teachers may create additional assessments as they feel necessary.
- Modules (student materials) may be printed and bound for students to use as a workbook.
- Common Core belief is to provide students with answer keys to practice correctly.
- The tests online have answer keys that are available to the public. It is suggested to use them for either reviews or correctives.

**Suggested Materials**

- Compass and straightedge
- Geometer’s Sketchpad or Geogebra Software
- Patty paper

**Summary of Year (from engageny)**

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.

Quarter 1

**Module 1: Congruence, Proof, and Constructions**

Module 1: In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions—translations, reflections, and rotations—and have strategically applied a rigid motion to informally show that two triangles are congruent. In this module, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They build upon this familiar foundation of triangle congruence to develop formal proof techniques. Students make conjectures and construct viable arguments to prove theorems—using a variety of formats—and solve problems about triangles, quadrilaterals, and other polygons. They construct figures by manipulating appropriate geometric tools (compass, ruler, protractor, etc.) and justify why their written instructions produce the desired figure.

**Topic A: Basic Constructions(G-CO.1)**

Module 1 embodies critical changes in Geometry as outlined by the Common Core. The heart of the module is the study of transformations and the role transformations play in defining congruence.

Students begin this module with Topic A, Constructions. Major constructions include an equilateral triangle, an angle bisector, and a perpendicular bisector. Students synthesize their knowledge of geometric terms with the use of new tools and simultaneously practice precise use of language and efficient communication when they write the steps that accompany each construction(G-CO.1).

Lesson / Description /

**Mathematical Practices**

1: Construct an Equilateral Triangle /

- Students learn to construct an equilateral triangle.
- Students communicate mathematic ideas effectively and efficiently.

2: Construct an Equilateral Triangle II /

- Students apply the equilateral triangle construction to more challenging problems.
- Students communicate mathematical concepts clearly and concisely.

3: Copy and Bisect an Angle /

- Students learn how to bisect an angle as well as how to copy an angle.
- Note: These more advanced constructions require much more consideration in the communication of the student’s steps.

MP.6

4: Construct a Perpendicular Bisector /

- Students learn to construct a perpendicular bisector and about the relationship between symmetry with respect to a line and a perpendicular bisector.

MP.6

5: Points of Concurrencies /

- Students become familiar with vocabulary regarding two points of concurrencies and understand why the points are concurrent.

**Topic B: Unknown Angles (G-CO.9)**

Constructions segue into Topic B, Unknown Angles, which consists of unknown angle problems and proofs. These exercises consolidate students’ prior body of geometric facts and prime students’ reasoning abilities as they begin to justify each step for a solution to a problem. Students began the proof writing process in Grade 8 when they developed informal arguments to establish select geometric facts (8.G.5).

Lesson / Description /

**Mathematical Practices**

6: Solve for Unknown Angles—Angles and Lines at a Point /

- Students review formerly learned geometry facts and practice citing the geometric justifications in anticipation of unknown angle proofs.

MP.7

7: Solve for Unknown Angles—Transversals /

- Students review formerly learned geometry facts and practice citing the geometric justifications in anticipation of unknown angle proofs.

8: Solve for Unknown Angles—Angles in a Triangle /

- Students review formerly learned geometry facts and practice citing the geometric justifications regarding angles in a triangle in anticipation of unknown angle proofs.
- Students recognize the relationship between angle measures and side lengths

9: Unknown Angle Proofs—Writing Proofs /

- Students write unknown angle proofs, which does not require any new geometric facts. Rather, writing proofs requires students to string together facts they already know to reveal more information.
- Note: Deduction is a mental disciplining by which we get more from our thinking.

MP.8

10: Unknown Angle Proofs—Proofs with Constructions /

- Students write unknown angle proofs involving auxiliary lines.

11: Unknown Angle Proofs—Proofs of Known Facts /

- Students write unknown angle proofs involving known facts.

**Topic C: Transformations/Rigid Motions (G-CO.2, G-CO.3, G-CO.4, G-CO.5, G-CO.6, G-CO.7, G-CO.12)**

Topic C, Transformations, builds on students’ intuitive understanding developed in Grade 8. With the help of manipulatives, students observed how reflections, translations, and rotations behave individually and in sequence (8.G.1, 8.G.2). In Grade 10, this experience is formalized by clear definitions (G.CO.4) and more in-depth exploration (G.CO.3, G.CO.5). The concrete establishment of rigid motions also allows proofs of facts formerly accepted to be true (G.CO.9). Similarly, students’ Grade 8 concept of congruence transitions from a hands-on understanding (8.G.2) to a precise, formally notated understanding of congruence (G.CO.6). With a solid understanding of how transformations form the basis of congruence, students next examine triangle congruence criteria. Part of this examination includes the use of rigid motions to prove how triangle congruence criteria such as SAS actually work (G.CO.7, G.CO.8).

Lesson / Description /

**Mathematical Practices**

12: Transformations—The Next Level /

- Students discover the gaps in specificity regarding their understanding of transformations.
- Students identify the parameters they need to complete any rigid motion.

MP.8

13: Rotations /

- Students manipulate rotations by each parameter—center of rotation, angle of rotation, and a point under the rotation.

MP.6

MP.7

14:Reflections /

- Students learn the precise definition of a reflection.
- Students construct the line of reflection of a figure and its reflected image. Students construct the image of a figure when provided the line of reflection.

MP.6

15:Rotations, Reflections, and Symmetry /

- Students learn the relationship between a reflection and a rotation.
- Students examine rotational symmetry within an individual figure.

16:Translations /

- Students learn the precise definition of a translation and perform a translation by construction.

17: Characterize Points on a Perpendicular Bisector /

- Students understand that any point on a line of reflection is equidistant from any pair of pre-image and image points in a reflection.

MP.6

MP.7

18: Looking More Carefully at Parallel Lines /

- Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180˚. They learn how to prove the alternate interior angles theorem using the parallel postulate and the construction.

19: Construct and Apply a Sequence of Rigid Motions /

- Students begin developing the capacity to speak and write articulately using the concept of congruence. This involves being able to repeat the definition of congruence and use it in an accurate and effective way.

20: Applications of Congruence in Terms of Rigid Motions /

- Students will understand that a congruence between figures gives rise to a correspondence between parts such that corresponding parts are congruent, and they will be able to state the correspondence that arises from a given congruence.
- Students will recognize that correspondences may be set up even in cases where no congruence is present, they will know how to describe and notate all the possible correspondences between two triangles or two quadrilaterals and they will know how to state a correspondence between two polygons.

21: Correspondence and Transformations /

- Students practice applying a sequence of rigid motions from one figure onto another figure in order to demonstrate that the figures are congruent.

**Mid-Module Assessment**

*Topics A through C (assessment 1 day, return 1 day, remediation or further applications 2 days)*

**Topic D: Congruence (G-CO.7, G-CO.8)**

In Topic D, Proving Properties of Geometric Figures, students use what they have learned in Topics A through C to prove properties—those that have been accepted as true and those that are new—of parallelograms and triangles (

**G.CO.10, G.CO.11**). The module closes with a return to constructions in Topic E (G.CO.13), followed by a review that of the module that highlights how geometric assumptions underpin the facts established thereafter (Topic F).

Lesson / Description /

**Mathematical Practices**

22: Congruence Criteria for Triangles—SAS /

- Students learn why any two triangles that satisfy the SAS congruence criterion must be congruent.

23: Base Angles of Isosceles Triangles /

- Students examine two different proof techniques via a familiar theorem.
- Students complete proofs involving properties of an isosceles triangle.

24: Congruence Criteria for Triangles—ASA and SSS /

- Students learn why any two triangles that satisfy the ASA or SSS congruence criteria must be congruent.

25: Congruence Criteria for Triangles—SAA and HL /

- Students learn why any two triangles that satisfy the SAA or HL congruence criteria must be congruent.
- Students learn why any two triangles that meet the AAA or SSA criteria are not necessarily congruent.

26: Triangle Congruency Proofs—Part 1 /

- Students complete proofs requiring a synthesis of the skills learned in the last four lessons.

27: Triangle Congruency Proofs—Part 2 /

- Students complete proofs requiring a synthesis of the skills learned in the last four lessons.

**Topic E: Proving Properties of Geometric Figures (G-CO.9, G-CO.10, G-CO.11)**

In Topic E, students extend their work on rigid motions and proof to establish properties of triangles and parallelograms. In Lesson 28, students apply their recent experience with triangle congruence to prove problems involving parallelograms. In Lessons 29 and 30, students examine special lines in triangles, namely mid-segments and medians. Students prove why a mid-segment is parallel to and half the length of the side of the triangle it is opposite from. In Lesson 30, students prove why the medians are concurrent.

Lesson / Description / Mathematical Practices

28: Properties of Parallelograms /

- Students complete proofs that incorporate properties of parallelograms.

29:Special Lines in Triangles /

- Students examine the relationships created by special lines in triangles, namely mid-segments.

30:Special Lines in Triangles /

- Students examine the relationships created by special lines in triangles, namely medians.

End-of-Module Assessment

*Topics A through G (F and G are in Honors only. Assessment 1 day, return 1 day, remediation or further applications 3 days)*

Focus Standards for Mathematical Practice for Module 1

MP.3 Construct viable arguments and critique the reasoning of others. Students articulate steps needed to construct geometric figures, using relevant vocabulary. Students develop and justify conclusions about unknown angles and defend their arguments with geometric reasons.

MP.4 Model with mathematics. Students apply geometric constructions and knowledge of rigid motions to solve problems arising with issues of design or location of facilities.

MP.5 Use appropriate tools strategically. Students consider and select from a variety of tools in constructing geometric diagrams, including (but not limited to) technological tools.

MP.6 Attend to precision. Students precisely define the various rigid motions. Students demonstrate polygon congruence, parallel status, and perpendicular status via formal and informal proofs. In addition, students will clearly and precisely articulate steps in proofs and constructions throughout the module.

Module 2: Similarity, Proof, and TrigonometryModule 2: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, make sense of and persevere in solving similarity problems, and apply similarity to right triangles to prove the Pythagorean Theorem. Students attend to precision in showing that trigonometric ratios are well defined, and apply trigonometric ratios to find missing measures of general (not necessarily right) triangles. Students model and make sense out of indirect measurement problems and geometry problems that involve ratios or rates.

Topic A: Similarity

Standards: G-SRT.1, 2, 3

Lesson / Description / Mathematical Practices

1: Ratios and Proportions /

- Students learn to construct an equilateral triangle.
- Students communicate mathematic ideas effectively and efficiently.

2: Similar Polygons /

- Students apply the equilateral triangle construction to more challenging problems.
- Students communicate mathematical concepts clearly and concisely.

3: Similar Triangles /

- Establish AA
- AA, SAS, SSS

4: Dilations /

- Students learn to construct a perpendicular bisector and about the relationship between symmetry with respect to a line and a perpendicular bisector.

Topic B: Proofs

Standards: G-SRT.1, 2, 3

Lesson / Description / Mathematical Practices

5: Triangle Proportionality Theorem /

- Triangle Proportionality
- A line parallel to one side of a triangle divides the other two proportionally

6: Prove Pythagorean Theorem /

- Prove Pythagorean Theorem through similarity
- Not application of Pythagorean theorem, but just deriving the formula from similar triangles

Topic C: Trigonometry

Standards: G-SRT.6, 7, 8

Lesson / Description / Mathematical Practices

7: Pythagorean Theorem /

- Solve problems using the Pythagorean Theorem

8: Pythagorean Theorem Converse /

- Apply the converse of the Pythagorean Theorem to classify triangles

9:Trig Ratios /

- Set up trig ratios given side lengths
- Complementary angle relationship with trig ratios

10:Solve for a missing side /

- Use trig ratios to solve for a missing side given an angle and a side length

11:Solve for a missing angle /

- Use trig ratios to solve for a missing angle given two side lengths

12: Special Right Triangles /

- Identify special right triangles in relation to trig.
- 30-60-90
- 45-45-90

Topic D: Modeling

Standards: G-MG.1, 2, 3

Lesson / Description / Mathematical Practices

13: Angles of Elevation and Depression /

- Apply trigonometric ratios to solve real-world problems.

14: Further Applications of Similarity and Trig /

- Apply similarity and trigonometric ratios to solve real-world problems.

Module 3: Extending to Three Dimensions

Module 3: Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line. They reason abstractly and quantitatively to model problems using volume formulas.

Topic A: Area and Surface Area

Lesson / Description / Mathematical Practices

1: Review of Area Formulas /

- Triangles, Parallelograms, Trapezoids, Rhombi, Kites, Circles

2: Area of Regular Polygons /

- Given a side length and an apothem
- HONORS: find apothem or side

3: Area of Regular Polygons Continued /

- Given a side length and an apothem
- HONORS: find apothem or side

4: Surface Area of Prisms and Cylinders /

- Make sure to include regular polygon bases

5: Surface Area of Pyramids and Cones /

- Make sure to include regular polygon bases

6: Surface Area of Spheres /

- Find the SA of a Sphere

Topic B: Volume

Standards: G-GMD.1, 3

Lesson / Description / Mathematical Practices

7: Volume of Prisms and Cylinders /

- Find the volume of prisms and cylinders

8: Volume of Pyramids and Cones /

- Find the volume of pyramids and cones

9: Volume of Spheres /

- Find the volume of spheres

Topic C: Relationships and Modeling

Standards: G-GMD.4; G-MG.1

Lesson / Description / Mathematical Practices

10: Relationships between 3-D figures /

- Students will solve problems involving relationships between 3-D figures

11: Modeling with 3-D figures /

- Students will model situations involving 3-D figures

Module 4: Coordinate Geometry

Module 4: Building on their work with the Pythagorean theorem in 8th grade to find distances, students analyze geometric relationships in the context of a rectangular coordinate system, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines, relating back to work done in the first module. Students attend to precision as they connect the geometric and algebraic definitions of parabola. They solve design problems by representing figures in the coordinate plane, and in doing so, they leverage their knowledge from synthetic geometry by combining it with the solving power of algebra inherent in analytic geometry.

Topic A: Coordinate Formulas

Standards: G-GPE.5, 6

Lesson / Description / Mathematical Practices

1: Midpoint Formula /

- Given two points, find the midpoint

2: Distance Formula /

- Given two points, find the distance

3: Distance Ratio /

- Find the point on a directed line segment between two given points that partitions the segment in a given ratio (example, 2/3)

4: Slope Formula /

- Given two points, find the slope

5: Comparing Slopes /

- Given two equations or graphs determine the relationship between the lines
- Parallel
- Perpendicular
- Coincident
- Intersecting

Topic B: Coordinate Proofs

Standards: G-GPE.4

Lesson / Description / Mathematical Practices

6: Prove 4 points make a specific quadrilateral /

- Prove parallelism using slope
- Use distance to prove congruence

7: Prove 4 Points Make a Specific Quadrilateral Continued /

- Continued from above.

Topic C: Areas and Perimeters in Coordinate Plane

Standards: G-GPE.7

Lesson / Description / Mathematical Practices

8: Area in Coordinate Plane /

- Given 3-4 points, calculate the area of the polygon on a coordinate plane

9: Perimeter in Coordinate Plane /

- Given 3-4 points, calculate the perimeter of a polygon

10: Solve Problems involving the Coordinate Plane /

- Students will solve problems that can be placed a coordinate plane.

Module 5: Circles

Module 5: In this module students prove and apply basic theorems about circles, such as: a tangent line is perpendicular to a radius theorem, the inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths and angle measures. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students explain the correspondence between the definition of a circle and the equation of a circle written in terms of the distance formula, its radius, and coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations.

Topic A: Circle Basics

Standards: G-C.1, 2, 3

Lesson / Description / Mathematical Practices

1: Definitions /

- Center, Radius, Diameter
- Chord, Secant, Tangent
- Central angle, Inscribed angle, Arc

2: Measure of Angles and Arcs /

- Find measure of angles and arcs given a circle with many radii and values

3: Arcs and Chords /

- Find measure of arcs and chords given angle or arc measures

4: Inscribed Angles /

- Calculate the value of inscribed angles
- An inscribed angle with endpoints on a diameter equal 90 degrees

5: Tangents /

- Identify and solve problems given tangents
- Solve problems given a tangent and a radius (right angle)

6: Secants, Tangents, and Angle Measures /

- Find measure of arcs or angles given tangents, secants and angle measures

7: Segments /

- Solve problems using:
- Perpendicular chords
- Secant lengths
- Chord lengths
- Tangent lengths

Topic B: Arc Length and Area of Sector

Standards: G-C.5

Lesson / Description / Mathematical Practices

8: Arc Length /

- Find the length of a circular arc

9: Area of a Sector /

- Find the area of a sector

Topic C: Coordinates of Circles & Proof

Standards: G-GPE.1, 4

Lesson / Description / Mathematical Practices

10: Derive the Equation of a Circle /

- Use the Pythagorean Theorem to derive the equation of a circle

11: Equations of Circles /

- Complete the Square to write the equation
- Write the equation of a circle given:
- Center and radius
- Center and point
- Radius and point

Topic D: Modeling

Standards: G-MG.1

Lesson / Description / Mathematical Practices

13: Modeling involving Circle Parts /

- Students will solve problems involving circle parts

14: Modeling involving Area and Sector Area /

- Students will solve problems involving area and sector area

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