Curriculum and Instruction – Office of Mathematics

4th Quarter GEOMETRY

Introduction

In 2014, the Shelby County Schools Board of Education adopted a set of ambitious, yet attainable goals for school and student performance. The District is committed to these goals, as further described in our strategic plan, Destination2025. By 2025,

80% of our students will graduate from high school college or career ready 90% of students will graduate on time 100% of our students who graduate college or career ready will enroll in a post-secondary opportunity

In order to achieve these ambitious goals, we must collectively work to provide our students with high quality, College and Career Ready standards-aligned instruction. The Tennessee State Standards provide a common set of expectations for what students will know and be able to do at the end of a grade. College and Career Ready Standards are rooted in the knowledge and skills students need to succeed in post-secondary study or careers. The TN State Standards represent three fundamental shifts in mathematics instruction: focus, coherence and rigor.

Focus Coherence Rigor

• The Standards call for a greater focus in mathematics. Rather than racing to cover topics in a mile-wide, inch-deep curriculum, the Standards require us to significantly narrow and deepen the way time and energy is spent in the math classroom. We focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom.

• For geometry, the major clusters, account for 53% of time spent on instruction.

• Supporting Content - information that supports the understanding and implementation of the major work of the grade.

• Additional Content - content that does not explicitly connect to the major work of the grade yet it is required for proficiency.

Thinking across grades:• The Standards are designed around coherent

progressions from grade to grade. Learning is carefully connected across grades so that students can build new understanding onto foundations built in previous years. Each standard is not a new event, but an extension of previous learning.

Linking to major topics:• Instead of allowing additional or supporting

topics to detract from course, these concepts serve the course focus. For example, instead of data displays as an end in themselves, they are an opportunity to do grade-level word problems.

Conceptual understanding: • The Standards call for conceptual understanding

of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives so that they are able to see math as more than a set of mnemonics or discrete procedures.

• Procedural skill and fluency: • The Standards call for speed and accuracy in

calculation. While the high school standards for math do not list high school fluencies, there are suggested fluency standards for algebra 1, geometry and algebra 2.

Application: • The Standards call for students to use math

flexibly for applications in problem-solving contexts. In content areas outside of math, particularly science, students are given the opportunity to use math to make meaning of and

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access content.

Our collective goal is to ensure our students graduate ready for college and career. The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation and connections.

The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations) procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics and sensible, useful and worthwhile, coupled with a belief in diligence and one’s own efficacy). Throughout the year, students should continue to develop proficiency with the eight Standards for Mathematical Practice.

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Mathematical Practices

Make sense of problems and persevere in solving them

Reason abstractly and quatitatively

Construct viable arguments and

crituqe the reasoning of

others

Model with mathematics

Use appropriate tools

strategically

Attend to precision

Look for and make use of

structure

Look for and express

regularity in repeated reasoning

Problem Solving

Reasoning and Proof

CommunicationRepresentation

Connection

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How to Use the Mathematic Curriculum Maps

This curriculum map is designed to help teachers make effective decisions about what mathematical content to teach so that ultimately our students can reach Destination 2025. To reach our collective student achievement goals, we know that teachers must change their instructional practice in alignment with the three College and Career Ready shifts in instruction for Mathematics. We should see these shifts in all classrooms:

1) Focus2) Coherence3) Rigor

Throughout this curriculum map, you will see resources as well as links to tasks that will support you in ensuring that students are able to reach the demands of the standards in your classroom. In addition to the resources embedded in the map, there are some high-leverage resources around each of the three shifts that teachers should consistently access:

The TNCore Mathematics StandardsThe Tennessee Mathematics Standards:https://www.tn.gov/education/article/mathematics-standards

Teachers can access the Tennessee State standards, which are featured throughout this curriculum map and represent college and career ready learning at each respective grade level.

Mathematical ShiftsFocushttp://achievethecore.org/shifts-mathematics

The standards are focused on fewer topics so students can learn more

Coherencehttp://achievethecore.org/shifts-mathematics

Topics within a grade are connected to support focus, and learning is built on understandings from previous grades

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Rigorhttp://achievethecore.org/shifts-mathematics

The standards set expectations for a balanced approach to pursuing conceptual understanding, procedural fluency, and application and modeling

Curriculum Maps:

Locate the TDOE Standards in the left column. Analyze the language of the standards and match each standard to a learning target in the second column.

Consult your McGraw-Hill/Glencoe Teachers’ Edition (TE) and other cited references to map out your week(s) of instruction. Plan your weekly and daily objectives, using the standards' explanations provided in the second column. Best practices tell us that

making objectives measureable increases student mastery. Carefully review the web-based resources provided in the 'Content and Tasks' column and use them as you introduce or assess a

particular standard or set of standards. The additional resources provided are supplementary and should be used as needed for content support and differentiation.

Review the Literacy Connections found in the right column. Make plans to address the content vocabulary, utilizing the suggested literacy strategies, in your instruction.

Examine the other standards and skills you will need to address in order to ensure mastery of the indicated standard. Using your McGraw-Hill/Glencoe TE and other resources cited in the curriculum map, plan your week using the SCS lesson plan

template. Remember to include differentiated activities for small-group instruction and math stations.

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TN State Standards Essential Understandings Content & Tasks Literacy Connections

Unit 6: Properties of Circles (continued)Angles and Segments in Circles

(Allow 2.5 weeks for instruction, review and assessments)

G-C CirclesUnderstand and apply theorems about circles

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.

The properties of polygons, lines, and angles can be used to understand circles; the properties of circles can be used to solve problems involving polygons, lines and angles.

Students will Identify and describe relationships

involving inscribed angles; Prove properties of angles for a

quadrilateral inscribed in a circle.

Essential QuestionHow can the properties of circles, polygons, lines and angles be useful when solving geometric problems?

Lesson 10-4 – Inscribed Angles, pp. 709-715Additional ResourcesThe resources below are supplementary and should be used as needed for additional content support and differentiation.Glencoe Video Lessons Use this link to access online video links to textbook lessons.Engage ny Geometry Module 5, Topic A, Lesson 5:Inscribed Angle Theorem and its Applications • Students prove the inscribed angle theorem: The

measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle.

• Students recognize and use different cases of the inscribed angle theorem embedded in diagrams. This includes recognizing and using the result that inscribed angles that intersect the same arc are equal in measure.

Writing in MathCompare and contrast inscribed angles and central angles of a circle. If they intercept the same arc, how are they related?

VocabularyInscribed angle, intercepted arc

G-C Circles Understand and apply theorems about circles

G-C.A.2 G-C.A.3 Construct a tangent line from a

point outside a given circle to the circle.G-CO CongruenceMake geometric constructions

G.CO.D.12 Make formal geometric

Students will Identify and describe relationships

among tangents and radii; Identify and describe relationships

among circumscribed angles and central angles;

Construct a tangent line from a point outside a circle to the circle.

Lesson 10-5 – Tangents, pp.718-725Tangent Lines and the Radius of a Circle Task

Mica Items

Writing in MathHow many tangents can be drawn from a point outside a circle, from a point on a circle, and from a point inside a circle? Explain your reasoning. VocabularyTangent, point of tangency, common tangent

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TN State Standards Essential Understandings Content & Tasks Literacy Connections

constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G-C.A.2 Question #41 ID #44090 G-C.A.2 Question #42 ID #44343G-C.A.2 Question #43 ID #44341 G-C.A.2 Question #44 ID #43578 G-C.A.2 Question #45 ID #43832

Engage ny Geometry Module 5, Topic C, Lesson 11: Properties of Tangents • Students discover that a line is tangent to a circle

at a given point if it is perpendicular to the radius drawn to that point.

• Students construct tangents to a circle through a given point.

• Students prove that tangent segments from the same point are equal in length.

G-CO Congruence Make geometric constructions

G.CO.D.12 G.CO.D.13 Construct an equilateral

triangle, a square, and a regular hexagon inscribed in a circle.

G-C CirclesUnderstand and apply theorems about circles

G.C.A.3

Students will Construct an equilateral triangle, a

square, and a regular hexagon inscribed in a circle.

Construct the inscribed and circumscribed circles of a triangle

Extend Lesson 10-5 Geometry Lab: Inscribed and Circumscribed Circles, p. 726

Use geometry software or graphing calculator such as TI-Nspire or the Cabri Jr. APP on the TI-84 to investigate. A regular compass and straight edge can also be used.

Writing in MathWhy is the term “incenter” a good term for the intersection of the three angle bisectors? Explain your reasoning.

G-C CirclesUnderstand and apply theorems about circles

G-C.A.2

Students will Find measures of angles formed by lines

intersecting on or inside a circle; Find measures of angles formed by lines

intersecting outside the circle.

Lesson 10-6 Secants, Tangents, and Angle Measures, pp. 727-735

Additional ResourcesThe resources below are supplementary and should be used as needed for additional

Ticket Out the DoorSelect examples and ask students to name the segments in the figure as they leave.

Vocabulary

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TN State Standards Essential Understandings Content & Tasks Literacy Connectionscontent support and differentiation.Chords, Secants, and Tangents Tasks, pp. 56 & 69

Engageny

Geometry Module 5, Topic C, Lesson 16: Similar Triangles in Circle-

Secant (or Circle-Secant-Tangent) Diagrams

Students find “missing lengths” in circle-secant or circle-secant-tangent diagrams.

Secant

G-C CirclesUnderstand and apply theorems about circles

G-C.A.2

Students will Find measures of segments that intersect

in the interior of a circle; Find measures of segments that intersect

in the exterior of a circle.

Lesson 10-7 Special Segments in Circles, pp. 736-742

Writing in MathDescribe the relationship among segments in a circle when two secants intersect inside a circle.

Ask students to describe how the lesson on secants, tangents, and angles (10-6) helped them better understand the lesson on special segments in a circle.

VocabularyChord segment, secant segment, external secant segment, tangent segment

Unit 6: Properties of Circles Arc Length, Sector Area, and Equations of Circles

(Allow 2.5 weeks for instruction, review and assessments)

G-C CirclesUnderstand and apply theorems about circles

G-C.A.2

The properties of polygons, lines, and angles can be used to understand circles; the properties of circles can be used to solve problems involving polygons, lines and angles.

Lesson 10-2 – Measuring Angles and Arcs, pp. 692-700

Writing in MathDescribe the three different types of arcs in a circle and the method for finding the measure of each one.

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TN State Standards Essential Understandings Content & Tasks Literacy Connections

G-C Circles Find arc length and areas of sectors of circles

G.C.B.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.

Students will Derive and apply the formula for arc

length; Derive the fact that the length of the arc

intercepted by an angle is proportional to the radius;

Define and apply radian measure.

Essential QuestionHow can the properties of circles, polygons, lines and angles be useful when solving geometric problems?

Additional ResourcesThe resources below are supplementary and should be used as needed for additional content support and differentiation.Circles and Spheres Tasks

Circles and their Relationships among Central Angles, Arcs and Chords Task , p.15

Investigating Angle Relationships in Circles Tasks, p. 46 & p.52

Engage ny Geometry Module 5, Topic A, Lesson 4; Experiments with Inscribed AnglesStudents explore the relationship between inscribed angles and central angles and their intercepted arcs.

VocabularyCentral angle, arc, minor arc, major arc, semicircle, congruent arcs, adjacent arcs, arc length

G-C Circles Find arc length and areas of sectors of circles

G.C.B.5

Students will Derive a formula for the area of a

sector of a circle; Find the area of circles and sectors

of circles.

Lesson 11-3 – Areas of Circles, pp.782 - 788

Additional ResourcesThe resources below are supplementary and should be used as needed for additional content support and differentiation.Arc Length and Area of Sector Tasks, p. 82 & p.91Grain Storage Task

Mica ItemsG-C.B.5 Question #51 ID #43869

Writing in Mathp.787, # 49If the radius of a circle doubles, will the measure of a sector of that circle double? Will it double if the arc measure of that sector doubles?

VocabularySector of a circle, segment of a circle

Ticket Out the DoorHave students describe how to find the area of a circle, given its circumference.

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TN State Standards Essential Understandings Content & Tasks Literacy Connections

G-C.B.5 Question #52 ID #44342

Engage ny Geometry Module 3, Topic A, Lesson 4 Students use inscribed and circumscribed polygons for a circle (or disk) of radius r and circumference C to show that the area of a circle is 1/2Cr or as it is usually written, πr2.

G-GPE Expressing Geometric Properties with Equations Translate between the geometric description and the equation of a conic section

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.

Use coordinates to prove simple geometric theorems algebraically G.GPE.B.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).

Students will Derive the equation of a circle given

the center and the radius. Complete the square to find the

center and radius of a circle by an equation.

Lesson 10-8 – Equations of Circles and Graphing Technology Lab 10.8 (using TI-Nspire), pp.743 - 749

Equations of Circles Lesson

Mica ItemsG-GPE.A.1 Question #53 ID #44343 G-GPE.A.1 Question #54 ID #43578 G-GPE.B.4 Question #56 ID # 44066

Writing in Mathp.748 # 40Describe how the equation for a circle changes if the circle is translated a units to the right and b units down.

VocabularyCompound locus

Unit 7: Measurement and Modeling in Two and Three Dimensions (continued)Visualizing Solids

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TN State Standards Essential Understandings Content & Tasks Literacy Connections

G-MG Modeling with GeometryApply geometric concepts in modeling situations

G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★

G-GMD Geometric Measurement and Dimension Visualize relationships between two-dimensional and three-dimensional objects

G-GMD.B.4 Identify the shapes of two-dimensional cross- sections of three dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Geometric definitions, properties and theorems allow one to describe, model, and analyze situations in the real world.

Students will Investigate cross sections of three-

dimensional figures.

Essential QuestionIn what ways can geometric figures be used to understand real-world problems?

Lesson 12-1 – Representations of Three- Dimensional Figures, Lesson pp. 823-828

Additional ResourcesThe resources below are supplementary and should be used as needed for additional content support and differentiation.Volumes of Cylinders, Cones, Pyramids, and Spheres Videos

Volumes of Cylinders, Cones, Pyramids, and Spheres Task, p.98

Unit on Area, Perimeter, and Volume with multiple tasks

Boxing Basketballs p.5 Great Pyramid p.13 Walter and Juanita’s Water

Troughs p.17 Greenhouse p.23

Writing in MathWhen an object on a video game is viewed from only one side, what are some ways that the object can be made to appear three-dimensional?

VocabularyIsometric view, cross section

G-MG Modeling with GeometryApply geometric concepts in modeling situations

G-MG.A.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).★

Students will Find the lateral area and surface area of

prisms; Find the lateral area and surface area of

cylinders.

Lesson 12-2 – Surface Area of Prisms and Cylinders, pp.830-837

Additional ResourcesThe resources below are supplementary and should be used as needed for additional content support and differentiation.Cereal Box Project (Surface Area & Volume) Tasks

Mica ItemsG-MG.A.3 & G-MG.A.1 Question #66 ID #43681

Writing in Mathp. 836, #40 Compare and contrast finding the surface area of a prism and finding the surface area of a cylinder.

VocabularyLateral face, lateral edge, base edge, altitude, height, lateral area, axis, composite solid

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TN State Standards Essential Understandings Content & Tasks Literacy Connections

G-MG.A.3 Question #71 ID #13037

G-MG Modeling with GeometryApply geometric concepts in modeling situations

G-MG.A.3

Students will Find the lateral area and surface area of

pyramids. Find the lateral area and surface area of

cones.

Lesson 12-3 – Surface Area of Pyramids and Cones, pp.838-846

Writing in Mathp. 845, #41 Describe how to find the surface area of a regular polygonal pyramid with an n-gon base, height h units and an apothem of a units.

VocabularyRegular pyramid, slant height, right cone, oblique cone

G-MG Modeling with GeometryApply geometric concepts in modeling situations

G-MG.A.3

Students will Find the surface area of a sphere

Lesson 12-6 – Surface Areas of Spheres, pp.864-871

Writing in MathDescribe the difference between the surface area of a sphere and the volume of a sphere.

VocabularyGreat circle, pole, hemisphere

Unit 5: Trigonometry (continued)Trigonometry with All Triangles

(Allow 1.5 weeks for instruction, review and assessments)(Advanced Algebra & Trigonometry)G-AT Applied TrigonometryUse trigonometry to solve problems

Dilations, similarity, and the properties of similar triangles allow for the application of trigonometric ratios to solve real-world situations.

Essential QuestionHow might the features of one figure be useful when solving problems about a similar figure?

Lesson 8-6 – The Law of Sines and CosinesIntroduction – How Big is the Bermuda Triangle?

Engage ny Geometry Module 2, Topic E, Lesson 30 Students rewrite the Pythagorean Theorem in terms

of sine and cosine ratios, and use it in this form to solve problems.

Students write tangent as an identity in terms of sine and cosine, and use it in this form to solve

Writing in Mathp. 590, #57 Draw and label a triangle that can be solved: a. using only the Law of Sines; b. using only the Law of Cosines. Explain why each triangle cannot be solved using the other Law.

VocabularyLaw of Sines, Law of Cosines

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TN State Standards Essential Understandings Content & Tasks Literacy Connections

Students will Derive a trigonometric formula for the

area of a triangle; Prove and apply the Law of Sines; Prove and apply the Law of

Cosines.

problems.

Right Triangle Trigonometry Tasks

RESOURCE TOOLBOXTextbook ResourcesConnectED Site - Textbook and Resources Glencoe Video Lessons Hotmath - solutions to odd problems

Comprehensive Geometry Help: Online Math Learning (Geometry) I LOVE MATHNCTM IlluminationsNew Jersey Center for Teaching & Learning (Geometry)

CalculatorFinding Your Way Around TI-83+ & TI-84+ (mathbits.com) Texas Instruments Calculator Activity Exchange

StandardsCommon Core Standards - Mathematics Common Core Standards - Mathematics Appendix A TN CoreCCSS Flip Book with Examples of each Standard Geometry Model Curriculum http://www.ccsstoolbox.org/http://insidemathematics.org/index.php/high-school- geometry http://www.azed.gov/azcommoncore/mathstandards/hs math/ http://learnzillion.com/common_core/math/hs http://www.livebinders.com/play/play/454480https://www.livebinders.com/play/play?id=464831 http://www.livebinders.com/play/play?id=571735 North Carolina – Unpacking Common Core

Videos Math TV VideosThe Teaching Channel Teacher TubeKhan Academy Videos (Geometry)

NWEA MAP Resources:https://teach.mapnwea.org/assist/help_map/ApplicationHelp.htm#UsingTestResults/MAPReportsFinder.htm - Sign in and Click the Learning Continuum Tab – this resources will help as you

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Texas Instruments Math NspiredSTEM ResourcesCasio Education for Teachers*Graphing Calculator Note: TI tutorials are available through Atomic Learning and also at the following link: Math Bits - graphing calculator steps Some activities require calculator programs and/or applications.

Use the following link to access FREE software for your MAC. This will enable your computer and TI Calculator to communicate: Free TI calculator downloads

TasksTNCore TasksNYC tasksUT Dana CenterMars TasksInside Math TasksDan Meyer's Three-Act Math TasksIllustrative Math TasksGSE Analytical Geometry; Unit 3- Circles & Volume

http://thegeometryteacher.wordpress.com/the- geometry-course/ http://math termind.blogspot.com/2012/07/common-core- geometry.html Utah Electronic School - Geometry Ohio Common Core Resources Chicago Public Schools Framework and Tasks Mathy McMatherson Blog - Geometry in Common Core

plan for intervention, and differentiating small group instruction on the skill you are currently teaching. (Four Ways to Impact Teaching with the Learning Continuum)

https://support.nwea.org/khanrit - These Khan Academy lessons are aligned to RIT scores.  

Interactive ManipulativesGeoGebra – Free software for dynamic math and science learningNCTM Core Math Tools http://www.keycurriculum.com/products/sketchpad (Not free) Any activity using Geometer’s Sketchpad can also be done with any software that allows construction of figures and measurement, such as Cabri, Cabri Jr. on the TI-83 or 84 Plus,TI-92 Plus, or TI-Nspire

Mica Items

Literacy Resources Literacy Skills and Strategies for Content Area Teachers (Math, p. 22)Glencoe Reading & Writing in the Mathematics ClassroomGraphic Organizers (9-12) (teachervision.com)

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