HS Honors Geometry Semester 2 (Quarter 3) Unit 3 .

sphere and other solid figures.G.GMD.A.3A. Explain volume formulas and use them tosolve problemsEureka Math:Module 3 Lesson 5,8,9Use volume formulas for cylinders, pyramids, cones,and spheres to solve problems.G.GMD.B.4B. Visualize relationships between twodimensional and three dimensional objectsEureka Math:Module 3 Lesson 5,6,7,13Identify the shapes of two-dimensional cross-sectionsof three-dimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects.G.MG.A.1A. Apply geometric concepts in modelingsituationsEureka Math:Module 3 Lesson 5-13Use geometric shapes, their measures, and theirproperties to describe objects (e.g., modeling a treetrunk or a human torso as a cylinder).G.MG.A.2G.MG.A.3A. Apply geometric concepts in modelingsituationsApply concepts of density based on area and volume inmodeling situations (e.g., persons per square mile,BTUs per cubic foot).A. Apply geometric concepts in modelingsituationsEureka Math:Module 3 Lesson 5-13Eureka Math:Module 3 Lesson 5-13Apply geometric methods to solve design problems(e.g., designing an object or structure to satisfy physical6/9/2016Page 11 of 28

constraints or minimize cost; working with typographicgrid systems based on ratios).MP.1Make sense of problems and persevere in solvingthem.MP.2Reason abstractly and quantitatively.MP.3Construct viable arguments and critique thereasoning of others.6/9/2016Mathematically proficient students start by explaining to themselvesthe meaning of a problem and looking for entry points to its solution.They analyze givens, constraints, relationships, and goals. They makeconjectures about the form and meaning of the solution and plan asolution pathway rather than simply jumping into a solution attempt.They consider analogous problems, and try special cases and simplerforms of the original problem in order to gain insight into its solution.They monitor and evaluate their progress and change course ifnecessary. Mathematically proficient students check their answers toproblems using a different method, and they continually askthemselves, "Does this make sense?" They can understand theapproaches of others to solving complex problems and identifycorrespondences between different approaches.Mathematically proficient students make sense of quantities and theirrelationships in problem situations. They bring two complementaryabilities to bear on problems involving quantitative relationships: theability to decontextualize—to abstract a given situation and representit symbolically and manipulate the representing symbols as if they havea life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during themanipulation process in order to probe into the referents for thesymbols involved. Quantitative reasoning entails habits of creating acoherent representation of the problem at hand; considering the unitsinvolved; attending to the meaning of quantities, not just how tocompute them; and knowing and flexibly using different properties ofoperations and objects.Mathematically proficient students understand and use statedassumptions, definitions, and previously established results inconstructing arguments. They make conjectures and build a logicalprogression of statements to explore the truth of their conjectures.They are able to analyze situations by breaking them into cases, andcan recognize and use counterexamples. They justify their conclusions,communicate them to others, and respond to the arguments of others.They reason inductively about data, making plausible arguments thattake into account the context from which the data arose.Eureka Math:Module 3 Lesson 1,4, 10,12Eureka Math:Module 3 Lesson 4Eureka Math:Module 3 Lesson 1,3,4, 811Page 12 of 28

MP.6Attend to precision.MP.7Look for and make use of structure.MP.8Look for and express regularity in repeatedreasoning.6/9/2016Mathematically proficient students are also able to compare theeffectiveness of two plausible arguments, distinguish correct logic orreasoning from that which is flawed, and—if there is a flaw in anargument—explain what it is.Students will formalize definitions, using explicit language to defineterms such as right rectangular prism that have been informal andmore descriptive in earlier grade levels.The theme of approximation in this unit is an interpretation ofstructure. Students approximate both area and volume (curved twodimensional shapes and cylinders and cones with curved bases)polyhedral regions. They must understand how and why it is possibleto create upper and lower approximations of a figure’s area or volume.The derivation of the volume formulas for cylinders, cones, andspheres, and the use of Cavalieri’s principle is also based entirely onunderstanding the structure and sub-structures of these figures.Mathematically proficient students notice if calculations are repeated,and look both for general methods and for shortcuts.Eureka Math:Module 3 Lesson 6Eureka Math:Module 3 Lesson 2,4,7,8,11,12Eureka Math:Module 3 Lesson 3,9Page 13 of 28

HS Honors Geometry Semester 2 (Quarter 4)Module 5: Circles With and Without CoordinatesWith geometric intuition well established through Units 1, 2, 3, and 4, students are now ready to explore the rich geometry of circles. This module brings together the ideas ofsimilarity and congruence studied in Units 1 and 2, the properties of length and area studied in Units 3 and 4, and the work of geometric construction studied throughout the entireyear. It also includes the specific properties of triangles, special quadrilaterals, parallel lines and transversals, and rigid motions established and built upon throughout thismathematical story. This unit’s focus is on the possible geometric relationships between a pair of intersecting lines and a circle drawn on the page. If the lines are perpendicular andone passes through the center of the circle, then the relationship encompasses the perpendicular bisectors of chords in a circle and the association between a tangent line and aradius drawn to the point of contact. If the lines meet at a point on the circle, then the relationship involves inscribed angles. If the lines meet at the center of the circle, then therelationship involves central angles. If the lines meet at a different point inside the circle or at a point outside the circle, then the relationship includes the secant angle theoremsand tangent angle theorems. (G-C.A.2, G-C.A.3)Students build on their knowledge of circles from Unit 2 and prove that all circles are similar. Students develop a formula for arc length in addition to a formula for the area of asector and practice their skills solving unknown area problems (G-C.A.1, G-C.A.2, G-C.B.5). Students explore geometric relations in diagrams of two secant lines, or a secant andtangent line (possibly even two tangent lines), meeting a point inside or outside of a circle. They establish the secant angle theorems and tangent-secant angle theorems. Bydrawing auxiliary lines, students also notice similar triangles and thereby discover relationships between lengths of line segments appearing in these diagrams (G-C.A.2, G-C.A.3, GC.A.4). Coordinate geometry is used to establish the equation of a circle. Students solve problems to find the equations of specific tangent lines or the coordinates of specific pointsof contact. They also express circles via analytic equations (G-GPE.A.1, G-GPE.B.4). Big Idea:EssentialQuestions:VocabularyStandardG.C.A.1 Thales’ Theorem, diameter, radius, central angle, inscribed angle, inscribed angle theorem, chord, converse of Thales’ theorem, equidistant, arc, interceptedarc, minor and major arc, central angle, consequence of inscribed angle theorem, Inscribed angle theorem, arc, minor and major arc, semicircle, inscribedangle, central angle, intercepted arc of an angle, length of an arc, radian, sector, area of a sectorAZ College and Career Readiness StandardsUnderstand and apply theorems about circles.Prove that all circles are similar.6/9/2016Explanations & ExamplesResourcesEureka Math:Module 5 Lesson 7,8Page 14 of 28

G.C.A.2Understand and apply theorems about circles.Identify and describe relationships among inscribedangles, radii, and chords. Include3 the relationshipbetween central, inscribed, and circumscribed angles;inscribed angles on a diameter are right angles; theradius of a circle is perpendicular to the tangent wherethe radius intersects the circle.6/9/2016Eureka Math:Module 5 Lesson 1-8, 1316Page 15 of 28

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G.C.A.3Understand and apply theorems about circles.Construct the inscribed and circumscribed circles of atriangle, and prove properties of angles for aquadrilateral inscribed in a circle.6/9/2016Eureka Math:Module 5 Lesson 3, 1316,20Page 18 of 28

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G.C.A.4Honors Only6/9/2016Understand and apply theorems about circles.Construct a tangent line from a point outside a givencircle to the circle.Eureka Math:Module 5 Lesson 11-12Page 20 of 28

G.C.B.5Find arc lengths and areas of sectors of circles.Derive using similarity the fact that the length of the arcintercepted by an angle is proportional to the radius,and define the radian measure of the angle as theconstant of proportionality; derive the formula for thearea of a sector.6/9/2016Eureka Math:Module 5 Lesson 9-10Page 21 of 28

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G.GPE.A.1Translate between the geometric description andthe equation for a conic section.Eureka Math:Module 5 Lesson 17,19Derive the equation of a circle of given center andradius using the Pythagorean Theorem; complete thesquare to find the center and radius of a circle given byan equation.6/9/2016Page 23 of 28

G.GPE.B.4Use coordinates to prove simple geometrictheorems algebraically.Eureka Math:Module 5 Lesson 18,19Use coordinates to prove simple geometric theoremsalgebraically. For example, prove or disprove that afigure defined by four given points in the coordinateplane is a rectangle; prove or disprove that the point(1, 3) lies on the circle centered at the origin andcontaining the point (0,2).6/9/2016Page 24 of 28

G.GMD.A.1A. Explain volume formulas and use them tosolve problemsCavalieri’s principle is if two solids have the same height and the samecross-sectional area at every level, then they have the same volume.Eureka Math:Module 3 Lesson 1-5, 8,10-13Give an informal argument for the formulas for thecircumference of a circle, area of a circle, volume of acylinder, pyramid, and cone. Use dissection arguments,Cavalieri’s principle, and informal limit arguments.HS.MP.3. Construct viable arguments and critique thereasoning of others.HS.MP.4. Model with mathematics.HS.MP.5. Use appropriate tools strategically.6/9/2016Page 25 of 28

MP.16/9/2016Make sense of problems and persevere in solvingthem.Students solve a number of complex unknown angles and unknownarea geometry problems, work to devise the geometric construction ofgiven objects, and adapt established geometric results to new contextsand to new conclusions.Eureka Math:Module 5 Lesson 1,3,8,9,13, 18-19Page 26 of 28

MP.2Reason abstractly and quantitatively.MP.3Construct viable arguments and critique thereasoning of others.MP.4Model with mathematics.MP.6Attend to precision.6/9/2016Mathematically proficient students make sense of quantities and theirrelationships in problem situations. They bring two complementaryabilities to bear on problems involving quantitative relationships: theability to decontextualize—to abstract a given situation and representit symbolically and manipulate the representing symbols as if they havea life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during themanipulation process in order to probe into the referents for thesymbols involved. Quantitative reasoning entails habits of creating acoherent representation of the problem at hand; considering the unitsinvolved; attending to the meaning of quantities, not just how tocompute them; and knowing and flexibly using different properties ofoperations and objects.Students must provide justification for the steps in geometricconstructions and the reasoning in geometric proofs, as well as createtheir own proofs of results and their extensions.Mathematically proficient students can apply the mathematics theyknow to solve problems arising in everyday life, society, and theworkplace. Mathematically proficient students who can apply whatthey know are comfortable making assumptions and approximations tosimplify a complicated situation, realizing that these may need revisionlater. They are able to identify important quantities in a practicalsituation and map their relationships using such tools as diagrams,two-way tables, graphs, flowcharts and formulas. They can analyzethose relationships mathematically to draw conclusions. They routinelyinterpret their mathematical results in the context of the situation andreflect on whether the results make sense, possibly improving themodel if it has not served its purpose.Mathematically proficient students try to communicate precisely toothers. They try to use clear definitions in discussion with others and intheir own reasoning. They state the meaning of the symbols theychoose, including using the equal sign consistently and appropriately.They are careful about specifying units of measure, and labeling axes toclarify the correspondence with quantities in a problem. They calculateaccurately and efficiently, express numerical answers with a degree ofprecision appropriate for the problem context.Eureka Math:Module 5 Lesson 10Eureka Math:Module 5 Lesson 1,2,4,6,7,12,15,17,19Eureka Math:Module 5 Lesson 10Eureka Math:Module 5 Lesson 11Page 27 of 28

MP.7Look for and make use of structure.MP.8Look for and express regularity in repeatedreasoning.6/9/2016Students must identify features within complex diagrams (e.g., similartriangles, parallel chords, and cyclic quadrilaterals) which provideinsight as to how to move forward with their thinking.Mathematically proficient students notice if calculations are repeated,and look both for general methods and for shortcuts. As they work tosolve a problem, mathematically proficient students maintain oversightof the process, while attending to the details. They continually evaluatethe reasonableness of their intermediate results.Eureka Math:Module 5 Lesson1,2,4,5,8-10, 11,14, 20,21Eureka Math:Module 5 Lesson 1,4,9,11,16Page 28 of 28

HS Honors Geometry Semester 2 (Quarter 3) Unit 3: Connecting Algebra and Geometry Through Coordinates . In Unit 3, students explore and experience the utility of analyzing algebra and geometry challenges through the framework of coordin