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UNIT 1 Congruence, Proof, and Construction Total Number of Days: 26 days ESSENTIAL QUESTIONS 1. 2. How do you identify corresponding parts of congruent triangles? How can you make a conjecture and prove that triangles are congruent? How can you describe the attributes of a segment or angle? 3. Grade/Course: __Geometry 10th grade__ ENDURING UNDERSTANDINGS 4. If two triangles are congruent, then every pair of their corresponding parts is also congruent. 5. Given information, definitions, properties, postulates, and previously proven theorems can be used as reasons in a proof. 6. Number operations can be used to find and compare the lengths of segments. 7. The Ruler and Segment Addition Postulates can be used in reasoning about lengths. 8. The Protractor and Angle Addition Postulates can be used in reasoning about angle measures. RESOURCES STANDARDS CONTENT PACE SKILLS OTHER (CCSS/MP) Pearson LEARNING ACTIVITIES and ASSESSMENTS (e.g., tech) Square numbers when calculating areas of certain Getting Ready figures. for Geometry: 2 days Example: Basic Skills 1) The number you get when (Squaring you multiply an integer by numbers, itself. Evaluating 1. 8.EE.2 Students recognize perfect Pearson squares and cubes, understanding that non-perfect Chapter #1 squares and non-perfect cubes Get Ready are Irrational. Interactive website for exponent: Website related to step to solve http://www.mathsisfun equations: .com/exponent.html www.svmimac.org/images/MNM. 052913.stsec.pdf Interactive practice 1 Expressions, Finding Absolute Value, Solving 4 × 4 = 16, so 16 is a square Equations) number. 2) 32 = 9 3) 42 = 16 Lesson Check: 8.EE.7 games: Students solve one-variable Text book equations including those with page #1 the variables being on both sides of the equal sign. Students recognize that the solution to the equation is the value(s) of the variable, which make a true equality when substituted back into the equation. Equations shall include rational numbers, distributive property and combining like terms. Text book Get Ready page 1 classroom.jcschools.net/basic/mathexpon.html Video: www.mathplayground.c om/howto_algebraeq1. MP.1 Evaluate expressions by substituting given values. MP.2 -Example: Evaluate each using the values given: p Basic skills Use 30mn. Review: HSPA Ex m; use m , and p Apply Pythagorean Theorem G.SRT.8 Use trigonometric ratios Pearson and the Pythagorean Theorem in real life problems: to solve right triangles in applied Chapter #1 Example: Get Ready Web link for Pythagorean Theorem and the distance Pythagorean Theorem Power point presentation: www.jamestownpublicschools.o 2 PREP/PARCC/S problems. AT Michelle was fishing in her canoe at point A in the lake depicted above. After trying to fish there, she decided to paddle due east at a steady speed of 10 miles per hour. As she paddled, a wind blowing due south at 5 miles per hour caused a change in her direction. What is the speed of her canoe, measured to the nearest tenth of a mile per hour, which has a velocity represented by vector AC? formula. Text book page #1 www.mathscore.com/m ath/practice/Pythagore an%20Theorem rg/highschool/faculty/.../pythag thm.ppt Kuta software for worksheets www.kutasoftware.co m Michelle 1. Nets and Drawing for 1 day Visualizing Geometry Construct nets and drawings G.CO.1 of three- dimensional figures. Know precise definitions of Example: refer to link under angle, circle, perpendicular line, resources on discovering 3-D parallel line, and line segment, shapes. based on the undefined notions of point, line, distance along a Example: line, and distance around a Find the surface area of this circular arc. Pearson Chapter 1 Text book page #4-9 Interactive Animated polyhedron models: 1. Basic: problems 1-4 Exs www.mathsisfun.com/g 6-19 all, 20-26 even, 27, 28-36 even, eometry/polyhedron43-51 3-Dimentional shapes 2. Average: problems 1-4 videos and worksheets Exs. 7-19 odd, 20-38, 43-51 3 box below www.onlinemathlearnin 3. Advanced: Problems 1-4 g.com/3d-shapesExs. 7-19 odd, 20-51 nets MP.3 MP.7 Web link for 3D shapes: http://www.xtec.cat/mo nografics/cirel/pla_le/nil e/mrosa_garcia/worksh eets.pdf Basic Review: skills Identify congruent figures and their corresponding parts. G.SRT.5 Use congruence and similarity HSPA criteria for triangles to solve PREP/PARCC/SA Example: problems and to prove relationships in geometric T A design follows this pattern: figures. an equilateral triangle is divided into 4 congruent 30mn triangles as shown below in Stage 1. Then, the top triangle is divided into 4 congruent triangles and the pattern repeats for each stage. In Stage 2, what is the ratio of the area of the larger shaded triangle to the area of the Pearson Chapter 4-1 Web link for congruent Standardized Test Prep triangles: (SAT/HSPA) www.mathopenref.com Text book page 224 Q. 50-53 /congruenttriangles.ht Text book page #4-9 4 smaller shaded triangle? 1. Points, Lines, Planes - To understand basic terms and postulates of Geometry Example: 1 day G.CO.1 Pearson Know precise definitions of Chapter 1 angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions Text book of point, line, distance along a page #10-19 line, and distance around a circular arc. MP.3 HSPA 30mn PREP/PARCC/SAT To identify parallel and perpendicular lines. G-C.O.1 3. Pearson Chapter 1 Ex. How would you determine whether two lines are parallel See below 1. Basic: problems 1-2 Exs 8-14 all, 65-80. Problems 3-4 Exs. 15-26 all, 28-46 even, 51, 54-58 even http://coachmetz.files.w ordpress.com/2012/04/ geometry_point_lines_a nd_planes_worksheet_a 2. Average: problems 1-2 Exs. 9-13 odd, 65-80. Problems .pdf 3-4 Exs. 15-25 odd, 27-58 MP.6 Line Basic skills Review: Web link for points, lines and planes: Advanced: Problems 1-2 Exs.9-13 odd, 65-80. Problems 3-4 Exs. 15-25 odd, 27-64 Web link for parallel Standardized Test Prep and perpendicular lines: (SAT/HSPA) www.clackamasmiddlec ollege.org/.../Parallel+a Text book page 10 Q. 43-45 5 or perpendicular? 1. Measuring Segments 1 day Determine and compare length of segments. Example: G.CO.1 Text book page #10 nd+Perpendicular+lines. pdf Pearson Web link for segments and their measures: Know precise definitions of Chapter 1 angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions Text book of point, line, distance along a page #20-27 line, and distance around a circular arc. www.kutasoftware.com /FreeWorksheets/Geo Worksheets/2Line%20Seg 4. Basic: problems 1-4 Exs 8-22 all, 24-34 even, 35, 3739, 44-56 5. Average: problems 1-4 Exs. 9-21 odd, 23-41, 44-56 6. Advanced: Problems 1-4 Exs. 9-21 odd, 23-56 G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 6 MP.1, MP.3 Basic skills Review: 30 mn To find translation image of G-CO.6 Pearson figures: Use geometric descriptions of Chapter 9-1 Example: HSPA rigid motions to transform PREP/PARCC/S figures and to predict the effect Consider parallelogram ABCD of a given rigid motion on a AT Text book with coordinates A(2,-2), B(4,4), C(12,4) and D(10,-2). given figure; given two figures, page #552 use the definition of congruence Perform the following in terms of rigid motions to transformations. Make predictions about how the decide if they are congruent. lengths, perimeter, area and angle measure will change under each transformation. Weblink for practice with translation in coordinate geometry. Standardized Test Prep (SAT/HSPA) Text book page 552 www.mathsisfun.com/g Q. 36-39 eometry/translation.ht Weblink for Video on translation: www.brightstorm.com/math/geo metry/transformations/translatio ns a. A reflection over the x-axis. b. A rotation of 270about the origin. c. A dilation of scale factor 3 about the origin. d. A translation to the right 5 and down 3. Verify your predictions. Compare and contrast which transformations preserved the size and/or shape with those that did not preserve size and/or shape. Generalize, how could you determine if a 7 transformation would maintain congruency from the pre-image to the image? 1.4 Measuring Determine and compare the G.CO.1 measures of angles. Angles Know precise definitions of Identify special angle pairs angle, circle, perpendicular line, and use their relationships to parallel line, and line segment, 1-5 Exploring find angle measures. based on the undefined notions Angle Pairs of point, line, distance along a Example: line, and distance around a circular arc. Web link for angles and 1. Basic: (1.4) problems 1-2 Exs 6-17 all, Exs. 6-17 all, 41their measures: Chapter 1 49. http://aggiejots.tripod.c om/sitebuildercontent/s Problems 3-4 Ex. 18-23 all, 24itebuilderfiles/geo_0106 28 even, 29, 31-34, 41-49 Text book page # 28-47 _ans.pdf (1.5) problems 1-2 Exs. 7-23 all, 48-59. Problems 3-4 Pearson Exs. 7-26 all, 28-30 even, 31,34-38 even, 39-40 G.CO.12 1 day Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a 2. Average: 1) Section (1.4) problems 1-2 Exs. 7-17 odd, 41-49. 3. Problems 3-4 Exs. 19-23 odd, 24-32, 41-49. (1.5) Problems 1-2 Exs. 7-23 odd, 48-59. Problems 3-4 Exs. 25, 27-41 4. Advanced: 8 1) Section (1.4) Problems 1-4 Exs.7-23 odd, 24-49. point not on the line. 2) Section (1-5) Problems 1-4 Exs. 7-25 odd, 27-59 MP.1, MP.3 Basic skills Review: G.SRT.8 Pearson Use the Pythagorean Theorem to solve real life Use trigonometric ratios and the Chapter 5 problems: HSPA Pythagorean Theorem to solve Example: PREP/PARCC/S right triangles in applied AT A 16-ft ladder leans against problems. Text book 30mn 1.6 Basic 1 day Constructions a building. To the nearest foot, how far is the base of the ladder from the building? Sketch the diagram. page # 291 Perform basic constructions G.CO.12 using a straightedge and Make formal geometric compass. constructions with a variety of Example: tools and methods (compass and straightedge, string, Construct a circle reflective devices, paper folding, circumscribed about triangle Pearson Kuta software for worksheet: Standardized Test Prep (SAT/HSPA) www.kutasoftware.com Text book page 498 Q. 55-58 /FreeWorksheets/PreAlg Worksheets/Pythagorea n Interactive game link for Pythagorean Theorem: www.mathplay.com/PythagoreanTheorem-Game Chapter 1 Compass and 5. Basic: straightedge construction worksheets 1) problems 1-2 available on this site: Exs 7-12 all, 20, 39-47. www.mathopenref.com Text book page # 48-56 /worksheetlist 2) Problems 3-4 9 ABC 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. Exs. 13-16 all, 18, 19, 22, 24, 25, 26-30 even, 36-38 6. Average: 1) problems 1-2 Exs. 7-11 odd, 20, 39-47. 2) Problems 3-4 Exs. 13, 15, 17-19, 21-32, 36-38 MP.1, MP.3, MP.5, MP.7 7. Advanced: 1) Problems 1-2 Exs. 7-11 odd, 20, 39-47 2) Problems 3-4 Exs. 13, 15, 17-19, 21-38 Basic skills Review: Use different scale to solve real life applications HSPA Example: 30mn PREP/PARCC/S AT Jan is building a scale model of a house. If the actual house is 86 feet wide and 172 feet long, G.GMD.3 Pearson Use volume formulas for Chapter 10-1 cylinders, pyramids, cones, and spheres to solve problems. Text book page # 622 Web link for area and parallelogram: Standardized Test Prep (SAT/HSPA) Text book page 622 Q. 47-49 http://www.mathgoodie Kutasoftware for worksheet: s.com/lessons/vol1/area www.kutasoftware.com/FreeW _parallelogram.html orksheets/PreAlgWorksheets/ Area 10 what will be the length in inches of the scale model if it is 18 inches wide? G.GPE.4 1.7 Midpoint Find the midpoint of a and Distance in segment. the Coordinate Plane 1 day Pearson Use coordinates to prove simple Chapter 1 geometric theorems algebraically. For example, prove or disprove that a figure Text book defined page #49-66 Find the distance between two points in the coordinate by four given points in the plane. coordinate plane is a rectangle; Example: prove or disprove that the point (1, 3) lies on the circle centered The distance from the floor to at the origin and containing the the ceiling of a rectangular point (0, 2). room is 8 ft. The diagonals of two adjacent walls are 17 ft and ft, respectively. G.PE.7 How long is a diagonal of the Use coordinates to compute floor? perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. MP.1, MP.3 Web link for 8. Pythagorean theorem and the distance formula: Basic: 1) problems 1-2 Exs. 6-21 all, 62-67 http://users.manchester 2) Problems 3-4 .edu/Student/slmiller02 Exs. 22-35all, /ProfWeb/PythagoreanT 36-44 even, 45-47 all, 48-56 heoremNotes.pdf even 9. Average: 1) problems 1-2 Exs. 7-21 odd, 62-72 2) problems 3-4 Exs. 23-35 odd, 36-57 10. Advanced: 1) Problems 1-2 Exs. 7-21 odd, 62-72 2) Problems 3-4 11 Exs. 23-35 odd, 36-61 Basic skills Review: Apply trigonometric ratios. Example: G.SRT.8 Pearson Use trigonometric ratios and the Chapter 8-1 HSPA Pythagorean Theorem to solve Joshua is flying his kite at the PREP/PARCC/S right triangles in applied end of a100ft. string. The AT problems. Text book angle of the string to the page # 498 ground is 50 degrees. Find the 30 mn height, x, of the kite above the ground. Web link for trigonometry ratios: 11. Standardized Test Prep (SAT/HSPA) http://www.kutasoftwar Text book page 498 Q. 55-58 e.com/FreeWorksheets/ GeoWorksheets/9Trigonometric%20Ratios 12. Web link for video .pdf tutorial on trigonometric ratio www.youtube.com/watch? 12 2.1 Patterns and inductive reasoning 1 day Use inductive reasoning to make conjectures. Weblink for an 13. Basic: application of inductive Prove theorems about lines and Chapter 2-1 reasoning: 1) problems 1-3 Example: angles. Theorems include: Exs. 6-30 vertical angles are congruent; www.rohan.sdsu.edu/~i Show the conjecture is false 2) Problems 4-5 when a transversal crosses tuba/math303s08/mathi Text book by finding a counterexample. parallel lines, alternate interior page # 82-88 a i f Exs. 31-40, 50, 53,54, 59Conjecture: The sum of two angles are congruent and 66 corresponding angles are numbers is always greater 14. Average: congruent; points on a than the larger of the two perpendicular bisector of a line numbers. 1) problems 1-3 segment are exactly those qui i an fr g n’ Exs. 7-29, 38-49 endpoints. 2) problems 4-5 Prepare for: G.CO.9 Pearson Exs. 31-37 odd, 50-55, 59-66 15. Advanced: 1) Problems 1-3 Exs. 7-29 odd, 38-49 2) Problems 4-5 Exs. 31-37 odd, 50-66 13 Basic skills Review: Apply the surface area and the volume formulas: Example: 30 mn HSPA PREP/PARCC/S Find the approximate surface AT area of this can to the nearest square inch. The diameter of the top is about 6 inches and the height of the cylinder is 8 inches. G.GMD.3 Pearson Use volume formulas for Chapter 11 cylinders, pyramids, cones, and spheres to solve problems. Text book page #695 Web link for surface area and volume: 16. www.mathatube.com/c ylinder-volume-surfacearea-worksheets Standardized Test Prep (SAT/HSPA) Text book page 695 Q. 51-55 17. Web link for video tutorial on surface area and volume: www.khanacademy.org/.../c ylinder-volume-and-surfacearea 14 2.2 Conditional statements 1 day Compose and distinguish Prepares for: between the converse, inverse, and contrapositive G.CO.9 Prove theorems about of a conditional statement. lines and angles. Theorems include: vertical angles are Example: congruent; when a transversal crosses parallel lines, alternate Your classmate claims that interior angles are congruent the conditional and and corresponding angles are contrapositive of the congruent; points on a following statement are both perpendicular bisector of a line true. Is he correct? Explain segment are exactly those 2 qui i an fr g n’ If X = 2, then X = 4 endpoints. 1. Can you find a counterexample of the conditional? 2. Do you need to find a counter example of the contrapositive to know its truth table? Pearson Chapter 2-2 Web link for conditional 3. Basic: statement: 1) Problems 1-4 http://wwwregensprep. Exs. 5-24, 28-29, 35, 37 org/Regents/n Text book page #89-96 39-40, 47-58 Kuta software for worksheet: www.kutasoftware.co m 4. Average: 1) problems 1-4 Exs. 5-23, 25-42, 47-58 5. Advanced: you tube video tutorial: www.youtube.com/wat ch?v=undSZSratIA 1) Problems 1-4 Exs. 5-23 odd, 25-58 Math power point notes: Google search: ac r nric k va u … C n i titionalStatement/LECTURE2-1.ppt www a c rg … G ryPP Ts/2.1%20Conditional%20Statement www a a c … C a r% 20Powerpoints/2.2%20Intro%2 Math skills practice: 15 Intranet.asfa.k.12 a u … Law % f%20Logic 16 Basic skills Review: HSPA PREP/PARCC/S AT 30 mn To apply the Pythagorean G.SRT.8 Use trigonometric ratios Pearson Theorem and the perimeter and the Pythagorean Theorem to solve right triangles in applied Chapter 2-2 formula: problems. Example: The backyard behind Mr. J n n’ u i a rectangle. A sidewalk from one corner of the backyard to the opposite corner is 76 feet long. Both the backyard and the house are 40 feet wide. What is the approximate perimeter of the backyard? Kuta software for worksheet: 6. Standardized Test Prep (SAT/HSPA) Text book page 95 Q. 47-50 7. Interactive game link for Pythagorean Theorem: www.kutasoftware.co m Text book page #95 www.mathplay.com/PythagoreanTheorem-Game.html Sketch the figure. 17 Illustrate bi-conditionals and Prepares for Pearson recognize good definitions. Bi-conditionals G.CO.10 Chapter 2-3 Example: and Definitions Prove theorems about triangles. What are the two conditional Theorems include: measures of statements that form this bi- interior angles of a triangle sum Text book conditional? to 180°; base angles of isosceles page #99-105 triangles are congruent; the A ray is an angle bisector if segment joining midpoints of and only if it divides an angle two sides of a triangle is parallel into two congruent angles. to the third side and half the length; the medians of a triangle meet at a point. 2.3 1 day Web link for bi8. conditional statement: http://www.mathgoodie s.com/lesson/ Basic: 1) Problems 1-3 Exs. 7-30, 33, 35-36, 43, 45, 49-57 Kuta software for worksheet: www.kutasoftware.co m 9. Average: 1) problems 1-4 Exs. 7-30, 33, 35-36, 43, 45, 49-57 you tube video tutorial: www.youtube.com/wat ch?v=12CeL-hFky8 10. Advanced: 1) Problems 1-4 Exs. 7-29 odd, 30-57 Math power point notes: Google search: teachers.henric k va u … C n dititionalStatement/LECTURE21.ppt www a c rg … G f c fu i n u /get_group_file.phtml?... u ry… www.ohio.edu/people/melkonia/ math306/slides/logic2.ppt 18 Math skills practice: In ran a fa k 0of%20Logic a u … Law % PDF worksheet link: http://mycoursecan.com/Files/Sub jects/Ge 19 Basic skills Review: Apply reflections: Example: G.CO.1 Pearson Use the undefined notion of a Chapter 2-3 HSPA point, line, distance, along a line PREP/PARCC/S and distance around a circular arc to develop definitions for AT is the image when Text book angles, circles, parallel lines, point F is reflected over page #104 perpendicular lines and line the line and segments. then over the line The location of 30 mn is Which of the following is the location of point F ? Web link on reflection: 11. www.regentspre.org/Re gents/math/geometry/G T1/reflect.htm Standardized Test Prep (SAT/HSPA) Text book page 104 Q. 49-51 12. Interactive game link for Pythagorean Theorem: www.mangahigh.com/en_us /games/translar a. b. c. d. 20 2-4 Deductive reasoning 1 day Prepares for: Apply the law of detachment G.CO.11 and the law of syllogism: Prove theorems about Example: parallelograms. Theorems include: opposite sides are What can you conclude from congruent, opposite angles are the given true statement? congruent, the diagonals of a If a u n g an “A” n a parallelogram bisect each other, final exam, then the student and conversely, rectangles are parallelograms with congruent will pass the course. diagonals. Pearson Chapter 2-4 Text book page # 106112 Web links for bi-conditional statements: rewww.khanacademy.or g/.../geometry.../cageometry--deductive- 13. Basic: 1) Problems 1-3 Exs. 6-21, 26, 28, 30, 33-39 Average: www.sparknotes.com/... 14. /geometry3/inductivean 1) problems 1-3 ddeductivereasoning/se ct.. Exs. 7-17 odd, 18-30, Kuta software for worksheet: 33-39 15. Advanced: www.gobookee.net/ded uctive-reasoning-kuta/ 1) Problems 1-3 www.mybookezz.org/ge 17 odd, 18-39 Exs. 7- ometric-mean-kutasoftware-1344/ you tube video tutorial: Math power point notes: www.khanacademy.org/ Google search: .../geometry.../cawww.cecs.csulb.edu/~mopkins/cecs100/D geometry--deductive-... eductInduct.pptx www.youtube.com/wat ch?v=GluohfOedQE www.taosschools.org/.../GeometryPPTs/2 .3Deductive%20Reasoning.ppt 21 Math skills practice: www.csun.edu/~kme52026/Chapter4.pdf www.brighthubeducation.com › Lessons: Grades 9-12 › Math PDF worksheet link: www.frapanthers.com/.../Geometry( H)/worksheets/WorksheetDeductiv e www.matsuk12.us/cms/lib/AK010009 53/Centricity/.../geo2_1WS.pdf 22 Basic skills Review: G.CO.2-5 Develop and perform rigid HSPA transformations that include Triangle ABC is shown in the PREP/PARCC/S reflections, rotations, coordinate plane below. Draw translations and dilations using AT the result of the geometric software, graph transformation when triangle paper, tracing paper, and ABC is translated 6 units to geometric tools and compare the right and then rotated them to non-rigid clockwise about the transformations. origin. 30 Apply transformations: Pearson Section 2-4 More HSPA PREP Text book page #112 Q. 33-34 Kuta software for 16. Standardized Test Prep worksheet (SAT/HSPA) www.kutasoftware.com Text book page 112 Q. 33, /FreeWorksheets/GeoW 34 orksheets/12All%20Tran 17. Interactive game link for Pythagorean Theorem: www.kidsmathgamesonline. com/geometry/transformati on.html mn 23 2-5 Prepares for: Connect reasoning in G.CO.9 algebra and Prove theorems about lines and geometry. angles. Theorems include: Example: vertical angles are congruent; when a transversal crosses What is the name of the parallel lines, alternate interior property of equality or angles are congruent and congruence that justify going corresponding angles are from the first statement to congruent; points on a the second statement? perpendicular bisector of a line segment are exactly those qui i an fr g n’ 1. 2x + 9 =19 endpoints. Reasoning in Algebra and Geometry 1 day 2. and so 3. Pearson Chapter 2-5 Text book page # 113119 Web link for reasoning 4. Basic: in algebra and 1) Problems 1-3 geometry: Exs. 5-17, 20, 22, 23 www.mathplayground.c om/games.html 29-41 5. www.xpmath.com Average: 1) problems 1-3 Exs. 5-13 odd, 14-24, Kuta software for worksheet: www.gobookee.net/geo metry-review-5-kutaanswers/ 29-41 6. Advanced: 1) Problems 1-3 Exs. 5-13 odd, 14-41 Math power point notes: you tube video tutorial: Google search: www.cvsd.org/.../Geometry.../2www.youtube.com/wat 5%20Reasoning%20in%20Algebra ch?v=xkTgnN5pOks %20a vimeo.com/49030863 jcs.k12.oh.us/.../Geometry/PH_G eo_2a ning in A g ra Math skills practice: www.nhvweb.net/nhhs/math/ms chuetz/files/.../Section-2-5-and-224 6.pdf www.brighthubeducation.com › › Math Lessons: Grades 9-12 PDF worksheet link: www.frapanthers.com/.../Geome try(H)/worksheets/WorksheetDed uctive www.quia.com/files/quia/users/a lamed/Geoguide/Geoguide2.5 25 Basic skills Review: Use reflections, rotations, and HSPA transformations: PREP/PARCC/S Example: AT Triangle ABC and triangle LMN are shown in the coordinate plane below. G.CO.6-8 Pearson Use rigid transformations to determine, explain and prove congruence of geometric figures. Section 2-5 More HSPA PREP Text book page #119 Q. 29-33 Kuta software for worksheet: 7. www.kutasoftware.com /freeige.html Standardized Test Prep (SAT/HSPA) Text book page 119 Q. 29- 33 8. Interactive game link for Pythagorean Theorem: www.mathplayground.com/ ShapeMods/ShapeMods.ht ml 30 mn Part A: Explain why triangle ABC is congruent to triangle LMN using one or more reflections, rotations, and translations. Part B: Explain how you can use the transformations described in Part A to prove triangle ABC is congruent to triangle LMN by any of the criteria for triangle 26 congruence (ASA, SAS, or SSS). 27 2.6 Proving Angles Prove and apply theorems about angles: Congruent Example: Write a paragraph proof: Given: are supplementary. are supplementary. Prove: G.CO.9 Pearson Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those qui i an fr g n’ endpoints. Chapter 2-6 Text book page # 120127 Web link for proving 9. theorems about angles: www.mathwarehouse.c om/.../triangles/similartriangle-theorems.php Basic: 1) problems 1-2 Exs. 6-12, 46-48 2) Problems 3 Exs. 13-14, 20-21, 25, www.khanacademy.org/ 26, 28 .../angles/v/angleAverage: bisector-theorem-proo 10. 1) problems 1-2 Kuta software for worksheet: Exs. 7-11 odd, 36-48 2) problems 3 www.letspracticegeome try.com/free-geometryworksheets/ Exs. 13-30 11. 3 you tube video tutorial: www.youtube.com/wat ch?v=gq1B3ceW4TE 2 day Advanced: 1) Problems 1-2 Exs. 7-11 odd, 36-48 2) Problems 3 1 Exs. 13-35 www.youtube.com/wat ch?v=G_RsPC2dKHM Math power point notes: Google search: 2 www.taosschools.org/.../GeometryP PTs/4.4%20ASA%20AND%20AAS 28 www.dgelman.com/powerpoints/.../ 2.6%20Proving%20Statements%20a. . Math skills practice: Basic skills Review: HSPA PREP/PAR CC/SAT G.CO.6, 7, 8 Use transformations: Use rigid transformations to determine, explain and prove Quadrilateral PQRS is shown congruence of geometric below. Which of the following figures. transformations of triangle PTS could be used to show that triangle PTS is congruent to triangle QTR ? Pearson Section 2-6 More HSPA PREP Text book page #127 Kuta software for worksheet: 12. ww2.d155.org/.../Geom etry%20363%20.../Ch%2 % % Pack Standardized Test Prep (SAT/HSPA) Text book page 127 Q. 36- 39 13. Interactive game link for transformation www.onlinemathlearning.co m/transformation-ingeometry 30 mn 1) A reflection over segment QS 2) A reflection over segment PR 29 3) A reflection over line m 4) A reflection over line l 30 3.1 Lines and Angles Identify relationships between figures in space. Identify angles formed by two lines and a transversal. Example: Think of each segment in the diagram as part of a line. Which of the lines appears to fit the description? a. Parallel to line AB and contains D 1 day b. Perpendicular to line AB and contains D c. Skew to line AB and contains D d. Name the plane(s) that contain D and appear to be parallel to plane ABE. G.CO.1 Pearson Use the undefined notion of a point, line, distance, along a line and distance around a circular arc to develop definitions for angles, circles, parallel lines, perpendicular lines, and line segments. Chapter 3-1 Text book page # 140146 Web link for properties 14. Basic: and parallel lines: 1) problems 1-3 www.nexuslearning.net/ Exs. 11-29 all, 30-44 .../ML%20Geometry%20 3-1%20Lines% an % Even, 49-60 www.khanacademy.org/ 15. Average: .../angles/v/angle1) problems 1-3 bisector-theorem-proo Exs. 11-23 odd, 25, 45, 49-60 Kuta software for 16. Advanced: worksheet: www.kutasoftware.com 1) Problems 1-3 /.../13Exs. 11-23 odd, 25 - 60 Line%20Segment%20Co n ruc n f Math power point notes: you tube video tutorial: Google search: www.khanacademy.org/ www.eht.k12.nj.us/~Simmonsg/lines .../segments.../lines-- %20ppt.ppt line-segments--and-ra.. www2.carrollk12.org/instruction/el www.youtube.com/wat emcurric/.../line%20powerpoint. ppt c v nv wI Math skills practice: www.mathwarehouse.com/.../triangles/si milar-triangle-theorems.php 31 www.sanjuan.edu/webpages/john higgins/files/81%20Practice PDF worksheet link: tms6thgrade.weebly.com/uploads /8/5/1/7/8517785/practice_81 32 Basic skills Review: HSPA PREP/PAR CC/SAT G. CO.12, 13 Use perpendicular lines Example: Use paper folding to construct the perpendicular bisector of line segment shown below. Trace and label the line segment JK. Generate formal constructions with paper folding, geometric software and geometric tools to include, but not limited to, the construction of regular polygons inscribed in a circle. Pearson Section 3-1 More HSPA PREP Text book page #146 Kuta software for worksheet: 17. Standardized Test Prep (SAT/HSPA) Text book page 146 Q. 49- 52 www.kutasoftware.com /FreeWorksheets/Geo Worksheets/218. Interactive game link for Line%20Seg transformation www.sheppardsoftware.com/ mathgames/geometry/.../line _shoot.htm 30 mn 33 3.2 properties of parallel lines 2 days Pearson Prove theorems about parallel lines. G.CO.9 Chapter 3-2 Prove theorems about lines and Use properties of parallel angles. Theorems include: lines to find angle measures. vertical angles are congruent; Text book page # 147when a transversal crosses Example: parallel lines, alternate interior 155 angles are congruent and Complete the proof of the Consecutive Interior Angles corresponding angles are congruent; points on a Theorem. perpendicular bisector of a line GIVEN: p q segment are exactly those qui i an fr g n’ PROVE: 1 and 2 are endpoint. supplementary. Web link for reasoning 19. in algebra and geometry: www.slideshare.net/1co nejo/proving-lines-areparallel web.mnstate.edu/peil/g 20. eometry/.../6ExteriorAn gleR.htm Kutasoftware for worksheet: www.kutasoftware.com fr ig you tube video tutorial: 21. www.youtube.com/wat c v L G Basic: 1) problems 1-2 Exs. 7-11, 29-39 2) Problems 3-4 Exs. 12-18, 29-39 Average: 1) problems 1-2 Exs. 7-11 odd, 29-39 2) problems 3-4 Exs. 13-17 odd,18-26 Advanced: 1) Problems 1-2 Exs. 7-11 odd, 29-39 2) Problems 3-4 Exs. 13-17 odd,18-28 Math power point notes: Google search: teachers.henrico.k12.va.us/.../02Perp endicularParallel/...5ProvingLinesP a... 34 geometryf.mths.schoolfusion.us/modules/.../get_gro Math skills practice: www.nexuslearning.net/.../ML%20Ge ometry%203-3%20Parallel%20Lin PDF worksheet link: www.bowerpower.net/geometry/ch03/ Wksh%203.2B.pdf 35 Basic skills Review: HSPA PREP/PAR CC/SAT 12. Prove theorems about parallel lines. Example: Using the figure above and 30 the fact that line is parallel mn to segment prove that the sum of the angle measurements in a triangle is G.CO.9, 10, 11 Pearson Create proofs of theorems involving lines, angles, triangles, and parallelograms.* (Please note G.CO.10 will be addressed again in unit2 and G.CO.11 will be addressed again in unit 4) Section 3-2 More HSPA PREP Text book page #155 Kuta software for worksheet: 22. Standardized Test Prep (SAT/HSPA) Text book page 155 Q. 29- 32 23. Interactive game link for transformation www.kutasoftware.com fr ig www.onlinemathlearning.co m/proving-parallellines Use as many or as few rows in the table as needed. 36 3.3 Proving Lines are Parallel 2days G.CO.9 To determine whether two lines are parallel. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses Example: parallel lines, alternate interior In the diagram at the right, angles are congruent and corresponding angles are each step is parallel to the congruent; points on a step immediately below it perpendicular bisector of a line and the bottom step is parallel to the floor. Explain segment are exactly those g n’ why the top step is parallel to qui i an fr endpoint. the floor. Pearson Chapter 3-3 Text book page # 156163 Web link for proving 24. Basic: lines are parallel 1) problems 1-2 https://dionmath.wikisp Exs. 7-11, 18-28 even aces.com/.../3.5+Showi ng+Lines+are+Parallel.p 2) Problems 3-4 pt... Exs. 12-16 all, 30-34 https://s3.amazonaws.c Even, 35, 36,38, 47-57 om/engrademyfiles/.../03-0525. Average: Kutasoftware for worksheet: 1) problems 1-2 Exs. 7-11 odd, 17-28 www.kutasoftware.com 2) problems 3-4 37 /.../3Parallel%20Lines%20an % ran v r a Exs. 13-15 odd, 29-41, 47-57 26. Advanced: you tube video tutorial: 1) Problems 1-2 www.khanacademy.org/ Exs. 7-11 odd, 17-28 .../parallel...lines/.../ide 2) Problems 3-4 ntifying-parallelExs. 13-15 odd, 28-57 Math power point notes: Google search: teachers.henrico.k12.va.us/.../02 PerpendicularParallel/...5Proving LinesPa Math skills practice: www.regentsprep.org/Regents/ math/ALGEBRA/AC3/pracParallel PDF worksheet link: glencoe.mcgrawhill.com/sites/dl/free/007888484 5/634463/geohwp.pdf 38 Basic skills Review: In HSPA PREP/PAR CC/SAT 30 mn Proving that a quadrilateral is a parallelogram. In the quadrilateral below, and Prove that the quadrilateral is a parallelogram. G.CO.9 Pearson Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those qui i an fr g n’ endpoint Section 3-3 More HSPA PREP Text book page #163 Kuta software for worksheet: 27. www.kutasoftware.com /.../3Proving%20Lines%20Par allel f Standardized Test Prep (SAT/HSPA) Text book page 163 Q. 47- 51 28. Interactive game link for transformation www.mathplay.com/AnglesJeopardy/AnglesJeopardy.html Write an informal proof. __________________________ __________________________ __________________________ __________________________ __________________________ __________________________ __________________________ __________________________ __________________________ __________________________ __________________________ __________________________ __________________________ ___________________ 39 4.4 Using corresponding parts of congruent triangles. Apply triangle congruence and corresponding parts of congruent triangles. Prove that parts of two triangles are congruent. Example: 1day G.SRT.5 Pearson Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Chapter 4-4 Text book page # 244256 Web link for proving 29. Basic: that two parts of two 1) problems 1-2 triangles are congruent. www c a z n c › Geometry Concepts and Skills › Chapter 5 Exs. 5-8 all, 10-16 even 17, 20, 23-32 salinesports.org/mr_fre 30. Average: derick/GeomCS/Unit%2 1) problems 1-2 g an f Kutasoftware for worksheet: Exs. 5, 7, 9-20, 23-32 Advanced: www.kutasoftware.com 31. /.../41) Problems 1-2 Congruence%20and%20 Triangles f Exs. 5-7, 9-32 you tube video tutorial: Math power point notes: Google search: 40 www.khanacademy.org/ www.grossmont.edu/carylee/Ma .../congruent126/lectures/Chapter14.ppt triangles/...triangle/.../fi Math skills practice: n i www.regentsprep.org/Regents/ math/geometry/GP4/PracCongTr i PDF worksheet link: www.kutasoftware.com/FreeWo rksheets/GeoWorksheets/4Congruence 41 Basic skills Review: HSPA PREP/PAR CC/SAT Prove vertical angles are congruent: G.CO.9 Pearson Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those qui i an fr g n’ endpoint Section 4-4 More HSPA PREP Text book page #224 Kuta software for worksheet: 32. www.kutasoftware.com /.../4Right%20Triangle%20Co ngruence f Standardized Test Prep (SAT/HSPA) Text book page 224 Q. 5053 33. Interactive game link for transformation www.mathsisfun.com/geom etry/trianglescongruent.html 30 mn U Using the figure above, prove that vertical angles are congruent. Use as many or as few rows in the table as needed. 42 4.5 Isosceles and Equilateral Triangles 2 days G.CO.10 Pearson Apply properties of isosceles Prove theorems about triangles. Chapter 4-5 and equilateral triangles. Theorems include: measures of interior angles of a triangle sum Example: to 180°; base angles of isosceles Text book Rock Climbing triangles are congruent; the page # 250In one type of rock climbing, segment joining midpoints of 257 climbers tie themselves to a two sides of a triangle is parallel rope that is supported by to the third side and half the anchors. The diagram shows a length; the medians of a triangle red and a blue anchor in a meet at a point. horizontal slit in a rock face. Web link for 34. Basic: applying properties 1) problems 1-2 of isosceles and equilateral triangles. Exs. 6-12 all, 37-44 www.mathworksheet.org/isosceles -and-equilateraltriangles 2) Problems 3 Exs. 13-15, 16-24 even www.kutasoftware.com 28-32 even /FreeWorksheets/GeoW 35. Average: orksheets/4Isosceles% 1) problems 1-2 Kutasoftware for 43 worksheet: Exs. 7-11 odd, 37-44 www.kutasoftware.com 2) problem 3 /FreeWorksheets/GeoW Exs. 13-32 orksheets/436. Advanced: Isosceles% you tube video tutorial: 1) Problems 1-3 Exs. 7-13 odd, 14-44 www.khanacademy.org/ ...triangles/.../equilatera Math power point notes: l-and-isoscelesGoogle search: yourcharlotteschools.net/Schools /PCHS/Dubbaneh_site/4.5a.ppt Math skills practice: library.thinkquest.org/20991/tex tonly/quizzes g q PDF worksheet link: www.kutasoftware.com/FreeWo rksheets/GeoWorksheets/4Isosceles% 44 Basic skills Review: HSPA PREP/PAR CC/SAT 30 mn G.CO.11 Prove theorems about parallelograms. Theorems Prove that two angles or line include: opposite sides are segments are congruent: congruent, opposite angles are In isosceles ABC, the vertex congruent, the diagonals of a parallelogram bisect each other, angle is A. What can you and conversely, rectangles are prove? parallelograms with congruent 1. AB = CB diagonals. 2. Pearson Section 4-5 More HSPA PREP Text book page #256 3. 4. Kuta software for worksheet: 5. www.kutasoftware.com /FreeWorksheets/GeoW orksheets/4Isosceles% Standardized Test Prep (SAT/HSPA) Text book page 256 Q. 3740 6. Interactive game link for transformation www.mathsisfun.com/trian gle BC = AC INSTRUCTIONAL FOCUS OF UNIT In previous grades, students were asked to draw triangles based on given measurements. Students also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. Students use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about lines, angles, triangles, quadrilaterals, and other polygons. Students also apply reasoning to complete geometric constructions and explain why constructions work. PARCC FRAMEWORK/ASSESSMENT 7. PARCC EXEMPLARS: www.parcconline.org (copy & paste the URL or link into search engine) 8. Dollar Line: http://balancedassessment.concord.org/hs033.html Example: 45 Think of a situation which could be represented by the graph below, Write a full description of this situation (be sure to tell what each axis represents in your story.) 9. 10. How would you determine whether two lines are parallel or perpendicular? Consider parallelogram ABCD with coordinates A(2,-2), B(4,4), C(12,4) and D(10,-2). Perform the following transformations. Make predictions about how the lengths, perimeter, area and angle measure will change under each transformation. a. A reflection over the x-axis. b. A rotation of 270about the origin. c. A dilation of scale factor 3 about the origin. d. A translation to the right 5 and down 3. Verify your predictions. Compare and contrast which transformations preserved the size and/or shape with those that did not preserve size and/or shape. Generalize, how could you determine if a transformation would maintain congruency from the pre-image to the image? 11. Prove that any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the line. 46 12. A carpenter is framing a wall and wants to make sure his the edges of his wall are parallel. He is using a cross-brace. What are several different ways he could verify that the edges are parallel? Can you write a formal argument to show that these sides are parallel? Pair up with another student who created a different argument than yours, and critique their reasoning. Did you need to modify the diagram in anyway to help your argument? 13. Andy and Javier are designing triangular gardens for their yards. Andy and Javier want to determine if their gardens that they build will be congruent by looking at the measures of the boards they will use for the boarders, and the angles measures of the vertices. Andy and Javier use the following combinations to build their gardens. Will these combinations create gardens that enclose the same area? If so, how do you know? a. Each garden has length measurements of 12ft, 32ft and 28ft. b. Both of the gardens have angle measure of 110, 25and 45. c. One side of the garden is 20ft another side is 30ft and the angle between those two boards is 40. d. One side of the garden is 20ft and the angles on each side of that board are 60and 80. e. Two sides measure 16ft and 18ft and the non-included angle of the garden measures 30. Wiki page for Common Core Assessments: http://maccss.ncdpi.wikispaces.net/file/view/Geometry%20Unpacking062512%20.pdf/359510147/Geometry%20Unpacking062512%20.pdf PARCC Framework Assessment questions with Model Curriculum Website for all units: http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf 47 http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsGeo_Nov2012V3_FINAL.pdf http://www.nj.gov/education/assessment/hs/hspa_mathhb.pdf 21ST CENTURY SKILLS (4Cs & CTE Standards) 1. Career Technical Education (CTE) Standards 1. st 21 Century Life and Career Skills: All students will demonstrate the creative, critical thinking, collaboration, the problem-solving skills needed to function successfully as both global citizens and workers in diverse ethnic and organizational cultures. 9.1.12.B.1: Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using multiple perspectives. 2. Personal Financial Literacy: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 9.2.12.B.3: Construct a plan to accumulate emergency “rainy day” funds. 3. Career Awareness, Exploration, and Preparation: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age. 9.3.12.C.2: Characterize education and skills needed to achieve career goals, and take steps to prepare for postsecondary options, including making course selections, preparing for and taking assessments, and participating in extra-curricular 48 activities. 4. Career and Technical Education: All students who complete a career and technical education program will acquire academic and technical skills for careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees. 9.4.12.B.4: Perform math operations, such as estimating and distributing materials and supplies, to complete classroom/workplace tasks. Project Base Learning Activities: http://www.achieve.org/files/CCSS-CTE-Task-FramingaHouse-FINAL.pdf MODIFICATIONS/ACCOMMODATIONS 1. Group activity or individual activity 2. Review and copy notes from eno board/power point/smart board etc. 3. Group/individual activities that will enhance understanding. 4. Provide students with interesting problems and activities that extend the concept of the lesson 5. Help students develop specific problem solving skills and strategies by providing scaffolded guiding questions 49 Peer tutoring 1. Team up stronger math skills with lower math skills Use of manipulative 1. Eno or smart boards 2. Dry erase markers 3. Reference sheets created by special needs teacher 4. Pairs of students work together to make word cards for the chapter vocabulary 5. Use 3D shapes for visual learning 6. Reference sheets for classroom 7. Graphing calculators APPENDIX (Teacher resource extensions) 1. CCSS. Mathematical Practices: MP1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and 50 meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. MP2: Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. MP3: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can 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 that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are 51 not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. MP4: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. MP5: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 52 MP6: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. MP7: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MP8: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the 53 reasonableness of their intermediate results. 2. Kuta G 1: Kuta Software – Geometry (Free Worksheets) 3. Teacher Edition: Geometry Common Core by Pearson 4. Student Companion: Geometry Common Core by Pearson 5. Practice and Problem Solving Workbook: Geometry Common Core by Pearson 6. Teaching with TI Technology: Pearson Mathematics by Pearson 7. Progress Monitoring Assessments: Geometry Common Core by Pearson 8. http://www.jmap.org/JMAP_RESOURCES_BY_TOPIC.htm#Geo 9. http://www.mathopenref.com/ 10. http://www.mathisfun.com/ 11. http://www.mathwarehouse.com/ 12. http://www.regentsprep.org/Regents/math/geometry/math-GEOMETRY.htm 13. http://www.cpm.org/pdfs/state_supplements 14. http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqgeomapr15.pdf 15. http://illuminations.nctm.org 16. http://www.state.nj.us/education/cccs/standards/9/ Notes to teacher (not to be included in your final draft): 54 4 Cs Three Parts Objective Creativity: projects Behavior Critical Thinking: Math Journal Condition Collaboration: Teams/Groups/Stations Demonstration of Learning (DOL) Communication – Power points/Presentations UNIT 2 Similarity, Proof And Trigonometry Total Number of Days: 42 days ESSENTIAL QUESTIONS 17. Grade/Course: __Geometry 10th grade__ ENDURING UNDERSTANDINGS How do you identify corresponding parts of congruent 23. If two triangles are congruent, then every pair of their corresponding parts is also triangles? congruent. 55 18. 19. How do you show that two triangles are congruent? How do you use proportions to find side lengths in similar polygons? 20. How do you show two triangles are similar? 24. Two ways triangles can be proven to be congruent are by using three pairs of corresponding sides or by using two pairs of corresponding sides and the pair of corresponding angles included between those sides. 25. Two geometric figures are similar when corresponding lengths are proportional and corresponding angles are congruent. 21. How do you find a side length or angle measure in a right triangle? 26. Ratios and proportions can be used to prove whether two polygons are similar and to find unknown side lengths 22. How do trigonometric ratios relate to similar right triangles? 27. If the lengths of any two sides of a right triangle are known, the length of the third side can be found by using the Pythagorean Theorem. 28. Ratios can be used to find side lengths and angle measures of a right triangle when certain combinations of the side lengths and angle measures are known RESOURCES STANDARDS PACE CONTENT SKILLS (CCSS/MP) OTHER Pearson LEARNING ACTIVITIES/ASSESSMENTS (e.g., tech) To Relate Parallel and Perpendicular G.MG.3 Apply Text book geometric methods page 164lines to solve design 170 Example: problems (e.g., 3.4 1 day Parallel and Perpendicular lines designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Teacher made power points Page 170 Concept Byte – presentations. Perpendicular Lines and Plane http://www.summit.k12.co. Page 167: 1 - 5 us/cms/lib04/CO01001195/ Centricity/Domain/565/cgp0 3gad.pdf 29. Basic – Problems 1-2 Ex. 6 – 9, 12 – 16 even, 17 – 18, 3139 56 http://www.mathsisfun.com 30. Average – Problems 1 – 2 /algebra/line-parallelEx. 6 – 18 , 31 –39 perpendicular.html MP 1 MP 3 31. Advanced – Problems 1-2 Ex 6 -26, 31 - 39 To use properties of Parallel Lines G.MG.3 See below Text book page 167 http://www.regentsprep.org Standardized Test Prep /Regents/math/ALGEBRA/A Q. 27 - 30 C3/Lparallel.htm Question: Basic Skills Review 30 mins PARCC/HSPA PREP Using the given information, state the theorem that allows you to conclude that j || k http://www.mathsisfun.com /algebra/line-parallelperpendicular.html 57 Examine and identify that two figures are congruent and identify their corresponding parts Example: In the diagram of 4.1 Congruent 2 days Figures below, , and and . G.SRT.5 Use Text book congruence and page 218 similarity criteria 224 for triangles to solve problems and to prove relationships in geometric figures. MP 1 PowerGeometry.com Interactive Practice games Page 221: 1 - 7 Teacher made power points Building congruent triangles activity page 225 Practice problem solving exercises Mathopenref.com/tocs/con page 222 – 224. gruencetoc.html 32. Basic – Problems 1-2 Ex. 8 – 29, 50 – 61 http://jmap.org/htmlstandar d/Geometry/Informal_and_ 33. Average – Problems 1 – 4 Formal_Proofs/G.G.28.htm Ex. 9 – 29 odd, 50 –61 MP 3 MP 4 MP 6 34. Advanced – Problems 1-4 Ex 9-29 odd, 50-61 Basic Skills Review 30 mins PARCC/HSPA PREP To find angle measures of a triangle using Angle Sum Theorem G.SRT.5 See below Text book page 224 Mathopenref.com/tocs/con gruencetoc.html Page 224: Standardized Test Prep – Q. 50 - 53 Question: The measure of one angle In a triangle is 80˚ r w ang ar c ngru n What is the measure of each? 58 Conclude and defend that two triangles congruent using the SSS and SAS Postulates SSS stands for "side, side, side" and means that we have two triangles with all three sides equal. G.SRT.5 Use Text book congruence and page 226 similarity criteria 230 for triangles to solve problems and to prove relationships in geometric figures. SAS stands for "side, angle, side" and 4.2 Triangle MP 1 2 days Congruence by means that we have two triangles where we know two sides and the SSS and SAS MP 3 included angle are equal. PowerGeometry.com 35. Teacher made power points 36. 37. 38. http://mathopenref.com/co ngruentsss.html 39. http://www.mathsisfun.com /geometry/congruent.html 40. Page 230:1-7 Lesson Quiz Practice and Problem Solving Exercises page 230 Practice and Problem Solving Exercises Basic – Problems 1-2 Ex. 8 – 12, 35– 46 Average – Problems 1 – 2 Ex. 9, 35 –46 41. Advanced – Problems 1-3 Ex 9-13 odd, 15- 46 Basic Skills Review 30 mins PARCC/HSPA PREP To Find the coordinates of one endpoint using the midpoint formula. G.C0.1 Question: Find the point on a directed line segment between A segment has a midpoint at (2, 2) G.GPE.6 Text book page 233 http://jmap.org/htmlstand 42. Standardized Test Prep ard/Geometry/Coordinate Questions 35 – 38 _Geometry/G.G.67.htm 59 and an endpoint at (-2, 4). What are two given points the coordinates of the other endpoint that partitions the segment in a given of the segment? ratio. Construct a proof defending that two triangles congruent using the ASA and AAS Postulates G.SRT.5 Use Text book congruence and page 234 similarity criteria 240 for triangles to ASA an f r "ang , i , ang ” solve problems and means that we have two and to prove triangles where we know two angles relationships in and the included side are equal. geometric figures. 4.3 Triangle 2 days Congruence by ASA and AAS For example: the 2 triangles below are congruent MP 1 * PowerGeometry. Com 44. Page 238: 1 - 7 * Teacher made power point presentations 45. Lesson Quiz page 241A 46. Basic – Problems 1-2: Ex. 8 43. http://www.mathsisf – 12, 32- 39 un.com/geometry/congru 47. Average – Problems 1 – 2: ent.html Ex. 9 – 11 odd, 32-39 48. Advanced – Problems 1-2: Ex 9-11 odd, 32 – 39 MP 3 AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal. 60 For example: the 2 triangles below are congruent To write converses, inverses and contrapositives of conditionals. Basic Skills Review 30 mins PARCC/HSPA PREP 1 day Question: ri c nv r f “ If y u ar than 18 years old, then you are too y ung v in ni Sa ” To prove right triangles congruent 4.6 Congruence using the Hypotenuse Leg Theorem in Right Triangles Example: Prepares for G.CO.11 Prove Text book page 241 theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. G.SRT.5 Use Text book congruence and page 258 similarity criteria 264 for triangles to solve problems http://www.cpm.org/pdfs/st Standardized Test Prep Questions ate_supplements/Logical_St 32 - 35 atements.pdf http://www.finneytown.or g/Downloads/GETE04062.p Page 261: 1 - 7 df 61 and to prove relationships in geometric figures. 49. http://www.mathopenref.co Ex. 8 – 16, 20,22,25,32 - 36 m/congruenthl.html 50. Average – Problems 1 – 2 MP 1 Ex. 9 , 11– 28, 32 –36 MP 3 51. Question: Basic Skills Review 30 mins PARCC/HSPA PREP Advanced – Problems 1-2 http://www.regentsprep.org /Regents/math/geometry/G Ex 9, 11-28, 32 - 36 P4/Ltriangles.htm Solution To determine whether given triangles are congruent. Basic – Problems 1-2 G.SRT.5 See above Text book page 264 http://www.regentsprep.o 52. Standardized Test Prep rg/Regents/math/geometr Questions 29 – 31 y/GP4/Ltriangles.htm Determine whether the triangles are congruent. If they are, write a congruent statement. 62 To identify congruent overlapping triangles, prove two triangles congruent using other congruent triangles For example: G.SRT.5 Use Text book congruence and page 265 – similarity criteria 271 for triangles to solve problems and to prove relationships in geometric figures. Separate and redraw DFG and 4-7 2days EHG. Identify the common angle. MP 1 Congruence in Overlapping Triangles 30 mins PARCC/HSPA PREP Page 268: 1 - 7 Basic – Problems 1-2: Ex. 8 – 16, 17 – 20, 33 - 37 http://www.youtube.com/w 55. atch?v=wlEYuPhShig Average – Problems 1 – 2: Ex. 9– 15 odd, 17 , 19 -26, 33 37 56. Teacher made power point presentation Advanced – Problems 1-2: Ex 9-15 odd, 17, 19 – 28, 33 - 37 MP 3 http://www.sophia.org/over lappingtriangles/overlappingtriangles--2tutorial?topic=congruenttriangles Solution Basic Skills Review http://www.finneytown.org 53. /Downloads/GETE04072.pdf 54. To prove congruent segments in overlapping triangles. Question: G.SRT.5 See above Text book page 271 http://www.sophia.org/over 57. Standardized Test Prep lappingQuestions 29 – 32 triangles/overlappingtriangles--2tutorial?topic=congruenttriangles 63 To use properties of midsegments to G.CO.10 solve problems G.CO.12 For example: 1 day 5-1 Midsegments of Triangles G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Text book http://www.finneytown.org 58. page 284 – /Downloads/GETE05012.pdf 291 59. 60. http://www.regentsprep.org /Regents/math/geometry/G 61. P10/MidLineL.htm 62. http://www.mathopenref.co m/trianglemidsegment.html Page 284 Concept Byte – Investigating Midsegments Page 288: 1 - 6 Basic – Problems 1-3: Ex. 7 – 26, 28 – 30, 32 – 42 even Average – Problems 1 – 3: Ex. 7– 25 odd, 26 -45, Advanced – Problems 1-3: Ex 7-25 odd, 26 – 48, 53 - 57 MP 1 MP 3 MP 5 64 To find the midsegment of a triangle G.CO.10 Question: G.CO.12 Text book page 291 http://www.regentsprep.org 63. Standardized Test Prep /Regents/math/geometry/G Questions 49 - 52 P10/MidLineL.htm G.SRT.5 See above Basic Skills Review 30 mins PARCC/HSPA PREP 1. The triangular face of the rock and Roll Hall of Fame in Cleveland, Ohio is isosceles. The length of the base is 229ft 6in. What is the length of the highlighted segment? 2. Explain your reasoning. 65 To use properties of perpendicular bisectors and angle bisectors G.CO.9 5-2 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Perpendicular and Angle Bisectors MP 1 For example: Text book http://163.150.89.242/YHS/ 64. page 292 – Faculty/AB/bagg/Geometry/ images/Geometry%20text% 299 65. 20PDFs/5.6.pdf 66. http://www.youtube.com/w 67. atch?v=wxsr8egcq0M 68. http://www.mathopenref.co m/bisectorperpendicular.ht ml 1 day Page 284 Concept Byte – Investigating Midsegments Page 288: 1 - 6 Basic – Problems 1-3: Ex. 7 – 26, 28 – 30, 32 – 42 even Average – Problems 1 – 3: Ex. 7– 25 odd, 26 -45, Advanced – Problems 1-3: Ex 7-25 odd, 26 – 48, 53 - 57 MP 3 MP 5 Basic Skills Review 30 mins PARCC/HSPA PREP To construct perpendicular bisector G.CO.10 of a triangle G.CO.12 Question: G.SRT.5 Text book page 299 http://www.mathopenref.co 69. Standardized Test Prep m/bisectorperpendicular.ht Questions 39 - 42 ml See above 66 A company plans to build a warehouse that is equidistant from each of its three stores, A, B, and C. Where should the warehouse be built? Discriminate - interior and exterior angles of polygons and their sums. Determine and justify angle measures using Polygon Angle-Sum Theorems 6-1 2days The Polygon Angle-Sum For example: Theorems Find the number of degrees in each interior angle of a regular dodecagon. It is a regular polygon, so we can use the formula. In a dodecagon, n = 12. G.SRT.5 Use Text book congruence and page 353 similarity criteria 358 for triangles to solve problems and to prove relationships in geometric figures. MP 1 http://mathopenref.com/co 70. ngruentsss.html 71. http://www.regentsprep.org 72. /Regents/math/geometry/G G3/LPoly2.htm 73. Page 356: 1 - 6 Basic – Problems 1-2: Ex. 7 – 14, 49 - 54 Average – Problems 1 – 2: Ex. 7 – 13 odd, 22 – 25, 49 -54 Advanced – Problems 1-4: Ex 7-21 odd, 22 – 44, 49 -54 Activity on Exterior Angles of polygon page 352 MP 3 67 To find the sum of the measures of interior and exterior angles of polygon. Question: Basic Skills Review 30 mins PARCC/HSPA What is m∠ x ? G.SRT.5 Use Test boot congruence and page 358 similarity criteria for triangles to solve problems and to prove relationships in geometric figures. http://www.cde.ca.gov/ta/t Standardized Test Prep Questions g/sr/documents/cstrtqgeom 45 - 48 apr15.pdf PREP 2 days Determine and justify sides and Text book G.SRT.5 angles through relationships among page 359 G.CO.11 Prove 6-2 parallelograms 366 theorems about Properties of For example: parallelograms. Parallelograms Theorems include: opposite sides are In the accompanying diagram of congruent, http://jmap.org/htmlstandar 74. d/Geometry/Informal_and_ 75. Formal_Proofs/G.G.38.htm Page 363: 1 - 8 Basic – Problems 1-4: Ex. 9 – 24, 38- 41, 49- 54 76. Average – Problems 1 – 4: Ex. 9 – 23 odd, 25 - 41, 49 -54 77. Advanced – Problems 1-4: 68 parallelogram ABCD, and degrees in . Find the number of . opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Ex 9-23 odd, 25 – 44, 49 -54 MP 1 MP 3 Basic Skills Review 30 mins PARCC/HSPA PREP To apply the relationship among sides, angels and diagonals of parallelograms. G.CO.11 See aboveText book page 366 http://www.cde.ca.gov/ta/t Standardized Test Prep Questions g/sr/documents/cstrtqgeom 45 - 48 apr15.pdf Question: What values of a and b make quadrilateral MNOP a parallelogram? 69 1 day G.SRT.5 Justify that a quadrilateral is a Test book parallelogram using the properties of page 367 G.CO.11 Prove parallelogram 374 theorems about parallelograms. Theorems include: For example: opposite sides are congruent, opposite The accompanying diagram shows 6-3 angles are quadrilateral BRON, with diagonals congruent, the Proving that a and , which bisect each other diagonals of a Quadrilateral Is parallelogram bisect at X. a Parallelogram each other, and conversely, rectangles are parallelograms with congruent diagonals. http://jmap.org/htmlstandar 78. d/Geometry/Informal_and_ 79. Formal_Proofs/G.G.27.htm Page 372: 1 - 6 Basic – Problems 1-3: Ex. 7 – 16, 18- 20, 22- 28, 32-44 80. Average – Problems 1 – 3: Ex. 7 – 15, odd, 17 - 28, 32 - 44 81. Advanced – Problems 1-3: Ex 7-15 odd, 17 – 28, 32 -44 Prove: MP 1, MP 3 To determine whether a quadrilateral is a parallelogram Basic Skills Review 30 mins PARCC/HSPA PREP Question: Based on the markings, determine if the figure is a parallelogram. If so, justify your answer. Text book page 374 G.CO.11 See above G.SRT.5 http://www.cde.ca.gov/ta/t Standardized Test Prep Questions g/sr/documents/cstrtqgeom 29 - 31 apr15.pdf http://www.jmap.org/Static Files/PDFFILES/WorksheetsB yPI/Geometry/Informal_and _Formal_Proofs/Drills/PR_G. G.38_2.pdf 70 I day Analyze parallelograms to determine G.SRT.5 Text book page 375 special types G.CO.11 Prove 382 theorems about parallelograms. For example: Theorems include: opposite sides are Rectangle: congruent, parallelogram opposite angles are 6-4 with 4 right angles congruent, the Properties of Rhombus: diagonals of a Rhombus, parallelogram parallelogram Rectangle, with all 4 sides bisect each other, Square congruent and conversely, rectangles are Square: parallelograms with a rectangle with congruent all 4 sides congruent diagonals. http://www.regentsprep.org 82. /Regents/math/geometry/G 83. P9/LRectangle.htm Page 379: 1 - 6 Basic – Problems 1-3: Ex. 7 – 23, 24 – 40 even, 41, 43, 46, 47, 60 -69. 84. Average – Problems 1 – 3: Ex. 7 – 23, odd, 24 - 54, 60 - 69 85. Advanced – Problems 1-3: Ex 7-23 odd, 24 – 54, 60-69 MP 1 MP 3 71 To find a side length of a parallelogram Basic Skills Review Text book page 382 G.CO.11 See above G.SRT.5 Question: What is the height of this rectangle? http://www.jmap.org/Static Files/PDFFILES/WorksheetsB yPI/Geometry/Informal_and _Formal_Proofs/Drills/PR_G. G.38_2.pdf 30 mins PARCC/HSPA PREP Determine if a parallelogram is a Rhombus, Rectangle or Square G.SRT.5 Text book page 383 388 G.CO.11 Prove theorems about Conditions for parallelograms. 2 days Rhombus, For example: Theorems include: Rectangle, Which reason could be used to prove opposite sides are Square that a parallelogram is a rhombus? congruent, opposite angles are congruent, the 6-5 http://www.cde.ca.gov/ta/t Standardized Test Prep Questions g/sr/documents/cstrtqgeom 55 - 58 apr15.pdf http://www.jmap.org/htmls 86. tandard/Geometry/Informal 87. _and_Formal_Proofs/G.G.39 .htm Page 386: 1 - 7 Basic – Problems 1-3: Ex. 8 – 18, 24 – 31, 36 -43 88. Average – Problems 1 – 3: Ex. 9 – 13, odd, 15 - 31, 36 - 43 89. Advanced – Problems 1-3: Ex 9 -13 odd, 15 – 31, 36 - 43 72 1) Diagonals are congruent. 2) Opposite sides are parallel. 3) Diagonals are perpendicular. 4) Opposite angles are congruent. diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. MP 1 MP 3 Basic Skills Review 30 mins PARCC/HSPA PREP To determine whether a given parallelogram is a rhombus or rectangle Text book page 388 G.CO.11 See above G.SRT.5 http://www.cde.ca.gov/ta/t Standardized Test Prep Questions g/sr/documents/cstrtqgeom 32- 35 apr15.pdf Question: Each diagonal of a quadrilateral bisects a pair of opposite angles of the quadrilateral. What is the most http://www.jmap.org/Static Files/PDFFILES/WorksheetsB yPI/Geometry/Informal_and _Formal_Proofs/Drills/PR_G. 73 precise name for the quadrilateral? 1. parallelogram 2. rhombus 3. rectangle 4. not enough information To determine, verify and apply the G.SRT.5 Use Text book properties of trapezoids and Kite to congruence and page 389 similarity criteria 397 solve problems. for triangles to Example: solve problems and to prove Exa : In the diagram below of isosceles relationships in trapezoid DEFG, , geometric figures. , , , and . Find the value of 6-6 x. MP 1 2 days Trapezoids and Kites MP 3 G.38_2.pdf http://www.jmap.org/htmls 5. tandard/Geometry/Informal 6. _and_Formal_Proofs/G.G.40 .htm Page 393: 1 - 6 Basic – Problems 1-4: Ex. 7 – 24, 26 – 34 even, 46 – 49, 71 - 76 7. Average – Problems 1 – 4: Ex. 7 – 23, odd, 25 - 62, 71- 76 8. Advanced – Problems 1-4: Ex 7 -23 odd, 25 – 66, 71 - 76 74 To verify and apply properties of trapezoids and kites G.SRT.5 See above Question: Text book page 397 http://www.cde.ca.gov/ta/t Standardized Test Prep Questions g/sr/documents/cstrtqgeom 67 - 70 apr15.pdf Figure ABCD is a kite. http://www.jmap.org/Static Files/PDFFILES/WorksheetsB yPI/Geometry/Informal_and _Formal_Proofs/Drills/PR_G. G.38_2.pdf Basic Skills Review 30 mins PARCC/HSPA PREP What is the area of figure ABCD, in square centimeters? 7.1 2 days Ratios and Proportions Conclude from evidence provided which sides correspond in similar triangles and identify appropriate ratios to establish proportions and solve for a missing side length For example: If ∆ABC ∼∆ E , n i n ify appropriate ratios, establish a G.SRT.5 Use Text book 9. PowerGeometry.Co congruence and page 432 m similarity criteria 438 10. Teacher made for triangles to power points solve problems presentations and to prove relationships in geometric figures. MP 1 11. Page 436: 1 - 8 12. Test prep page 438 13. Basic – Problems 1-3: Ex. 9 – 16, 61- 69 14. Average – Problems 1 – 3: Ex. 9 – 15 odd, 33-34, 61-69 15. Advanced – Problems 1-3: Ex 9-15 odd, 33- 34, 61-69 75 proportion and solve for side length MP 3 x. MP 7 Basic Skills Review 30 mins PARCC/HSPA PREP To apply ratio and proportion to solve real life problem Prepares for Test book G.SRT.5 See below page 438 Standardized Test Prep: Questions 61 - 65 Questions: The ratio of the width to the height of i ’ c u r ni r cr n i 6: If the screen is 12 inches high, how wide is it? 76 2 days 7.2 Similar Polygons Examine similar polygons and utilize G.SRT.5 Use Text book traits of similar polygons to solve congruence and page 440 similarity criteria 446 problems for triangles to solve problems and to prove For example: relationships in geometric figures. http://jmap.org/htmlstandar d/Geometry/Informal_and_ Formal_Proofs/G.G.45.htm 16. If 19. Advanced – Problems 1-2: Teacher-made Power Point Ex 9-17 odd, 51-64 presentation , , and length of , . What is the ? , MP 1 Page 444: 1 - 8 17. Basic – Problems 1-2: Ex. 9 – 17, 51- 64 http://mathopenref.com/si 18. Average – Problems 1 – 2: milarpolygons.html Ex. 9 – 17 odd, 51-64 MP 3 Basic Skills Review 30 mins PARCC/HSPA PREP To apply scale factor to find the length of a segment G.SRT.5 Use Text book congruence and page 447 similarity criteria Question: for triangles to ∆P S ~ ∆JKL wi a ca fac r f solve problems 4:3, QR = 8cm. What is the value of KL?and to prove relationships in geometric figures. http://jmap.org/htmlstandar Standardized Test Prep: Questions d/Geometry/Informal_and_ 51 - 54 Formal_Proofs/G.G.45.htm 77 To apply the AA Similarity Postulate and G.SRT.5 Use the SAS and SSS Similarity Theorems. congruence and To use the similarity to determine and justify indirect measurements. For example: 2 days Text book page 450 similarity criteria 458 for triangles to solve problems and to prove relationships in geometric figures. Given that the triangles below are similar - If two of their angles are equal, then the third angle must also be equal, because angles of a triangle always add to 180. In G.GPE.5 7.3 Proving Triangle Similar this case the missing angle is 180 - (72 + MP 1 35) = 73 MP 3 20. Page 455: 1 - 6 http://jmap.org/htmlstandar 21. d/Geometry/Informal_and_ Formal_Proofs/G.G.45.htm 22. Basic – Problems 1-2: Ex. 7 – 12, 37 - 52 http://mathopenref.com/si 23. milarpolygons.html Advanced – Problems 1-4: Ex 7-17 odd, 18 - 52 Average – Problems 1 – 2: Ex. 7 – 11 odd, 37 - 52 Teacher-made Power Point presentations MP 3 78 Basic Skills Review 30 mins PARCC/HSPA PREP To apply the Pythagorem Theorem to find the missing side of a right triangle Question: G.SRT.8 Use Text book trigonometric page 458 ratios and the Pythagorean Theorem to solve right triangles in applied problems. http://jmap.org/htmlstandar 24. d/Geometry/Informal_and_ Formal_Proofs/G.G.48.htm 25. Standardized Test Prep: Questions 37- 40 Worksheet from the stated website A 17 ft ladder leans against a wall, if the ladder is 8ft from the base of the wall, how far is it from the bottom of the wall to the top of the ladder. 79 UNIT #3 Extending to three dimensions 80 Grade/Course: __Geometry 10th grade__ Total Number of Days: 19 days ESSENTIAL QUESTIONS 26. 27. ENDURING UNDERSTANDINGS How do you find the area of a polygon or 28. find the circumference and area of a circle? 29. How can you determine the intersection of a solid and a plane? 30. The area of a regular polygon is a function of the distance from the center to a side and the perimeter. A three-dimensional figure can be analyzed by describing the relationships among its vertices, edges, and faces. The surface area of a three-dimensional figure is equal to the sum of the areas of each surface of the figure. RESOURCES STANDARDS PACING CONTENT SKILLS LEARNING ACTIVITIES and ASSESSMENTS OTHER (CCSS/MP) Pearson (e.g., tech) 10.1 Areas of parallelograms and triangles 2 days To find the area of parallelograms and triangles. G.MG.1 Use geometric shapes, their measures, and their properties to Example: describe objects (e.g., modeling a tree trunk The piece of stained glass or a human torso as a at the bottom is made up cylinder) of eight congruent parallelograms. Each parallelogram has a base of 8 centimeters and a height of 3 centimeters. Find the MP.3 Pearson Chapter #10.1 Get Ready Web link for the area of parallelograms and 1. triangles: 1) problems 1-2 www.wyzant.com/help/math/.../a reas/parallelograms_and_tria Exs 8-13 all, 47-62 ngles Text book page #616622 2) Problems 3-4 www.virtualnerd.com/geometry/l ength-area/parallelogramtriangles-area Kuta software for worksheet: Basic Exs. 14-17 all, 19, 22, 23, 30-40 even, 37-38 2. Average: www.kutasoftware.com/.../6- 81 area of the entire piece. MP.5 Area%20of%20Triangles%20a 1) problems 1-2 Exs. 9-13 odd, 47-62. www.kutasoftware.com/.../Area %20of%20Squares,%20Rectan 2) Problems 3-4 Exs. 15-17 odd, 18-43 you tube video tutorial: www.youtube.com/watch?v=Fac www.youtube.com/watch?v=FG LWKWcg0Vo 3. Advanced: 1) Problems 1-4 Exs. 9-17 odd, 18-62 Math power point notes: Google search: granicher.wikispaces.com/.../b)+Area+of+a+Parallelog ram+%26+Triangl.. Math skills practice: www.finneytown.org/Downloads/wk10.pdf 82 Basic skills Apply distance in Use Review: the coordinate plane. Ex Example: HSPA PREP/PAR CC/SAT G.GPE.6 Pearson Find the point on a Chapter #10 directed line segment Get Ready between two given points that partitions the segment in a given ratio. Kuta software for worksheets Text book page # 622 Standardized Test Prep book page 622 Q. 47-49 (SAT/HSPA) Text www.kutasoftware.co m/.../Area%20of%20S quares,%20Rectangles 30 mn Ex On the directed line segment from R to S on the coordinate plane above, what are the coordinates of the point that partitions the segment in the ratio 2 to 3? 83 84 10.2 Areas of trapezoids, rhombuses, and kites Pearson Find the area of a G.MG.1 trapezoid, Use geometric Chapter rhombus, and kite. shapes, their #10.2 Get measures, and their Ready Example: properties to describe The roof on the bridge objects (e.g., modeling below consists of four a tree trunk or a sides, two congruent Text book human torso as a trapezoids and two page #623cylinder) congruent triangles. 628 MP.1- 6 Web link for the area 1. of a trapezoid, rhombus, and kite: 2) Problems 3-4 Exs. 20-25 all, 26-38 even www.khanacademy.org/.../area.. ./areas_of_trapezoids_rhombi _and_kites 2. -Use the diagram above to find the combined area of the two triangles. -What is the area of the entire roof? Average: 1) problems 1-2 Kuta software for worksheet: www.mybookezz.org/kutasoftware-infinite-geometryfinding-total-area -Find the combined area of the two trapezoids. 1) problems 1-2 Exs 11-19 all, 45-53 www.slideshare.net/.../112areas-of-trapezoidsrhombuses-and-kites Exs. 11-19 odd, 45-53. 2) Problems 3-4 www.kutasoftware.com/FreeW orksheets/.../Area%20of%20Tra pezoids 2 days Basic: Exs. 21-25 odd, 26-41 3. Advanced: 1) Problems 1-4 you tube video tutorial: Exs. 11-25 odd, 26-53 www.youtube.com/watch?v=1N DXo8nnRUE Math power point notes: www.youtube.com/watch?v=V2x Google - search: nehsmath.wikispaces.com/.../74+PPT+Areas+of+Trapezoids,+Rhombus Math skills practice: 85 www.khanacademy.org/.../area.../areas_of_trapezoids _rhombi_and_kites 86 Basic skills Review: Apply distance in HSPA the coordinate PREP/PARCC/S plane. AT Example: G.GPE.6 Pearson Find the point on a Chapter #10 directed line segment Get Ready between two given points that partitions the segment in a given ratio. Kuta software for worksheets Text book page # 628 30 mn www.kutasoftwar e.com/FreeWorks heets/GeoWorks heets/3Points%20in... Standardized Test Prep (SAT/HSPA) Text book page 628 Q. 45-47 87 10.3 Areas of Regular Polygons Find the area of a regular polygon. G.MG.1 Use geometric Chapter shapes, their #10.3 Get Example: measures, and their Ready The gazebo in the photo is properties to describe built in the shape of a objects (e.g., modeling regular octagon. Each side a tree trunk or a Text book is 8 ft long, and its human torso as a page #629apothem is 9.7 ft. What is cylinder) 634 the area enclosed by the gazebo? MP.1 1 day Pearson MP.3 Web link for the area 4. of a regular polygon: www.mathwords.com/a/area_re gular_polygon.htm Basic: 1) problems 1-3 Exs 8-25 all, 26-30 even 31-33 all, 35, 44-52 www.kutasoftware.com/.../6Area%20of%20Regular%20Pol 5. ygons.pdf 1) problems 1-3 Kuta software for worksheet: Exs. 9-25 odd, 26-41, 44-52 6. www.kutasoftware.com/FreeW orksheets/.../Area%20of%20Tra pezoids Average: Advanced: 1) Problems 1-3 Exs. 9-25 odd, 26-52 MP.4 MP.7 www.gobookee.net/kutasoftware-area-of-regularpolygons-answ Math power point notes: Google search: you tube video tutorial: jcs.k12.oh.us/teachers/.../PH_Geo_103_Areas_of_Regular_Polygons.pp www.youtube.com/watch?v=eQ hgozrRiYI Math skills practice: www.youtube.com/watch?v=Hlc www.finneytown.org/Downloads/GETE1003.pdf Hd-psOWs 88 Basic skills Review: 30 mn Partition line segment given ratio. HSPA PREP/PARCC/S Example: AT Point p lies on the direct line segment from A(2,3) to B(8,0)and partitions the segment in the ratio 2 to 1. What are the coordinates of point P? G.GPE.6 Pearson Kuta software for worksheets Find the point on a Chapter #10 directed line segment Get Ready schoolwires.henry.k12. between two given ga.us/.../415_Partitioning%20a points that partitions the segment in a given ratio. Standardized Test Prep (SAT/HSPA) Text book page 634 Q. 44-47 Text book page # 634 89 10.4 Perimeters and areas of similar figures Find the perimeters and areas of similar polygons. Community Service During the summer, a group of high school students used a plot of city land and harvested 13 bushels of vegetables that they gave to a food pantry. Their project was so successful that next summer the city will let them use a larger, similar plot of land. In the new plot, each dimension is 2.5 times the corresponding dimension of the original plot. G.MG.3 Pearson Apply geometric Chapter methods to solve #10.3 Get design problems (e.g., Ready designing an object or structure to satisfy physical constraints or minimize cost; working Text book with typographic grid page #629634 systems based on Web link for finding the perimeters and areas of similar polygons: 7. 1) problems 1-2 Exs 9-16 all, 52-62 www.onemathematicalcat.org/M ath/...obj/per_area_similar_figu 2) Problems 3-4 res.htm Exs. 17-24 all, 26-30 even 31-33 all, 34-44 even www.kutasoftware.com/freeige. 8. Average: ratios). Kuta software for worksheet: MP.1 1) problems 1-2 Exs. 9-15 odd, 52-62. www.kutasoftware.com/freeige. 2) Problems 3-4 2 days MP.5 MP.8 you tube video tutorial: Exs. 17-23 odd, 25-47 www.youtube.com/watch?v=ae- 9. Advanced: 1) Problems 1-4 www.youtube.com/watch?v=ML How many bushels can they expect to harvest next year? Basic: AdoSrJfi0 Exs. 9-23 odd, 25-62 Math power point notes: Google search: www.villagechristian.org/media/2322216/lesson%2086.ppt Math skills practice: 90 www.mathwarehouse.com/.../similar/triangles/areaand-perimeter-of-simi 91 Basic skills Review: Partition line segment given ratio. G.GPE.6 Pearson Kuta software for worksheets Find the point on a Chapter #10 directed line segment Get Ready schoolwires.henry.k12. HSPA between two given ga.us/.../4PREP/PARCC/S Example: 15_Partitioning%20a points that partitions AT Point R lies on the the segment in a given Text book directed line segment ratio. page # 641 from L(-8, -10) to Standardized Test Prep (SAT/HSPA) Text book page 641 Q. 52-55 M(4, -2) and 30 mn partitions the segment in the ratio 3 to 5. What are the coordinates of point R? 92 Recognize polyhedral and Spaces Figures their parts. and Cross Section -To visualize cross sections of space figures. 11.1 Example: Julie incorrectly identified the solid below as a pyramid with a square base. 1. Correctly identify the solid. 2 days 2. What would you say to Julie to help her tell the difference between this solid and a pyramid? G.GMD.4 Pearson Identify the shapes of two-dimensional cross-sections of three- dimensional objects, and identify three-dimensional objects generated by rotations of twodimensional objects. Chapter #11.1 Get Ready MP.1 MP.3 MP.4 MP.5 Web link for solid figures: 10. Basic: 1) problems 1-3 www › Geometry Concepts and Skills › Chapter 9 Exs 6-17 all, 51-62 2) Problems 4-5 Text book page #688695 www.superteacherworksheets. com/solid-shapes Kuta software for worksheet: Exs. 18-23 all, 24-34 even, 38 11. Average: 1) problems 1-3 www.kutasoftware.com/FreeW orksheets/GeoWorksheets/10- Exs. 7-17 odd, 51-62. 2) Problems 4-5 you tube video tutorial: Exs. 19-23 odd, 24-40 www.youtube.com/watch?v=DB S-8meBgZs 12. MP.7 www.youtube.com/watch?v=q2 Advanced: 1) Problems 1-3 Exs. 7-17 odd, 51-62 Math power point notes: Google search: www.clintweb.net/ctw/ppsspowerpointsolidshapes.pp t Math skills practice: www.ixl.com/math/grade-5/identify-planar-and-solid- 93 figures 94 Basic skills Review: Use the Pythagorean theorem to solve problems. HSPA PREP/PARCC/S Example: AT G.SRT.8 Pearson Use trigonometric ratios Chapter #11 and the Pythagorean Get Ready Theorem to solve right triangles in applied problems. Text book page # 695 30 mn Kuta software for worksheets Standardized Test Prep (SAT/HSPA) Text book page 695 Q. 51-55 www.gobookee.net/kut a-software-righttriangles-andpythagorean-theor Triangle JKL above represents the boundary of a state wilderness area. An access road will be constructed that intersects side JK at a 90 degrees angle and extends to point L. The road will intersect side JK 24 miles from point K, and the length of side KL is 30 miles. Part A: What is the length, in miles, of the access road? Show your work. 95 Part B: What is the length, in miles, of side JK? Show your work. 96 97 Find the surface area of prism and Surface Areas cylinder. of Prisms and Cylinders Example: 11.2 G.MG.1 Pearson Chapter #11.2 Get Ready Use geometric shapes, their Architecture In this exercise measures, and their below use the following properties to describe information. Suppose a objects (e.g., modeling Text book skyscraper is a prism that a tree trunk or a page #699is 415 meters tall and each human torso as a 707 base is a square that cylinder). measures 64 meters on a side. MP.1 2 days MP.3 MP.7 Web link for finding the surface area of prism and cylinder: 13. Basic: 1) problems 1-2 Exs 7-13 all, 44-45 hotmath.com/help/gt/genericpre alg/section_9_4.html 2) Problems 3-4 Exs. 14-20 all, 22-30 even, 37 www.virtualnerd.com/geometry/ surface-area.../prismscylinders-area 14. Kuta software for worksheet: Average: 1) problems 1-2 Exs. 7-13 odd, 44-45. www.kutasoftware.com/.../10Surface%20Area%20of%20Pri sms 2) Problems 3-4 Exs. 15-19 odd, 21-38 you tube video tutorial: www.youtube.com/watch?v=DB 1. What is the lateral area of this skyscraper? 2. Challenge What is the surface area of this skyscraper? (Hint: The ground is not part of the surface area of the skyscraper.) MP.8 S-8meBgZs 15. Advanced: 1) Problems 1-4 www.youtube.com/watch?v=q2 Exs. 7-19 odd, 21-55 Math power point notes: Google search: https://www.madison.k12.al.us/.../Surface%20Area%2 0of%20Prisms Math skills practice: www.ixl.com/math/grade-8/surface-area-of-prisms- 98 and-cylinders 99 Basic skills Review: Use proportions. G.SRT.5 Example: Use congruence and Chapter #11 similarity criteria for Get Ready triangles to solve problems and to prove relationships in geometric figures. Text book page # 707 HSPA PREP/PARCC/S AT 30 mn The figure above represents a swing set. The supports on each side of the swing set are constructed from two 12-foot poles connected by a brace at their midpoint. The distance between the bases of the two poles is 5 feet. Pearson Kuta software for worksheets Standardized Test Prep (SAT/HSPA) Text book page 707 Q. 44-47 www.gobookee.net/kut a-software-righttriangles-andpythagorean-theor Part A: What is the length of each brace? Part B: Which theorem about triangles did you apply to find the solution in Part A? 100 101 Find the surface area of a pyramid Surface areas and cone. and pyramids and cones Example: 11.3 G.MG.1 Pearson Chapter #11.3 Get Ready Use geometric shapes, their Veterinary Medicine A cone- measures, and their properties to describe shaped collar, called an Elizabethan collar, is used to objects (e.g., modeling Text book a tree trunk or a page #708prevent pets from aggravating a healing wound. human torso as a 715 cylinder). 2 days Web link for finding 3. the surface area of a pyramid and cone: Basic: 1) problems 1-4 Exs 9-15 all, 44-53 www.virtualnerd.com/geo metry/surfacearea.../pyramids-codes- 2) Problems 1-4 area Exs. 16-21 all, 22, 25, 26-36 Kuta software for worksheet: even MP.1 www.kutasoftware.com/. ../10Surface%20Area%20of %20Pyramids%20a.. MP.3 you tube video tutorial: MP.6 www.youtube.com/watch 4. Average: 1) problems 1-4 Exs. 9-15 odd, 44-53. 2) Problems 1-4 Exs. 17-21 odd, 22-38 MP.7 5. Advanced: 1) Problems 1-4 Exs. 9-21 odd, 21-53 Math 1. Find the lateral area of the entire cone shown above. power point notes: 2. Find the lateral area of the small cone that has a radius of 3 inches and a www.marianhs.org/.../12.3%20Surface%20 Area%20of%20Pyramids Google search: 102 height of 4 inches. Use your answers to E rci “a” an “ ” find the amount of material needed to make the Elizabethan collar shown. Math skills practice: www.ixl.com/math/grade-8/surface-area-ofpyramids-and-cones 103 Apply trigonometric G.SRT.8 Pearson Kuta software for Standardized Test Prep (SAT/HSPA) ratios: worksheets Use trigonometric ratios Chapter #11 Text book page 715 Q. 44-48 HSPA and the Pythagorean Get Ready www.kutasoftware.co m/.../10PREP/PARCC/S Theorem to solve right Surface%20Area%20o triangles in applied AT f%20Pyramids problems. Basic skills Review: 30 mn The figure above represents a plan for a wheelchair ramp to a step that has a height of 10 inches. Jodi and Kevin each used righttriangle trigonometry to determine the length of the ramp. Both solutions are shown below. Explain why both solutions resulted in the same answer. Text book page # 715 104 105 11.4 Volume of prisms and cylinders Find the volume of a G.GMD.1 prism and the volume Give an informal of a cylinder. argument for the Example: formulas for the circumference of a a. How do the radius and circle, area of a circle, height of the mug compare volume of a cylinder, to the radius and height of pyramid, and cone. the dog bowl? Use dissection b) How many times greater arguments, Cavalieri’s is the volume of the bowl principle, and informal limit arguments. than the volume of the mug? 2 days Pearson Chapter #11.4 Get Ready Web link for finding 6. the volume of a prism and the volume of a cylinder. hotmath.com/help/gt/gen ericprealg/section_9_6.ht Kuta software for worksheet: Exs 6-13 all, 46-53 2) Problems 3-4 7. www.kutasoftware.com/... /10Volume%20of%20Prism s%20and%20Cyli you tube video tutorial: MP.3 www.khanacademy.org/m ath/.../volume.../solidgeometry-volu MP.7 1) problems 1-2 Exs. 14-21 all, 24, 30- 32 all, 38 Text book page #717724 MP.1 MP.6 Basic: Average: 1) problems 1-2 Exs. 7-13 odd, 46-53. 2) Problems 3-4 Exs. 15-19 odd, 21-42 8. Advanced: 1) Problems 1-4 Exs. 17-19 odd, 21-53 Math power point notes: Google search: www.lms.stjohns.k12.fl.us/.../8th%20Std%20 Chapter%209%20PowerPoi Math skills practice: 106 www.ixl.com/math/grade-8/volume-ofprisms-and-cylinders 107 Basic skills Review: Use trigonometric ratios. G.SRT.8 Pearson Use trigonometric ratios Chapter #11 Example: HSPA and the Pythagorean Get Ready PREP/PARCC/S Theorem to solve right A 12-foot ladder that is triangles in applied AT leaning against a wall problems. makes a 75.50 30 mn Angle with the level ground. Which of the equations below can be used to determine the height, y, above the ground, in feet, that the ladder touches the wall? (Sketch the diagram) Kuta software for worksheets Standardized Test Prep (SAT/HSPA) Text book page 724 Q. 46-49 www.kutasoftware.com/.../10Volume%20of%20Prisms%20a nd%20Cyli Text book page # 724 108 109 Find the volume of a pyramid and the Volumes of volume of a cone. pyramids and Example: cones 11.5 Popcorn A movie theater G.GMD.3 Use volume formulas Chapter for cylinders, #11.5 Get pyramids, cones, and Ready spheres to solve problems. serves a small size of popcorn in a conical G.MG.1 container and a large size of Use geometric popcorn in a cylindrical container. shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 2 days a) What is the volume of the small container? What is the MP.1 volume of the large MP.3 container? MP.7 b) How many small containers of popcorn do you have to buy to equal the amount of popcorn in a large container? c) Which container gives you more popcorn for your money? Explain your Pearson Text book page #726732 Web link for finding 9. the volume of a pyramid and cone: Basic: 1) problems 1-2 Exs 5-14 all, 39-46 www.glencoe.com/sec/m ath/prealg/mathnet/pr01/p 2) Problems 3-4 Exs. 15-21 all, 24-32, even Kuta software for worksheet: www.mybookezzz.com/k 10. uta-software-volume-ofpyramids-and-cones Average: 1) problems 1-2 you tube video tutorial: Exs. 5-13 odd, 39-46. www.youtube.com/watch 2) Problems 3-4 Exs. 15-19 odd, 20-34 11. Advanced: 1) Problems 1-4 Exs. 5-19 odd, 20-46 Math power point notes: Google search: jcs.k12.oh.us/.../PH_Geo_115_Volumes_of_Pyramids_and_Cones.ppt Math skills practice: 110 reasoning hotmath.com/help/gt/genericprealg/section_ 111 Basic skills Review: Use the Pythagorean Theorem: Example: HSPA PREP/PARCC/S AT G.SRT.8 Pearson Use trigonometric ratios Chapter #11 and the Pythagorean Get Ready Theorem to solve right triangles in applied problems. Kuta software for worksheets Standardized Test Prep (SAT/HSPA) Text book page 732 Q. 39-42 www.kutasoftware.com/... /10Volume%20of%20Pyram ids%20and%20C.. Text book page # 732 30 mn The figure above represents a plot of land that Susan has measured to use as a fenced garden. Before she builds the fence, she wants to make sure that the plot is rectangular. She measures the length of one of the diagonals. If Susan's plot is rectangular, what will be the length, in feet, of the diagonal she measured? 112 113 Find the surface area G.GMD.3 Pearson and volume of a Use volume formulas Chapter Surface areas sphere. for cylinders, and volume of #11.6 Get Example: pyramids, cones, and Ready spheres spheres to solve The entrance to the Civil problems. Rights Institute in Birmingham, Alabama, G.MG.1 Text book includes a hemisphere page #733that has a radius of 25.3 Use geometric 740 shapes, their feet. measures, and their properties to describe objects (e.g., modeling a tree trunk or a 2 days human torso as a cylinder). 11.6 a) Find the volume of the hemisphere. MP.1 b) Find the surface area of the hemisphere, not MP.3 including its base. MP.6 c) The walls of the hemisphere are 1.3 feet MP.7 thick. So, the rounded surface inside the building is a hemisphere with a radius of 24 feet. Find its surface area, not Web link for finding 12. the surface area and volume of a sphere. Basic: 1) problems 1-2 Exs 6-16 all, 60-71 www.murrieta.k12.ca.us/c ms/lib5/CA01000508/Cen tricity/.../T9.6 2) Problems 3-4 Exs. 17-26 all, 29-31, 34-42 Kuta software for worksheet: Even, 50 www.kutasoftware.com/Fr 13. eeWorksheets/GeoWork Average: 1) problems 1-2 sheets/10-Spheres.pdf Exs. 7-15 odd, 60-71. you tube video tutorial: 2) Problems 3-4 www.khanacademy.org/m ath/.../volume.../v/volum e-of-a-sphere Exs. 17-25 odd, 26-54 14. Advanced: 1) Problems 1-4 Exs. 7-25 odd, 26-71 Math power point notes: Google search: www.dgelman.com/powerpoints/.../12.6%2 0Surface%20Area 114 including its base. Math skills practice: www.ixl.com/math/grade-8/volume-andsurface-area-of-spheres 115 Basic skills Review: Use the inverse of trigonometric ratios. Example: HSPA PREP/PARCC/S AT The diagram below shows a model of a staircase in which all the riser heights are equal and all the tread lengths are equal. G.SRT.8 Pearson Use trigonometric ratios Chapter #11 and the Pythagorean Get Ready Theorem to solve right triangles in applied problems. Kuta software for worksheets Standardized Test Prep (SAT/HSPA) Text book page 732 Q. 39-42 www.kutasoftware.com/FreeWo rksheets/GeoWorksheets/10Spheres.pdf Text book page # 740 30 mn A carpenter wants to build a staircase in which each riser has a height of 6 inches and each tread has a length of 11 inches. Which of the following expressions is equal to the stair angle? 116 117 Compare and find the G.MG.1 Pearson areas and volumes of Use geometric Areas and Chapter similar solids. shapes, their Volumes of #11.7 Get Example: measures, and their Ready Similar Solids properties to describe Spheres in Architecture In objects (e.g., modeling Exercises a–d, refer to a tree trunk or a the information below Text book human torso as a about The Rose Center cylinder). page #742for Earth and Space at 749 New York City’s American Museum of G.MG.2 Natural History. The sphere has a diameter of Apply concepts of 87 feet. The glass cube density based on area 2 days surrounding the sphere and volume in is 95 feet long on each modeling situations edge. (e.g., persons per square mile, BTUs per cubic foot). 11.7 MP.3 15. Web link for comparing and finding the areas and volumes of similar solids. Basic: 1) problems 1-2 Exs 5-14 all, 42-54 2) Problems 3-4 www.ck12.org/geometry/ Area-and-Volume-ofSimilar-Solids Exs. 15-26 all, 28-29, 34-38 Even 16. Kuta software for worksheet: Average: 1) problems 1-2 Exs. 5-13 odd, 42-54. www.kutasoftware.com/Fr eeWorksheets/.../10Similar%20Solids 2) Problems 3-4 Exs. 15-23 odd, 24-38 you tube video tutorial: 17. www.youtube.com/watch - Advanced: 1) Problems 1-4 Exs. 5-23 odd, 24-54 Math power point notes: MP.7 Google search: MP.8 a) Find the surface area of the sphere. cs.k12.oh.us/.../PH_Geo_117_Areas_and_Volumes_of_Similar_Solid s 118 b) Find the volume of the sphere. c) Find the volume of the glass cube. Math skills practice: www.ck12.org/geometry/Area-and-Volumeof-Similar-Solids d) Find the approximate amount of glass used to make the cube. (Hint: Do not include the ground or roof in your calculations) 119 Basic skills Review: Apply the formula to find area of oblique triangles. HSPA PREP/PARCC/S Example: AT G.SRT.9 Pearson Derive and use the Chapter #11 formula for the area of Get Ready an oblique triangle (A = 1/2 ab sin (C)). Kuta software for worksheets Standardized Test Prep (SAT/HSPA) Text book page 749 Q. 42-46 www.kutasoftware.com/FreeWo rksheets/.../10Similar%20Solids Text book page # 749 30 mn The figure above shows a triangle drawn over a map of Honduras. Use the measurements of the triangle to approximate the area, in square kilometers, of Honduras. Show your work. INSTRUCTIONAL FOCUS OF UNIT 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. 120 PARCC FRAMEWORK/ASSESSMENT Square and circles: http://balancedassessment.concord.org/hs012.html Examples: A cookie factory is making cookies in a pyramid shape with equilateral triangle as a base. We know that the lateral edge of the cookie is 2 cm long and the base edge of the cookie is 3 cm long. 1. Prove that the height of the cookie is 1 cm. 2. Find the volume of the cookie. 3. Each cookie is wrapped totally in an aluminum foil. Prove that the minimum surface of foil necessary to wrap 100 cookies is greater than 960 cm2. Example 2: Diana’s Christmas present is placed into a cubic shaped box. The box is wrapped in a golden paper. a. Are 3 m of golden paper enough for wrapping? 2 b. If 1 m of golden paper cost $3, how much would the wrapping material cost? 2 Could you pour 1 liter of juice in the box? (Know that 1 l=0.001 m3) Wiki page for Common Core Assessments: http://maccss.ncdpi.wikispaces.net/file/view/Geometry%20Unpacking062512%20.pdf/359510147/Geometry%20Unpacking062512%20.pdf PARCC Framework Assessment questions with Model Curriculum Website for all units: 121 http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsGeo_Nov2012V3_FINAL.pdf http://www.nj.gov/education/assessment/hs/hspa_mathhb.pdf 21ST CENTURY SKILLS (4Cs & CTE Standards) 4. Career Technical Education (CTE) Standards 1. 21 Century Life and Career Skills: All students will demonstrate the creative, critical thinking, collaboration, the problem-solving skills needed to function successfully as both global citizens and workers in diverse ethnic and organizational cultures. 2. Personal Financial Literacy: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 3. Career Awareness, Exploration, and Preparation: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age. 4. Career and Technical Education: All students who complete a career and technical education program will acquire academic and technical skills for careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees. st 122 1. Career and Technical Education: All students who complete a career and technical education program will acquire academic and technical skills for careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees. 9.4.12.B.4: Perform math operations, such as estimating and distributing materials and supplies, to complete classroom/workplace tasks. Project Base Learning Activities: http://www.achieve.org/files/CCSS-CTE-Task-IvySmith-GrowsUp-FINAL.pdf MODIFICATIONS/ACCOMMODATIONS Group activity or individual activity 1. Review and copy notes from eno board/power point/smart board etc. 2. Group/individual activities that will enhance understanding. 3. Provide students with interesting problems and activities that extend the concept of the lesson 4. Help students develop specific problem solving skills and strategies by providing scaffolded guiding questions 123 Peer tutoring 1. Team up stronger math skills with lower math skills Use of manipulative 2. Eno or smart boards 3. Dry erase markers 4. Reference sheets created by special needs teacher 5. Pairs of students work together to make word cards for the chapter vocabulary 6. Use 3D shapes for visual learning 7. Reference sheets for classroom 8. Graphing calculators APPENDIX 124 (Teacher resource extensions) 9. CCSS. Mathematical Practices: MP1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. MP2: Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations 125 and objects. MP3: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can 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 that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. MP4: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 126 MP5: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. MP6: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. MP7: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and 127 can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MP8: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. UNIT 4 Connecting Algebra And Geometry Coordinates Total Number of Days: 11 days Grade/Course: __Geometry 10th grade__ ESSENTIAL QUESTIONS 10. How do you prove that two lines are parallel or ENDURING UNDERSTANDINGS 14. A line can be graphed and its equation written when certain facts about the line 128 11. 12. 13. perpendicular? such as the slope and a point on the line are known. How do you write an equation of a line in the coordinate 15. plane? The equations of a line can be written in various forms such as the Slope-intercept form and the Point-slope form. 16. How can you classify quadrilaterals? How can you use coordinate geometry to prove general 17. relationship? Comparing the slopes of two lines can show whether the lines are parallel, perpendicular, or neither. The relationship between parallel or perpendicular lines can sometimes be used to write the equation of a line. 18. The formulas for slope, distance and midpoint can be used to classify and to prove geometric relationships for figures in the coordinate plane. 19. Using variables to name the coordinates of a figure allows relationships to be shown to be true for a general case 20. Geometric relationships can be proven using variable coordinates for figures in the coordinate plane. 129 PACING CONTENT SKILLS STANDARDS RESOURCES LEARNING ACTIVITIES/ASSESSMENTS OTHER Pearson (e.g., tech) Prepares for Text book http://www.regentsprep G.GPE.5 Prove page 189- .org/Regents/math/geo Example: the slope criteria metry/GCG1/EqLines.ht Page 193: 1-7 196 for parallel and m The slope of a line is the rate of change perpendicular and is represented by m lines and Basic – Problems 1-2 Graph and write linear equations 2 days 3.7 Equations of Lines in the Coordinate Plane When a line passes through the points (x1, y1) and (x2, y2), the slope (m) is uses them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). http://www.mathwareh Ex. 8 – 29, 50 – 63, 69 -77 ouse.com/algebra/linear Average – Problems 1 – 4 _equation/slope-of-a-21. line.php Ex. 9 – 29 odd, 50 –63, 669 - 77 22. Advanced – Problems 1-4 Ex 9-29 odd, 50-63, 69 - 77 Equations of line can take on several MP 1 forms: Slope Intercept Form: [used when you know, or can find, the slope, m, and the y-intercept, b.] y = mx + b MP 3 Point Slope Form: [used when you know, or can find, a point on the line (x1, y1), and the slope, m.] 130 y – y1 = m(x – x1) Write the equation of a line Basic Skills Review 30 mins PARCC/ HSPA PREP G.GPE.5 See above Text book page 196 http://www.regentsprep.o Standardized Test Prep rg/Regents/math/geometr Q 64 – 68 y/GCG1/EqLines.htm Question: What is the equation of the line in slope-intercept form of the line parallel to y = 5x + 2 that passes through the point with coordinates (-2, 1) http://www.mathwarehou Worksheets from stated websites. se.com/algebra/linear_equ ation/slope-of-a-line.php 131 Relate slope to parallel and perpendicular lines Example: 2 days G.GPE.5 Prove Text book the slope page 197criteria for 204 parallel and If we look at both equations, we notice perpendicular that they both have slopes of 2. Since lines and both lines "rise" two units for every one uses them to unit they "run," they will never intersect. Thus, they are parallel lines. solve geometric The graph of these equations is shown problems (e.g., 3.8 below. find the equation of a Slopes of line Parallel and 25. http://www.regentsprep 26. .org/Regents/math/ALG EBRA/AC3/Lparallel.htm Page 201: 1-6 Basic – Problems 1-2: Ex. 7 – 14, 27, 53 - 62 Average – Problems 1 – 2: Ex. 7 - 13 odd, 27, 53 - 62 Advanced – Problems 1-2: Ex 7-21 odd, 27, 53 - 62 Kuta Software for worksheets. Perpendicular Lines http://www.wyzant.com 23. /help/math/geometry/li 24. nes_and_angles/parallel _and_perpendicular parallel or perpendicular to a given line that passes through a given point). MP 1 MP2 30 mins Basic Skills Classify lines as parallel, perpendicular G.GPE.5 or neither. Review See above Question: PARCC/ Text book page 204 http://www.regentsprep Standardized Test Prep .org/Regents/math/geo metry/GCG1/EqLines.ht Q 48 - 52 m HSPA PREP Classify each of the following pairs of lines as parallel, perpendicular or 132 neither Parallel/ http://www.mathwareh Worksheets from stated websites. ouse.com/algebra/linear _equation/slope-of-a-27. line.php Lines Perpendicular /Neither 3y= -5x -5 (y – 7) = 0.6(x – 5) 2x + 3y = 4 4x + 5y = 6 y = 4x + 1 (y – 2) = 4(x – 3) y = -3x + 5 9x + 3y = 2 133 134 6.7 1 day Polygons in the Coordinate Plane Classify polygons in the coordinate Text book Teacher made power28. G.GPE.7 plane applying the formulas for slope, page 400 – point presentations Use coordinates 29. distance and midpoint. 405 to compute perimeters of Example: polygons and 30. Is parallelogram WXYZ a rhombus? areas of triangles and rectangles, e.g., 31. Explain using the distance formula. Page 403: 1-4 Basic – Problems 1-3, Ex. 5– 18, 21 - 24, 31, 35 – 44, 49 - 54 Average – Problems 1 – 3, Ex. 5 - 15 odd, 17 – 44, 49 - 54 Advanced – Problems 1-3, Ex 5-15 odd, 17 - 44, 49 - 54 MP 1 MP 3 MP 8 135 Determine if a given polygon is a triangle, parallelogram or a quadrilateral G.GPE.5 G.GPE 7 Use coordinates to compute perimeters of Question: polygons and In the coordinate plane, quadrilateral areas of ABCD has vertices with triangles and coordinates rectangles, e.g., A(1, -1), B(-5, 3), C(-3, 6), and using the Basic Skills D(3, 2). distance Review 1) Compute the lengths of the sides of formula. 30 mins PARCC/ HSPA PREP Text book page 405 Teacher made power point presentations Kuta software for worksheets Standardized Test Prep Q 45 - 48 Worksheets from stated websites. quadrilateral ABCD. AB = ____ BC = ___ CD = ____ DA = ___ 2) Compute the slopes of the sides AB and AD . Slope of AB = _____ Slope of AD = _____ 136 3. Indicate in the table below whether ABCD is an example of each shape listed. Explain why it is or is not. Shape Yes or Explain No Parallelogram Rhombus Rectangle Square 137 Name coordinates of special figures by Prepares for using their properties G.GPE.4 Question 6.8 2 days 30 mins SQRE is a square where SQ = 2a. The axes bisect each side, what are the coordinates of the vertices of SQRE? Teacher made Power32. point presentations 33. Use coordinates to Text book prove simple geometric theorems page 402 algebraically. For 412 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). Applying Coordinate Geometry Find the coordinates of vertices of a G.GPE.4 Basic Skills polygon given coordinates of two Review vertices and a point of intersection of See above the diagonals. PARCC/ Kuta software worksheets. Page 403: 1-3 Basic – Problems 1-3, Ex. 7– 13, 14, 17, 19, 23, 24, 28, 42-49 34. Average – Problems 1 – 3, Ex. 7- 13 odd, 14 – 31, 42 - 49 35. http://www.mathopenr ef.com/coordsquare.ht ml Advanced – Problems 1-3, Ex 7-13 odd, 14 - 41, 42 - 49 http://www.youtube.co m/watch?v=EZtXevirdes Text book page 412 http://www.mathopenr Standardized Test Prep ef.com/coordsquare.ht Q 38 - 41 ml Worksheets from stated websites. HSPA PREP Question: A parallelogram has two vertices at (1, http://www.youtube.co m/watch?v=EZtXevirdes 138 1) and (0, 7) and its diagonals cross at the point (4, 3). Where are the other two vertices of the parallelogram 139 Prove theorems using figures in the coordinate plane. G.GPE.4 Use coordinates to prove simple Text book geometric page 414theorems 418 algebraically. For example, prove or disprove that a figure defined 6.9 2 days Proofs Using Coordinate Geometry Example: 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). MP 1 MP 3 MP 7 http://www.youtube.com 36. /watch?v=EZtXevirdes Page 416: 1-3 37. Basic – Problems 1-2 Ex. 4, 6– 20 even, 21, 23, 33 40 38. http://on.aol.com/video/ how-to-write-coordinateproofs-516909807 Average – Problems 1 – 2, Ex. 4 - 14, 15 – 26, 33 – 40. 39. Advanced – Problems 1-2, Ex 4 – 28, 33 - 40 http://www.regentsprep. org/Regents/math/geome try/GCG4/CoordinatepRA CTICE.htm http://hotmath.com/hot math_help/topics/coordin ate-proofs.html http://www.whiteplainsp ublicschools.org/cms/lib5/ NY01000029/Centricity/D omain/360/Coordinate%2 0Geometry%20Proofs%20 Packet%202012.pdf 140 Use Coordinate Geometry to Prove Right Triangles and Parallelograms Basic Skills Review Question: 30 mins PARCC/ Daniel and Isaiah see a drawing of quadrilateral ABCD, A(2,2), B(5,-2), HSPA PREP C(9,1) and D(6,5). Daniel says the figure is a rhombus, but not a square. Isaiah says the figure is a square. Write a proof to show who is making the correct observation. G.G.PE 4 See above Text book page 418 http://www.regentsprep Standardized Test Prep .org/Regents/math/geo metry/GCG4/Coordinate Q 29 - 32 pRACTICE.htm Worksheets from stated websites. http://www.whiteplains publicschools.org/cms/li b5/NY01000029/Centrici ty/Domain/360/Coordin ate%20Geometry%20Pr oofs%20Packet%202012 .pdf 141 To write the equation of a circle and find the center and radius of a circle. Example: Definition: A circle is a locus (set) of points in a plane equidistant from a fixed point. Circle whose center is at the origin 12.5 1 day Equation: Circles in the Coordinate plane Example: Circle with center (0,0), radius 4 G.GPE 1 Derive Text book the equation of page 798 a circle of given 803 center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. http://www.regentsprep 40. .org/Regents/math/algtr 41. ig/ATC1/circlelesson.ht m 42. http://www.mathwareh ouse.com/geometry/circ 43. le/equation-of-acircle.php Page 800: 1-7 Basic – Problems 1-3 Ex. 8 – 30 all, 31 – 52 even, 58 65 Average – Problems 1 – 3, Ex. 9 – 29 odd, 315– 56, 58 – 65. Advanced – Problems 1-3, Ex 9 – 29 odd, 31 - 65 http://www.mathsisfun. com/algebra/circleequations.html http://www.mathopenr ef.com/coordgeneralcirc le.html 142 Graph: Circle whose center is at (h, k) ( i wi r f rr ra iu f r ” I ay a f r ”) r f rr a “c n r- a “ an ar Equation: 143 Example: Circle with center (2,-5), radius 3 Graph: 144 G.GPE 2 Derive Text book http://www.mathsisfun. Concept Byte Page 804 the equation of page 804 – com/geometry/parabola Example Activity 1 a parabola given 805 .html a focus and Activity 2 12.5 Definition: A parabola is a curve where directrix. any point is at an equal distance from: Circles in the http://www.purplemath Activity 3 Coordinate 1. a fixed point (the focus), and .com/modules/parabola. Ex: 17 - 23 plane 2. a fixed straight line htm To write the equation of a parabola. 1 day (the directrix) http://hotmath.com/hot math_help/topics/findin g-the-equation-of-a145 parabola-given-focusand-directrix.html * the axis of symmetry (goes through the focus, at right angles to the directrix) * the vertex (where the parabola makes its sharpest turn) is halfway between the focus and directrix. 146 G.GPE 1 Derive Text book the equation of page 803 Question: a circle of given In the coordinate plane, the circle center and radius using the with radius r centered at h, k Pythagorean consists of all the points x, y that Theorem; are r units from h, k . Use the complete the square to find Basic Skills Pythagorean theorem and the figure the center and Review below to find an equation of the circle radius of a circle with radius r and center PARCC/ given by an equation. Explain your answer. HSPA PREP Find the equation of a circle 30 mins http://www.mathwareh Standardized Test Prep ouse.com/geometry/circ Q 58 - 60 le/equation-of-acircle.php Worksheets from stated websites. http://www.mathsisfun. com/algebra/circleequations.html INSTRUCTIONAL FOCUS OF UNIT 3. 4. Building on their knowledge and work with the Pythagorean theorem, students will find distances in the coordinate plane. Students will use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines. 147 5. Students will continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola. PARCC FRAMEWORK/ASSESSMENT PARCC EXEMPLARS www.parcconline.org 1. The coordinates are for a quadrilateral, (3, 0), (1, 3), (-2, 1), and (0,-2). Determine the type of quadrilateral made by connecting these four points? Identify the properties used to determine your classification. You must give confirming information about the polygon. 2. If Quadrilateral ABCD is a rectangle, where A(1, 2), B(6, 0), C(10,10) and D(?, ?) is unknown. a. Find the coordinates of the fourth vertex. b. Verify that ABCD is a rectangle providing evidence related to the sides and angles. 3. Derive the equation of the parabola that has the focus (1, 4) and the directrix x=-5. 4. Derive the equation of the parabola that has the focus (2, 1) and the directrix y=-4. 5. Derive the equation of the parabola that has the focus (-3, -2) and the vertex (1, -2). 6. Given the midpoint of a segment and one endpoint. Find the other endpoint. a. Midpoint: (6, 2) endpoint: (1, 3) b. Midpoint: (-1, -2) endpoint: (3.5, -7) 148 7. Investigate the slopes of each of the sides of the rectangle ABCD (shown below). What do you notice about the slopes of the sides that meet at a right angle? What do you notice about the slopes of the opposite sides that are parallel? Can you generalize what happens when you multiply slopes of perperpendicular lines? 8. If general points N at (a,b) and P at (c,d) are given. Why are the coordinates of point Q (a,d)? Can you find the coordinates of point M? 149 9. Jennifer and Jane are best friends. They placed a map of their town on a coordinate grid and found the point at w ic u i a ( , ) an Jan ’ u i a ( , ) an y wan in i , w a ar c r ina f ac f ir u i If J nnif r’ ace they should meet? 10. John was visiting three cities that lie on a coordinate grid at (-4, 5), (4, 5), and (-3, -4). If he visited all the cities and ended up where he started, what is the distance in miles he traveled? 11. Suppose a line k in a coordinate plane has slope c/d a. What is the slope of a line parallel to k? Why must this be the case? b. What is the slope of a line perpendicular to k? Why does this seem reasonable? 12. Two points A(0, -4) , B(2, -1) determines a line, AB. a. What is the equation of the line AB? b. What is the equation of the line perpendicular to AB passing through the point (2,-1)? 150 Wiki page for Common Core Assessments http://maccss.ncdpi.wikispaces.net/file/view/Geometry%20Unpacking062512%20.pdf/359510147/Geometry%20Unpacking062512%20.pdf PARCC Framework Assessment questions with Model Curriculum Website for all units http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsGeo_Nov2012V3_FINAL.pdf http://www.nj.gov/education/assessment/hs/hspa_mathhb.pdf 21ST CENTURY SKILLS (4Cs & CTE Standards) Career Technical Education (CTE) Standards 1. 21st Century Life and Career Skills: All students will demonstrate the creative, critical thinking, collaboration, the problem-solving skills needed to function successfully as both global citizens and workers in diverse ethnic and organizational cultures. 9.1.12.B.1 Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using multiple perspectives. 2. Personal Financial Literacy: All students will develop skills and strategies that promote personal and financial responsibility related to financial 151 planning, savings, investment, and charitable giving in the global economy. 9.2.12.B.1 Prioritize financial decisions by systematically considering alternatives and possible consequences. 3. Career Awareness, Exploration, and Preparation: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age. 9.3.12.C.2 Characterize education and skills needed to achieve career goals, and take steps to prepare for postsecondary options, including making course selections, preparing for and taking assessments, and participating in extra-curricular activities. 4. Career and Technical Education: All students who complete a career and technical education program will acquire academic and technical skills for careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees. 9.4.12.B.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and opportunities. career Project Base Learning Activities: http://www.achieve.org/files/CCSS-CTE-Task-Stairway-FINAL.pdf MODIFICATIONS/ACCOMMODATIONS Group activity or individual activity - Review and copy notes from eno board/power point/smart board etc. - Group/individual activities that will enhance understanding. - Provide students with interesting problems and activities that extend the concept of the l 152 - Help students develop specific problem solving skills and strategies by providing scaffolding guiding questions Peer tutoring - Team up stronger math skills with lower math skills Use of manipulative - Eno or smart boards - Dry erase markers - Reference sheets created by special needs teacher - Pairs of students work together to make word cards for the chapter vocabulary - Use 3D shapes for visual learning - Reference sheets for classroom - Graphing calculators APPENDIX (Teacher resource extensions) MP1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than 153 simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continua y a k v ,“ i ak n ” y can un r an a r ac f r ving c r an i ntify correspondences between different approaches. MP 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. MP 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can 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 that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. MP 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe 154 how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. MP 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts MP 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. MP 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for 155 solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MP 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x– 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 +x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 5. Kuta G 1: Kuta Software – Geometry (Free Worksheets) 6. Teacher Edition: Geometry Common Core by Pearson 7. Student Companion: Geometry Common Core by Pearson 8. Practice and Problem Solving Workbook: Geometry Common Core by Pearson 9. Teaching with TI Technology: Pearson Mathematics by Pearson 10. Progress Monitoring Assessments: Geometry Common Core by Pearson 11. http://www.jmap.org/JMAP_RESOURCES_BY_TOPIC.htm#Geo 12. http://www.mathopenref.com/ 13. http://www.mathisfun.com/ 14. http://www.mathwarehouse.com/ 156 15. http://www.regentsprep.org/Regents/math/geometry/math-GEOMETRY.htm * http://www.cpm.org/pdfs/state_supplements * http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqgeomapr15.pdf * http://illuminations.nctm.org * http://www.state.nj.us/education/cccs/standards/9/ *http://www.whiteplainspublicschools.org/cms/lib5/NY01000029/Centricity/Domain/360/Coordinate%20Geometry%20Proofs%20Packet%202012.pdf NOTE: Standards alignment in accordance with Appendix A of Common Core State Standards and Pear n’ G 1 and 2 ry C nC r ac r’ E i i n V u Notes to teacher (not to be included in your final draft): 4 Cs Three Part Objective 157 Creativity: projects Behavior Critical Thinking: Math Journal Condition Collaboration: Teams/Groups/Stations Demonstration of Learning (DOL) Communication – Powerpoints/Presentations UNIT 5 Circles With and Without Coordinates Total Number of Days: 11days Grade/Course: __Geometry 10th grade__ ESSENTIAL QUESTIONS ENDURING UNDERSTANDINGS 16. How do you solve problems that involve measurements20. of triangles? Angle bisectors and segment bisectors can be used in triangles to determine various angle and segment measures. 17. How do you find the area of a polygon or find the circumference and area of a circle? ng in the circle. 18. How can you prove relationships between angles and arcs 22. in a circle? The ar a f ar radius is known 19. How do you find the equation of a circle in the coordinate 23. plane? Angles formed by intersecting lines have a special relationship to the arcs the intersecting lines intercept 21. f ar f a circ ’ circu f r nc can f a circ f r f un y ra ii an arc can y r a ing i f un w n an ang circ ’ 158 24. The information in the equation of a circle allows the circle to be graphed. The equation of a circle can be written if its center and radius are known. RESOURCES STANDARDS PACING CONTENT SKILLS (CCSS/MP) OTHER Pearson LEARNING ACTIVITIES/ASSESSMENTS (e.g., tech) 1 day 5.3 Bisectors in -To identify properties of perpendicular bisectors and angles G.C.3 Construct the Text book www.pkwy.k12.mo.us/ho page 300- mepage/ataylor1/file/2.2. pdf Page 300 - Concept Byte: Paper 159 Triangles inscribed and circumscribed Example: circles of a Technology Use geometry software to triangle, and prove draw ABC. Construct the angle properties of bisector of BAC. Then find the angles for a midpoint of . Drag any of the points. quadrilateral Does the angle bisector always pass inscribed in a through the midpoint of the opposite circle. side? Does it ever pass through the midpoint? bisectors. 307 Folding Bisectors http://www.jmap.org/Static Files/PDFFILES/Workshee tsByTopic/ANGLES/Drills Page 304: 1 - 6 /PR_Measuring_Angles_3. pdf Basic – Problems 1-3 Ex. 7– 20, 23, 26 – 29, 33 - 40 25. Average – Problems 1 – 3 Ex. 7 – 17 odd, 18–29, 33 - 40 26. Advanced – Problems 1-3 Ex 7-17 odd, 18-40. 160 30 mins To write an argument for the formulas G.GMD.1 Text book for the volume of a pyramid. page 307 Give an informal argument for the formulas for Basic Skills Question: the Review circumference A cube in the xyz-coordinate system PARCC/HSPA (not shown) centered at the origin has of a circle, area PREP of a circle, vertices at the points volume of a cylinder, pyramid, and cone. Use dissection and arguments, www.pkwy.k12.mo.us/ho Standardized Test Prep mepage/ataylor1/file/2.2. Q 33 – 36 pdf http://www.jmap.org/Static Worksheets from stated websites. Files/PDFFILES/Workshee tsByTopic/ANGLES/Drills /PR_Measuring_Angles_3. pdf 161 , where . If lines are drawn from the center of the cube to the 8 vertices of the cube, 6 pyramids are formed. Explain how a pyramid with height a and square Cava i ri’ principle, and informal limit arguments. MP 1 3 base of side length 2a has a volume a 1 day MP 3 To find the measures of central angles G.CO.1, G.C.1, Text book and arcs, find the circumference and G.C.2 page 649arc length 658 Identify and describe Example: relationships Challenge Engineers reduced the lean among inscribed of the Leaning Tower of Pisa. If they angles, radii, moved it back 0.46�, what was the arc and chords. length of the move? Round your Include the 10.6 answer to the nearest whole number. relationship between Circles and central, Arcs inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the http://www.regentsprep Page 658 - Concept Byte: Circle .org/Regents/math/geo Graphs metry/GP15/CircleArcs. htm 27. Page 654: 1-8 28. www.cpm.org/pdfs/skillB uilders/GC/GC_Extra_Pr actice_Section18.pdf Basic – Problems 1-4: Ex. 9 – 35, 36 – 50 even, 64 - 71 29. Average – Problems 1 – 4: Ex. 9- 35 odd, 36 – 56, 64 - 71 www.robertfant.com/Geo 30. metry/PowerPoint/Chapt er11.ppt Advanced – Problems 1-4: Ex 9-35 odd, 36 – 56, 64 - 71 162 radius intersects the circle. MP 1 MP 3 MP 8 30 mins To find the measure of an angle in a G.C.2 circle See above Basic Skills Question: Review In the figure below, is tangent to PARCC/HSPA the circle with center O at point A. If PREP has a measure of 68 degrees, what is the measure, in degrees, Text book page 657 163.150.89.242/yhs/F Standardized Test Prep aculty/AB/bagg/Geom etry/images/.../11.3.pd Q 60 - 63 f Worksheets from stated websites. http://www.mathopenr 31. ef.com/arccentralanglet heorem.html of 163 164 10.7 1 day Areas of Circles and Sectors To find the areas of circles, sectors and G.C.5 Derive Text book Teacher made power using similarity page 659 – point presentations segments of circles. the fact that the 667 Example: length of the arc intercepted by Landscaping The diagram shows the an angle is area of a lawn covered by a water proportional to sprinkler. Round your answer to the the radius, and nearest whole number. define the 32. 1. What is the area of the lawn that is radian measure covered by the sprinkler? of the angle as 33. the constant of 2. Suppose the water pressure is proportionality; weakened so that the radius is 12 feet. derive the What is the area of lawn that will be 34. formula for the covered? area of a sector. Page 659 - Concept Byte: Exploring the Area of a Circle 35. Advanced – Problems 1-3: Ex 7-25 odd, 26 – 50, 55 - 63 MP 1 Page 667 – Concept Byte: Inscribed and Circumscribed Figures Page 663: 1-6 Basic – Problems 1-3: Ex. 7 – 25, 26 – 34 even, 35 – 36, 55 - 63 Average – Problems 1 – 3: Ex. 7 - 25 odd, 26 – 44, 55 - 63 MP 3 MP 6 MP 8 165 166 To find the length of a chord in a circle G.C.5 Question: See above Basic Skills Review 30 mins Text book page 666 http://www.mathopenr Standardized Test Prep ef.com/chord.html Q 51 - 54 http://www.regentsprep .org/Regents/math/geo Worksheets from stated websites. metry/GP14/CircleSegm ents.htm A circle with center O and radius 5 has PARCC/HSPA central angle XOY. If mXY = 600, what is PREP the length of chord XY? A circle w To use properties of a tangent to a circle 2 days G.C.2 Identify and describe Example: relationships You are standing at C, 8 feet from a silo. among inscribed The distance to a point of tangency is angles, radii, 12.1 16 feet. What is the radius of the silo? and chords. Include the Tangent Lines relationship between central, inscribed, and circumscribed angles; inscribed Text book Teacher made power page 762 – point presentations 36. 769 37. http://www.murrieta.k1 2.ca.us/cms/lib5/CA010 38. 00508/Centricity/Domai n/1830/T11.2.pdf 39. Page 766: 1-5 Basic – Problems 1-5: Ex. 6 – 22, 26 – 31, 36 – 44, Average – Problems 1 – 5: Ex. 7 - 19 odd, 20 – 31, 36 - 44 Advanced – Problems 1-5: Ex 7-19 odd, 20 – 31, 36 - 44 http://jmap.org/htmlsta ndard/Geometry/Inform al_and_Formal_Proofs/ G.G.50.htm 167 angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. MP 1 MP 3 30 mins To visualize the relation between two- G.GMD.4 Text book dimensional and three-dimensional page 769 Identify the objects. shapes of twodimensional Basic Skills Question: cross-sections of Review A three dimensional object is created threePARCC/HSPA by rotating a circle about one of its dimensional diameters. What is the shape of the PREP objects, and resulting object? Give as much detail as identify threepossible. dimensional objects generated by rotations of two- Teacher made power Standardized Test Prep point presentation. Q 32 - 35 168 dimensional objects. A circle To use congruent chords, arcs and central angles and also apply perpendicular bisectors to chords. Example: 12.2 2 days Chords And Arcs Find the Length of a Chord G.C.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 Text Book page 771 779 http://163.150.89.242/y Page 770 - Concept Byte: Paper hs/Faculty/AB/bagg/Geo Folding With Circles metry/images/Geometr y%20text%20PDFs/11.4. pdf 40. Page 776: 1-5 41. http://www.jmap.org/ht mlstandard/Geometry/I nformal_and_Formal_Pr 42. oofs/G.G.52.htm 43. Teacher made power point presentation Basic – Problems 1-4: Ex. 6– 16, 18, 23 – 25, 29, 44 52 Average – Problems 1 – 4: Ex. 7 - 15 odd, 16 – 34, 44 - 52 Advanced – Problems 1-4: Ex 7-15 odd, 16 – 39, 44 - 52 169 right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. MP 1 MP 3 To use the properties of tangent to construct a line tangent to a circle. 30 mins G.C.4 Construct Text book a tangent line page 779 from a point Question: outside a given Basic Skills Construct a line through point P that is circle to the Review tangent to circle O below. Leave all circle. PARCC/HSPA construction marks. PREP Construct a line through point. Teacher made power Standardized Test Prep point presentation. Q 40 - 43 http://www.mathopenr ef.com/consttangent.ht ml http://mathbits.com/Ma thBits/GSP/TangentCircl e.htm 170 http://www.youtube.co m/watch?v=IT52gEoGe9 A A circle 12.3 2 days Inscribed Angles To find the measures of an inscribed G.C.2, G.C.3, angle, measure of an angle formed by Construct the a tangent and a chord. inscribed and circumscribed circles of a Example: triangle, and Find the measure of the inscribed angle prove properties of or the intercepted arc. angles for a quadrilateral inscribed in a circle. Text book page 780 787 http://163.150.89.242/y hs/Faculty/AB/bagg/Geo 44. metry/images/Geometr y%20text%20PDFs/11.5. 45. pdf Page 784: 1-5 Basic – Problems 1-3: Ex. 6– 19, 18, 20 – 24 even, 28 29, 44 - 51 Teacher made power46. point presentations. Average – Problems 1 – 3: Ex. 7 - 17odd, 19 – 34, 44 - 51 47. Advanced – Problems 1-3: Ex 7-17odd, 19 - 39, 44 - 51 http://www.jmap.org/ht mlstandard/Geometry/I nformal_and_Formal_Pr 171 MP 1 oofs/G.G.51.htm MP 3 http://www.youtube.co m/watch?v=DjgPtK0_Qh 0 http://www.mathopenr ef.com/circleinscribed.h tml 30 mins To find the measure of an inscribed Basic Skills angle. Review Question: PARCC/HSPA Quadrilateral WXYZ is inscribed in a PREP circle. If radians G.C.2, G.C.3 Above Text book page 787 Teacher made power Standardized Test Prep point presentation. Q 40 - 43 http://www.mathopenr ef.com/circleinscribed.h tml 172 and mX 4 radians, what are 5 the measures, in radians, of the other two angles in the quadrilateral? 1 day To find measures of angles formed by G.C.2 Text book chords, secants, tangents, and also page 789 Identify and find the lengths of segments 797 describe associated with circles relationships among inscribed Example: angles, radii, and chords. Include the 12.4 relationship Angle between Measures and central, Segment inscribed, and Lengths circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the Teacher made power point presentation http://www.finneytown. 48. org/Downloads/GETE12 49. 042.pdf http://www.youtube.co 50. m/watch?v=Ax33G6YdS v0 51. Page 789 - Concept Byte: Exploring Chords and Secants Page 794: 1-7 Basic – Problems 1-3: Ex. 8– 20, 22 = 26 even, 27 – 31 odd, 48 – 55 Average – Problems 1 – 3: Ex. 9 - 19 odd, 21 – 39, 48 - 55 Advanced – Problems 1-3: Ex 9-19 odd, 21 – 43, 48 - 55 http://www.jmap.org/ht mlstandard/Geometry/I nformal_and_Formal_Pr oofs/G.G.51.htm 173 radius intersects the circle. MP 1 MP 3 To find the measure of an angle. Question: 30 mins In the figure below, ABC is Basic Skills circumscribed about the circle centered Review at O. If the measure of AOC is PARCC/HSPA radians, what is the measure, in PREP radians, of G.C.2, Above Text book page 797 Teacher made power Standardized Test Prep point presentation. Q 44- 47 http://www.jmap.org/ht mlstandard/Geometry/I nformal_and_Formal_Pr oofs/G.G.51.htm 174 1 day To write the equation of a circle and to G.GPE.1 Derive Text book find the center and radius of a circle. the equation of page 798 a circle of given 803 center and radius using the Example: Pythagorean Using the Center and a Point on a Theorem; Circle. complete the 12.5 square to find Write the standard equation of the the center and Circles in the circle with center (1, -3) that passes radius of a circle Coordinate through the point (2, 2). given by an Plane equation. http://www.finneytown. org/Downloads/GETE 52. 12052.pdf 53. http://www.jmap.org/ht 54. mlstandard/Geometry/ Coordinate_Geometry/ G.G.71.htm 55. Page 800: 1-7 Basic – Problems 1-3: Ex. 8– 30, 31 - 52 even, 61 – 65 Average – Problems 1 – 3: Ex. 9 - 29 odd, 31 – 56, 61 - 65 Advanced – Problems 1-3: Ex 9-29 odd, 31 – 56, 61 - 65 http://www.ck12.org/g eometry/Circles-in-theCoordinate-Plane/ MP 1 MP 3 MP 7 175 To find the equation of a circle Basic Skills Question: Review 30 mins PARCC/HSPA PREP G.GPE 1 Derive Text book the equation of page 803 a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and http://www.mathwareh Standardized Test Prep ouse.com/geometry/circ Q 58 - 60 le/equation-of-acircle.php Worksheets from stated websites. http://www.mathsisfun. com/algebra/circleequations.html 176 radius of a circle given by an equation. In the coordinate plane, the circle with radius r centered at h, k consists x, y h, k . of all the points that are r units Use from the Pythagorean theorem and the figure below to find an equation of the circle with radius r and center Explain your answer. 177 INSTRUCTIONAL FOCUS OF UNIT 56. In this unit, students will prove basic theorems about circles, with particular attention to perpendicularity and inscribed angles, in order to see symmetry in circles and as an application of triangle congruence criteria. 57. They will study relationships among segments on chords, secants, and tangents as an application of similarity. 58. In the Cartesian coordinate system, students will use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they will draw the graph in the coordinate plane, and apply techniques for solving quadratic equations to determine intersections between lines and circles or parabolas and between two circles. PARCC FRAMEWORK/ASSESSMENT PARCC EXEMPLARS www.parcconline.org 1. An archeologist dug up an edge piece of a circular plate. He wants to know what the original diameter of the plate was before it broke. However, the piece of pottery does not display the center of the plate. How could he find the original dimensions? 2. Jessica works at a daycare center and she is watching three rambunctious toddlers. One of the toddlers is in a crib at point A, another toddler is in her high chair at point B and the third toddler is in a play-station at point C. Where can Jessica position herself so that she is equidistant from each of the children? Construct an argument using concrete referents such as objects, drawing, diagrams and actions. 3. Since all circles are similar, the ratio of the will be the scale factor for any circle. Two students use different reasoning to find the length of an arc with central angle measure of 45° in a circle with radius=3cm. Compare the effectiveness of these two plausible arguments: 178 Sv ana ay : “I kn w Since ic a a 36 0 = 2 radians = , the equivalent measure of the length of the arc will be (2 * 3) or ay : “36 0 = 2 radians therefore 1 = radians. 45 =( c ” ) radians. Therefore the measure of the arc will be 45 =( Can y u u Sv ana’ r a ning fin an arc ng a cia wi a c n ra ang f º f a circ wi a ra iu f f to find the arc length of a circle with a radius of 6m and a central angle of 120º? Which method do you prefer and why? 4. ) radians * 3cm = Can y u u c ” ic a ’ r a ning Given a coordinate and a distance from that coordinate develop a rule that shows the locus of points that is that given distance from the given point (based on the Pythagorean theorem). If the coordinate of point H in the diagram below is (x,y) and the length of DH is 4 units. Can you write a rule that represents the relationship of the x value, the y value and the radius? Why is this relationship true? As point H rotates around the circle, does this relationship stay true? 5. In the diagram below, circle D translated 4 units to the right to create circle E. Why is the equation of this new circle (x − 4)2 + y2 = 42 . Why is the equation for circle I x2 + (y − 4)2 = 42 . Using similar reasoning, could you right and equation for a circle with the center at (-4, 0) and a radius of 4? Center of (0, -4) and radius of 4? What is the equation of a circle with center at (-8,11) and a radius of 5 ? Can you generalize this equation for a circle with a center at (h,k) and a radius of r? 179 ion of a a directrix. Add limitations for course 2 and 3. G-GPE.2 Given a focus and directrix, derive the equation of a parabola. Parabola is defined as “the set of all points P in a plane equidistant from a fixed line and a fixed point in the plane.” The fixed lineCommon is called Core the directrix, and the fixed point is called the focus. Wiki page for Assessments (Level II) http://maccss.ncdpi.wikispaces.net/file/view/Geometry%20Unpacking062512%20.pdf/359510147/Geometry%20Unpacking062512%20.pdf Ex. Derive the equation of the parabola that has the focus (1, 4) and the directrix x=-5. Ex. Derive the equation of the parabola that has the focus (2, 1) and the directrix y=-4. Ex. Derive the equation of the parabola that has the focus (-3, -2) and the vertex (1, -2). PARCC Framework Assessment questions with Model Curriculum Website for all units quations of http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf G-GPE.3 Given the foci, derive the equation of an ellipse, noting that the sum of the distances from the foci to any en the foci, fixed point on the ellipse is constant, identifying the major and minor axis. or m the foci G-GPE.3 Given the foci, derive the equation of a hyperbola, noting that the absolute value of the differences of the distances from the foci to a point on the hyperbola is constant, and identifying the vertices, center, transverse axis, http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsGeo_Nov2012V3_FINAL.pdf conjugate axis, and asymptotes. http://www.nj.gov/education/assessment/hs/hspa_mathhb.pdf 21ST CENTURY SKILLS 180 (4Cs & CTE Standards) Career Technical Education (CTE) Standards 5. 21st Century Life and Career Skills: All students will demonstrate the creative, critical thinking, collaboration, the problem-solving skills needed to function successfully as both global citizens and workers in diverse ethnic and organizational cultures. 9.1.12.B.1 Present resources and data in a format that effectively communicates the meaning of the data and its implications for solving problems, using multiple perspectives. 6. Personal Financial Literacy: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 9.2.12.B.1 Prioritize financial decisions by systematically considering alternatives and possible consequences. 7. Career Awareness, Exploration, and Preparation: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age. 9.3.12.C.2 Characterize education and skills needed to achieve career goals, and take steps to prepare for postsecondary options, including making course selections, preparing for and taking assessments, and participating in extra-curricular activities. 8. Career and Technical Education: All students who complete a career and technical education program will acquire academic and technical skills for careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees. 9.4.12.B.2 Demonstrate mathematics knowledge and skills required to pursue the full range of postsecondary education and career opportunities. Project Base Learning Activities: http://www.achieve.org/files/CCSS-CTE-Spread-of-Disease-FINAL.pdf 181 MODIFICATIONS/ACCOMMODATIONS Group activity or individual activity - Review and copy notes from eno board/power point/smart board etc. - Group/individual activities that will enhance understanding. - Provide students with interesting problems and activities that extend the concept of the l - Help students develop specific problem solving skills and strategies by providing scaffolding guiding questions Peer tutoring - Team up stronger math skills with lower math skills Use of manipulative - Eno or smart boards - Dry erase markers - Reference sheets created by special needs teacher - Pairs of students work together to make word cards for the chapter vocabulary - Use 3D shapes for visual learning - Reference sheets for classroom - Graphing calculators 182 APPENDIX (Teacher resource extensions) MP1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, an y c n inua y a k v ,“ i ak n ” y can un r an a r ac f r ving c r an i n ify c rr n nc w en different approaches. MP 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. MP 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can 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 that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there 183 is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. MP 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. MP 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts MP 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying 184 units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. MP 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MP 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x– 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 +x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 9. Kuta G 1: Kuta Software – Geometry (Free Worksheets) 10. Teacher Edition: Geometry Common Core by Pearson 11. Student Companion: Geometry Common Core by Pearson 12. Practice and Problem Solving Workbook: Geometry Common Core by Pearson 13. Teaching with TI Technology: Pearson Mathematics by Pearson 185 14. Progress Monitoring Assessments: Geometry Common Core by Pearson 15. http://www.jmap.org/JMAP_RESOURCES_BY_TOPIC.htm#Geo 16. http://www.mathopenref.com/ 17. http://www.mathisfun.com/ 18. http://www.mathwarehouse.com/ 19. http://www.regentsprep.org/Regents/math/geometry/math-GEOMETRY.htm * http://www.cpm.org/pdfs/state_supplements * http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqgeomapr15.pdf * http://illuminations.nctm.org * http://www.state.nj.us/education/cccs/standards/9/ *http://www.whiteplainspublicschools.org/cms/lib5/NY01000029/Centricity/Domain/360/Coordinate%20Geometry%20Proofs%20Packet%202012.pdf 186