CCGPS Analytic Geometry Unit 1: Similarity, Congruence .

Georgia Department of EducationCommon Core Georgia Performance Standards Framework Student EditionAnalytic Geometry Unit 1RELATED STANDARDSStandards for Mathematical PracticeThe Standards for Mathematical Practice describe varieties of expertise that mathematicseducators at all levels should seek to develop in their students. These practices rest on important“processes and proficiencies” with longstanding importance in mathematics education. The firstof these are the NCTM process standards of problem solving, reasoning and proof,communication, representation, and connections. The second are the strands of mathematicalproficiency specified in the National Research Council’s report Adding It Up: adaptivereasoning, strategic competence, conceptual understanding (comprehension of mathematicalconcepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly,accurately, efficiently and appropriately), and productive disposition (habitual inclination to seemathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’sown efficacy).1. Make sense of problems and persevere in solving them. High school students start toexamine problems by explaining to themselves the meaning of a problem and looking forentry 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 solutionpathway rather than simply jumping into a solution attempt. They consider analogousproblems, and try special cases and simpler forms of the original problem in order to gaininsight into its solution. They monitor and evaluate their progress and change course ifnecessary. Older students might, depending on the context of the problem, transformalgebraic expressions or change the viewing window on their graphing calculator to getthe information they need. By high school, students can explain correspondences betweenequations, verbal descriptions, tables, and graphs or draw diagrams of important featuresand relationships, graph data, and search for regularity or trends. They check theiranswers to problems using different methods and continually ask themselves, “Does thismake sense?” They can understand the approaches of others to solving complex problemsand identify correspondences between different approaches.2. Reason abstractly and quantitatively. High school students seek to make sense ofquantities and their relationships in problem situations. They abstract a given situationand represent it symbolically, manipulate the representing symbols, and pause as neededduring the manipulation process in order to probe into the referents for the symbolsinvolved. Students use quantitative reasoning to create coherent representations of theproblem at hand; consider the units involved; attend to the meaning of quantities, not justhow to compute them; and know and flexibly use different properties of operations andobjects.3. Construct viable arguments and critique the reasoning of others. High schoolstudents understand and use stated assumptions, definitions, and previously establishedresults in constructing arguments. They make conjectures and build a logical progressionof statements to explore the truth of their conjectures. They are able to analyze situationsMATHEMATICS ANALYTIC GEOMETRY UNIT 1: Similarity, Congruence, and ProofsGeorgia Department of EducationDr. John D. Barge, State School SuperintendentMay, 2012 Page 6 of 73All Rights Reserved

Georgia Department of EducationCommon Core Georgia Performance Standards Framework Student EditionAnalytic Geometry Unit 1by breaking them into cases, and can recognize and use counterexamples. They justifytheir conclusions, communicate them to others, and respond to the arguments of others.They reason inductively about data, making plausible arguments that take into accountthe context from which the data arose. High school students are also able to compare theeffectiveness of two plausible arguments, distinguish correct logic or reasoning from thatwhich is flawed, and—if there is a flaw in an argument—explain what it is. High schoolstudents learn to determine domains to which an argument applies, listen or read thearguments of others, decide whether they make sense, and ask useful questions to clarifyor improve the arguments.4. Model with mathematics. High school students can apply the mathematics they know tosolve problems arising in everyday life, society, and the workplace. By high school, astudent might use geometry to solve a design problem or use a function to describe howone quantity of interest depends on another. High school students making assumptionsand approximations to simplify a complicated situation, realizing that these may needrevision later. They are able to identify important quantities in a practical situation andmap their relationships using such tools as diagrams, two-way tables, graphs, flowchartsand formulas. They can analyze those relationships mathematically to draw conclusions.They routinely interpret their mathematical results in the context of the situation andreflect on whether the results make sense, possibly improving the model if it has notserved its purpose.5. Use appropriate tools strategically. High school students consider the available toolswhen solving a mathematical problem. These tools might include pencil and paper,concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebrasystem, a statistical package, or dynamic geometry software. High school students shouldbe sufficiently familiar with tools appropriate for their grade or course to make sounddecisions about when each of these tools might be helpful, recognizing both the insight tobe gained and their limitations. For example, high school students analyze graphs offunctions and solutions generated using a graphing calculator. They detect possible errorsby strategically using estimation and other mathematical knowledge. When makingmathematical models, they know that technology can enable them to visualize the resultsof varying assumptions, explore consequences, and compare predictions with data. Theyare able to identify relevant external mathematical resources, such as digital contentlocated on a website, and use them to pose or solve problems. They are able to usetechnological tools to explore and deepen their understanding of concepts.6. Attend to precision. High school students try to communicate precisely to others byusing clear definitions in discussion with others and in their own reasoning. They statethe meaning of the symbols they choose, specifying units of measure, and labeling axesto clarify the correspondence with quantities in a problem. They calculate accurately andefficiently, express numerical answers with a degree of precision appropriate for theproblem context. By the time they reach high school they have learned to examine claimsand make explicit use of definitions.MATHEMATICS ANALYTIC GEOMETRY UNIT 1: Similarity, Congruence, and ProofsGeorgia Department of EducationDr. John D. Barge, State School SuperintendentMay, 2012 Page 7 of 73All Rights Reserved

Georgia Department of EducationCommon Core Georgia Performance Standards Framework Student EditionAnalytic Geometry Unit 17. Look for and make use of structure. By high school, students look closely to discern apattern or structure. In the expression x2 9x 14, older students can see the 14 as 2 7and the 9 as 2 7. They recognize the significance of an existing line in a geometricfigure and can use the strategy of drawing an auxiliary line for solving problems. Theyalso can step back for an overview and shift perspective. They can see complicatedthings, such as some algebraic expressions, as single objects or as being composed ofseveral objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive numbertimes a square and use that to realize that its value cannot be more than 5 for any realnumbers x and y. High school students use these patterns to create equivalent expressions,factor and solve equations, and compose functions, and transform figures.8. Look for and express regularity in repeated reasoning. High school students notice ifcalculations are repeated, and look both for general methods and for shortcuts. Noticingthe 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 ageometric series. As they work to solve a problem, derive formulas or makegeneralizations, high school students maintain oversight of the process, while attending tothe details. They continually evaluate the reasonableness of their intermediate results.Connecting the Standards for Mathematical Practice to the Standards for MathematicalContentThe Standards for Mathematical Practice describe ways in which developing studentpractitioners of the discipline of mathematics should engage with the subject matter as they growin mathematical maturity and expertise throughout the elementary, middle and high school years.Designers of curricula, assessments, and professional development should all attend to the needto connect the mathematical practices to mathematical content in mathematics instruction.The Standards for Mathematical Content are a balanced combination of procedure andunderstanding. Expectations that begin with the word “understand” are often especially goodopportunities to connect the practices to the content. Students who do not have an understandingof a topic may rely on procedures too heavily. Without a flexible base from which to work, theymay be less likely to consider analogous problems, represent problems coherently, justifyconclusions, apply the mathematics to practical situations, use technology mindfully to workwith the mathematics, explain the mathematics accurately to other students, step back for anoverview, or deviate from a known procedure to find a shortcut. In short, a missing mathematicalknowledge effectively prevents a student from engaging in the mathematical practices.In this respect, those content standards which set an expectation of understanding arepotential “points of intersection” between the Standards for Mathematical Content and theStandards for Mathematical Practice. These points of intersection are intended to be weightedtoward central and generative concepts in the school mathematics curriculum that most merit thetime, resources, innovative energies, and focus necessary to qualitatively improve theMATHEMATICS ANALYTIC GEOMETRY UNIT 1: Similarity, Congruence, and ProofsGeorgia Department of EducationDr. John D. Barge, State School SuperintendentMay, 2012 Page 8 of 73All Rights Reserved

Georgia Department of EducationCommon Core Georgia Performance Standards Framework Student EditionAnalytic Geometry Unit 1curriculum, instruction, assessment, professional development, and student achievement inmathematics.ENDURING UNDERSTANDINGS Given a center and a scale factor, verify experimentally, that when dilating a figure in acoordinate plane, a segment of the pre-image that does not pass through the center of thedilation, is parallel to its image when the dilation is preformed. However, a segment thatpasses through the center remains unchanged.Given a center and a scale factor, verify experimentally, that when performing dilationsof a line segment, the pre-image, the segment which becomes the image is longer orshorter based on the ratio given by the scale factor.Use the idea of dilation transformations to develop the definition of similarity.Given two figures determine whether they are similar and explain their similarity basedon the equality of corresponding angles and the proportionality of corresponding sides.Use the properties of similarity transformations to develop the criteria for proving similartriangles: AA.Use AA, SAS, SSS similarity theorems to prove triangles are similar.Prove a line parallel to one side of a triangle divides the other two proportionally, and itsconverse.Prove the Pythagorean Theorem using triangle similarity.Use similarity theorems to prove that two triangles are congruent.Use descriptions of rigid motion and transformed geometric figures to predict the effectsrigid motion has on figures in the coordinate plane.Knowing that rigid transformations preserve size and shape or distance and angle, usethis fact to connect the idea of congruency and develop the definition of congruent.Use the definition of congruence, based on rigid motion, to show two triangles arecongruent if and only if their corresponding sides and corresponding angles arecongruent.Use the definition of congruence, based on rigid motion, to develop and explain thetriangle congruence criteria: ASA, SSS, and SAS.Prove vertical angles are congruent.Prove when a transversal crosses parallel lines, alternate interior angles are congruent andcorresponding angles are congruent.Prove points on a perpendicular bisector of a line segment are exactly those equidistantfrom the segment’s endpoints.Prove the measures of interior angles of a triangle have a sum of 180º.Prove base angles of isosceles triangles are congruent.Prove the segment joining midpoints of two sides of a triangle is parallel to the third sideand half the length.Prove the medians of a triangle meet at a point.MATHEMATICS ANALYTIC GEOMETRY UNIT 1: Similarity, Congruence, and ProofsGeorgia Department of EducationDr. John D. Barge, State School SuperintendentMay, 2012 Page 9 of 73All Rights Reserved

Georgia Department of EducationCommon Core Georgia Performance Standards Framework Student EditionAnalytic Geometry Unit 1 Prove properties of parallelograms including: opposite sides are congruent, oppositeangles are congruent, diagonals of a parallelogram bisect each other, and conversely,rectangles are parallelograms with congruent diagonals.Copy a segment and an angle.Bisect a segment and an angle.Construct perpendicular lines, including the perpendicular bisector of a line segment.Construct a line parallel to a given line through a point not on the line.Construct an equilateral triangle so that each vertex of the equilateral triangle is on thecircle.Construct a square so that each vertex of the square is on the circle.Construct a regular hexagon so that each vertex of the regular hexagon is on the circle.CONCEPTS/SKILLS TO MAINTAINSome students often do not recognize that congruence is a special case of similarity.Similarity with a scale factor equal to 1 becomes a congruency. Students may not realize thatsimilarities preserve shape, but not size. Angle measures stay the same, but side lengths changeby a constant scale factor. Some students often do not list the vertices of similar triangles inorder. However, the order in which vertices are listed is preferred and especially important forsimilar triangles so that proportional sides can be correctly identified. Dilations and similarity,including the AA criterion, are investigated in Grade 8, and these experiences should be builtupon in high school with greater attention to precise definitions, careful statements and proofs oftheorems and formal reasoning.The Pythagorean Theorem and its converse are proved and applied in Grade 8. In highschool, another proof, based on similar triangles, is presented. The alternate interior angletheorem and its converse, as well as properties of parallelograms, are established informally inGrade 8 and proved formally in high school.Properties of lines and angles, triangles and parallelograms are investigated in Grades 7and 8. In high school, these properties are revisited in a more formal setting, giving greaterattention to precise statements of theorems and establishing these theorems by means of formalreasoning.The theorem about the midline of a triangle can easily be connected to a unit onsimilarity. The proof of it is usually based on the similarity property that corresponding sides ofsimilar triangles are proportional.Students should be expected to have prior knowledge/experience related to the concepts andskills identified below. Pre-assessment may be necessary to determine whether instructionaltime should be spent on conceptual activities that help students develop a deeper understandingof these ideas. Understand and use reflections, translations, and rotations.Define the following terms: circle, bisector, perpendicular and parallel.Solve multi-step equations.MATHEMATICS ANALYTIC GEOMETRY UNIT 1: Similarity, Congruence, and ProofsGeorgia Department of EducationDr. John D. Barge, State School SuperintendentMay, 2012 Page 10 of 73All Rights Reserved

Georgia Department of EducationCommon Core Georgia Performance Standards Framework Student EditionAnalytic Geometry Unit 1 Understand angle sum and exterior angle of triangles.Know angles created when parallel lines are cut by a transversal.Know facts about supplementary, complementary, vertical, and adjacent angles.Solve problems involving scale drawings of geometric figures.Draw geometric shapes with given conditions.Understand that a two-dimensional figure is congruent to another if the second can beobtained from the first by a sequence of rotations, reflections, and translations.Draw polygons in the coordinate

Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis and problem solving. The first unit of Analytic Geometry involves similarity, congruence, and proofs. Students will understand similarity in terms of similarity transformations, prove