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Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Milwaukee Public Schools PACING GUIDE for High School GEOMETRY Foundation Level Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Dear Milwaukee Public School Geometry Teachers: With the adoption of a new textbook series for the Milwaukee Public Schools in the Spring of 2008, it became necessary to design pacing guides for these textbooks. The following is Draft One of the Pacing Guide for Discovering Geometry. Just like the headers to this document state, this is a draft. Our district is relying on your good judgment and teaching skills to provide us with insights, suggestions, modifications, corrections, and plans with the intent of incorporating these into Draft 2. Members of the Key Curriculum Press have told us that a first year pacing guide must be very tentative – we will not have a clear picture of just what our students can accomplish and what our teachers can complete until the full one year teaching and learning cycle is complete. Therefore, we not only suggest, but highly recommend, that you use this as a guide – modify to fit the needs of your style of teaching and the needs of your students. This means that you might have to add more time to one section of the text and then reduce time spent on another section. This also gives you the ability to eliminate a section or sections because time is not available. However, the Discovering Series is a series that builds on investigations – please make sure these are a definite part of your lesson plans, and teaching and learning in your classrooms. Usually there is more than one investigation in a section – at least one should be tried. We also hope that you are using the cooperative learning skills and strategies so important in this math curriculum. Many of our schools are on different school calendars – some are blocked; some start earlier in August; some are year-round schools. This pacing guide is set for 36 weeks. Time is allotted for semester exams – review and assessment. All explorations and algebra skills activities are also included. These are places where adjustments can be made. Six chapters are covered in Semester One and seven chapters are covered in Semester Two. We have heard that some teachers may be struggling to cover the first six chapters in one semester – especially in the blocked schools. This is OK! The first semester is loaded with vocabulary, concepts and skills that will be needed for semester two – taking more time on the first five chapters may prove beneficial when you approach the last chapters of the book. From Week 28 on, fewer sections are covered each week and more time is allowed for chapter assessments. These may be the weeks during which you can catch-up. If time does run out, we suggest that perhaps Chapter 13, Geometry as a Mathematical System, be one that you eliminate, at least for this first year. Please remember that all of the above are suggestions – the ultimate decision of how to cover the material in the text and to follow the pacing guide resides with the classroom teacher. We ask that all chapters are covered (with the possibility that you will not cover Chapter 13 and some sections may be eliminated). Your reflections and feedback will periodically be requested by your MTL and/or department chair to share with our Math Specialists, FTF Math Instructional Coach, Key Curriculum Press, and others who are part of the mathematics leadership in our district. Thank you in advance for your participation. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Drafted by: Sherrie K. Akinsanya, FTF Mathematics Instructional Coahc Time (week) Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 0 Geometric Art CHAPTER 0 LT2*: Use a variety of mathematical and technological tools to determine measurements 0.1 Geometry directly (ruler, protractor, compass) and to check reasonableness of answers. When students apply in Art and Nature technology, e.g. computers and graphing calculators, they will interpret results 0.2 Line appropriately. Designs 0.3 Circle Designs 0.4 Op Art 0.6 Islamic Tiles LT16: Apply transformations and their compositions to analyze geometric figures and create designs. Benchmarks Embedded in Chapter Lessons District Model CABS CHAPTER 0 2.2* Measure lengths using a metric/English ruler (to the appropriate degree of tenths of cm and sixteenths of an inch). 16.4 Determine whether a figure has point symmetry, line symmetry or neither. 16.5 Identify tessellations in art, construction and nature. 16.6 Design a real-world application of transformations and communicate the process used in the design. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Symmetry, axis of symmetry, compass, straightedge Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback Reflection Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity Chapter 1 Introducing Geometry 1.1 Building Blocks of Geometry Algebra Review Midpoint 1.2 Poolroom Math 1.3 Widgets MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Benchmarks Embedded in Chapter Lessons District Suggested Model CABS vocabulary CHAPTER 1 CHAPTER 1 LT2*: Use a variety of mathematical and 2.3* Measure angles using a technological tools to determine measurements protractor (to appropriate degrees). directly (ruler, protractor, compass) and to check reasonableness of answers. When students apply technology, e.g. computers and graphing calculators, they will interpret results appropriately. LT8*: Use the two-dimensional coordinate plane to analyze, generalize and represent algebraic properties in mathematical and real-world applications. LT12*: Identify and apply the definitions related to lines and angles in order to use them to solve problems and perform basic geometric constructions. 8.3* Use a coordinate plane and the distance and midpoint formulas to determine lengths and midpoints of segments. 12.1* Identify and name points, lines, rays, segments and planes. 12.2* Name and classify angles and line segments. 12.3* Identify and use angle-pair relationships in mathematical and real-world applications. 12.4* Classify and communicate the differences between pairs of lines. 12.6 * Prove statements about segments, lines, angles and angle relationships by using properties, postulates, and theorems. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Point, line, plane, collinear, coplanar, line segment, endpoint, congruent, midpoint, protractor, angle, vertex, counterexample , right angle, acute angle, obtuse angle, complementary angles, supplementary angles, vertical angles, linear pair, dimension Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback and Reflection Week 2 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity Chapter 1: continued MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. LT13*: Classify and describe polygons and 3dimensional figures. Assessment 1.4 Polygons 1.5 Triangles 1.6 Special Quadrilaterals Assessment 1.7 Circles 1.8 3-D Geometry LT14: Use and analyze geometric relationships and properties of polygons and 3-dimensional figures to solve mathematical and real-world applications. LT15: Apply the basic theorems of congruence and similarity to solve mathematical and realworld problems. LT18: Use perimeter, area, volume, and surface area to solve real-world applications. Benchmarks Embedded in Chapter Lessons District Model CABS 13.1* Classify polygons by sides and angles. 13.2* Classify triangles by sides and angles. 13.3* Classify quadrilaterals by their parts. 13.5* Describe properties of quadrilaterals. 14.7 Make a two-dimensional representation of a 3-dimensional object. 15.1 Identify congruent and similar polygons. 18.8 Make a model of a 3-dimensional figure from a 2-dimensional drawing. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Polygon, right triangle, acute triangle, isosceles triangle, scalene triangle, equilateral triangle, obtuse triangle, quadrilateral, circle, chord, diameter, radius, tangent, arcs in circles Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback Reflection Week 3 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Benchmarks Embedded in Chapter Lessons District Model CABS Chapter1: continued 1.9 Pictures Chapter 1: Learning Targets Covered: LT12 – Lines and Angles LT 13 – Classify Polygons Exploration: LT 14 – Analyze Polygons Geometric LT 15 – Congruence and Similarity Probability LT 18 – Geometric Formulas Review Assessment Chapter 2 Reasoning in Geometry 2.1 Inductive Reasoning 2.2 Nth Term CHAPTER 2 CHAPTER 2 LT10*. Evaluate information, analyze patterns ,identify relationships, and represent them using algebraic expressions, functions, and equations, e.g., linear, exponential and quadratic, recognizing that a family of functions can model a variety of real-world applications. 10.7* Identify a pattern in a numerical sequence and complete the pattern to a given term. LT19: Use increased rigor in justifying conclusions to mathematical and real-world applications and in verifying the truth of geometric theorems. 19.3 Write conjectures using characteristics, postulates and theorems of geometric figures. 19.4 Use and interpret Venn diagrams. 19.6 Use valid forms of deductive reasoning, including the laws of detachment and syllogism, to prove conjectures. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Venn diagram, probability, justification, inductive logic, deductive logic, line of best fit, conjecture, pattern, strategy, randomness, reasonableness, representations Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Reflection and Feedback: Week 4 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Benchmarks Embedded in Chapter Lessons District Model CABS Suggested vocabulary Transversal Chapter 2: continued LT10*. Evaluate information, analyze patterns, 10.7* Identify a pattern in a identify relationships, and represent them using numerical sequence and complete 2.3 Modeling algebraic expressions, functions, and equations, the pattern to a given term. e.g., linear, exponential and quadratic, recognizing Assessment that a family of functions can model a variety of 2.4 realDeductive world applications. Logic 12.3* Identify and use angle-pair LT12*: Identify and apply the definitions related to relationships in mathematical and 2.5 Angle Relationships lines and angles in order to use them to solve real-world applications. problems and perform basic geometric 12.5* Classify the types of angles 2.6 Angles constructions. formed by two lines and a and Parallel transversal. Lines 14.6 Solve problems involving real-world applications using the LT14: Use and analyze geometric relationships and relationships between pairs of properties of polygons and 3-dimensional figures to angles. solve mathematical and real-world applications. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback and Reflection: Week 5 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity Chapter 2: continued Algebra Review: Slope Review Assessment MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Benchmarks Embedded in Chapter Lessons District Model CABS Slope, coordinate system, rate of change, bisector, perpendicular Learning Targets Covered: Chapter 2 LT 12 – Lines and Angles LT 14 – Analyze Polygons LT 19 – Proofs CHAPTER 3 LT2*: Use a variety of mathematical and technological tools to determine measurements directly (ruler, protractor, compass) and to check reasonableness of answers. When students 3.1 apply technology, e.g. computers and graphing Duplicating calculators, they will interpret results Segments and appropriately. Angles CHAPTER 3 2.2* Measure lengths using a metric/English ruler (to the appropriate degree of tenths of cm and sixteenths of an inch). 2.3* Measure angles using a protractor (to appropriate degrees). 3.2 Construct Perpendicular. Bisectors 12.7* Perform basic geometric constructions. Chapter 3: Using Tools of Geometry LT12*: Identify and apply the definitions related to lines and angles in order to use them to solve problems and perform basic geometric constructions. Suggested vocabulary Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback Reflection Week 6 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Benchmarks Embedded in Chapter Lessons District Model CABS Chapter 3: continued LT2*: Use a variety of mathematical and 2.3* Measure angles using a technological tools to determine measurements protractor (to appropriate directly (ruler, protractor, compass) and to check degrees). 3.3 Construct reasonableness of answers. When students apply Perpendicular. technology, e.g. computers and graphing calculators, Lines they will interpret results appropriately. Assessment LT12*: Identify and apply the definitions related to 12.7* Perform basic geometric lines and angles in order to use them to solve problem constructions. and perform basic geometric constructions. 3.4 Angle Bisectors 3.5 Parallel Lines Algebra Review Slopes of perpendicular and parallel lines LT16: Apply transformations and their compositions to analyze geometric figures and create designs. 16.1 Plot and label geometric figures using a coordinate plane. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Altitude, angle bisector, parallel Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback Reflection Week 7 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Benchmarks Embedded in Chapter Lessons Chapter 3: continued LT12*: Identify and apply the definitions related to lines and angles in order to use them to solve problems and perform basic 3.6 geometric constructions. Constructions 12.7* Perform basic geometric constructions. LT18: Use perimeter, area, volume, and surface area to solve real-world Exploration: applications. 18. 7 Solve real-world application problems using scale drawings, perspective drawings, blueprints, or computer drawings as models of 3dimensional objects. Assessment Perspective Drawing Learning Targets Covered: Chapter 3 LT 12 – Lines and Angles 3.7 LT 16 – Transformations Concurrency LT 18 – Geometric Formulas District Model CABS 3.8 Centroid Review Assessment Feedback Reflection Week 8 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Perspective, concurrency, circumcener, incenter, centroid, orthocenter Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity Chapter 4: Discovering and Proving Triangle Properties MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Benchmarks Embedded in Chapter Lessons CHAPTER 4 LT12*: Identify and apply the definitions related to lines and angles in order to use them to solve problems and perform basic geometric constructions. CHAPTER 4 12.7* Perform basic geometric constructions. LT14: Use and analyze geometric relationships and properties of polygons and 3-dimensional figures to 4.2 Isosceles solve mathematical and real-world applications. Triangle 14.1 Determine if lengths of 14.2 three segments can form a triangle. 14.3 Determine the range in which the length of the third side of a triangle must be given the lengths of two sides. 4.1 Triangle Sum Algebra Review Solving Equations 4.3 Triangle Inequalities Assessment LT18: Use perimeter, area, volume and surface area to solve real-world applications. District Model CABS 18.5 Compare perimeters and areas of similar 2-dimensional figures using proportions. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Equation Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback Reflection Week 9 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity Chapter 4, continued 4.4 Congruence Shortcuts MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. LT15: Apply the basic theorems of congruence and similarity to solve mathematical and real-world problems. 4.5 Congruence Shortcuts 4.6 Congruence Shortcuts 4.7 Flow Charts 4.8 Extra Triangle Proofs LT19: Use increased rigor in justifying conclusions to mathematical and real-world applications and in verifying the truth of geometric theorems. Benchmarks Embedded in Chapter Lessons District Model CABS 15.2 Justify congruency or similarity of polygons by using formal and informal proofs. 15.3 Prove two triangles are congruent using coordinate methods (e.g. distance and slope formulas). 15.4 Solve real-world applications using the properties of congruency and similarity. 19.7 Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures. Feedback and Reflection: Week 10 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Inequality Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity Review Assessment Algebra Review: Systems of Equations MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Learning Targets Covered in Chapter 4: Benchmarks Embedded in Chapter Lessons District Model CABS LT12 – Lines and Angles LT 14 – Analyze Polygons LT 15 – Congruency and Similarity LT 18 – Geometric Formulas LT 19 - Proofs WKCE Prep Feedback Reflection Week 11 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Benchmarks Embedded in Chapter Lessons District Model CABS WKCE prep NOTE: MPS High Schools choose their own dates within a specific window for the WKCE test. Two weeks are given in this pacing guide for WKCE test WKCE review. Each school will decide who many days for review they want to take. The remainder of the days can be used to catch-up on the pacing guide if four chapters have NOT been covered and/or to start on Chapter 5. Feedback Reflection Week 12 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 5: CHAPTER 5 Discovering and Proving LT12. Identify and apply the definitions related to lines and angles in order to use them to Polygon Properties solve problems and perform basic geometric constructions. 5.1 Polygon Sum Conjecture LT13*: Classify and describe polygons and 3dimensional figures. 5.2 Exterior Angles of LT14: Use and analyze geometric relationships Polygons and properties of polygons and 3-dimensional figures to solve mathematical and real-world Assessment applications. 1 Benchmarks Embedded in Chapter Lessons District Model CABS CHAPTER 5 12.6 Prove statements about segments, lines, angles and angle relationships by using properties, postulates, and theorems. 13.1* Classify polygons by sides and angles. 14.4 Find the measures of each interior and exterior angle of a regular polygon. 14.5Find the sum of the measures of the interior and exterior angles of a convex polygon. 14.6 Solve problems involving real-world applications using the relationships between pairs of angles.. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Exterior angles Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback Reflection: Week 13 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. LT2. Use a variety of mathematical and technological tools to determine measurements directly (ruler, protractor, compass) and to 5.3 Kite and check reasonableness of answers. When Trapezoid students apply technology, e.g. computers and Properties graphing calculators, they will interpret results appropriately. 5.4 Properties of LT12*: Identify and apply the definitions related Midsegments to lines and angles in order to use them to solve problems and perform basic geometric 5.5 Properties of constructions. Parallelograms LT13*: Classify and describe polygons and 3dimensional figures. ASSESSMENT Chapter 5, continued 5.6 Special Parallelograms LT19* Use increased rigor in justifying Algebra conclusions to mathematical and real-world Review: Linear equations applications and in verifying the truth of geometric theorems. LT9. Model and solve mathematical and realworld applications that can be represented by linear equations and systems of equations, linear inequalities, and quadratic equations. Benchmarks Embedded in Chapter Lessons District Model CABS 2.2* Measure lengths using a metric/English ruler (to the appropriate degree of tenths of cm and sixteenths of an inch). 2.3* Measure angles using a protractor (to appropriate degrees). 12.7* Perform basic geometric constructions. 13.4 Describe properties of a given polygon. 13.5 Describe properties of quadrilaterals. 13.6 Describe properties of other polygons. 19.3 Write conjectures using characteristics, postulates and theorems of geometric figures 9.8 Solve linear equations, inequalities, quadratic equations and systems of linear equations related to real-world applications. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Midsegment, Intercept Perpendicular Parallel Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback and Reflection: Week 14 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 5: continued 5.7 Proving Quadrilateral Properties Review Chapter Assessment Benchmarks Embedded in Chapter Lessons LT19. Use increased rigor in justifying conclusions to mathematical and real-world applications and in verifying the truth of geometric theorems. 19.6 Use valid forms of inductive and deductive reasoning to draw conclusions and make conjectures. Learning Targets Covered in Chapter 5: LT12 Lines and Angles LT13 – Classify Polygons LT14 – Analyze Polygons LT19 - Proofs 19.7 Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures. District Model CABS Feedback Reflection Week 15 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. CHAPTER 6 Discovering and Proving Circle Properties 6.1 Tangent 6.2 Chord Assessment 6.3 Arcs/Angles Benchmarks Embedded in Chapter Lessons CHAPTER 6 CHAPTER 6 LT12. Identify and apply the definitions related to lines and angles in order to use them to solve problems and perform basic geometric constructions. LT17: Use properties of circles to solve mathematical and real-world applications. 12.7 Perform basic geometric constructions. District Model CABS 17.2 Determine measures of inscribed and central angles. 17.3 Determine arc lengths and areas of sectors of circles. 17.4 Write and solve equations to find the measures of angles formed by chords, secants, and tangents. 17.5 Calculate lengths of chord, tangent and secant segments. Feedback Reflection Week 16 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 6, continued LT17: Use properties of circles to solve mathematical and real-world applications. 6.4 Proving Circle Conjectures 6.5 Circles and Diameter LT19: Use increased rigor in justifying conclusions to mathematical and real-world applications and in verifying the truth of geometric theorems. Benchmarks Embedded in Chapter Lessons District Model CABS 17.1 Determine circumference and area of a circle. 17.5 Calculate lengths of chord, tangent and secant segments. 17.7 Solve real-world applications of circles. 19. 7 Use valid forms of inductive and deductive reasoning to draw conclusions and make conjectures. 19.8 Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures. Feedback Reflection Week 17 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 6, continued LT17: Use properties of circles to solve mathematical and real-world applications. 6.6 Around the World Algebra Review: Systems of Equations 6.7 Arc length Review Chapter Assessment Benchmarks Embedded in Chapter Lessons District Model CABS 17.1 Determine circumference and area of a circle. 17.3 Determine arc lengths and areas of sectors of circles. Learning Targets Covered in Chapter 6: LT12 – Lines and Angles LT17 – Circles LT19 - Proofs Feedback Reflection Week 18 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Review for Final Exam Final Exam Semester One – MA 301 LT2. Use a variety of mathematical and technological tools to determine measurements directly (ruler, protractor, compass) and to check reasonableness of answers. When students apply technology, e.g. computers and graphing calculators, they will interpret results appropriately. LT12*: Identify and apply the definitions related to lines and angles in order to use them to solve problems and perform basic geometric constructions. LT13*: Classify and describe polygons and 3-dimensional figures. LT14: Use and analyze geometric relationships and properties of polygons and 3-dimensional figures to solve mathematical and real-world applications. LT15: Apply the basic theorems of congruence and similarity to solve mathematical and real-world problems. LT17: Use properties of circles to solve mathematical and real-world applications. MPS Learning Targets Supported in Semester One: MA301 District Model CABS Semester One – MA 301 LT16 Apply transformations and their compositions to analyze geometric figures and create designs. LT18 Use perimeter, area, volume and surface area to solve real-world applications. LT19 Use increased rigor in justifying conclusions to mathematical and real-world applications and in verifying the truth of geometric theorems. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested Vocabulary Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback Reflection: Semester 1 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity Chapter 7: Transformations and Tesselations 7.1 Transformations/ Symmetry 7.2 Properties of Isometrics 7.3 Compositions of Transformations Assessment MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. CHAPTER 7 LT16: Apply transformations and their compositions to analyze geometric figures and create designs. Benchmarks Embedded in Chapter Lessons District Model CABS CHAPTER 7 16.2 Perform transformations using triangles in a coordinate plane. 16.3 Identify, given an image and pre-image, the transformation that has taken place as a reflection, rotation, or translation. 16.4 Determine whether a figure has point symmetry, line symmetry or neither. 16.5 Design a real-world application of transformations and communicate the process used in the design. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Central angles, inscribed angles Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback and Reflection Week 20 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 7, continued 7.4 Tessellations of regular polygons 7.5 Tessellations of non-regular polygons Assessment 7.6 Tessellations with translations 7.7 Tessellations with rotations LT16: Apply transformations and their compositions to analyze geometric figures and create designs. Benchmarks Embedded in Chapter Lessons District Model CABS 16.6 Identify tessellations in art, construction and nature. 16.7 Design a real-world application of transformations and communicate the process used in the design. Exploration: Tessellating with Conway Criterion Review Chapter Assessment Learning Target covered in Chapter 7: LT 16: Transformations Feedback and Reflection: Week 21 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Systems of equations Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 8: Area 8.1 Area of rectangles and parallelograms 8.2 Area of kites, trapezoids and triangles Assessment CHAPTER 8 LT18: Use perimeter, area, volume and surface area to solve real-world applications. Benchmarks Embedded in Chapter Lesson District Model CABS CHAPTER 8 18.1 Determine the perimeter/circumference and area of quadrilaterals, triangles and other polygons, using real-world applications. 8.3 Area Problems Algebra Review: Products, factors, quadratic equations 8.4 Area of regular polygons Exploration: Pick’s formula for area Feedback and Reflection: Week 22 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Area, kite trapezoid Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Benchmarks Embedded in Chapter Lessons District Model CABS Sector of a circle, segment of a circle, annulus Chapter 8, continued 8.5 Area of circles Assessment LT17: Use properties of circles to solve mathematical and real-world applications. 8.6 Any Way you Slice it: Area of Circle segments Exploration: Geometric Probability LT18: Use perimeter, area, volume and surface area to solve real-world applications. 8.7 Surface Area Chapter 8 Assessment Suggested vocabulary Learning Targets covered in Chapter 8: LT 17: Circles LT 18: Formulas 17.1 Determine circumference and area of a circle. 17.3 Determine arc lengths and areas of sectors of circles. 18.1 Determine the perimeter/circumference and area of quadrilaterals, triangles and other polygons, using real-world applications. Feedback and Reflection: Week 23 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 9: Pythagorean Theorem 9.1 The Theorem of Pythagoras 9.2 The Converse of the Pythagorean Theorem Algebra Review: Radical Expressions CHAPTER 9 LT11: Explain and use the Pythagorean Theorem and right triangle trigonometry as models to solve a variety of right triangle problems in real-world applications. Benchmarks Embedded in Chapter Lessons District Model CABS CHAPTER 9 11.1 Apply the Pythagorean Theorem to find side lengths of a right triangle in mathematical and real-world applications. 11.2 Determine if a triangle is right, obtuse or acute by using the Pythagorean Theorem. 9.3 Two Special Right Triangles 9.4 Story Problems Feedback and Reflection: Week 24 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Surface area, Pythagorean Formula Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 9, continued LT12: Classify and describe polygons and 3-dimensional figures. Assessment 9.5 Distance in a Coordinate Plane Exploration: Ladder Climb 9.6 Circles and the Pythagorean Theorem LT11: Explain and use the Pythagorean Theorem and right triangle trigonometry as models to solve a variety of right triangle problems in real-world applications. LT18: Use appropriate measurements such as perimeter, area, volume, surface area and degrees to solve real-world applications. Review Chapter 9 Assessment Benchmarks Embedded in Chapter Lessons District Model CABS 12.7 Classify geometric properties by using the coordinate system (e.g. distance formula, slope). 11.1 Apply the Pythagorean Theorem to find side lengths of a right triangle in mathematical and real-world applications. 18.4 Solve real-world application problems for missing lengths in special right triangles. Learning Targets covered in Chapter 9: LT 11: Pythagorean Theorem LT 12: Classify polygons LT 18: Formulas Feedback and Reflection: Week 25 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 10: Volume 10.1 Geometry of solids CHAPTER 10 Benchmarks Embedded in Chapter Lessons CHAPTER 10 LT18: Use perimeter, area, volume and surface area to solve real-world applications. 18.2 Determine the volume and lateral area and total surface area of spheres, prisms, pyramids, cylinders and cones. 18.3 Use volume, lateral area and surface area of 3dimensional figures to solve real-world application problems. 18.8 Make a model of a 3dimensional figure from a 2dimensional drawing. LT12: Classify and describe polygons and 3dimensional figures. 12.8 Identify 3-dimensional objects (e.g. prisms, pyramids, cones, cylinders and spheres). LT13: Use and analyze geometric relationships and properties of polygons and 3-dimensional figures to solve mathematical and real-world applications. 13.7 Make a twodimensional representation of a 3-dimensional object. Exploration: Euler’s formula for polyhedra 10.2 Volume of Prisms and cylinders 10.3 Volume of pyramids and cones Assessment Exploration: The Five Platonic Solids District Model CABS Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Volume, polyhedron, prism, pyramid, cone, sphere, lateral face, surface area, cylinder Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Reflection and Feedback: Week 26 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 10, continued 10.4 Volume LT18: Use perimeter, area, volume and surface area to solve real-world applications. 10.5 Displacement and Density LT19: Use increased rigor in justifying conclusions to mathematical and real-world applications and in verifying the truth of Exploration Orthographic Drawing geometric theorems. Assessment 10.6 Volume of a sphere Benchmarks Embedded in Chapter Lessons District Model CABS 18.2 Determine the volume and lateral area and total surface area of spheres, prisms, pyramids, cylinders and cones. 18.3 Use volume, lateral area and surface area of 3-dimensional figures to solve real-world application problems. 18.7 Solve real-world application problems using scale drawings, perspective drawings, blueprints, or computer drawings as models of 3-dimensional objects. 19.1 Identify the converse, inverse and contra positive of a conditional statement. 10.7 Surface area of a sphere Reflection and Feedback Week 27 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Displacement, density, orthographic drawing, isometric drawing Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity Chapter 10, continued Algebra: Solving for any variable MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models LT19: Use increased rigor in justifying conclusions to mathematical and real-world applications and in verifying the truth of geometric theorems. Benchmarks Embedded in Chapter Lessons District Model CABS 19.2 Translate short verbal arguments into symbolic form Assessment Exploration Sherlock Holmes and Forms of Reasoning Review Chapter 10 Assessment Feedback and Reflection Week 28 Learning Targets covered in Chapter 10: 12: Classify Polygons 13: Analyze Polygons 18: Circles 19: Proofs 19.3 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Negation Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 11: Similarity Algebra Skills: Proportion 11.1 Similar polygons 11.2 Similar triangles Assessment Exploration: Constructing a Dilation Design CHAPTER 11 LT13: Use and analyze geometric relationships and properties of polygons and 3-dimensional figures to solve mathematical and real-world applications. LT15: Apply the basic theorems of congruence and similarity to solve mathematical and realworld problems. Benchmarks Embedded in Chapter Lessons District Model CABS CHAPTER 11 13.3 Find the measures of segments in similar polygons. 15.1 Identify congruent and similar polygons. 15.2 Justify congruency or similarity of polygons by using formal and informal proofs. 15.4 Solve real-world applications using the properties of congruency and similarity. LT18: Use perimeter, area, volume and surface 18.5 Compare perimeters and area to solve real-world applications. areas of similar 2-dimensional figures using proportions. 18.6 Describe how a change in one measurement affects other measurements of the object. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Proportion, ratio, similar, dilation, scale Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Reflection and Feedback Week 29 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 11, continued 11.3 Indirect measurement with similar triangles 11.4 Corresponding parts of similar triangles Assessment LT15: Apply the basic theorems of congruence and similarity to solve mathematical and realworld problems. LT18: Use perimeter, area, volume and surface area to solve real-world applications. . Benchmarks Embedded in Chapter Lessons District Model CABS 15.5 Solve real-world applications using the properties of congruency and similarity. 18.5 Compare perimeters and areas of similar 2-dimensional figures using proportions. 18.6 Describe how a change in one measurement affects other measurements of the object. 11.5 Proportions with area 11.6 Proportions with volume Exploration: Why Elephants Have Big Ears Reflection and Feedback Week 30 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 11, continued 11.7 Proportion and parallel lines Exploration: Two more forms of Valid Reasoning Review Benchmarks Embedded in District Chapter Lessons Model CABS 14.3 Find the measures of LT14: Identify and apply the definitions related segments in similar to lines and angles in order to use them to solve polygons. problems and perform basic geometric constructions. LT 19: Use reasoning and logic to draw conclusions to mathematical and real-world applications. Chapter 11 Assessment Learning Targets covered in Chapter 11: LT13: Classify Polygons LT14: Analyze Polygons LT15: Congruence and Similarity LT18: Formulas LT19: Proofs 19.5 Diagram conditional statements to solve logic puzzles. 19.7 Use valid forms of inductive and deductive reasoning to draw conclusions and make conjectures. 19.8 Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures. Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Converse, inverse, contrapositive Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Feedback and Reflection: Week 31 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Benchmarks Embedded in Chapter Lessons District Model CABS Chapter 12: Trigonometry LT11: Explain and use the Pythagorean Theorem and right triangle trigonometry as models to solve a variety of right triangle 12.2 Solving with right problems in real-world applications. 12.1 Trigonometric ratios triangles Assessment Exploration: Indirect measurement . 11.3 Solve right triangle problems using trigonometric ratios. 11.4 Use trigonometric ratios to find the angle of elevation or the angle of depression in real world applications 12.3 Law of Sines 12.4 Law of Cosines Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Trigonometry functions, clinometer Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Reflection and Feedback: Week 32 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 12, continued 12.5 Problem solving with Trigonometry LT11: Explain and use the Pythagorean Theorem and right triangle trigonometry as models to solve a variety of right triangle problems in real-world applications. 11.3 Solve right triangle problems using trigonometric ratios. 11.4 Use trigonometric ratios to find the angle of elevation or the angle of depression in real world applications. LT16: Apply transformations and their compositions to analyze geometric figures and create designs. 16.6 Design a real-world application of transformations and communicate the process used in the design. LT19: Use reasoning and logic to draw conclusions to mathematical and real-world applications 19.5 Diagram conditional statements to solve logic puzzles. 19.6 Use valid forms of inductive and deductive reasoning to draw conclusions and make conjectures. 19.7 Select and use various types of reasoning and methods to prove Assessment Exploration: Trigonometric Ratios and the Unit Circle Algebra Skills: Transforming Functions Exploration: Three Types of Proof Review Benchmarks Embedded in Chapter Lessons Learning Targets covered in Chapter 12: District Model CABS Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Chapter 12: Assessment LT11: Pythagorean Theorem LT16: Transformations LT19: Proofs characteristics and properties of algebraic models and geometric figures. Reflection and Feedback: Week 33 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 13: Geometry as a Mathematical System 13.1 The Premises of Geometry 13.2 Planning a Geometry Proof CHAPTER 13 LT14: Identify and apply the definitions related to lines and angles in order to use them to solve problems and perform basic geometric constructions. LT15: Apply the basic theorems of congruence and similarity to solve mathematical and realworld problems. 13.3 Triangle proofs Assessment 13.4 Quadrilateral proofs LT19: Use reasoning and logic to draw conclusions to mathematical and real-world applications. Benchmarks Embedded in Chapter Lessons District Model CABS CHAPTER 13 14.6 Prove statements about segments, lines, angles and angle relationships by using properties, postulates, and theorems. 15.2 Justify congruency or similarity of polygons by using formal and informal proofs. 15.3 Prove two triangles are congruent using coordinate methods (e.g. distance and slope formulas). 19.4 Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures. Feedback and Reflection: Week 34 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Commutative property, associative property, transitivity, distributive property Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity MPS Learning Targets LT1: The student will communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 13, continued 13.5 Indirect Proof 13.6 Circle Proofs 13.7 Similarity Proofs Chapter 13: Assessment District Model CABS 15.3 Prove two triangles are LT15: Apply the basic theorems of congruencecongruent using coordinate methods (e.g. distance and slope and similarity to solve mathematical and realformulas). world problems. LT19: Use reasoning and logic to draw conclusions to mathematical and real-world applications. Algebra Skills: Coordinate Proof Review Benchmarks Embedded in Chapter Lessons Learning Targets covered in Chapter 13: LT14: Analyze Polygons LT15: Congruence and Similarity LT19: Proofs 19.3 Write conjectures using characteristics, postulates and theorems of geometric figures. 19.5 Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures. Feedback and Reflection: Week 35 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 – Spring 2009 Time Lesson/ Activity Final Review: Semester Two Final Assessment: Semester Two MPS Learning Targets: Semester Two MPS Learning Targets Supported in Semester 2: MA 311 Learning Targets Covered in Semester 2: MA 311 LT1: Communication LT2: Measurement LT11: Right Triangles LT15: Congruence and Similarity LT16: Transformations LT18: Geometric Formulas LT19: Proofs Learning Targets Supported in Semester 2: MA 311: District Model CABS LT12: Lines and Angles LT13: Classify Polygons LT14: Analyze Polygons Final Reflection Semester Two MA 311 Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898. Draft 10/05/08 Suggested vocabulary