Foundation Level. Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08 Fall 2008 Spring 2009

Transcription

1 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Milwaukee Public Schools PACING GUIDE for High School GEOMETRY Foundation Level

2 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry 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, 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.

3 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Drafted by: Sherrie K. Akinsanya, FTF Mathematics Instructional Coahc Time (week) Lesson/ Activity Chapter 0 Geometric Art 0.1 Geometry in Art and Nature 0.2 Line Designs MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. CHAPTER 0 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. Benchmarks Embedded in Chapter Lessons CHAPTER 0 2.2* Measure lengths using a metric/english ruler (to the appropriate degree of tenths of cm and sixteenths of an inch). Symmetry, axis of symmetry, compass, straightedge 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 Determine whether a figure has point symmetry, line symmetry or neither Identify tessellations in art, construction and nature Design a real-world application of transformations and communicate the process used in the design.

4 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Reflection

5 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry 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 and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. CHAPTER 1 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. 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. Benchmarks Embedded in Chapter Lessons CHAPTER 1 2.3* Measure angles using a protractor (to appropriate degrees). 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 * Prove statements about segments, lines, angles and angle relationships by using properties, postulates, and theorems. 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

6 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry and Reflection Week 2

7 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry

8 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity Chapter 1: continued 1.4 Polygons 1.5 Triangles 1.6 Special Quadrilaterals 1.7 Circles D Geometry MPS Learning Targets and reasoning using the 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 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 realworld problems. Benchmarks Embedded in Chapter Lessons 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 Make a two-dimensional representation of a 3-dimensional object Identify congruent and similar polygons. Polygon, right triangle, acute triangle, isosceles triangle, scalene triangle, equilateral triangle, obtuse triangle, quadrilateral, circle, chord, diameter, radius, tangent, arcs in circles LT18: Use perimeter, area, volume, and surface area to solve real-world applications Make a model of a 3-dimensional figure from a 2-dimensional drawing.

9 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Reflection Week 3

10 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity Chapter1: continued 1.9 Pictures Exploration: Geometric Probability Review Chapter 2 Reasoning in Geometry 2.1 Inductive Reasoning 2.2 Nth Term MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 1: Learning Targets Covered: LT12 Lines and Angles LT 13 Classify Polygons LT 14 Analyze Polygons LT 15 Congruence and Similarity LT 18 Geometric Formulas 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. 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 CHAPTER * Identify a pattern in a numerical sequence and complete the pattern to a given term Write conjectures using characteristics, postulates and theorems of geometric figures Use and interpret Venn diagrams Use valid forms of deductive reasoning, including the laws of detachment and syllogism, to prove conjectures. Venn diagram, probability, justification, inductive logic, deductive logic, line of best fit, conjecture, pattern, strategy, randomness, reasonableness, representations

11 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Reflection and : Week 4

12 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity Chapter 2: continued 2.3 ing 2.4 Deductive Logic 2.5 Angle Relationships 2.6 Angles and Parallel Lines MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. 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 realworld 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. Benchmarks Embedded in Chapter Lessons 10.7* Identify a pattern in a numerical sequence and complete the pattern to a given term. 12.3* Identify and use angle-pair relationships in mathematical and real-world applications. 12.5* Classify the types of angles formed by two lines and a transversal. Transversal LT14: Use and analyze geometric relationships and properties of polygons and 3-dimensional figures to solve mathematical and real-world applications Solve problems involving real-world applications using the relationships between pairs of angles.

13 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry and Reflection: Week 5

14 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity Chapter 2: continued Algebra Review: Slope Review Chapter 3: Using Tools of Geometry 3.1 Duplicating Segments and Angles MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. 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 apply technology, e.g. computers and graphing calculators, they will interpret results appropriately. Benchmarks Embedded in Chapter Lessons 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). Slope, coordinate system, rate of change, bisector, perpendicular 3.2 Construct Perpendicular. Bisectors LT12*: Identify and apply the definitions related to lines and angles in order to use them to solve problems and perform basic geometric constructions. 12.7* Perform basic geometric constructions.

15 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Reflection Week 6

16 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity Chapter 3: continued 3.3 Construct Perpendicular. Lines 3.4 Angle Bisectors 3.5 Parallel Lines Algebra Review Slopes of perpendicular and parallel lines MPS Learning Targets and reasoning using the 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 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 problem and perform basic geometric constructions. LT16: Apply transformations and their compositions to analyze geometric figures and create designs. Benchmarks Embedded in Chapter Lessons 2.3* Measure angles using a protractor (to appropriate degrees). 12.7* Perform basic geometric constructions Plot and label geometric figures using a coordinate plane. Altitude, angle bisector, parallel

17 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Reflection Week 7

18 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity Chapter 3: continued 3.6 Constructions Exploration: Perspective Drawing 3.7 Concurrency MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. LT12*: Identify and apply the definitions related to lines and angles in order to use them to solve problems and perform basic geometric constructions. LT18: Use perimeter, area, volume, and surface area to solve real-world applications. Learning Targets Covered: Chapter 3 LT 12 Lines and Angles LT 16 Transformations LT 18 Geometric Formulas Benchmarks Embedded in Chapter Lessons 12.7* Perform basic geometric constructions Solve real-world application problems using scale drawings, perspective drawings, blueprints, or computer drawings as models of 3- dimensional objects. Perspective, concurrency, circumcener, incenter, centroid, orthocenter 3.8 Centroid Reflection Week 8 Review

19 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity Chapter 4: Discovering and Proving Triangle Properties 4.1 Triangle Sum 4.2 Isosceles Triangle Algebra Review Solving Equations MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. 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. LT14: Use and analyze geometric relationships and properties of polygons and 3-dimensional figures to solve mathematical and real-world applications. Benchmarks Embedded in Chapter Lessons CHAPTER * Perform basic geometric constructions Determine if lengths of 14.2 three segments can form a triangle Determine the range in which the length of the third side of a triangle must be given the lengths of two sides. Equation 4.3 Triangle Inequalities LT18: Use perimeter, area, volume and surface area to solve real-world applications Compare perimeters and areas of similar 2-dimensional figures using proportions.

20 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Reflection Week 9

21 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time and Reflection: Week 10 Lesson/ Activity Chapter 4, continued 4.4 Congruence Shortcuts 4.5 Congruence Shortcuts 4.6 Congruence Shortcuts 4.7 Flow Charts 4.8 Extra Triangle Proofs MPS Learning Targets and reasoning using the 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. 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 15.2 Justify congruency or similarity of polygons by using formal and informal proofs Prove two triangles are congruent using coordinate methods (e.g. distance and slope formulas) Solve real-world applications using the properties of congruency and similarity Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures. Inequality

22 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity Review Algebra Review: Systems of Equations MPS Learning Targets and reasoning using the 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: LT12 Lines and Angles LT 14 Analyze Polygons LT 15 Congruency and Similarity LT 18 Geometric Formulas LT 19 - Proofs Benchmarks Embedded in Chapter Lessons WKCE Prep Reflection Week 11

23 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity WKCE prep WKCE test MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. 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 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. Benchmarks Embedded in Chapter Lessons Reflection Week 12

24 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity Chapter 5: Discovering and Proving Polygon Properties 5.1 Polygon Sum Conjecture 5.2 Exterior Angles of Polygons MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. CHAPTER 5 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. 1 Benchmarks Embedded in Chapter Lessons CHAPTER Prove statements about segments, lines, angles and angle relationships by using properties, postulates, and theorems. 13.1* Classify polygons by sides and angles 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 Solve problems involving real-world applications using the relationships between pairs of angles.. Exterior angles

25 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Reflection: Week 13

26 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 5, continued 5.3 Kite and Trapezoid Properties 5.4 Properties of Midsegments 5.5 Properties of Parallelograms ASSESSMENT 5.6 Special Parallelograms 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. Benchmarks Embedded in Chapter Lessons 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 Describe properties of a given polygon Describe properties of quadrilaterals Describe properties of other polygons. Midsegment, Intercept Perpendicular Parallel Algebra Review: Linear equations LT19* Use increased rigor in justifying conclusions to mathematical and real-world applications and in verifying the truth of geometric theorems Write conjectures using characteristics, postulates and theorems of geometric figures LT9. and solve mathematical and realworld applications that can be represented by linear equations and systems of equations, linear inequalities, and quadratic equations. 9.8 Solve linear equations, inequalities, quadratic equations and systems of linear equations related to real-world applications.

27 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry and Reflection: Week 14

28 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Reflection Week 15 Chapter 5: continued 5.7 Proving Quadrilateral Properties Review Chapter LT19. Use increased rigor in justifying conclusions to mathematical and real-world applications and in verifying the truth of geometric theorems. Learning Targets Covered in Chapter 5: LT12 Lines and Angles LT13 Classify Polygons LT14 Analyze Polygons LT19 - Proofs Benchmarks Embedded in Chapter Lessons 19.6 Use valid forms of inductive and deductive reasoning to draw conclusions and make conjectures Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures.

29 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. CHAPTER 6 Reflection Week 16 CHAPTER 6 Discovering and Proving Circle Properties 6.1 Tangent 6.2 Chord 6.3 Arcs/Angles 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. Benchmarks Embedded in Chapter Lessons CHAPTER Perform basic geometric constructions Determine measures of inscribed and central angles Determine arc lengths and areas of sectors of circles Write and solve equations to find the measures of angles formed by chords, secants, and tangents Calculate lengths of chord, tangent and secant segments.

30 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Reflection Week 17 Chapter 6, continued 6.4 Proving Circle Conjectures 6.5 Circles and Diameter LT17: Use properties of circles to solve mathematical and real-world applications. 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 17.1 Determine circumference and area of a circle Calculate lengths of chord, tangent and secant segments Solve real-world applications of circles Use valid forms of inductive and deductive reasoning to draw conclusions and make conjectures Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures.

31 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 6, continued 6.6 Around the World Algebra Review: Systems of Equations 6.7 Arc length Review Chapter LT17: Use properties of circles to solve mathematical and real-world applications. Learning Targets Covered in Chapter 6: LT12 Lines and Angles LT17 Circles LT19 - Proofs Benchmarks Embedded in Chapter Lessons 17.1 Determine circumference and area of a circle Determine arc lengths and areas of sectors of circles. Reflection Week 18

32 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, Review for Final Exam Final Exam pictures, charts, tables, diagrams, graphs, and models. 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 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. Vocabulary

33 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Reflection: Semester 1

34 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity Chapter 7: Transformations and Tesselations 7.1 Transformations/ Symmetry 7.2 Properties of Isometrics 7.3 Compositions of Transformations MPS Learning Targets and reasoning using the 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 CHAPTER Perform transformations using triangles in a coordinate plane Identify, given an image and pre-image, the transformation that has taken place as a reflection, rotation, or translation Determine whether a figure has point symmetry, line symmetry or neither Design a real-world application of transformations and communicate the process used in the design. Central angles, inscribed angles

35 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry and Reflection Week 20

36 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, Chapter 7, continued 7.4 Tessellations of regular polygons 7.5 Tessellations of non-regular polygons 7.6 Tessellations with translations 7.7 Tessellations with rotations Exploration: Tessellating with Conway Criterion pictures, charts, tables, diagrams, graphs, and models. LT16: Apply transformations and their compositions to analyze geometric figures and create designs. Benchmarks Embedded in Chapter Lessons 16.6 Identify tessellations in art, construction and nature Design a real-world application of transformations and communicate the process used in the design. Systems of equations and Reflection: Week 21 Review Chapter Learning Target covered in Chapter 7: LT 16: Transformations

37 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 8: Area CHAPTER Area of rectangles and parallelograms 8.2 Area of kites, trapezoids and triangles LT18: Use perimeter, area, volume and surface area to solve real-world applications. Benchmarks Embedded in Chapter Lesson CHAPTER Determine the perimeter/circumference and area of quadrilaterals, triangles and other polygons, using real-world applications. Area, kite trapezoid and Reflection: Week Area Problems Algebra Review: Products, factors, quadratic equations 8.4 Area of regular polygons Exploration: Pick s formula for area

38 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 8, continued 8.5 Area of circles 8.6 Any Way you Slice it: Area of Circle segments Exploration: Geometric Probability 8.7 Surface Area Chapter 8 LT17: Use properties of circles to solve mathematical and real-world applications. LT18: Use perimeter, area, volume and surface area to solve real-world applications. Learning Targets covered in Chapter 8: LT 17: Circles LT 18: Formulas Benchmarks Embedded in Chapter Lessons 17.1 Determine circumference and area of a circle Determine arc lengths and areas of sectors of circles Determine the perimeter/circumference and area of quadrilaterals, triangles and other polygons, using real-world applications. Sector of a circle, segment of a circle, annulus and Reflection: Week 23

39 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. CHAPTER 9 Chapter 9: Pythagorean Theorem 9.1 The Theorem of Pythagoras 9.2 The Converse of the Pythagorean Theorem Algebra Review: Radical Expressions 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 CHAPTER Apply the Pythagorean Theorem to find side lengths of a right triangle in mathematical and real-world applications Determine if a triangle is right, obtuse or acute by using the Pythagorean Theorem. Surface area, Pythagorean Formula and Reflection: Week Two Special Right Triangles 9.4 Story Problems

40 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. and Reflection: Week 25 Chapter 9, continued 9.5 Distance in a Coordinate Plane Exploration: Ladder Climb 9.6 Circles and the Pythagorean Theorem Review Chapter 9 LT12: Classify and describe polygons and 3-dimensional figures. 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. Learning Targets covered in Chapter 9: LT 11: Pythagorean Theorem LT 12: Classify polygons LT 18: Formulas Benchmarks Embedded in Chapter Lessons 12.7 Classify geometric properties by using the coordinate system (e.g. distance formula, slope) Apply the Pythagorean Theorem to find side lengths of a right triangle in mathematical and real-world applications Solve real-world application problems for missing lengths in special right triangles.

41 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 10: Volume CHAPTER Geometry of solids Exploration: Euler s formula for polyhedra 10.2 Volume of Prisms and cylinders 10.3 Volume of pyramids and cones Exploration: The Five Platonic Solids LT18: Use perimeter, area, volume and surface area to solve real-world applications. Benchmarks Embedded in Chapter Lessons CHAPTER Determine the volume and lateral area and total surface area of spheres, prisms, pyramids, cylinders and cones Use volume, lateral area and surface area of 3- dimensional figures to solve real-world application problems Make a model of a 3- dimensional figure from a 2- dimensional drawing. Volume, polyhedron, prism, pyramid, cone, sphere, lateral face, surface area, cylinder LT12: Classify and describe polygons and 3- dimensional figures 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 Make a twodimensional representation of a 3-dimensional object.

42 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Reflection and : Week 26

43 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, Reflection and Week 27 Chapter 10, continued 10.4 Volume 10.5 Displacement and Density Exploration Orthographic Drawing 10.6 Volume of a sphere 10.7 Surface area of a sphere pictures, charts, tables, diagrams, graphs, and models. 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. Benchmarks Embedded in Chapter Lessons 18.2 Determine the volume and lateral area and total surface area of spheres, prisms, pyramids, cylinders and cones Use volume, lateral area and surface area of 3-dimensional figures to solve real-world application problems Solve real-world application problems using scale drawings, perspective drawings, blueprints, or computer drawings as models of 3-dimensional objects Identify the converse, inverse and contra positive of a conditional statement. Displacement, density, orthographic drawing, isometric drawing

44 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models Chapter 10, continued Algebra: Solving for any variable 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 19.2 Translate short verbal arguments into symbolic form Negation and Reflection Week 28 Exploration Sherlock Holmes and Forms of Reasoning Review Chapter 10 Learning Targets covered in Chapter 10: 12: Classify Polygons 13: Analyze Polygons 18: Circles 19: Proofs 19.3

45 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. Chapter 11: CHAPTER 11 Similarity Algebra Skills: Proportion 11.1 Similar polygons 11.2 Similar triangles Exploration: Constructing a Dilation Design 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 CHAPTER Find the measures of segments in similar polygons Identify congruent and similar polygons Justify congruency or similarity of polygons by using formal and informal proofs Solve real-world applications using the properties of congruency and similarity. Proportion, ratio, similar, dilation, scale LT18: Use perimeter, area, volume and surface area to solve real-world applications Compare perimeters and areas of similar 2-dimensional figures using proportions Describe how a change in one measurement affects other measurements of the object.

46 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Reflection and Week 29

47 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the 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 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 15.5 Solve real-world applications using the properties of congruency and similarity Compare perimeters and areas of similar 2-dimensional figures using proportions Describe how a change in one measurement affects other measurements of the object Proportions with area 11.6 Proportions with volume Reflection and Week 30 Exploration: Why Elephants Have Big Ears

48 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the 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 Chapter 11 LT14: Identify and apply the definitions related to lines and angles in order to use them to solve problems and perform basic geometric constructions. LT 19: Use reasoning and logic to draw conclusions to mathematical and real-world applications. Learning Targets covered in Chapter 11: LT13: Classify Polygons LT14: Analyze Polygons LT15: Congruence and Similarity LT18: Formulas LT19: Proofs Benchmarks Embedded in Chapter Lessons 14.3 Find the measures of segments in similar polygons Diagram conditional statements to solve logic puzzles Use valid forms of inductive and deductive reasoning to draw conclusions and make conjectures Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures. Converse, inverse, contrapositive

49 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry and Reflection: Week 31

50 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, Chapter 12: Trigonometry 12.1 Trigonometric ratios 12.2 Solving with right triangles Exploration: Indirect measurement 12.3 Law of Sines 12.4 Law of Cosines pictures, charts, tables, diagrams, graphs, and models. 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 11.3 Solve right triangle problems using trigonometric ratios Use trigonometric ratios to find the angle of elevation or the angle of depression in real world applications Trigonometry functions, clinometer

51 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Reflection and : Week 32 Time Lesson/ Activity MPS Learning Targets and reasoning using the 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 Exploration: Trigonometric Ratios and the Unit Circle Algebra Skills: Transforming Functions Exploration: Three Types of Proof Review 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. LT16: Apply transformations and their compositions to analyze geometric figures and create designs. LT19: Use reasoning and logic to draw conclusions to mathematical and real-world applications Learning Targets covered in Chapter 12: Benchmarks Embedded in Chapter Lessons 11.3 Solve right triangle problems using trigonometric ratios Use trigonometric ratios to find the angle of elevation or the angle of depression in real world applications Design a real-world application of transformations and communicate the process used in the design Diagram conditional statements to solve logic puzzles Use valid forms of inductive and deductive reasoning to draw conclusions and make conjectures Select and use various types of reasoning and methods to prove

52 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Chapter 12: LT11: Pythagorean Theorem LT16: Transformations LT19: Proofs characteristics and properties of algebraic models and geometric figures. Reflection and : Week 33

53 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the of mathematics in a variety of ways, e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. CHAPTER 13 and Reflection: Week 34 Chapter 13: Geometry as a Mathematical System 13.1 The Premises of Geometry 13.2 Planning a Geometry Proof 13.3 Triangle proofs 13.4 Quadrilateral proofs 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. LT19: Use reasoning and logic to draw conclusions to mathematical and real-world applications. Benchmarks Embedded in Chapter Lessons CHAPTER Prove statements about segments, lines, angles and angle relationships by using properties, postulates, and theorems Justify congruency or similarity of polygons by using formal and informal proofs Prove two triangles are congruent using coordinate methods (e.g. distance and slope formulas) Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures. Commutative property, associative property, transitivity, distributive property

54 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity MPS Learning Targets and reasoning using the 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 LT15: Apply the basic theorems of congruence and similarity to solve mathematical and realworld problems. Benchmarks Embedded in Chapter Lessons 15.3 Prove two triangles are ruent using coordinate methods (e.g. distance and slope formulas). and Reflection: Week Circle Proofs 13.7 Similarity Proofs Algebra Skills: Coordinate Proof Review Chapter 13: LT19: Use reasoning and logic to draw conclusions to mathematical and real-world applications. 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 Select and use various types of reasoning and methods to prove characteristics and properties of algebraic models and geometric figures.

55 Milwaukee Public Schools Curriculum Pacing Guide for Discovering Geometry Time Lesson/ Activity Final Review: Semester Two Final : Semester Two MPS Learning Targets: Semester Two 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 MPS Learning Targets Supported in Semester 2: MA 311 Learning Targets Supported in Semester 2: MA 311: LT12: Lines and Angles LT13: Classify Polygons LT14: Analyze Polygons Final Reflection Semester Two MA 311