MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide GEOMETRY HONORS Course Code:

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1 Topic V: Quadrilaterals Properties Pacing Date(s) Traditional 22 days 10/24/13-11/27/13 Block 11 days 10/24/13-11/27/13 COMMON CORE STATE STANDARD(S) & MATHEMATICAL PRACTICE (MP) NEXT GENERATION SUNSHINE STATE STANDARD(S) ESSENTIAL CONTENT OBJECTIVES MACC.912.G-CO.3.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. (MP.2, MP.3, MP.5) MACC.912.G-GPE.2.4: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). (MP.2, MP.3, MP.7) MACC.912.G-GPE.2.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). (MP.3, MP.8) Standard 3: Quadrilaterals MA.912.G.3.1: Describe, classify, and compare relationships among quadrilaterals including the square, rectangle, rhombus, parallelogram, trapezoid, and kite.(wi) MA.912.G.3.2: Compare and contrast special quadrilaterals on the basis of their properties. (WI) MA.912.G.3.3: Coordinate geometry is used while students prove quadrilaterals to be congruent, similar, or regular. Coordinate geometry is used to prove properties of quadrilaterals. (EOC). MA.912.G.3.4: Prove theorems involving quadrilaterals. (WI, EOC) Standard 8: Mathematical Reasoning and Problem Solving MA.912.G.8.2: Use a variety of problem solving strategies, such as drawing a diagram, guess and check, solving a simpler problem, writing an equation, and working backwards. MA.912.G.8.4: Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. (FI, WI, EOC) Key: FI - Fall Interim WI - Winter Interim EOC Geometry EOC MA.912.G.8.5: Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two column, and indirect proofs. (WI) MA.912.G.8.6: Perform basic constructions using straightedge and compass, and/or drawing programs describing and justifying the procedures used. Distinguish between sketching, constructing, and drawing geometric figures. A. Special Quadrilaterals Definitions 1. Trapezoid and Kite 2. Parallelogram, Rhombus, Rectangle, and Square. B. Kites Properties 1. Angles 2. Diagonals 3. Diagonal Bisector 4. Angle Bisector C. Trapezoids Properties 1. Consecutive Angles 2. Isosceles Trapezoid a. Base Angles b. Diagonals D. Mid-segment Properties 1. Triangle Mid-segments 2. Trapezoid Mid-segment E. Properties of Parallelograms 1. Angles, Sides, and Diagonals F. Special Parallelograms Properties 1. Rectangle 2. Rhombus 3. Square G. Proving Quadrilaterals Properties H. Applications in the Real-World NGSSS State the properties of kites, trapezoids, parallelograms, rectangles, and rhombi Calculate the length of a mid-segments in triangles and trapezoids Write flowchart and paragraph proofs explaining the relationship between the sides and angles in a polygon Describe and compare the relationship among parallelograms, rectangles, rhombi, squares, kites, and trapezoids Use coordinate Geometry to prove properties of quadrilaterals. Practice construction skills Develop reasoning, problem-solving skills, and cooperative behavior tools for construction Office of Academics and Transformation Page 1 of 10

2 Core Text Book: Discovering Geometry (4 th Edition) INSTRUCTIONAL TOOLS Benchmark Suggested Lessons Teacher Notes MA.912.G.3.1 MA.912.G.3.2 MA.912.G.3.3 MA.912.G.3.4 MA.912.G.8.2 MA.912.G.8.4 MA.912.G.8.5 MA.912.G UYAS page 135 UYAS page 167 Notes: Use Geometry EOC Item Specifications as a resource document to help define grade-level content. Vocabulary: Kite, Vertex Angles, Non-vertex Angles, Trapezoid, Bases, Base Angles, Isosceles Trapezoid, Rhombus, Rectangle, Square, Midsegment. Instructional Strategies: Example/Non-Example, Definition Section of Notebook, Similarities/Differences, and Graphic Organizer Some students may believe that a construction is the same as a sketch or drawing. Emphasize the need for precision and accuracy when doing constructions. Stress the idea that a compass and straightedge are identical to a protractor and ruler. Explain the difference between measurement and construction. Discuss the role of algebra in providing a precise means of representing a visual image. Use slopes and the Euclidean distance formula to solve problems about figures in the coordinate plane such as: Given three points, are they vertices of an isosceles, equilateral, or right triangle? Given four points, are they vertices of a parallelogram, a rectangle, a rhombus, or a square? Office of Academics and Transformation Page 2 of 10

3 COMMON CORE STATE STANDARDS MATHEMATICAL PRACTICES DESCRIPTION MACC.K12.MP.1 Make sense of problems and persevere in solving them. Explain the meaning of a problem and looking for entry points to its solution. Analyze givens, constraints, relationships, and goals. Make conjectures about the form and meaning of the solution and plan a solution pathway. Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. Monitor and evaluate their progress and change course if necessary. Explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Check answers to problems using a different method, and continually ask, Does this make sense? Identify correspondences between different approaches. MACC.K12.MP.2 Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations. Decontextualize to abstract a given situation and represent it symbolically. Contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols Create a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them. Know and be flexible using different properties of operations and objects. MACC.K12.MP.3 Construct viable arguments and critique the reasoning of others. Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Make conjectures and build a logical progression of statements to explore the truth of their conjectures. Analyze situations by breaking them into cases, and can recognize and use counterexamples. Justify their conclusions, communicate them to others, and respond to the arguments of others. Reason inductively about data, making plausible arguments that take into account the context from which the data arose. Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Determine domains to which an argument applies. MACC.K12.MP.4 Model with mathematics. Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Apply what they know and feel comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. Analyze relationships mathematically to draw conclusions. Interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Office of Academics and Transformation Page 3 of 10

4 COMMON CORE STATE STANDARDS MATHEMATICAL PRACTICES DESCRIPTION MACC.K12.MP.5 Use appropriate tools strategically. Consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Make sound decisions about when each of the tools appropriate for their grade or course might be helpful, recognizing both the insight to be gained and their limitations. Example: High school students analyze graphs of functions and solutions using a graphing calculator. Detect possible errors by strategically using estimation and other mathematical knowledge. Know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. Use technological tools to explore and deepen their understanding of concepts MACC.K12.MP.6 Attend to precision. Communicate precisely to others. Use clear definitions in discussion with others and in their own reasoning. State the meaning of the symbols they choose, including using the equal sign consistently and appropriately. Be careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. MACC.K12.MP.7 Look for and make use of structure. Discern a pattern or structure. Example: In the expression x 2 + 9x + 14, students can see the 14 as 2 7 and the 9 as Recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. Step back for an overview and shift perspective. See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. Example: They can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MACC.K12.MP.8 Look for and express regularity in repeated reasoning. Notice if calculations are repeated, and look both for general methods and for shortcuts. Example: Noticing the regularity in the way terms cancel when expanding (x-1)(x+1),(x-1)(x2+x+1),and(x-1)(x3 +x2+x+1)might lead them to the general formula for the sum of a geometric series. Maintain oversight of the process, while attending to the details as they work to solve a problem. Continually evaluate the reasonableness of their intermediate results. Office of Academics and Transformation Page 4 of 10

5 NEXT GENERATION SUNSHINE STATE STANDARDS GEOMETRY BODY OF KNOWLEDGE Standard 3: Quadrilaterals Classify and understand relationships among quadrilaterals (rectangle, parallelogram, kite, etc.). Relate geometry to algebra by using coordinate geometry to determine regularity, congruence, and similarity. Use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas, and prove theorems involving quadrilaterals.. BENCHMARK CODE MA.912.G.3.1 BENCHMARK Describe, classify, and compare relationships among quadrilaterals including the square, rectangle, rhombus, parallelogram, trapezoid, and kite. Remarks/Examples: This benchmark examines properties of quadrilaterals one at a time. Example: Explore a trapezoid through manipulatives, drawings, and/or technology. Draw the diagonals and determine whether they are perpendicular. Give a convincing argument that your judgment is correct. Cognitive Complexity/Depth of Knowledge Rating:: Moderate Clarification (EOC): Assessed with MA.912.G.3.4 Content Limits (EOC): Assessed with MA.912.G.3.4 MA.912.G.3.2 Compare and contrast special quadrilaterals on the basis of their properties. Remarks/Examples: This benchmark examines similarities and differences between different types of quadrilaterals. Example: Explain the similarities and differences between a rectangle, rhombus, and kite. Create a Venn diagram to match your explanation. Cognitive Complexity/Depth of Knowledge Rating: Moderate Clarification (EOC): Assessed with MA.912.G.3.4 Content Limits (EOC): Assessed with MA.912.G.3.4 MA.912.G.3.3 MC Use coordinate geometry to prove properties of congruent, regular, and similar quadrilaterals. Remarks/Examples: Coordinate geometry is used while students prove quadrilaterals to be congruent, similar, or regular. Coordinate geometry is used to prove properties of quadrilaterals. Example 1: Is rectangle ABCD with vertices at A(0, 0), B(4, 0), C(4, 2), D(0, 2) congruent to rectangle PQRS with vertices at P( 2, 1), Q(2, 1), R(2, 1), S( 2, 1)? Justify your answer. Cognitive Complexity/Depth of Knowledge Rating: High Clarification (EOC): Students will use coordinate geometry and geometric properties to justify measures and characteristics of congruent, regular, and similar quadrilaterals. Content Limits (EOC) Items may include statements and/or justifications to complete formal and informal proofs. Items may include the use of coordinate planes. Office of Academics and Transformation Page 5 of 10

6 Standard 3: Quadrilaterals Classify and understand relationships among quadrilaterals (rectangle, parallelogram, kite, etc.). Relate geometry to algebra by using coordinate geometry to determine regularity, congruence, and similarity. Use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas, and prove theorems involving quadrilaterals.. BENCHMARK CODE MA.912.G.3.4 MC/ FR Prove theorems involving quadrilaterals Remarks/Examples:. Example: Prove that the diagonals of a rectangle are congruent. Cognitive Complexity/Depth of Knowledge Rating: High BENCHMARK Clarification (EOC): Students will use geometric properties to justify measures and characteristics of quadrilaterals. (Also assesses: MA.912.D.6.4, MA.912.G.3.1, MA.912.G.3.2, and MA.912.G.8.5.) Content Limits (EOC) Items may require statements and/or justifications to complete formal and informal proofs. Standard 8: Mathematical Reasoning and Problem Solving In a general sense, mathematics is problem solving. In all mathematics, students use problem-solving skills: they choose how to approach a problem, they explain their reasoning, and they check their results. At this level, students apply these skills to making conjectures, using axioms and theorems, constructing logical arguments, and writing geometric proofs. They also learn about inductive and deductive reasoning and how to use counterexamples to show that a general statement is false. BENCHMARK CODE MA.912.G.8.2 BENCHMARK Use a variety of problem solving strategies, such as drawing a diagram, making a chart, guess and check, solving a simpler problem, writing an equation, and working backwards. Remarks/Examples: Example: How far does the tip of the minute hand of a clock move in 20 minutes if the tip is 4 inches from the center of the clock? Cognitive Complexity/Depth of Knowledge Rating: Moderate Clarification (EOC): Embedded throughout. Content Limits (EOC): Embedded throughout Office of Academics and Transformation Page 6 of 10

7 Standard 8: Mathematical Reasoning and Problem Solving In a general sense, mathematics is problem solving. In all mathematics, students use problem-solving skills: they choose how to approach a problem, they explain their reasoning, and they check their results. At this level, students apply these skills to making conjectures, using axioms and theorems, constructing logical arguments, and writing geometric proofs. They also learn about inductive and deductive reasoning and how to use counterexamples to show that a general statement is false. MA.912.G.8.4 MC Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. Remarks/Examples: Example: Calculate the ratios of side lengths in several different-sized triangles with angles of 90, 50, and 40. What do you notice about the ratios? How might you prove that your observation is true (or show that it is false)? Cognitive Complexity/Depth of Knowledge Rating: High Clarification (EOC): Students will provide statements and/or reasons in a formal or informal proof or distinguish between mere examples of a geometric idea and proof of that idea. Content Limits (EOC) Items must adhere to the content limits stated in other benchmarks. Items may include proofs about congruent/similar triangles and parallel lines. MA.912.G.8.5 Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two column, and indirect proofs. Remarks/Examples: Example 1: Prove that the sum of the measures of the interior angles of a triangle is 180. Example 2: Prove that the perpendicular bisector of line segment AB is the set of all points equidistant from the endpoints A and B. Example 3: Prove that two lines are parallel if and only if the alternate interior angles the lines make with a transversal are equal. Cognitive Complexity/Depth of Knowledge Rating: High Clarification (EOC): Assessed with MA.912.G.3.4 and MA.912.G.4.6. Content Limits (EOC): Assessed with MA.912.G.3.4 and MA.912.G.4.6 MA.912.G.8.6 Not Assessed Perform basic constructions using straightedge and compass, and/or drawing programs describing and justifying the procedures used. Distinguish between sketching, constructing, and drawing geometric figures. Remarks/Examples: Example: Construct a line parallel to a given line through a given point not on the line, explaining and justifying each step. Cognitive Complexity/Depth of Knowledge Rating: High Office of Academics and Transformation Page 7 of 10

8 TECHNOLOGY TOOLS FLORIDA FOCUS Sign-in and Password: First 4 letters of Last Name and Last 4 Digits Employee Number. Example: abcd1234 MA.912.G.3.3 MA.912.G.3.4 GIZMO CORRELATION GIZMO TITLE Classifying Quadrilaterals - Activity B Parallelogram Conditions Special Parallelograms GEOMETER S SKETCHPAD ACTIVITIES SKETCHPAD TITLE TITLE Properties of Kites Properties of the Midsegment of a Trapezoid Properties of Parallelograms Properties of Special Parallelograms VIDEO TITLE Section A: General Definitions Quadrilaterals Coordinate Geometry Problems Coordinate Geometry and Quantitative Comparisons Review TOPIC V VIDEO TITLE IMAGE TOPIC V VIDEO TITLE Quadrilaterals DISCOVERY EDUCATION CORRELATION Office of Academics and Transformation Page 8 of 10

9 TOPIC V DISCOVERY EDUCATION CORRELATION MODEL LESSON Congruence and Proof: Session 4: Prove Theorems about Parallelograms Coordinate Geometry and How It's Used: Session 2: Parallel and Perpendicular Lines Coordinate Geometry and How It's Used: Session 3: Classifying Polygons Coordinate Geometry and How It's Used: Session 4: Equidistant to a Point MATH OVERVIEW Geometry: Special Parallelograms Geometry: Properties of Parallelograms Geometry: Quadrilaterals Geometry: Trapezoids and Kites MATH EXPLANATION TITLE Geometry: Properties of Parallelograms: Opposite Angles in Parallelograms Geometry: Properties of Parallelograms: Parallelogram Proofs Geometry: Properties of Parallelograms: Making Conclusions About Parallelograms Geometry: Properties of Parallelograms: Variables in Parallelograms Geometry: Properties of Parallelograms: Determining a Parallelogram Geometry: Proving that a Quadrilateral is a Parallelogram: Proofs for Parallelograms Geometry: Proving that a Quadrilateral is a Parallelogram: Parallelograms and Flow Proofs Geometry: Special Parallelograms: Finding Rhombus Angles Office of Academics and Transformation Page 9 of 10

10 Date Pacing Guide Standards Data Driven Standard(s) Activities Assessment(s) Strategies Traditional 22 days Standard 3: Quadrilaterals MA.912.G.3.1 Block 11 days 10/24/13-11/27/13 MA.912.G.3.2 MA.912.G.3.3 MA.912.G.3.4 Standard 8: Mathematical Reasoning and Problem Solving MA.912.G.8.2 MA.912.G.8.4 MA.912.G.8.5 MA.912.G.8.6 Office of Academics and Transformation Page 10 of 10