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

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1 Topic VIII: Similarity Pacing Date(s) Traditional 14 days 01/21/14 02/10/14 Block 7 days 01/21/14 02/10/14 COMMON CORE STATE STANDARD(S) & MATHEMATICAL PRACTICE (MP) NEXT GENERATION SUNSHINE STATE STANDARD(S) ESSENTIAL CONTENT OBJECTIVES MACC.912.G-SRT.1.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. (MP.3, MP.5, MP.7) MACC.912.G-SRT.1.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (MP.3) MACC.912.G-SRT.2.4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. (MP.3, MP.5) MACC.912.G-SRT.2.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (MP.3, MP.4, MP.6, MP.7) MACC.912.G-GPE.2.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (MP.2, MP.7, MP.8) Standard 2: Polygons MA.912.G.2.3: Use properties of congruent and similar polygons to solve mathematical or real world problems. (WI, EOC) Standard 4: Triangles MA.912.G.4.4: Use properties of congruent and similar triangles to solve problems involving lengths and areas. (WI) MA.912.G.4.5: Apply theorems involving segments divided proportionally. MA.912.G.4.6: Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. (WI, EOC) Standard 5: Right Triangles MA.912.G.5.2: State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. Standard 8: Mathematical Reasoning and Problem Solving MA.912.G.8.3: Determine whether a solution is reasonable in the context of the original situation. 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) A. Similar Polygons 1. Proportions and Reasoning 2. Dilations and Scale Factor B. Proving Similarity in Triangles 1. Similarity Shortcuts a. AA Similarity b. SSS Similarity c. SAS Similarity 2. Proportional Relationships a. Indirect Measurement b. Corresponding Parts of Similar Triangles i. Proportional Parts ii. iii. Opposite Side Ratios Altitude to hypotenuse Ratios C. Proportional Segments by Parallel Lines 1. Parallels and Proportionality 2. Extended Parallel/ Proportionality D. Applications in the Real-World NGSSS Define dilation Define similar polygons Explain why two polygons are similar Use the definition of similar triangles and other polygons to solve problems Identify and describe shortcut methods for determining similar triangles Practice using proportions to find measures in similar figures Determine the measure of corresponding parts of similar triangles Calculate the ratio of the parts into which an angle bisector of a triangle divides the opposite side State the relationship between the ratios of the parts into which parallel lines cut the sides of a triangle Dividing a segment into proportional parts using a compass and straightedge Calculate the length of the hypotenuse of a right triangle given the length of the altitude and the length of one leg Develop reading comprehension, problem-solving skills, and cooperative behavior Key: FI - Fall Interim WI - Winter Interim EOC Geometry EOC 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.2.3 MA.912.G.4.4 MA.912.G.4.5 MA.912.G.4.6 MA.912.G.5.2 MA.912.G.8.3. MA.912.G Notes: Use Geometry EOC Item Specifications as a resource document to help define grade-level content. Vocabulary: Ratios, proportions, similar polygons, similarity shortcuts, indirect measurement,, corresponding parts of similar triangles,. Instructional Strategies: Patty Paper Investigations, Compass/Straightedge Investigations, Geometer s Sketchpad, Problem Creation, Note Taking Using the theorem that the angle sum of a triangle is 180, verify that the AA criterion is equivalent to the AAA criterion. Given two triangles for which AA holds, use rigid motions to map a vertex of one triangle onto the corresponding vertex of the other in such a way that their corresponding sides are in line. Then show that dilation will complete the mapping of one triangle onto the other. Some students often do not recognize that congruence is a special case of similarity. Similarity with a scale factor equal to 1 becomes a congruency. Students may not realize that similarities preserve shape, but not size. Angle measures stay the same, but side lengths change by a constant scale factor. Some students often do not list the vertices of similar triangles in order. However, the order in which vertices are listed is preferred and especially important for similar triangles so that proportional sides can be correctly identified. 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 2: Polygons Identify and describe polygons (triangles, quadrilaterals, pentagons, hexagons, etc.), using terms such as regular, convex, and concave. Find measures of angles, sides, perimeters, and areas of polygons, justifying the methods used. Apply transformations to polygons. Relate geometry to algebra by using coordinate geometry to determine transformations. Use algebraic reasoning to determine congruence, similarity, and symmetry. Create and verify tessellations of the plane using polygons. BENCHMARK CODE MA.912.G.2.3 MC/FR BENCHMARK Use properties of congruent and similar polygons to solve mathematical or real world problems. Example: Suppose a building is in the shape of a regular hexagon. The architect wants to put walkways as indicated. Show that the triangles formed are equal in size and shape. Cognitive Complexity/Depth of Knowledge Rating: High Clarification (EOC): Students will use properties of congruent and/or similar polygons to solve problems. (Also assesses: MA.912.G.2.1, MA.912.G.4.1, MA.912.G.4.2, MA.912.G.4.4, and MA.912.G.4.5.) Content Limit (EOC): All angle measurements will be in degrees. Items may require statements and/or justifications to complete formal and informal proofs. Standard 4: Triangles Students identify and describe various kinds of triangles (right, acute, scalene, isosceles, etc.). They define and construct altitudes, medians, and bisectors, and triangles congruent to given triangles. They prove that triangles are congruent or similar and use properties of these triangles to solve problems involving lengths and areas. They relate geometry to algebra by using coordinate geometry to determine regularity, congruence, and similarity. They understand and apply the inequality theorems of triangles. BENCHMARK CODE MA.912.G.4.4 BENCHMARK Use properties of congruent and similar triangles to solve problems involving lengths and areas. Example: Of two similar triangles, the second has sides half the length of the first. The area of the first triangle is 20 cm2. What is the area of the second triangle? Cognitive Complexity/Depth of Knowledge Rating: Moderate Clarification (EOC): Assessed with MA.912.G.2.3 Content Limit (EOC): Assessed with MA.912.G.2.3 Office of Academics and Transformation Page 5 of 10

6 Standard 4: Triangles Students identify and describe various kinds of triangles (right, acute, scalene, isosceles, etc.). They define and construct altitudes, medians, and bisectors, and triangles congruent to given triangles. They prove that triangles are congruent or similar and use properties of these triangles to solve problems involving lengths and areas. They relate geometry to algebra by using coordinate geometry to determine regularity, congruence, and similarity. They understand and apply the inequality theorems of triangles. MA.912.G.4.5 Apply theorems involving segments divided proportionally. Example: In triangle ABC, PQ is parallel to BC. What is the length of AQ? Cognitive Complexity/Depth of Knowledge Rating: Moderate Clarification (EOC): Assessed with MA.912.G.2.3 Content Limit (EOC): Assessed with MA.912.G.2.3 MA.912.G.4.6 MC Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. Prove that triangles ABC and APQ are similar. Cognitive Complexity/Depth of Knowledge Rating: High Clarification (EOC): Students will use geometric properties to justify measures and characteristics of triangles. (Also assesses: MA.912.D.6.4 and MA.912.G.8.5) Content Limit (EOC): Items may require statements and/or justifications to complete formal and informal proofs. Standard 5: Right Triangles Apply the Pythagorean Theorem to solving problems, including those involving the altitudes of right triangles and triangles with special angle relationships. Use special right triangles to solve problems using the properties of triangles. BENCHMARK CODE MA.912.G.5.2 BENCHMARK State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. Example: Find the value of x in the right triangle shown. Cognitive Complexity/Depth of Knowledge Rating: Moderate Clarification (EOC): Assessed with MA.912.G.5.4 Content Limit (EOC): Assessed with MA.912.G.5.4 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, use problem solving skills, choose how to approach a problem, explain the reasoning, and check the results. At this level, apply these skills to making conjectures, using axioms and theorems, constructing logical arguments, and writing geometric proofs. Learn about inductive and deductive reasoning and how to use counterexamples to show that a general statement is false Apply the Pythagorean Theorem to solving problems, including those involving the altitudes of right triangles and triangles with special angle relationships. Use special right triangles to solve problems using the properties of triangles. MA.912.G.8.3 Determine whether a solution is reasonable in the context of the original situation. Example: The area of a circle is 49π and George determined that the diameter is 7. Is his answer reasonable? Why or why not? Cognitive Complexity/Depth of Knowledge Rating: Moderate Clarification (EOC): Embedded throughout. Content Limits (EOC): Embedded throughout 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. 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. 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.2.3 MA.912.G.4.6 MA.912.G.8.4 GIZMO CORRELATION GIZMO TITLE Similar Figures - Activity A Similar Figures - Activity B Similar Polygons Similarity in Right Triangles GEOMETER S SKETCHPAD ACTIVITIES UNIT/ TITLE Similarity TOPIC VIII VIDEO TITLE Hairs Found at Crime Scene Are Microscopically Similar to O.J. Simpson's, Says FBI Expert Witness Science of NHL Hockey: Hockey Geometry Science of NFL Football: Pythagorean Theorem Here, Hold This Tarantula: A Guide to Spiders and Webs IMAGE TOPIC VIII VIDEO TITLE DISCOVERY EDUCATION CORRELATION Defining Ratio and Proportion Reviewing Ratio Problems Review of Prerequisites for the Theorem of Pythagoras MODEL LESSON Office of Academics and Transformation Page 8 of 10

9 TOPIC VIII DISCOVERY EDUCATION CORRELATION Building Bridges with Similar Triangles: Session 1: Identifying Similar Triangles Using Angles Coordinate Geometry and How It's Used: Session 1: Dividing Segments MATH EXPLANATION TITLE Geometry: Geometric Mean: Using Similar Triangles Geometry: Determining and Using Similar Triangles: Real World Applications with Similar Triangles Geometry: Properties of Similar Triangles: Altitudes of Similar Triangles Geometry: Properties of Similar Triangles: Altitudes of Similar Triangles Geometry: Properties of Similar Triangles: Using Corresponding Sides to Determine Ratios Geometry: Determining and Using Similar Triangles: Proving with Similar Triangles Geometry: Properties of Similar Triangles: Determining Whether Statements are Always, Sometimes, or Never True Geometry: Properties of Similar Triangles: Determining True or False Statements Geometry: Properties of Similar Triangles: Using Corresponding Sides to Determine Ratios Geometry: Properties of Similar Triangles: Medians of Similar Triangles MATH OVERVIEW Geometry: Understanding and Applying Ratios and Proportions Geometry: Understanding Similar Polygons Geometry: Properties of Similar Triangles Geometry: Determining and Using Similar Triangles Office of Academics and Transformation Page 9 of 10

10 Date Pacing Guide Standards Data Driven Standard(s) Activities Assessment(s) Strategies Traditional 14 days Standard 2: Polygons MA.912.G.2.3 Block 7 days Standard 4: Triangles MA.912.G /21/14 02/10/14 MA.912.G.4.5 MA.912.G.4.6 Standard 5: Right Triangles MA.912.G.5.2 Standard 8: Mathematical Reasoning and Problem Solving MA.912.G.8.3. MA.912.G.8.4 Office of Academics and Transformation Page 10 of 10