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Geometry Unit 4 Title Suggested Time Frame 3rd Similarity 4th & Six Weeks Suggested Duration: 16 days Guiding Questions Big Ideas/Enduring Understandings Module 11 Similarity and Transformation can be used to solve real-world problems Module 11 How does a dilation transform a figure? How can similarity transformations be used to find out how two figures are similar? If you know two figures are similar, what can you determine about measures of corresponding angles and lengths? How can you show that two triangles are similar? Module 12 Similar Triangles can be used to solve real-world problems. Module 12 When a line parallel to one side of a triangle intersects the other two sides, how does it divide those sides? How do you find the point on a directed line segment that partitions the given segment in a given ratio? How can you use similar triangles to solve problems? How does the altitude to the hypotenuse of a right triangle help you use similar right triangles to solve problems? Vertical Alignment Expectations TEA Vertical Alignment Chart Grades 5-8, Geometry Sample Assessment Question Coming Soon... Geometry--Unit 4 Updated November 12, 2015 The resources included here provide teaching examples and/or meaningful learning experiences to address the District Curriculum. In order to address the TEKS to the proper depth and complexity, teachers are encouraged to use resources to the degree that they are congruent with the TEKS and research-based best practices. Teaching using only the suggested resources does not guarantee student mastery of all standards. Teachers must use professional judgment to select among these and/or other resources to teach the district curriculum. Some resources are protected by copyright. A username and password is required to view the copyrighted material. Portions of the District Specificity and Examples are a product of the AAMS found within the Region XI Mathematics Support pages. Ongoing TEKS Math Processing Skills G.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: • (A) apply mathematics to problems arising in everyday life, society, and the workplace; (B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution; • (C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems; Focus is on application Students should assess which tool to apply rather than trying only one or all (D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; (E) create and use representations to organize, record, and communicate mathematical ideas; • (F) analyze mathematical relationships to connect and communicate mathematical ideas; and • Students are expected to form conjectures based on patterns or sets of examples and non-examples (G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication • Precise mathematical language is expected. Geometry--Unit 4 Updated November 12, 2015 • Students should evaluate the effectiveness of representations to ensure they are communicating mathematical ideas clearly Students are expected to use appropriate mathematical vocabulary and phrasing when communicating ideas Knowledge and Skills with Student Expectations G.2 Coordinate and Transformational Geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one- and twodimensional coordinate systems to verify geometric conjectures. (A) The student is expected to determine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in oneand twodimensional coordinate systems, District Specificity/ Examples G.2A* A big idea for this student expectation (SE) is that there are an infinite amount of points on a line between two endpoints and students will need to find any point that is a fractional distance away. One-dimensional: For any fractional distance, students need to multiply the difference of one coordinate by the fractional distance (k). Students will need to use this formula to find any fractional distance: (x2 - x1) *k + x1 Examples: On a number line, point A is located at -3 and point B is located at 21. Find points that are located one-fourth, onethird, three-fifths, etc. of the distance between the endpoints. Two-dimensional: For any other fractional distance, students need to multiply the difference of each set of coordinates by the fractional distance (k) and then add that distance to the original coordinates (x1, y1). Students will need to use this new formula to find any fractional distance: ((x2 - x1) *k + x1 RUN, (y2 - y1) * k + y1) RISE Geometry--Unit 4 Updated November 12, 2015 Vocabulary • • • • • • coordinate distance line segment midpoint point endpoint Suggested Resources Resources listed and categorized to indicate suggested uses. Any additional resources must be aligned with the TEKS. HMH Geometry Unit 4 A&M Consolidated HS 03-04 Fall Unit 08 – Similar Polygons Example Reference: Distance between points example Partition a Segment Video including finding the midpoint. Instead of using this to find only the midpoint: ((x1 + x2) / 2, (y1 + y2) / 2) Examples: Point P has coordinates (-8, 5) and Point Q has coordinates (4, -1). Find points that are located one-fourth, one-third, three-fifths, etc. of the distance between the endpoints. Teachers should initially make a connection between the midpoint formula and general form which can apply to any ratio. Example Reference: Construct a partition of a segment (see video in resources column to the right) G.3 Coordinate and Transformational Geometry. The student uses the process skills to generate and describe rigid transformations (translation, reflection, and rotation) and nonrigid transformations (dilations that preserve similarity and reductions and enlargements that do not preserve similarity). • • • • • • G.3C* Students should be able to identify the process and the order of how the image was changed. Teachers need to show examples on the coordinate plane and off the plane. Off the plane could be on patty paper and real world examples. Off Coordinate Plane - Not limited to graphs Geometry--Unit 4 Updated November 12, 2015 • • • • • • dilation enlargement (scale factor >1) image point of rotation pre-image reduction (0<scale factor<1) reflection rotation scale factor similarity transformation translations Web Resources Dilations and Similarity Example Dilations Activity/Worksheet Dilations and Scale Factor Worksheet (C) The student is expected to identify the sequence of transformations that will carry a given pre-image onto an image on and off the coordinate plane; and G.7 Similarity, Proof, and Trigonometry. The student uses the process skills in applying similarity to solve problems. (A) The student is expected to apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles; and How It Could Be Assessed: Given pre image & image - students must list the compositions applied Misconception: Students assume that performing the transformations in any order will result in same result G.7A • Big Idea - identify similar figures, their proportional sides and congruent angles • • Prior Knowledge - refer back to dilations covered through transformation (w/ any point as center of dilation), similar figures • Teachers should show: • Characteristics of similar figures • Using proportions to find missing sides (Cross products are equal) • Identifying corresponding angles Students should: • Know key attributes similar figures (congruent angles & corresponding sides are proportional) • Find missing angles and side lengths • Know how to apply a scale factor ( > 1 and <1) Misconceptions: • When naming figures order matters. Geometry--Unit 4 Updated November 12, 2015 • • • • • • • Angle-Angle Criterion congruent congruent corresponding angles corresponding angles corresponding sides dilation proportional proportionality similar figures similar triangles similarity Web Resources Similarity, Congruency, and Transformations Activity Set Khan Academy Resource – Similarity Postulates Proofs with Similar Triangles Examples Similarity Activities and Lessons from NSA.gov • Corresponding angles should be in the same order in a figure. Examples: Use properties of triangles and the similarity statement to find the measures of the numbered angles. Using the given measurements, find the scale factor (similarity ratio) of each of the following sets of similar figures, comparing smaller to larger; then comparing larger to smaller. G.7B (B) The student is expected to apply the Angle- Big Idea - If triangles have two congruent angles, then they are similar triangles therefore sides are proportional. Angle criterion to verify similar Geometry--Unit 4 Updated November 12, 2015 **This is an extension of 7.A Using the AA Theorem to Determine Similar Triangles triangles and apply the proportionality of the corresponding sides to solve problems. Teacher should: • Show example of similar triangles with AA • Show examples where congruent angles might be from vertical angles, parallels cut by a transversal (prior knowledge) Mirror Mirror Project Students should: • Identify corresponding angles • Understand the AA theorem • Similar triangles - proportional sides • Set up proportions and solve for missing sides • Find the 3rd angle for both triangles Examples: Misconceptions The sides in SSS and SAS must be proportional now (and not congruent like they were with proving triangles congruent) G.8 Similarity, Proof, G.8A and Trigonometry. The student uses the process skills with deductive reasoning to prove and apply Big Idea - prove theorems & apply to solve Teachers should: Geometry--Unit 4 Updated November 12, 2015 • • • altitude Angle-Angle Similarity Theorem Angle-SideAngle theorems by utilizing a variety of methods such as coordinate, transformational, axiomatic and formats such as twocolumn, paragraph, flow chart. (A) The student is expected to prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems; and (B) The student is expected to identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to solve problems. Complete proofs of AA Similarity Postulate, SSS Similarity Theorem, SAS Similarity Theorem, and the Triangle Proportionality Theorem. Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally • • • • • • • Students need to: • Complete proofs of AA Similarity Postulate, SSS Similarity Theorem, SAS Similarity Theorem, and the Triangle Proportionality Theorem. • Find missing sides and angles Misconceptions • Must have the included angle to be true for SAS • The sides in SSS and SAS must be proportional now (and not congruent like they were with proving triangles congruent) Geometry--Unit 4 Updated November 12, 2015 Similarity Theorem geometric mean hypotenuse right triangle Side-Side-Side Similarity Theorem similar triangles theorem Triangle Proportionality theorem Mirror Mirror Project IXL – Triangle Proportionality Theorem Stating and Proving the Triangle Proportionality Theorem Similarity in Right Triangles - Kuta Similarity in Right Triangles - Anderson Proofs Involving Similar Triangles Examples Geometry--Unit 4 Updated November 12, 2015 Give the appropriate theorem or postulate that can be used to prove the triangles similar and write the similarity statement. G.8B Big Idea - identify and apply relationships of an altitude drawn to the hypotenuse to solve problems Teachers need to Show: Geometric Mean - It is the nth root of the product of n numbers. That means you multiply the numbers together, and then take the nth root, where n is the number of values you just multiplied. Geometry--Unit 4 Updated November 12, 2015 Altitude on a Right Triangle- The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse. In terms of our triangle, this theorem simply states what we have already shown: since AD is the altitude drawn from the right angle of our right triangle to its hypotenuse, and CD and DB are the two segments of the hypotenuse. Students Need To: • Be able to calculate geometric mean • Work backwards from geo mean to a side • Draw the altitude • Identify similar triangles created Geometry--Unit 4 Updated November 12, 2015 Examples: Geometry--Unit 4 Updated November 12, 2015