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Course Name: Math iii Congruence: (part 1) Unit # 2 Unit Title: Geometry Enduring understanding (Big Idea): Students will understand that a) you can determine if 2 figures are congruent by comparing corresponding parts b) triangles can be proven congruent without having to compare all corresponding parts c) the angles and sides of Isosceles and Equilateral triangles have special relationships. Essential Questions: 1) How do you identify corresponding parts of congruent triangles? 2) How do you show that 2 triangles are congruent? 3) How can you tell whether a triangle is isosceles or equilateral? 4) How do you solve problems that involve measurements of triangles? BY THE END OF THIS UNIT: Students will be able to… 1) use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and Students will know… only if corresponding pairs of sides and corresponding 1) that two figures are congruent if a series of rigid motion pairs of angles are congruent. carries one onto the other. 2) explain how the criteria for triangle congruence (ASA, 2) that two triangles are congruent if all corresponding pairs SAS, and SSS) follow from the definition of congruence of sides are congruent and all corresponding pairs of in terms of rigid motions. angles are congruent. 3) Use properties of midsegments to solve problems 4) Use properties of perpendicular and angle bisectors to solve problems Vocabulary: 5) Use properties of medians and altitudes to solve Throughout standards problems Unit Resources: Throughout standards Suggested Pacing: (11 days total)? Part 1 (3 days) - G.CO. 1, 9, 10, 11, 12 Part 2 (3 days) - G.SRT. 2, 3, 4, 5 Part 3 ( 3 days) - G.C. 1, 2, 3, 5 G.MG.3 (throughout entire unit) Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Mathematical Practices in Focus: 1) Make sense of problems and persevere in solving them 2) Model with mathematics 3) Attend to precision 4) Model with mathematics. 5) Look for and make use of structure. CCSS-M Included: G.CO. 1, 9, 10, 11, 12 (part 1) G.SRT. 2, 3, 4, 5 (part 2) G.C. 1, 2, 3, 5 (part 3) G.MG. 3 (throughout) Course Name: Math iii CORE CONTENT Unit # 2 Unit Title: Geometry Cluster Title: Experiment with transformations in the plane Standard: Standard G.CO.1 Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Concepts and Skills to Master: Define angle, circle, perpendicular line, parallel line, and line segment Use precise definitions to identify and model an angle, circle, perpendicular line, parallel line, and line segment Demonstrate mathematical notation for each term. Apply the segment addition postulate and the angle addition postulate. Apply properties of lines and transversals. Identify special angle pairs. SUPPORTS FOR TEACHERS Critical Background Knowledge Solve algebraic equations; Understanding the undefined terms: point, line, and plane; Understand distance is a nonnegative quantity. Academic Vocabulary Angle, circle, perpendicular line, parallel line, line segment, distance, arc Suggested Instructional Strategies: Resources: Have students write their own understanding of a given term Textbook Correlation: Give students formal and informal definitions of each term 1-2 points, lines and plane 1-3 measuring segments and compare them Develop precise definitions through use of examples and 1-4 measuring angles non-examples 1-5 exploring angle pairs Discuss the importance of having precise definitions 3-1 lines and angles Start line segment addition with integers, move to labeled segments then expressions. Do the same with angle addition postulate. Sample Assessment Tasks Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Skill-based task State the definition of a circle, perpendicular lines, parallel lines, and line segment. Unit # 2 Unit Title: Geometry Problem Task CORE CONTENT Cluster Title: Prove Geometric Theorems Standard: Standard: G.CO.9 -- Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistace from the segment’s endpoints. Concepts and Skills to Master: Identify parallel, perpendicular, and skew lines and parallel planes Classify/determine angle relationships formed by two lines and a transversal Prove that vertical angles are congruent. Prove that when parallel lines are cut by a transversal, pairs of alternate interior angles are congruent, pairs of alternate exterior angles are congruent, and pairs of same-side interior angles are supplementary. When given parallel lines, use properties of parallel lines to calculate angle measures and solve for variables When given angle measures, determine which lines, if any, are parallel SUPPORTS FOR TEACHERS Critical Background Knowledge Solve multi-step equations; know properties of supplementary, complementary, vertical, & adjacent angles (7.G.5) Academic Vocabulary Vertical angles, Parallel lines & planes, Transversal, Alternate Interior/Exterior angles (AIA and AEA), Corresponding angles (CA), Same-side Interior angles (SSI), Skew lines Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Unit Title: Geometry Suggested Instructional Strategies: Resources: -Use multiple formats to write justifications: narrative Textbook Correlation: 2-6, 3-1, 3-2, 3-3, 3-4 paragraphs, flow diagrams, two-column format, and Online Teacher Resource Center 3-1 Game: Name It – diagrams without words. Claim It (Suggestion, enlarge the four diagrams on an 8 x 10 -When teaching skew lines, highlight all segments in a and allow students more room to play with their die or rectangular prism that are not skew; thus all other segments number cube) are skew. Online Teacher Resource Center 3-2 Performance Task -Use dynamic geometry software to explore angle Activity (Suggestion, enlarge the four diagrams on an 8 x 10 relationships and allow students more room to play with their die or -Connect angle relationships to the creation of tessellation number cube) patterns Angle Hunter Video -Formal and informal proofs do not have to be introduced at this point. Students can argue with justification by completing short answer/open-ended questions such as “John says that consecutive interior angles are congruent when lines are parallel. How would you convince John that that these angles are actually supplementary?” Focus on the validity of the underlying reasoning of the justifications -The “Build a City” project or similar activity is suggested as a real-world application follow-up assignment to assess student understanding. Sample Assessment Tasks Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Unit Title: Geometry Skill-based task I. Lines a and b are parallel. Solve for x. Solve for y. Find the measures of angles 1-5. Based on this information, are lines c and d parallel? In complete sentences, explain why or why not. Problem Task Marie is building a sandbox in her back yard. Only equipped with the tools to measure angles, how can Marie determine whether both pairs of opposite sides are parallel? If both pairs of opposite sides are parallel, and one of the angles measures 85 degrees, what are the measures of the remaining angles? II. Suppose a || b and c || d. 1. If m∠ 6 = 50, then find m∠ 11. 2. If m∠ 2 = 70, then find m∠ 6. 3. If m∠ 7= 110, then find m∠ 10. 4. If m∠ 4= 45, then find m∠ 12. 5. Which angle could you show is congruent to ∠ 11 to prove a || b? 6. What relationship between ∠ 6 and ∠ 11 shows c || d? Find as many angle relationships as possible in this pattern: CORE CONTENT Cluster Title: Make geometric constructions Standard: G.CO.12 -- Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Concepts and Skills to Master: Perform the following constructions using a variety of tools and methods: copying a segment, copying an angle, bisecting a segment, bisecting an angle, constructing perpendicular lines, including the perpendicular bisector of a line segment, and constructing a line parallel to a given line through a point not on the line. Explain why these constructions result in the desired objects. SUPPORTS FOR TEACHERS Critical Background Knowledge Definitions of the following terms: circle, bisector, perpendicular, and parallel Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Unit Title: Geometry Academic Vocabulary Segment, Angle, Bisect, Perpendicular, Parallel, Circle, Construction, Transversal, Alternate Interior/Exterior angles, Corresponding angles, Same-side Interior angles, Skew lines, Parallel planes Suggested Instructional Strategies: Resources: Completing constructions can be used as a follow up activity Textbook Correlation: 1-6, 3-6 or as an investigation. Investigation Activity (from CMS Curriculum Guide) If used as an investigation, provide students with openended follow up questions to help them draw accurate http://math.springbranchisd.com/high/classes/algebra conclusions. For example, after giving students _one/Laying%20The%20Foundation/Lessons/Paralle l%20and%20Perpendicular%20Lines%20187instructions for constructing an angle bisector, pose the following: “Fold your patty paper along this line. What 188.pdf can you conclude about each angle? Which of the vocabulary terms from this section have you just constructed?” Have students explore how to make a variety of constructions using different tools. Ask students to justify how they know their method results in the desired construction. Discuss the underlying principles that different tools rely on to produce desired constructions (e.g. compass:circle; miro: reflections) Sample Assessment Tasks Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Skill-based task Using patty/tracing paper, pencil, straight edge, and compass, complete the following constructions Parallel lines Perpendicular lines Perpendicular bisector Angle bisector Unit # 2 Unit Title: Geometry Problem Task Jessica is studying architecture at the University of North Carolina at Charlotte. For homework, she must find the center point of a regular pentagon by connecting all of the angle bisectors. Unfortunately, Jessica has her straight edge, but has lost her protractor. What step by step instructions would you give to Jessica to help her complete the assignment? CORE CONTENT Cluster Title: Congruence-Prove Theorems about Triangles Standard: G-CO.10 Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of 2 sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Concepts and Skills to Master: Use parallel lines to prove a theorem about triangles Find measures of angles in triangles Prove right triangles congruent using the Hypotenuse-Leg Theorem SUPPORTS FOR TEACHERS Critical Background Knowledge The sum of the angle measures of triangles is always the same i.e. 180 degrees Understanding the parts of a right triangle Academic Vocabulary Exterior angle, remote interior angle, auxiliary line, hypotenuse, legs Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Suggested Instructional Strategies: Use a number of examples with exterior angles and show how the 2 remote angles add up to the exterior angle. Draw the correlation between triangle interior angles totaling 180° and an exterior angle and its linear pair inside the triangle totaling 180° Sample Assessment Tasks Skill-based task Pearson website Lesson 3.5 Enrichment Pearson Solve it! 3.5 Pearson website Lesson 4.5 Enrichment (Swan Puzzle) Unit Title: Geometry Resources: Textbook Correlation: Pearson 3.5, 4.1, 4.2, 4.3, 4.5, 4.6, 5.1, 5.4 Video resource: Introduction Congruent Triangles Problem Task Pearson website Activities, Games and Puzzles 3.5 CORE CONTENT Cluster Title: Polygon Angle Sum Theorems Standard: G-CO.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. Concepts and Skills to Master: Identify, verify, and classify properties of quadrilaterals. Define and classify special types of parallelograms SUPPORTS FOR TEACHERS Critical Background Knowledge Academic Vocabulary equiangular polygon, equilateral polygon, regular polygon, irregular polygon, n-gon, diagonal of a polygon, convex polygon, overlapping triangles, interior angles, exterior angles, consecutive angles Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Unit Title: Geometry Suggested Instructional Strategies: Have students sketch regular polygons (3 sided to 8 sided shapes). Then have students make a conjecture about the number of overlapping triangles in each after drawing diagonals that connect all vertices – remember: triangles cannot overlap. See if students can discover the Polygon Angle Sum Theorem. Resources: Interactive Learning: 6-1 Solve It (Dynamic Activity Online Teacher Resources –Interactive Digital Path) www.pearsonsuccessnet.com Wiki Exploration: Exterior Angle Sum Theorem http://www.geogebra.org/en/wiki/index.php/Angles (Click on Polygon Exterior Angle Sum Theorem – explore – click next in the top right hand corner to Extension: Have students label all interior angles of the continue exploration) sketched polygons, extend the vertices, and label the Concept Byte Exploration Activity: p.352 Exterior Angles of measures of each exterior angle formed. Ask students Polygons Essential Question #2 – What conjecture can be made Cluster Review – Use Links Below: about the sum of the exterior angles of any convex polygon? http://freedownload.is/ppt/3-4-the-polygon-angle-sumtheorems-ppt Click on 3.4 (also labeled 3.5 in description) The Polygon Angle Sum Theorems http://www.mathwarehouse.com/geometry/polygon/ -Scroll entire web page to see questions.- Sample Assessment Tasks Skill-based task Problem Task Question I: A polygon has n sides. An interior angle of the polygon and an adjacent exterior angle form a straight angle. a. Use an algebraic expression to represent the sum of the measures of the n straight angles? b. Use an algebraic expression to represent the sum of the measures of the n interior angles? c. Using your answers above, what is the sum of the measures of the n exterior angles? d. What theorem do the steps above prove? 1. For each regular polygon, state the sum of the measures of the interior angles and give the measure of an interior angle. Question II: A triangle has two congruent interior angles and an exterior angle that measures 100. Find two possible sets of interior angle measures for the triangle? 2. For each regular polygon, state the sum of the measures of the exterior angles and give the measure of an exterior angle. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Unit Title: Geometry Similarity: (part 2) Enduring understanding (Big Idea): Students will be able to apply their previous knowledge of transformations to determine similarity. They will focus on proving and using the AA similarity theorem and will apply all information to solve problems, which include finding angle measures and side lengths. Essential Questions: 1. How can you determine whether two figures are similar using similarity transformations, angle measures, and side lengths? 2. What is the AA Similarity theorem and why does it sufficiently determine whether two triangles are similar or not? 3. How can you prove that a line parallel to one side of a triangle divides the other two sides proportionally? How can this information be used to solve problems? BY THE END OF THIS UNIT: Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Students will know… · AA Similarity Theorem · A line parallel to one side of a triangle divides the other two proportionally, and conversely Vocabulary: · Similarity, similar triangles, AA Similarity theorem, transformation, corresponding angles, corresponding sides (corresponding parts), proportion Students will be able to… · Prove the AA Similarity Theorem · Determine whether two figures and/or triangles are similar using similarity transformations · Apply knowledge of similar triangles to solve problems, which include setting up proportions, and finding angle measures & side lengths. Unit Resources Learning Task: 1. 7.2 Think About a Plan; 2. 7.3 Think About a Plan *Both located at www.pearsonsuccessnet.com * Performance Task: Ch 7 Performance Tasks: www.pearsonsuccessnet.com Project: Visit link below. Note: This project will need to be edited and personalized before implementation. https://vasicek.wikispaces.com/file/view/The+Similar+Triang les+Project.pdf Mathematical Practices in Focus: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 6. Attend to precision Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Unit Title: Geometry Course Name: Math iii Unit # 2 Unit Title: Geometry CORE CONTENT Cluster Title: Understand similarity in terms of similarity transformations Standard: G-SRT.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. Concepts and Skills to Master: Determine if two figures are similar using properties of transformations Determine if two triangles are similar, given their angle measures and side lengths Given angle measures and side lengths, determine if two triangles are similar SUPPORTS FOR TEACHERS Critical Background Knowledge 8.G.4 – Understand similarity as a sequence of transformations Knowledge of the different types of transformations (rotations, reflections, translations, and dilations). Academic Vocabulary Similarity, Transformation, Congruence, Corresponding Parts Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Unit Title: Geometry Suggested Instructional Strategies: Resources: · Give students pairs of triangles, some of which are similar and some of which are not. Have students verify 1. Textbook Correlation: or disprove similarity using transformations and the o 7.2 Similar Polygons definition of similarity. · Use geometry software and patty paper to explore properties of similarity. Sample Assessment Tasks Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Skill-based task Unit # 2 Unit Title: Geometry Problem Task 1. Jan uses an overhead projector to enlarge a picture 5 in. high and 7 in. wide. She projects the picture on a 1. blackboard 4 ft 2 in. high and 12 ft wide. What are the dimensions of the largest picture that can be projected 2. on the blackboard? Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. http://illustrativemathematics.org/illustrations/603 – “Are they similar” activity Under what conditions do two lines intersected by two transversals form similar triangles? Course Name: Math iii Unit # 2 Unit Title: Geometry CORE CONTENT Cluster Title: Understand similarity in terms of similarity transformations Standard: G-SRT.3 – Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Concepts and Skills to Master: Prove using the properties of similarity transformations that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar (AA). SUPPORTS FOR TEACHERS Critical Background Knowledge · 8.G.4 – Understand similarity as a sequence of transformations · 8.G.5 – Use informal arguments to establish facts about the angle sum…of triangles and…the angle-angle criterion for similarity of triangles · Knowledge of the different types of transformations (rotations, reflections, translations, and dilations). · If two angles of a triangle are congruent to two corresponding angles of a second triangle, then the third pair of corresponding angles must be congruent. Academic Vocabulary Similarity, Transformation, Angle-Angle Similarity Theorem Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Suggested Instructional Strategies: · Review the Triangle Angle Sum Theorem and angle relationships with parallel lines prior to addressing this standard. Sample Assessment Tasks Skill-based task Unit # 2 Unit Title: Geometry Resources: 1. Textbook Correlation a. 7.3 Proving Triangles are Similar 2. Puzzle: Similarity Search a. Located at www.pearsonsuccessnet.com under the “Activities, Games, and Puzzle” link for section 7.3 Problem Task 1. Determine whether the two triangles are similar. Justify 1. Given two different-sized triangle cutouts with two corresponding angles congruent, allow the students to your conclusion. show that the third angle is congruent, and find a dilation that produced the two triangles. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Unit Title: Geometry Teacher Created Argumentation Tasks · Your classmate provides the following solutions to the problems below. In complete sentences, identify and explain the error in each explanation, and tell me how you would help your classmate reach an accurate conclusion. (From www.pearsonsuccessnet.com – Chapter 7: Find the Errors! for sections 7.2-7.3) CORE CONTENT Cluster Title: Prove theorems involving similarity Standard: G-SRT.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. Concepts and Skills to Master: · Prove that if a line intersecting a triangle is parallel to one of the sides, then it divides the other two sides of that triangle proportionally. · Prove that if a line divides two sides of a triangle proportionally, then it is parallel to the third side. SUPPORTS FOR TEACHERS Critical Background Knowledge · Understand Angle-Angle Similarity · Ability to set up and solve proportions Academic Vocabulary Parallel Lines, Similar triangles, Angle-Angle Similarity, Proportion Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Unit Title: Geometry Suggested Instructional Strategies: Resources: Help students understand that the sides are proportional for · Textbook Correlation: any segment parallel to the base, not just the midsegment. o 7.3 Proving Triangles are Similar o 7.4 Similarity in Right Triangles o 7.5 Proportions in Triangles • 7.5 Enrichment located at www.pearsonsuccessnet.com · Section B: Right Triangles, Altitudes, the Pythagorean Theorem, and You (Located at www.discoveryeducation.com) http://player.discoveryeducation.com/index.cfm?guidAssetId= A42740C7-7915-4A90-A1CB-DD891170FFF5 Sample Assessment Tasks Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Skill-based task 1. Unit # 2 Unit Title: Geometry Problem Task Is segment SU parallel to segment RV? Explain why or 1. Given: T is the midpoint of , U is the midpoint of why not. and V is the midpoint of . Prove: ∆QRS ~ ∆VUT *This problem found at www.pearsonsuccessnet.com under the Enrichment task for section 7.3* Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. , Course Name: Math iii CORE CONTENT Unit # 2 Unit Title: Geometry Cluster Title: Prove theorems involving similarity Standard: G-SRT.5 – Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Concepts and Skills to Master: Apply knowledge of congruent and/or similar triangles to find scale factor, angle measures, side lengths, and other measurements such as perimeter and area. Apply knowledge of congruent and/or similar triangles to determine the similarity of triangles based on various given information. SUPPORTS FOR TEACHERS Critical Background Knowledge · Students should be able to identify corresponding parts of triangles · Set up and solve proportions · Understand that similar figures have congruent corresponding angles and proportional corresponding sides Academic Vocabulary Similar triangles, proportions, scale factor Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Unit Title: Geometry Suggested Instructional Strategies: Resources: · Challenge students to find real-world examples of · Textbook Correlation: similar triangles on their own. Encourage them to prove o 7.3 Proving Triangles are Similar how they were able to conclude that the triangles were o 7.4 Similarity in Right Triangles similar. · Videos: o “Section A: Proving the Similarity of Triangles” located at www.discoveryeducation.com http://player.discoveryeducation.com/index.cfm?guidAssetId= F6D5649D-C548-40B4-9546DB85CA892F02&blnFromSearch=1&productcode=US Sample Assessment Tasks Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Skill-based task Unit # 2 Unit Title: Geometry Problem Task 1. At 4:00 pm, Karl stands next to his house and measures his shadow and the house’s shadow. Karl’s shadow is 8 ft. long and the house’s shadow is 48 ft. long. If Karl is 6 ft tall, how tall is his house? 2. From www.pearsonsuccessnet.com, 7.4 Enrichment assignment (specifically #6-7) Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Conic Sections: (part 3) Unit # 2 Unit Title: Geometry Enduring understanding (Big Idea): Students will understand that concepts related to circles and conic sections are applicable in real world scenarios as they explore properties of tangent lines, represent circles as equations based on the center and radius, calculate the measures of central angles, inscribed angles, and their intercepted arc measures and lengths. Essential Questions: 1. Discuss what it means to “go off on a tangent”? 2. How do you find the equation of a circle in a coordinate plane? 3. When lines intersect a circle or intersect within a circle, how do you find the measure of resulting angles, arcs, and segments? 4. How can you prove relationships between angles and arcs in a circle? 5. What is the intersection of a cone and a plane parallel to a line along side of the cone? 6. How can you derive the equation for a parabola, given a focus and directrix? BY THE END OF THIS UNIT: Students will know… · Properties of tangent lines as it relates to a circle · Concepts of chords, arcs, and angle measures as it relates to a circle · Arc Length and Segment Lengths as it relates to circles · Equations of a Circle and Parabola · Conic Sections Vocabulary: arc measure, arc length, inscribed angle, intercepted arc, chord, point of tangency, tangent line (tangent to a circle), secant, standard form of the equation of a circle, conic sections, directrix, focus, parabola, ellipse, hyperbola Unit Resources Learning Task:click on Circle Formulas – download file, print, and copy http://www.mathworksheetsgo.com/sheets/geometry/circles/circl e-formula-graphic-organizer.php Performance Task:Have students view the power point presentation. In writing, allow students to describe how well the presentation reflects what was learned in class. Be sure to include what concepts were discussed and which were left out.www.btinternet.com/~mathsanswers/CircleTheorems.ppt Unit Review Game:Jeopardy Review Gamehttp://www.superteachertools.com/jeopardyx/jeopardy-reviewgameconvert.php?gamefile=../jeopardy/usergames/May201221/jeopardy133 7974033.txt Students will be able to… · Identify a tangent and use properties of tangent as it relates to a circle · Compute chord, arc, and angle measures · Find arc length given the arc’s central angle and the circle’s diameter or radius · Find lengths of segments related to circles and its intersecting lines · Write the equation of a circle given its center and radius · Identify conic sections · Write the equation of a parabola given its directrix and focus. Mathematical Practices in Focus: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 6. Attend to precision. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. NOTE: For Unit Resources, the Performance Task Activity can also be a Project. Course Name: Math iii Unit # 2 Unit Title: Geometry Cluster Title: Find arc lengths and areas of sectors of circles Standard: G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality Concepts and Skills to Master • • • Identify major and minor arcs and semicircles Find the measure of a central angle and its intercepted arc Compute the circumference of a circle and arc length (i.e. distances along circular paths) SUPPORTS FOR TEACHERS Critical Background Knowledge • Circumference of a Circle • Exact Circumference (leave your answer in terms of pi) • Congruent circles have congruent radii Academic Vocabulary circle, center, diameter, radius, congruent circles, central angle, semicircle, minor arc, major arc, adjacent arcs, intercepted arc, circumference, pi, concentric circles, arc length, congruent arcs, exact circumference Suggested Instructional Strategies Resources · Be sure to highlight for students that an arc is · Textbook Correlation: 10-6 Circles and Arcs measured by the central angle that defines it. The central angle captures within its rays the · Online Teacher Resource Center: intercepted arcs. www.pearsonsuccessnet.com · Error Prevention: Students may benefit from Activities, Games, and Puzzles (10-6 Circles and Arcs tracing the cited arc(s) of the figure(s) with colored Crossword) pencils · Explain to students that as it relates to standard · Commonly Confused: Arc Measure & Arc Length G.C.5, the length of an arc can be found by Bright storm Video – use the link below multiplying the ratio of the arc’s measure to 360 degrees by the circle’s circumference. www.brightstorm.com/math/geometry/.../arc-length/ · Students often confuse arc measure with arc length. Be sure to note that one is measured in degrees and the other is measured in units. Sample Formative Assessment Tasks Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Skill-based task Find the arc measure and arc length of each darkened arc. Leave your answer in terms of π. Unit # 2 Unit Title: Geometry Problem Task Task: It is 5:00. What is the measure of theminor arc formed by the hands of an analog clock hanging on a classroom wall? What is the arc length if the radius of the clock is 6 inches? Sketch a wall clock to support your answer. Cluster Title: Understand and apply theorems about circles Standard: G.C.2Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Concepts and Skills to Master · · · · Tangent Lines Chord and Arc Measures Central and Inscribed Angles Angle Measures and Segment Lengths SUPPORTS FOR TEACHERS Critical Background Knowledge · Students will use understanding of congruent triangles and right triangles to prove statements about tangent lines. · Prior knowledge of a circle and its common features are needed: center, radius, diameter, chord, arc. · Triangle Angle Sum Theorem · Pythagorean Theorem · Perimeter of Polygons · Congruence Academic Vocabulary tangent to a circle, point of tangency, inscribed circles, chord, arc, semicircle, inscribed angles, circumscribed polygons, secant Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Suggested Instructional Strategies • Students sometimes get confused identifying segments of a circle. Have students create a vocabulary sheet that includes definitions and diagrams of each type of segment. • Students sometimes get confused identifying central and inscribed angles and, therefore, use the wrong formula to compute angle measures. Perhaps making a connection that a central angle has its vertex in the center of the circle will help students distinguish between the two. • Paper folding activities offer students a good way to develop key concepts related to central angles, chords, and arcs. • Have students to organize all the theorems taught in sections 12.1 to 12.4 in an effort to increase learning. Unit # 2 Unit Title: Geometry Resources • Cluster Review http://library.thinkquest.org/20991/geo/circles.html • Circle Concept Interactive Math Site http://www.mathopenref.com/chordsintersecting.html (Explore circle concept by scrolling down and clicking from the selection on the bottom left of the screen) • Concept Byte Exploration Activity: p.770 - Paper Folding With Circles Critical Background Knowledge · Domain and Range · Graphing on a Coordinate Plane · Lines of Symmetry Academic Vocabulary conic sections (parabolas, circles, ellipses, and hyperbolas), lines of symmetry, focus, directrix, focal length Suggested Instructional Strategies Resources · At this point, do not make graphing the conic · Textbook Correlation: Algebra II Textbook sections a more difficult task by having students 10-1 Exploring Conic Sections solve for x and or y. Instead, simply have (www.pearsonsuccessnet.com) student graph conic sections using a table of 10-2 Parabolas (www.pearsonsuccessnet.com) values that range from -5 to +5; substituting for · Conic Sections Explained whichever variable is easier. [Note: If you have http://math2.org/math/algebra/conics.htm a classroom set of graphing calculators, you · Parabolas and Their Equations Powerpoint https://docs.google.com/viewer?a=v&q=cache:epOo8GE may want students to practice solving for y in order to use the equation editor and table of PeOIJ:princemath.wikispaces.com/file/view/parabolas.pp values.] (Also, note that more emphasis will t+parabola+and+its+equations+powerpoint&hl=en&gl=us be placed on conic sections in further math &pid=bl&srcid=ADGEESh5fKhyjqpZxcMuqaQOU5kouLH LYDR4TuYHy5eWBU8yqGviMzQqb_iESTO7MRFVXhc3 courses.) mKlAOn· Be sure that students understand that a conic c0nbIFTkIgQggy6EXbwLGEzz1vJAfGo1wYmUlIynOQgD section is simply the intersection of a plane and tEreV1tKGzC4yU9RT&sig=AHIEtbRUvWX5ZWLTFr68Jn 4HTR3RP-aRLQ a cone. (Use resource: Conic Sections Explained as a teaching aid if needed.) Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Unit # 2 Unit Title: Geometry Cluster Title: Understand and apply theorems about circles (i.e. Circle Similarity) Standard: G.C.1 Prove that all circles are similar. Concepts and Skills to Master • Prove Similarity in Circles SUPPORTS FOR TEACHERS – NOTE:This concept is not in the textbook and limited information appropriate for HS students is available online. Critical Background Knowledge • Definition of Similarity • Applications with Circle Formulas and Right Triangles Academic Vocabulary similarity of circles Suggested Instructional Strategies Resources · Recall: being similar means having corresponding · Textbook Correlation: none congruent angles but proportional corresponding · Online Resource A sides. See Online Resource A. Core Challenge – Standard G.C.1 – Prove all circles · In general, two figures are similar if there is a set of are similar. Click on ‘download file’. transformations that will move one figure exactly http://app.corechallenge.org/learningobjects/7878 covering the other. To view proof, see Online Resource B. · Online Resource B – All Circles are Similar · To prove any two circles are similar, only a Examples.pdf translation (slide) and dilation (enlargement or www.cpm.org/pdfs/state_supplements/Similar_Circle reduction) are necessary. Using the differences in s.pdf the center coordinates to determine the translation and determining the quotient of the radii for the · Online Resource C – YouTube Video – dilation can always do this. For further All circles are similar demonstration explanation, see Online Resource C. http://www.youtube.com/watch?v=jTvlvLFZQPY · Problem Task:Take students to the lab if possible to view the you-tube video that teaches the lesson on circle similarity. If students do not have access to the site, save the link elsewhere so that students can view it – or make it a homework assignment. (Honor and IB Classes only) If the video is used for Standard classes, teacher explanation and modeling is necessary. Sample Formative Assessment Tasks Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Skill-based task Unit # 2 Unit Title: Geometry Problem Task(see suggested instructional strategies – item 4) Use the link below to view the 32min 20 sec you-tube video that discusses circle similarity. Take notes during the video. After viewing the video, complete a written assignment, documenting what you have learned. Link: http://www.youtube.com/watch?v=2QOj02EKDTE CORE CONTENT Cluster Title: Apply geometric concepts in modeling situations Standard: G.MG.3- Apply geometric methods to solve design problems (e.g. designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios. Concepts and Skills to Master Students will be able to solve design problems by designing an object or structure that satisfies certain constraints. SUPPORTS FOR TEACHERS Critical Background Knowledge Parts of a right triangle and congruent corresponding parts Academic Vocabulary congruent triangles, hypotenuse, legs of a right triangle Suggested Instructional Strategies: Resources: · Textbook Correlation: Pearson Chapter 11.2, 3, 4, 5, 6 · Perfume Packaging –Dana Center chapter 5 pg 1 Sample Assessment Tasks Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Course Name: Math iii Skill-based task Unit # 2 Problem Task Pearson Additional Problems Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes. Unit Title: Geometry