Geometry Ch 1

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1 Geometry Ch 1 Lesson 1 What are the three undefined terms in Geometry? (Section 1.1) HW #1: P5 8: 1, 5, 6, 8, 15, 19, 33, 35, 36 Objectives: name a segment, a line, a ray, a plane know the terms collinear, parallel, skew and coplanar know the intersection of planes and lines Lesson 2 Aim: How do we prove two line segments are congruent? (Section 1.2) HW #2: P12 14: 1, 13 14, 21 26, 28, 29. Objectives: know the segment addition postulate Know the difference between congruent and equal Lesson 3&4 Aim: How do we find the midpoint when given two points? (Section 1.3) HW #3-4: P19 20: 4, 5, 11, 13, 17 19, 25 27, 47 Objectives: know the midpoint divides a segment into two equal pieces Know how to find the coordinates of the midpoint when given two endpoints Know how to find the coordinates of an endpoint when given an endpoint and the midpoint. Lesson 5 Aim: How do we find the distance between two points? (Section 1.3) 2 Days HW #5: P20 22: 20, 23, 30 32, 41, 44, 45. Objectives: know that distance between two points is the same as the hypotenuse of a right triangle. know how to express the length or distance in simplest radical form or decimals know how to find the perimeter of a triangle when given the three vertices. Lesson 6 Aim: How do we classify angles? (Section 1.4) HW #6: P28 30: 3 5, 11 14, 21 23, 25 27, Objectives: know how to name an angle Know the different types of angles: acute, right, obtuse and straight Know the angle addition postulate Know that linear pairs are supplementary Lesson 7 Aim: How do we describe angle pair relationships? (Section 1.5) HW #7: P38 39: 3 7, 17, 19, 31, 32, 45* Objectives: know the terms adjacent Know the complementary angles add up to 90 degrees and supplementary angles add up to 180 degrees. Lesson 8 Aim: How do we classify polygons? (Sections ) HW #8: P44 46: 1, 3 6, 8 13, 15 17, 28, 30

2 Objectives: know what is a triangle, quadrilateral, pentagon or hexagon Know what is a convex or concave polygon Know the terms equiangular, equilateral and regular Ch 1 Review/Uniform Unit Test Common Core Standards Addressed G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G-CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 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 equidistant from the segment s endpoints. G-PFE.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). G-PFE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. A-CED.1 Create equations in one variable and use them to solve problem. A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as solving equations. Methods of Differentiated Instructions Revolution K-12, a student-driven online learning platform, is used to differentiate instruction for all students in class and subject areas. All students are held to the Common Core-aligned standards. Teacher performs formative assessments through lessons to check student understanding, including cold-calling, student board work, and exit

3 tickets. Teacher circulates room to perform further formative assessments and to guide small groups and individuals. Lower-performing students are grouped heterogeneously with higher-performing students using assessment data, interests, and work habits, to encourage student-to-student engagement and learning. They are also provided guided notes and graphic organizers to build schemas. English Language Learners are paired with students who are fluent in English, and given extra time. Vocabulary is pronounced several times and accompanied by alternative words or phrases that are simplified. Visual aids, including pictures, Smart boards, and manipulatives, help students make clear connections to the text. Students with Special Needs are grouped with helpers and given instructions or assessments with simplified language or extra time. Color-coding on the Smart board help illustrate steps required to solve a problem. Hands-on activities are provided to help construct student learning. Gifted/Honor students are given challenge problems during lessons, homework, and summative assessments, which earn extra credits. They are expected to complete the whole worksheet, and are given challenge (e.g. open-ended) problems that develop higher-level thinking.

4 Geometry Ch 2 Lesson 10 Aim: How do we use inductive reasoning and deductive reasoning? (Section 2.1/2.2) 2 Days HW #10: P75: 5 7, 11, 13 14, 17, 20, 22. Objectives: Know what inductive reasoning, conjecture, and counterexample are Know how to use inductive reasoning to form conjectures Know how to find a counterexample to disapprove a conjecture Know what deductive reasoning is Lesson 11 Aim: How do we evaluate the truth value of a conjunction or disjunction? HW #11: Worksheet Objectives: know what is a conjunction and a disjunction Know when a conjunction is true and when a disjunction is true Lesson 12 Aim: How do we analyze conditional statements? (Section 2.2) HW #12: P83 84: 9 10, 12, 13, 19, 21, 26. *find a news article that contains any of the statements we learned in class Objectives: know how to write a conditional statement and its converse, inverse and Contrapositive Know how to write the negation of a statement and determine its truth value Know the term logically equivalent and that the contrapositive is always logically equivalent to the conditional statement. Lesson 13 Aim: How do we use deductive reasoning? (Section 2.2) HW #13: p90-91: 1, 4 12, 15 Objective: Know what deductive reasoning is Know or be able to apply the Laws of Syllogism and Detachment Lesson 14&15 Aim: How do we use postulates and diagrams? (Section 2.4) HW #14&15: P99-100: 3 5, Objectives: Know, when given a diagram, what to assume and what not to assume Know how to answer questions about planes and lines. Know the symbol for perpendicular Lesson 16 Aim: How do we prove statements about segments and angles? (Section 2.6)

5 HW #16: P : 1, 5 11, 17 Objectives: know the reflexive property, symmetric property and the transitive property. Know how to write a short two-column proof Lesson 17 Aim: What are the angle pair relationships? (Section 2.7) HW #17: P : 4 5, 12, 28, Objectives: know the congruent complements theorem and the congruent supplements theorem Lesson 18 Aim: What are the angle pair relationships? (Section 2.7) HW #18: P128: Objectives: know the vertical angles congruence theorem and the linear pair postulate Ch 2 Review/Uniform Unit Test Common Core Standards Addressed 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 equidistant from the segment s endpoints. 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. Methods of Differentiated Instructions Revolution K-12, a student-driven online learning platform, is used to differentiate instruction for all students in class and subject areas. All students are held to the Common Core-aligned standards. Teacher performs formative assessments through lessons to check student understanding, including cold-calling, student board work, and exit tickets. Teacher circulates room to perform further formative assessments and to guide small groups and individuals. Lower-performing students are grouped heterogeneously with higher-performing students using assessment data, interests, and work habits, to encourage student-to-student engagement and learning. They are also provided guided notes and graphic organizers to build schemas.

6 English Language Learners are paired with students who are fluent in English, and given extra time. Vocabulary is pronounced several times and accompanied by alternative words or phrases that are simplified. Visual aids, including pictures, Smart boards, and manipulatives, help students make clear connections to the text. Students with Special Needs are grouped with helpers and given instructions or assessments with simplified language or extra time. Color-coding on the Smart board help illustrate steps required to solve a problem. Hands-on activities are provided to help construct student learning. Gifted/Honor students are given challenge problems during lessons, homework, and summative assessments, which earn extra credits. They are expected to complete the whole worksheet, and are given challenge (e.g. open-ended) problems that develop higher-level thinking.

7 Geometry Ch 3 Class website: For graph paper, ask Mr. Lee after class, or visit To receive credit, you must copy the questions and all diagrams, and use a straightedge for lines. (Long word problems may be paraphrased, i.e. summarized.) Some homework will be distributed in class as handouts, with some differences in assigned problems. Lesson 20 Aim: What are the properties of parallel lines cut by a transversal? (Section 3.2) 2 Days HW #20: P : 1 3, 9 19, 24. P138: 3 4. HW #21: P : 3, 5, 7, 8, 10 15, 22, 23. P138: 7, 15. Objectives: Know the terms parallel and skew Know the Alternate Interior Angles theorem, Corresponding Angles Postulate, Alternate Exterior Angles theorem and Consecutive Interior Angles theorem. Lesson 22 Aim: How do we prove that two lines are parallel? (Day 1) (Section 3.3) HW #22: P : 3, 5, 7, 8, 10 15, 22, 23. P138: 8, 15. Objectives: Know how to prove two lines cut by a transversal are parallel by using the Alternative Interior Angles Converse, Corresponding Angles Converse, Alternative exterior Angles Converse and the Consecutive Interior Angles Converse. Lesson 22 Aim: How do we prove two lines are parallel? (Day 2) (Section 3.3) HW #22b: P : 4, 8, 19 20, 23, 28, 34, 35*(CHALLENGING) Objectives: Know how to prove two lines cut by a transversal are parallel by using the Alternative Interior Angles Converse, Corresponding Angles Converse, Alternative Exterior Angles Converse and the Consecutive Interior Angles Converse in twocolumn proofs Lesson 23 Aim: How do we find and use the slopes of lines? (Day 1) (Section 3.4) HW #23: P : 2 10, P138: 9, 10, 21 Objectives: Know how to find the slope of a line determined by two points Know how to graph a line when given a slope and a point Lesson 24 Aim: How do we find and use the slopes of lines? (Day 2): (Section 3.4) HW #24: P : 13 16, 23 25, 27 29, 42. P150: 3 6.

8 Objectives: know when two lines are parallel, perpendicular or neither using their slopes Lesson 25 Aim: How do we find and use the slopes of lines? (Day 3): (Section 3.4) HW #25: FINSIH WORKSHEET Objectives: know how to graph a line through a point that is parallel or perpendicular to a given line Lesson 26 Aim: How do we write and graph the equations of lines? (Day 1) (Section 3.5) HW #26: P184: 3 5, P166: 21, 24, 31. P : 7, Objectives: know what is the slope-intercept form and turn any given linear equation into this form Know how to graph a linear equation Know how to write the equation of a line when the graph is given Lesson 27 Aim: How do we write and graph the equations of lines? (Day 2) (Section 3.5) HW #27: P : 17 18, 20, 23, 28, 31, 34, 36, 41 P176: 31 Objectives: how to write the equation of a line when two points are given How to write an equation that is parallel or perpendicular to a given line. Lesson 28 Aim: How do we prove theorems about perpendicular lines (day 1)? (Section 3.6) HW #28: P : 1-7, 16-17, 26, 32; P185: 49, 51 Lesson 29 Aim: How do we prove theorems about perpendicular lines (day 2)? (Section 3.6) HW #29: P : 8-10, 13, 14, 29, Objectives: how to reason and write proof about theorems on parallel and perpendicular lines. Ch 3 Review/ Uniform Unit Test Common Core Standards Addressed G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 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 equidistant from the segment s endpoints. 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. G-GPE.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

9 G-GM.3 line that passes through a given point). 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). Methods of Differentiated Instructions Revolution K-12, a student-driven online learning platform, is used to differentiate instruction for all students in class and subject areas. All students are held to the Common Core-aligned standards. Teacher performs formative assessments through lessons to check student understanding, including cold-calling, student board work, and exit tickets. Teacher circulates room to perform further formative assessments and to guide small groups and individuals. Lower-performing students are grouped heterogeneously with higher-performing students using assessment data, interests, and work habits, to encourage student-to-student engagement and learning. They are also provided guided notes and graphic organizers to build schemas. English Language Learners are paired with students who are fluent in English, and given extra time. Vocabulary is pronounced several times and accompanied by alternative words or phrases that are simplified. Visual aids, including pictures, Smart boards, and manipulatives, help students make clear connections to the text. Students with Special Needs are grouped with helpers and given instructions or assessments with simplified language or extra time. Color-coding on the Smart board help illustrate steps required to solve a problem. Hands-on activities are provided to help construct student learning. Gifted/Honor students are given challenge problems during lessons, homework, and summative assessments, which earn extra credits. They are expected to complete the whole worksheet, and are given challenge (e.g. open-ended) problems that develop higher-level thinking.

10 Geometry Ch 4 Class website: To receive full credits, you must copy or summarize the questions, copy all diagrams, and construct all line drawings using a straightedge. Graphing must be done on graph paper. Attend tutoring (and have logs signed) for extra credits. Lesson 30 Aim: How do we apply the triangle sum properties? (Section 4.1) HW #30: P221: 7, 11, 13, 15, 18, 27, 32, 34. P144: 4; P214: 8 Lesson 31 Aim: What are the properties of congruent triangles? (Section 4.2) HW #31: P228: 2 3, 5 10, 16, 17, 20, 26, 32*(Challenging) P221: 1 6. Lesson 32 Aim: How do we prove triangles are congruent using SSS congruence postulate? (Section 4.3) HW #32: P236: 1 7, 13 14**, 17, 18, **Read P236 Lesson 33 Aim: How do we prove triangles are congruent using SAS congruence postulate? (Section 4.4) HW #33: P : 2 5, 10 13, 21, 26, 36**Challenging Lesson 34 Aim: How do we prove triangles are congruent using HL congruence theorem? (Section 4.4) HW #34: P : 14, 22, 25, 27, 34, 37, 38. Lesson 35&36 Aim: How do we prove triangles are congruent using ASA congruence postulate and AAS congruence theorem? (Section 4.5) HW #35&36: P : 25, 34 P244: 35 Lesson 37 Aim: How do we prove triangles are congruent? HW #37: FINISH WORKSHEET. Lesson 38 Aim: How do we prove corresponding parts of congruent triangles are congruent? (Section 4.6) HW #38: P259: 3 6, 10 11, 22, 31 Lesson 39 Aim: How do we prove corresponding parts of congruent triangles are congruent? (Section 4.6) HW #39: P259: 8, 16, 23. P238: 27

11 Lesson 40 Aim: How do we apply the properties of isosceles and equilateral triangles? (Section 4.7) HW #40: P267: 1 3, 6, 15 17, 45; P263: 5 At the end of this unit, you should be able to 1) know how to prove the Triangle Sum Theorem CH 4 Review/Uniform Unit Test 2) know and apply the Exterior Angles Theorem and Triangle Sum Theorem 3) know that if two polygons are congruent, then their corresponding parts are congruent, and vice versa. 4) use shortcut SSS Congruence Postulate to prove two triangles are congruent. 5) use shortcut SAS Congruence Postulate to prove two triangles are congruent. 6) use shortcut ASA Congruence Postulate to prove two triangles are congruent. 7) use shortcut AAS Congruence Theorem to prove two triangles are congruent. 8) use shortcut HL Congruence Theorem to prove two triangles are congruent. 9) know that we can t prove two triangles are congruent by using SSA SSA or AAA AAA. 10) know that in order to prove the corresponding angles or corresponding sides of two triangles are congruent, it is necessary to prove the two triangles are congruent first using the properties and shortcuts in steps ) prove two corresponding angles or corresponding sides are congruent using CPCTC. 12) know the Base Angles Theorem and the Converse of the Base Angles Theorem. 13) know vertical angles are congruent, angles or segments that get bisected result in two smaller congruent angles or segments, perpendicular lines form right angles, 14) apply the reflexive property. 15) know to apply the properties of two parallel lines cut by a transversal in your proof. Common Core Standards Addressed G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

12 G-CO.8 G-CO.9 G-CO.10 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. 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 equidistant from the segment s endpoints. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Methods of Differentiated Instructions Revolution K-12, a student-driven online learning platform, is used to differentiate instruction for all students in class and subject areas. All students are held to the Common Core-aligned standards. Teacher performs formative assessments through lessons to check student understanding, including cold-calling, student board work, and exit tickets. Teacher circulates room to perform further formative assessments and to guide small groups and individuals. Lower-performing students are grouped heterogeneously with higher-performing students using assessment data, interests, and work habits, to encourage student-to-student engagement and learning. They are also provided guided notes and graphic organizers to build schemas. English Language Learners are paired with students who are fluent in English, and given extra time. Vocabulary is pronounced several times and accompanied by alternative words or phrases that are simplified. Visual aids, including pictures, Smart boards, and manipulatives, help students make clear connections to the text. Students with Special Needs are grouped with helpers and given instructions or assessments with simplified language or extra time. Color-coding on the Smart board help illustrate steps required to solve a problem. Hands-on activities are provided to help construct student learning. Gifted/Honor students are given challenge problems during lessons, homework, and summative assessments, which earn extra credits. They are expected to complete the whole worksheet, and are given challenge (e.g. open-ended) problems that develop higher-level thinking.

13 Geometry Ch 5 Class website: To receive full credits, you must copy/summarize questions, copy diagrams, and construct using a straightedge and compass. You will need your own ruler and compass for the Regents, so buy them now. 5 full tutoring sessions = 10% on a test, or 2 HW passes A review booklet will be provided to you. You may use it on the test, so fill it out. Lesson 42 Aim: What is the mid-segment theorem? (Section 5.1) 2 Days HW #42: P298: 3, 5, 7, 9, 11, 25 27, 29*Challenging Lesson 43 Aim: What are the properties of perpendicular bisectors of a triangle? (Section 5.2) 2 Days HW #43: P : 3 5, 11 14, 26 Lesson 44 Aim: What is the circumcenter? (Section 5.2) HW #44: P306: Lesson 45 Aim: What are the angle bisector theorem and its applications? (Section 5.3) HW #45: P313: 12 13, construct the angle bisector any obtuse angle. construct the angle bisector any acute angle. Bring a triangle cutout. It must be straight sides and big. Lesson 46 What is the incenter of a triangle? HW #46: P314: 19, 20, 23, 24, construct the incenter of any triangle Lesson 47 Aim: What are the medians of a triangle? (Section 5.4) HW #47: P : 7, 10, 33 35, 41* Lesson 48 Aim: What are the altitudes of a triangle? (Section 5.4) HW #48: Construct an altitude to AB from a point P above AB Construct the orthocenter of an obtuse triangle. Lesson 49 Aim: What is the Triangle Inequality? (Section 5.5) HW #49: P331: 2, 7 9, 16 18, 21 23, 31**. **Need to use calculator Review booklet can be used on the test, but must be completed before the test Ch 5 Review/Uniform Unit Test At the end of this unit, you should be able to

14 1) know what is a mid-segment; know and apply the Mid-segment Theorem on triangles. 2) know what the perpendicular bisector does to a segment; know and apply the Perpendicular Bisector Theorem and its converse. 3) know that the concurrency of three perpendicular bisectors is the circumcenter in a triangle and solve problems involving the circumcenter. 4) know and apply the Triangle Inequality Theorem: a) which three side lengths can (and cannot) make up a triangle b) the possible length for the third side 5) know what an angle bisector does to an angle; know and apply the Angle Bisector Theorem and its converse. 6) know that the concurrency of three angle bisectors is the incenter and solve problems involving the incenter. 7) know the concurrency of three medians is the centroid and solve problems involving the centroid. 8) know the concurrency of the three altitudes forms the orthocenter. 9) construct a perpendicular bisector, a median, an angle bisector and an altitude using a compass and a straightedge. 10) know that within a triangle, the longest side is always opposite the largest angle, the second longest side is always opposite the second largest angle and the shortest side is always opposite the smallest angle. Common Core Standards Addressed G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 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. G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G-C.3 Construct the inscribed and circumscribed circles of a triangle. 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)._

15 Methods of Differentiated Instructions Revolution K-12, a student-driven online learning platform, is used to differentiate instruction for all students in class and subject areas. All students are held to the Common Core-aligned standards. Teacher performs formative assessments through lessons to check student understanding, including cold-calling, student board work, and exit tickets. Teacher circulates room to perform further formative assessments and to guide small groups and individuals. Lower-performing students are grouped heterogeneously with higher-performing students using assessment data, interests, and work habits, to encourage student-to-student engagement and learning. They are also provided guided notes and graphic organizers to build schemas. English Language Learners are paired with students who are fluent in English, and given extra time. Vocabulary is pronounced several times and accompanied by alternative words or phrases that are simplified. Visual aids, including pictures, Smart boards, and manipulatives, help students make clear connections to the text. Students with Special Needs are grouped with helpers and given instructions or assessments with simplified language or extra time. Color-coding on the Smart board help illustrate steps required to solve a problem. Hands-on activities are provided to help construct student learning. Gifted/Honor students are given challenge problems during lessons, homework, and summative assessments, which earn extra credits. They are expected to complete the whole worksheet, and are given challenge (e.g. open-ended) problems that develop higher-level thinking.

16 Corrected Geometry Ch 6 Homework Sheet Class website (lesson handout and homework answers): To receive full credits, you must copy/summarize questions,and copy diagrams using rulers. You will want a good ruler and compass (Safe-T compass $2 at Staples) for the Regents. 5 full tutoring sessions = 10% on a test, or 3 HW passes Saturday Academy on Jan. 11 and 25, 08:30 to 11:30 (1 tutoring credit per half an hour! That s 10% test grade in one day.) Lesson 50 Aim: How do we solve problems involving ratios, proportions and the geometric mean? (Section 6.1) HW #50: P : 18 21, 23, 25, 30, 31, 33, 35, 53, 56*(Challenging) Lesson 51 Aim: What are the properties of similar polygons (Section 6.3) HW #51: P : 1, 3, 5, 6, 8 10, Lesson 52 Aim: How do we prove triangles similar by AA Similarity Postulate? (Section 6.4) HW #52: P : 1, 3 12, 15, 16, 20. Lesson 53 Aim: How do we prove triangles similar by AA Similarity Postulate? (Section 6.4) HW #53: Finish worksheet Lesson 54 Aim: How do we prove triangles similar by SSS Similarity Theorem? (Section 6.5) HW #54: P : 3 6, 10, 15, 26*(Challenging) Lesson 55 Aim: How do we prove triangles similar by SAS Similarity Theorem? (Section 6.5) HW #55: P391: 7 9, 11 12, 14, 31. Lesson 56 Aim: What is the Triangle proportionality Theorem? (Section 6.6) HW #56: Finish worksheet Lesson 57 Aim: How do we compare the perimeters and areas of similar figures? (Section 11.3) HW #57: P : 3 5, 9, 11, 13, 14, 22. Regents Review Mid-term Final Ch 6 Review/Uniform Unit Test At the end of this unit, you should be able to

17 1) know how to set up and solve an equation involving ratios. 2) know how to calculate the geometric mean. 3) know the properties of similar polygons: corresponding angles are congruent but corresponding side lengths are proportional. 4) know how to set up a proportion to solve the length of a corresponding side or a corresponding angle in similar polygons 5) know that two triangles are similar if two pairs of corresponding angles are congruent: AA AA or AA Similarity Theorem; prove two triangles are similar by AA Similarity Theorem. 6) find the missing angle measure to show that two triangles are similar by AA Similarity Theorem. 7) know that two triangles are similar if three pairs of corresponding sides have the same ratio: SSS Similarity Theorem; prove two triangles are similar by SSS Similarity Theorem. 8) find the missing length to show that two triangles are similar by SSS Similarity Theorem. 9) know that two triangles are similar if the lengths of two corresponding sides have the same ratio and the included angles are congruent: SAS Similarity Theorem. 10) find the missing length to show that two triangles are similar by SSS Similarity Theorem. 11) know the relationship between the ratio of the perimeters and the ratio of corresponding sides of similar polygons, and solve problems involving these ratios. 12) know the relationship between the ratio of the areas and the ratio of corresponding sides of similar polygons, and solve problems involving these ratios. 13) know how to prove and apply the Triangle Proportionality Theorem and its converse. Common Core Standards Addressed G-CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

18 G-SRT.2 G-SRT.3 G-SRT.4 G-SRT.5 G-C.1 G-MG.3 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. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 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. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove that all circles are similar. 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). Methods of Differentiated Instructions Revolution K-12, a student-driven online learning platform, is used to differentiate instruction for all students in class and subject areas. All students are held to the Common Core-aligned standards. Teacher performs formative assessments through lessons to check student understanding, including cold-calling, student board work, and exit tickets. Teacher circulates room to perform further formative assessments and to guide small groups and individuals. Lower-performing students are grouped heterogeneously with higher-performing students using assessment data, interests, and work habits, to encourage student-to-student engagement and learning. They are also provided guided notes and graphic organizers to build schemas. English Language Learners are paired with students who are fluent in English, and given extra time. Vocabulary is pronounced several times and accompanied by alternative words or phrases that are simplified. Visual aids, including pictures, Smart boards, and manipulatives, help students make clear connections to the text. Students with Special Needs are grouped with helpers and given instructions or assessments with simplified language or extra time. Color-coding on the Smart board help illustrate steps required to solve a problem. Hands-on activities are provided to help construct student learning. Gifted/Honor students are given challenge problems during lessons, homework, and summative assessments, which earn extra credits. They are expected to complete the whole worksheet, and are given challenge (e.g. open-ended) problems that develop higher-level thinking.

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