Multiple Pathways To Success Quarter 1 Learning Module Geometry Unit 1: Congruence, Constructions, and Proofs

Transcription

1 Multiple Pathways To Success Quarter 1 Learning Module Geometry Unit 1: Congruence, Constructions, and Proofs Copyright July 31, 2014 Drafted October 29, 2015 Prince George s County Public Schools Board of Education of Prince George s County, Maryland

2 Dear Scholars, As you move through the Geometry curriculum, the level of academic rigor will increase. This could potentially lead to gaps in your understanding. Therefore, this learning module has been designed to assist you in acquiring and strengthening the essential skills needed for successful completion of Geometry Common Core. Your experiences with this module will also help to remediate misconceptions, confusion, and rebuild areas of weakness. Sincerely, Writers of the Multiple Pathways to Success Modules

3 Part I: Understanding Congruence in Terms of Rigid Motion Student Learning Objectives Use descriptions of rigid motion and transformed geometric figures to predict the effects rigid motion has on figures in the coordinate plane. Determine if two given figures are congruent using the concept of rigid transformations. Determine if triangles are congruent by determining if the corresponding parts are congruent Explain how the criteria for triangle congruence (ASA, SAS, SSS) follow from the definition of congruence in terms of rigid motions. Use triangle congruence postulates and theorems to prove triangles congruent. Mathematical Practices MP1: Make sense of a problem and persevere in solving them MP2: Reason abstractly and quantitatively MP3: Construct a viable argument and critique the reasoning of others MP4: Model with mathematics MP6: Attend to precision MP7: Look for and make use of structure Resources/Websites The following Khan Academy Playlist contains videos that explain all the concepts learned in quarter 1 that deal with congruence and proofs as well as providing helpful exercises to work through. The following link to Mathbitsnotebook contains an excellent page on Rigid motions and how they are used to show congruence of segments, angles, and shapes: This link to the NY regents prep center contains notes that will help you with the triangle congruence shortcuts.

4 1) a) Reflect triangle ABC over the line y=x and label the image A B C. b) Rotate triangle A B C counter clockwise around the origin and label the image A B C.

5 2) Tasha and Jose are trying to decide how much information they need to know about two triangles before they can convince themselves that the two triangles are congruent. They are wondering if knowing that two angles and the included side of one triangle are congruent to the corresponding two and the included side of another triangle (a set of criteria their teacher refers to as ASA) is enough to know that the two triangles are congruent. They are trying to justify that this would be so. To start reasoning about the congruence of the two triangles, Tasha and Jose have created the following diagram in which they have marked an ASA relationship between the triangles. a. Based on the diagram, which angles have Tasha and Jose indicated are congruent? Which sides? b. To convince themselves that the two triangles are congruent, what else would Tasha and Jose need to know.

6 Jose s Argument I know what to do, said Jose. We can translate point A until it coincides with point R, then rotate till it coincides with. Finally, we can reflect across and then everything coincides so the triangles are congruent. [Jose and Tasha s teacher has suggested they use the word coincides when they want to say that two points or line segments occupy the same position on the plane. They like the word, so they plan to use it a lot.] c. What do you think about Jose s argument? Does it convince you that the two triangles are congruent? Does it leave out any essential ideas that you think need to be included? Write a paragraph explaining your reaction to Jose s argument d. Tasha isn t sure that Jose s argument is really convincing. She asks Jose, How do you know point C coincides with point T after you reflect the triangle? How do you think Jose might answer Tasha s question? 3) Explain whether or not the triangles are congruent, similar, or neither based on the markings that indicate congruence.

7 a. b. c.

8 Part II: Prove Geometric Theorems Student Learning Outcomes: Prove vertical angles are congruent. Prove when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angels are congruent. Likewise, if these angle pairs are congruent, then the lines are parallel (converse). Prove that when a transversal crosses parallel lines same-side interior angles are supplementary and conversely that if that angle pair is supplementary, then lines are parallel. Given the definition of parallelogram, student will use triangle congruence to prove that opposites sides are congruent, opposite angles are congruent and diagonals bisect each other. Given a quadrilateral with opposite sides congruent, students will prove that it is a parallelogram; likewise for opposite angles congruent and bisected diagonals. Mathematical Practices: MP1: Make Sense of a Problem and Persevere MP2: Reason Abstractly and Quantitatively MP3: Construct a viable argument and critique the reasoning of others MP4: Model with Mathematics MP6: Attend to Precision MP7: Make Use of Structure Resources/Websites: The following Khan Academy Playlist contains videos that explain all the concepts learned in quarter 1 that deal with congruence and proofs as well as providing helpful exercises to work through. This page from the regents prep center details how to construct a proof This page from the regents prep center details what a proof is This article argues why you should care about proofs and how they are applicable to your life

9 1) Fill in all the missing blanks in the proof of the theorem below. Theorem: Vertical Angles are Congruent (We show that.) Statements Reasons A pair of angles that make up one side of a line are supplementary. Subtract from both sides

10 2) Fill in all the missing blanks in the proof of the theorem below. Theorem: Alternate interior angles are congruent. Statements Reasons

11 3) Create a proof using the picture below for the following theorem. Theorem: Opposite sides and opposite angles in a parallelogram are congruent. A B D C Statements Reasons

12 Maryland College and Career Readiness Standards Understand Congruence in Terms of Rigid Motions [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. (major) [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. (major) [G.CO.8] Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (major) Prove Geometric Theorems [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. (major) [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. (major) [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. (major)

13 Scoring Rubric / Success Criteria Conceptual Understanding Part I: Understanding Congruence in terms of rigid motions. (1a, 1b, 2a, 2b, 2c, 2d, 3a, 3b, 3c) Part II: Prove Geometric Theorems (1a, 1b, 1c, 1d, 2a, 2b, 2c, 3 one point per line correct) 22 Total Points 9 One point for each part of each problem 13 One point for each part of each problem Section Total /22 Execution of Mathematical Practices MP1: Make sense of a problem and persevere in solving them Analyze and explain the meaning of the problem Actively engage in problem solving (Develop, carry out, and refine a plan) MP2: Reason abstractly and quantitatively Represent a problem with symbols Explain their thinking Examine the reasonableness of their answers/calculations MP3: Construct a viable argument and critique the reasoning of others Justify solutions and approaches 12 Total Points 2 one point per bullet 3 one point per bullet 1 MP4: Model with mathematics Use representations to solve real life problems Apply formulas and equations where 2 one point per bullet

14 appropriate MP6: Attend to precision Calculate accurately and efficiently Explain their thinking using mathematics vocabulary Use appropriate symbols and specify units of measure 3 one point per bullet MP7: Look for and make use of structure Use knowledge of properties to efficiently solve problems 1 Section Total /12 Final Score /34