Multiple Pathways To Success Quarter 2 Learning Module Aligned with Maryland State Standards

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1 Multiple Pathways To Success Quarter 2 Learning Module Aligned with Maryland State Standards Geometry Units 2&3: Connecting Algebra and Geometry through Coordinates & Similarity and Proofs Copyright July 31, Written December 15, 2015 Prince George's County Public Schools Board of Education of Prince George's County, Maryland PGCPS 9,1e4r ekierx

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: Using Coordinate Geometry as a Tool Student Learning Objectives Find the midpoint of a line segment Derive the midpoint formula and a general formula for finding a point that partitions a directed line segment. Use coordinate geometry and the distance formula to find the perimeters of polygons and the areas of triangles and rectangles. 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 website provides the midpoint formula and examples of finding the midpoint and the point on a directed line segment that partitions it into a given ratio. /020School/Courses/CC VM/020CCGPS%20Coordinate/020Algebra/Curriculum/Unit%206/U6 PartitionLineSe gmentareaperimeter/u6 PartitionLineSegmentAreaPerimeterhtml The following website provides a proof of the midpoint formula using the distance formula that will be used in question 2. The following Khan Academy playlist provides videos, examples, and exercises that deal with all of analytical geometry. The following website shows the box method for finding the area of a triangle. Use on problem 5. Formulas needed in this section for two points TI i v 1) Ti, and a given ratio V2 then 'Midpoint 4-x2 1/14-1/2 2 2 Distance = x2 X1) +(y2-02 Point that divides a segment into a given ratio: = (x i + k(x 2 xi), y1+k(y2 y1))

4 Find the midpoint of segment AB e A B 1

5 2) Find the point that divides the segment from (31-7) to (-5,5) into a ratio of 3:1.

6 Use the following picture to answer questions 3 and 4. 3) Find the perimeter of AABC units. 4) Find the area of AABC in square units.

7 Part II: Coordinate Proofs Student Learning Outcomes: Given a collection of lines, or points on the coordinate plane demonstrate or verify relationships between these objects. Prove that non-vertical parallel lines have equal slopes. Prove that perpendicular lines have opposite reciprocal slopes. Write equations of parallel and perpendicular lines. Prove that lines in the coordinate plane are parallel, perpendicular or neither. 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 ResourcesNVebsites: The following youtube video shows you how to verify if a point is on a line or not and this website shows how to find where two lines intersect: The following website provides 2 different proofs that non-vertical parallel lines have the same slope The following website provides a proof of the slope criteria for perpendicular lines. httios:// This Khan Academy Playlist provides examples of how to show when two lines are parallel, perpendicular, or neither and how to write equations of parallel and perpendicular lines. Explanation: (Proof by contradiction) In order to do a proof by contradiction add in one given which is the negation (put a not in front of the statement) of the conclusion and then prove that you get a pair of statements which are opposites like ra = n and. Then your original conclusion has to be true!

8 Use the following figure to answer all parts of question 1. 1a) Create a statement and reason chart that shows that lines a, and b intersect at point B. Statements Reasons

9 1b) Create a statement and reason chart that shows that point A is on line a. Statement Reason

10 2) Use the equations and a proof by contradiction to fill in the proof that two lines that are parallel have the same slope. (A paragraph proof of this type can be found in the resource section, use it to fill in the blanks in the proof below) =rni x+bi 12:y = in2s+b2 Statements Reasons 11 and /2 are parallel Given b b2 li and 12 are distinct lines Negation of the conclusion m2-740 Subtraction property of equality rri9x±b2= :mix + bi subtraction property of equality (nt2 7721)X = bi b2 Distributive property of equality 1)1 b2 Substitution 11 and 12 intersect at ( 61 b2 ( bi ) I mr2 mi m9 m i Contradicts the fact that 11 and /2 are parallel.

11 3) Write the equation for the lines described below. a) Parallel to Y = 3s +8 passing through the point (3:4). y 3 b) Perpendicular to 4x+2 passing through the point (-7/1). 4) Line 1 passes through the points (-5,11) and (4,-1) on the coordinate plane. Line m passes through the points (-5,6) and (4,w). For what value of w is the line m parallel to line 1?

12 5) Given the figure, write an expression that can replace t and will guarantee that lines j and k are parallel.

13 Part Ill: Similarity Transformations Student Learning Outcomes: Given a center and a scale factor, verify experimentally, that when dilating a figure in the coordinate plane, a segment of the pre-image that does not pass through the center of the dilation, is parallel to its image when the dilation is performed. However, a segment that passes through the center remains unchanged. Given a center and a scale factor, verify experimentally, that when performing dilations of a line segment, the pre-image, the segment which becomes the image is longer or shorter based on the ratio given by the scale factor. Use the idea of dilation transformations to develop the definition of similarity. Given two figures determine whether they are similar and explain. Students will demonstrate why AA is a similarity postulate. 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! Webs ites : The following Khan Academy playlist contains video tutorials and exercises with hints and explanations for problems involving similarity. The following Learnzillion video shows how to verify the properties of dilations given a center and a scale factor. httos://learnzil lion. com/resou rces/72563-verify-experimental ly-the-properties-of-d Natio ns-given-by-a-center-and-a-scale-factor The following Learnzillion playlist shows how to decide if two figures are similar or not. rces/72642-decide-if-two-fig u res-are-similar-and-exolain-t he-meaning-of-triangle-similarity The following EngageNY lesson contains the proof of the AA criterion on page org/file/47096/download/math-g8-m3-topic-b-lesson-10-tea cher.pdf?token=y5fjojdyaxgdjp16jbt7l7buevvvbxcda SzgntxHmGs

14 Use the following picture to complete the answers to problem 1. 1) a) Dilate AABC about the point (3,4) with scaling factor k=2 and graph the result below and label the triangle AAVai I

15 b) Calculate the equations of line AB and line A'B', and then classify them as parallel, perpendicular, same line, or neither. c) Calculate the equations of line ACand line A u, and then classify them as parallel, perpendicular, same line, or neither.

16 d) Find the length of segment BC and Bi. iength(etv) e) Calculate 41/04-T3C) and compare it to k. 2) Write a precise definition of similarity using the definition of congruence for an example. Definition (Congruence): Two figures A and B are congruent if there is a sequence of rigid motions (rotations, translations, and reflections) that takes figure A to figure B. Definition (Similar):

17 3 Are the two triangles similar or not? Explain? 19 io i i 12 13

18 4) Fill in the following statement and reason chart that proves that the following theorem. Theorem: If two triangles are related by AA (ie two of their angles are the same) then the triangles are similar. Statements Reasons LIABC and ADEF are related by AA -Ma--.L 4D and B --E. Since triangles are related by AA Define. k km0(de) -= ' length(ab) length(ab)#0 Dk(AB DE Ruler Postulate A(A), A', Dk(B)-. Bf, Dk(C)=C1/ AA' Ire' f.,-, AABC AA'Bie ADEF ASA Congruence AA' B`Cli ADEF Transitive property of similarity

19 Maryland College and Career Readiness Standards Using Coordinate Geometry as a Tool [G.GPE.6] Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (major) [G.GPE.7] Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* Coordinate Proofs [G.GPE.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. (major) [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 line that passes through a given point). (major) Similarity Transformations [G.SRT.1] Verify experimentally the properties of dilations given by a center and a scale factor. (major) [G.SRT.1 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. (major) [G.SRT.1 b] The dilation of a line segment is longer or shorter in the ratio given by the scale factor. (major) [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. (major) [G.SRT.3] Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (major)

20 Scoring Rubric / Success Criteria Conceptual Understanding Part I: Using Coordinate Geometry as a Tool (1, 2, 3, 4) Part II: Coordinate Proofs (1 a 3pts see rubric, lb 3 pts see rubric, 2 one point per blank, 3a, 3b, 4, 5) Part III: Similarity Transformations (la, 1 b, 1 c, Id, le, 2, 3 [2 points], 4 one point per blank) 32 Total Points 4 One point for each part of each problem 15 One point for each part of each problem unless otherwise specified 13 Section Total /32 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 12 Total Points 2 one point per bullet 3 one point per bullet 1

21 Justify solutions and approaches MP4: Model with mathematics Use representations to solve real life problems Apply formulas and equations where appropriate 2 one point per bullet 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 /44