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Geometry Opener(s) 2/5 2/5 It’s National Weatherperson’s Day, Wear Red Day, World Nutella Day, Primrose Day, Disaster Day, Bubble Gum Day and National Chocolate Fondue Day!!! Happy Birthday Bobby Brown, Laura Linney, Jennifer Jason Leigh, Barbara Hershey, Errol Morris, Hank Aaron, Red Buttons, William S. Burroughs and Adlai Stevenson!!! 2/5 What to do today: 1. Do the opener. 2. Watch a video. 3. Practice rotating with spinning/protractor. 4. Watch a video. 5. Practice rotating with 3 steps. 6. Take some double entry journal notes. 7. Take some PPT notes. 8. Practice rotational symmetry 9. Start HW. 10. Do the exit pass. TODAY’S OPENER 1. Agenda 1. Opener (15) 2. REGULAR: Model 1: Rotation with No Tracing Paper (5) https://www.youtube.com/watch?v=Foe8VFKAB8Y 3. Practice: With Spinning (15) 4. HONORS: Model 2: Rotation with Protractor & Point (5) https://www.youtube.com/watch?v=U4Hv494HwrQ 5. Practice: With Protractor (15) 6. Model 3: Rotation w/out Protractor & Point (5) https://www.youtube.com/watch?v=FqiGuTtjmMg 7. Practice: Rotation in 3 Steps (15) 8. Notes 1: Rotational Symmetry – Text, p. 478 Double Entry Journal (10) 9. Notes 2: Rotational Symmetry – PPT (5) 10. Practice: Wksht. 9-3, p. 492 (10) 11. HW Time: Selected problems 12. Exit Pass (15) Standard(s) CCSS-M-G-CO.2: Represent transformations in the plane using tracing paper, functions, etc. CCSS-M-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. CCSS-M-G-CO.6: Use descriptions of rigid motions to transform figures. Essential Question(s) How Do I (HDI) transform images? HDI transform images within the coordinate plane? Objective(s) Students will be able to (SWBAT) correlate anti-blobbiness with geometry. SWBAT transform images using measurement or other methods. SWBAT transform images within the coordinate plane. Decide on one person from your group to get a piece of graph paper for each member and your group’s tool box. 2. Create a BIG coordinate plane. 3. Plot the following triangle: B(0,0) I(10,0) G(10,8) 4. Label it with 3 non-collinear points on the vertices. 5. Draw a sketch on your opener paper and label it. 6. Now, rotate the triangle 90° and 180° counterclockwise around the origin. Use your formulas. 7. Draw a sketch on your opener paper and label it. 8. Draw, measure (in cm) and label the following segments with the measurements: ̅̅̅̅ 𝑰𝑶, ̅̅̅̅ 𝑰′𝑶 , ̅̅̅̅ 𝑮𝑶, ̅̅̅̅̅ 𝑮′𝑶 [ O = (0,0) … the origin ] 9. Get a protractor and measure the following angles and label them with degrees: IOI’ and GOG’ 10. Can you make a conjecture about spins / rotations? Write your conjecture on your opener paper. THE LAST OPENER 11. One person from your group get a piece of graph paper for each member and your group’s tool box. 12. Create a BIG coordinate plane. 13. Take out a protractor. 14. Put the straight edge of the inner semicircle along the positive xaxis. 15. Place the left ‘vertex’ of the semicircle on the origin. 16. Trace it and label it with 3 non-collinear points: 2 on the vertices and 1 at the top of the semicircle. 17. Draw a sketch on your opener paper and label it. 18. Now, spin your protractor 90° clockwise around the origin. 19. Trace it and label it. 20. Draw a sketch on your opener paper and label it. 21. Do the same for TWO (2) more 90° spins. 22. Can you make a conjecture about spins / rotations? Write your conjecture on your opener. (HINT: Use a ruler!) Exit Pass (12/11 – 13/14) For numbers 1-6 find the coordinates of each image. 1. Rx-axis (A) 2. Ry-axis (B) 3. Ry = x (C) 4. Rx = 2 (D) 5. Ry= -1 (E) 6. Rx = -3 (F) The Last Exit Pass HOMEWORK Period 1 Text ?s, p. 480, #14-15, #22-24 HOMEWORK Period 7 Kuta Rotations Wksht. [Redo p. 1 w/non-origin spin points: #1. (2,-2) #2. (-2,-2) #3. (-2,2) #4. (2,2) #5. (1,1) #6. (-1,1) HOMEWORK Period 3 Text ?s, p. 480, #22-24 HOMEWORK Period 5 and 8 Kuta Rotations Wksht. HOMEWORK Period 2A Text ?s, p. 480, #22-24 Extra Credit Period 1 Period 2A Period 3 Israel H. Jose C. Mirian S. (2x) 1. Jaime A. (2x) Nadia L. Anthony P. Griselda Z. Jaclyn C. Brandon S. Sonia T. Amanda S. Josue A. Period 5 Period 7 Antonio B. (2x) Rogelio G. (2x) Eraldy B. Anthony G. Alex A. Brianna T. Jose B. Carlos L. Adriana H. Jackie B. Jose R. (2x) Julian E. Jocelyn C. (2x) Jenny Q. Ruby L. Ana R. Watch a portion of RHYTHmetric video (“What are the 3 ‘laymen’s’ terms for 3 of our 4 transformations?”). http://www.youtube.com/watch?v=NKtJd1hkI9k http://www.youtube.com/watch?v=V9uYcnjlAks http://www.youtube.com/watch?v=X1xPZjItmDk%20 http://www.shodor.org/interactivate/activities/Transmographer/ Period 8 Jessica T. (2x) Jorge C. Gerardo L. Val R. Gerardo L. Esme V. Tessellation Rubric Name Period What I’m looking for What it’s worth Tessellation Image Construction Recognizable: 6 pts. Almost recognizable: 5 pts. Almost recognizable but incomplete: 4 Unrecognizable: 2 pts. Clearly identifiable figure due to markings: 6 pts. Identifiable but incomplete…or minimally detailed: 5 pts. Semi-identifiable figure: 4 pts. No figure: 3 pts. Multiple colors fill image: 6 pts. More than 1 color but primarily 1: 5 pts. Single color fills image/Alternating color pattern: 4 pts. Single color fills image: 3 pts. Bits of color spottily used: 2 pts. No color used: 1 pt. Images ‘lock’: 6 pts. Images partially ‘lock’: 2 pts. Images do not ‘lock’: 1 pts. Incredible: 6 pts. Intriguing…almost fascinating…adorable: 5 pts. Interesting…cute: 4 pts. Typical…unoriginal: 3 pts. Minimalist: 2 pts. 14 boxes by 20 boxes: 6 pts. < 14 boxes by 20 boxes: 2 pts. Tessellation Image Figure Tessellation Image Coloring Tessellation Image ‘Locking’ Tessellation Image Creativity Tessellation Size Total Tessellation Rubric 36 pts. Name Period What I’m looking for What it’s worth Tessellation Image Construction Recognizable: 6 pts. Almost recognizable: 5 pts. Almost recognizable but incomplete: 4 Unrecognizable: 2 pts. Clearly identifiable figure due to markings: 6 pts. Identifiable but incomplete…or minimally detailed: 5 pts. Semi-identifiable figure: 4 pts. No figure: 3 pts. Multiple colors fill image: 6 pts. More than 1 color but primarily 1: 5 pts. Single color fills image/Alternating color pattern: 4 pts. Single color fills image: 3 pts. Bits of color spottily used: 2 pts. No color used: 1 pt. Images ‘lock’: 6 pts. Images partially ‘lock’: 2 pts. Images do not ‘lock’: 1 pts. Incredible: 6 pts. Intriguing…almost fascinating…adorable: 5 pts. Interesting…cute: 4 pts. Typical…unoriginal: 3 pts. Minimalist: 2 pts. 14 boxes by 20 boxes: 6 pts. < 14 boxes by 20 boxes: 2 pts. Tessellation Image Figure Tessellation Image Coloring Tessellation Image ‘Locking’ Tessellation Image Creativity Tessellation Size Total What you get 36 pts. What you get Today’s Worksheets Rubrics Line of Symmetry Wksht. #2. 2 pts. 50 % at least 3 pts. #3. 2 pts. 64% at least 6 pts. #4. 2 pts. (Correctly identifying 5 letters with 76% at least 9 pts. lines of symmetry; correctly identifying 5 88% at least 12 pts. letters with NO lines of symmetry) 93% at least 15 #5. 6 pts. (1 for each image) 100% 18 #6. 4 pts. (1 for each image WITH the specified # of lines of symmetry) #7. 2 pts. (Yes with an example to support ‘yes’ or no with counterexample to support ‘no’) Reflections Wksht. #1. 8 pts. (1 for each ? or gap fill) #2. 7 pts. (1 for each instruction or request to write a statement) #3. 4 pts. (1 for each image) #4. 5 pts. (1 for each instruction; 2 for the entire path) 50 % at least 4 pts. 64% at least 8 pts. 76% at least 12 pts. 88% at least 16 pts. 93% at least 20 100% 24 Reflections on a Coordinate Plane Wksht. #2. 8 pts. (1 for each table entry and ?) 50 % at least 6 pts. #3. 7 pts. (1 for each table entry and ?) 64% at least 12 pts. #4. 9 pts. (1 for each table entry, ? and 76% at least 18 pts. labeling the image) 88% at least 24 pts. #5. 12 pts. (1 for each set of pre-image & 93% at least 30 image coordinates; 2 for each set of 100% 36 formulaic coordinates [in the boxes!]) Copying & Bisecting a Segment Copying a Segment 1. 2. 3. 4. 5. Draw a segment. Label its endpoints I and F. Draw another non-collinear point. Label it E. Draw an arc in your segment from 1 endpt. to the other. (Don’t change your compass’s width!) 6. Draw the same arc starting from point E. 7. Label a point on the second arc N. 8. Connect E to N. Bisecting a Segment 1. Place your compass point on E. 2. Place your pencil point at least ̅̅̅̅ and draw a halfway across 𝑬𝑵 half-circle arc. (Don’t change your compass’s width!) 3. Place your compass point on N. 4. Draw the exact same halfcircle arc. (Don’t change your compass’s width!) 5. Label the arc intersection points T and H. 6. Connect T to H. Copying & Bisecting an Angle Copying an Angle 1. Draw a ray. 2. Label its endpoint U. 3. Draw another non-collinear ray that shares the 1st ray’s endpt. 4. Draw another separate ray… far away from the angle. 5. Label its endpt. L 6. Draw an arc in your from 1 side to the other. (Don’t change your compass’s width!) 7. Label the intersection points B and G. 8. Draw the same arc on your 3rd ray. 9. Label its intersection point Y. 10.Draw a 2nd arc in your with compass pt. on G and pencil point on B. (Don’t change your compass’s width!) 11.Draw the same arc on your 3rd ray. 12.Label the intersection point F. 13.Connect L to F. Bisecting an Angle 1. Draw an angle. 2. Label its vertex A. 3. Draw an arc in your angle from one side to the other. 4. Label the intersection pts. W and P. 5. Draw an arc with your compass pt. on W, at least ½ -way across the interior. (Don’t change your compass’s width!) 6. Draw the same arc from your P. 7. Label the intersection point of those 2 arcs S. 8. Connect A to S. YOUR PROOF CHEAT SHEET IF YOU NEED TO WRITE A PROOF ABOUT ALGEBRAIC EQUATIONS…LOOK AT THESE: Reflexive Property Symmetric Property Transitive Property Addition & Subtraction Properties Multiplication & Division Properties Substitution Property Distributive Property IF YOU NEED TO WRITE A PROOF ABOUT LINES, SEGMENTS, RAYS…LOOK AT THESE: For every number a, a = a. Postulate 2.1 For all numbers a & b, if a = b, then b = a. For all numbers a, b & c, if a = b and b = c, then a = c. For all numbers a, b & c, if a = b, then a + c = b + c & a – c = b – c. For all numbers a, b & c, if a = b, then a * c = b * c & a ÷ c = b ÷ c. For all numbers a & b, if a = b, then a may be replaced by b in any equation or expression. For all numbers a, b & c, a(b + c) = ab + ac Postulatd 2.2 Postulate 2.3 Postulate 2.4 Postulate 2.5 Postulate 2.6 Postulate 2.7 The Midpoint Theorem IF YOU NEED TO WRITE A PROOF ABOUT THE LENGTH OF LINES, SEGMENTS, RAYS…LOOK AT THESE: Reflexive Property Symmetric Property Transitive Property Addition & Subtraction Properties Multiplication & Division Properties Substitution Property Segment Addition Postulate Through any two points, there is exactly ONE LINE. Through any three points not on the same line, there is exactly ONE PLANE. A line contains at least TWO POINTS. A plane contains at least THREE POINTS not on the same line. If two points lie in a plane, then the entire line containing those points LIE IN THE PLANE. If two lines intersect, then their intersection is exactly ONE POINT. It two planes intersect, then their intersection is a LINE. If M is the midpoint of segment PQ, then segment PM is congruent to segment MQ. IF YOU NEED TO WRITE A PROOF ABOUT THE MEASURE OF ANGLES…LOOK AT THESE: AB = AB (Congruence?) If AB = CD, then CD = AB If AB = CD and CD = EF, then AB = EF If AB = CD, then AB EF = CD EF If AB = CD, then AB */ EF = CD */ EF If AB = CD, then AB may be replaced by CD If B is between A and C, then AB + BC = AC If AB + BC = AC, then B is between A and C Reflexive Property Symmetric Property Transitive Property Addition & Subtraction Properties Multiplication & Division Properties Substitution Property m1 = m1 (Congruence?) If m1 = m2, then m2 = m1 If m1 = m2 and m2 = m3, then m1 = m3 If m1 = m2, then m1 m3 = m2 m3 DEFINITION OF CONGRUENCE Whenever you change from to = or from = to . If m1 = m2, then m1 */ m3 = m2 */ m3 If m1 = m2, then m1 may be replaced by m2 IF YOU NEED TO WRITE A PROOF ABOUT ANGLES IN GENERAL…LOOK AT THESE: Postulate 2.11 The Addition Postulate Theorem 2.5 The Equalities Theorem If R is in the interior of PQS, then mPQR + mRQS = mPQS. THE CONVERSE IS ALSO TRUE!!!!!! Q Congruence of s is Reflexive, Symmetric & Transitive P R S Theorem 2.8 Vertical s Theorem If 2 s are vertical, then they are . (1 3 and 2 4) IF YOU NEED TO WRITE A PROOF ABOUT COMPLEMENTARY or SUPPLEMENTARY ANGLES …LOOK AT THESE: Theorem 2.3 Supplement Theorem If 2 s form a linear pair, then they are supplementary s. Theorem 2.4 Complement Theorem If the non-common sides of 2 adjacent s form a right , then they are complementary s. Theorem 2.12 Supplementary Right s Therorem Theorem 2.6 R The Supplements Theorem S P Q Q P If 2 s are and supplementary, then each is a right . Theorem 2.7 The Complements R Theorem S Theorem 2.13 Linear Pair Right s Therorem s supplementary to the same or to s are . (If m1 + m2 = 180 and m2 + m3 = 180, then 1 3.) s complementary to the same or to s are . (If m1 + m2 = 90 and m2 + m3 = 90, then 1 3.) If 2 s form a linear pair, then they are right s. YOUR PROOF CHEAT SHEET (continued) IF YOU NEED TO WRITE A PROOF ABOUT RIGHT ANGLES or PERPENDICULAR LINES…LOOK AT THESE: Theorem 2.9 Perpendicular lines Theorem 3-4 If a line is to the 1st of two || lines, Perpendicular Transversal Theorem 4 Right s Theorem intersect to form 4 right s. then it is also to the 2nd line. Theorem 2.10 Postulate 3.2 All right s are . 2 non-vertical lines are if and only if the PRODUCT of their Right Congruence Theorem Slope of Lines slopes is -1. (In other words, the 2nd line’s slope is the 1st line’s slope flipped (reciprocal) with changed sign.) Theorem 2.11 Perpendicular lines Postulate 3.2 If 2 lines are to the same 3rd line, then thhose 2 Adjacent Right s Theorem form adjacent s. and || Lines Postulate lines are || to each other. Theorem 4-6 Theorem 4-7 If the 2 legs of one right are to If the hypotenuse and acute of one right Leg-Leg (LL) Congruence Hypotenuse-Angle the corresponding parts of another are to the corresponding parts of (HA) Congruence right , then both s are . another right , then both s are . Theorem 4-8 Postulate 4-4 If the hypotenuse and one leg of one right If the leg and acute of one right are Leg-Angle (LA) to the corresponding parts of another Hypotenuse-Leg (HL) are to the corresponding parts of Congruence Congruence another right , then both s are . right , then both s are . IF YOU NEED TO WRITE A PROOF ABOUT ICCE ANGLES or PARALLEL LINES…LOOK AT THESE: Postulate 3.1 If 2 || lines are cut by a Postulate 3.4 If 2 lines are cut by a transversal Corresponding Angles transversal, then each pair of CO Corresponding Angles/|| Lines so that each pair of CO s is , Postulate (CO s Post.) s is . Postulate (CO s/|| Lines Post.) then the lines are ||. Theorem 3.1 If 2 || lines are cut by a Theorem 3.5 If 2 lines are cut by a transversal so Alternate Interior Angles transversal, then each pair Alternate Exterior Angles/|| Lines that each pair of AE s is , then the Theorem (AI s Thm.) of AI s is . Theorem (AE s/|| Lines Thm.) lines are ||. Theorem 3.2 If 2 || lines are cut by a Theorem 3.6 If 2 lines are cut by a transversal Consecutive Interior Angles transversal, then each pair Consecutive Interior Angles/|| Lines so that each pair of CI s is Theorem (CI s Thm.) of CI s is supplementary. Theorem (CI s/|| Lines Thm.) supplementary, the lines are ||. Theorem 3.3 If 2 || lines are cut by a Theorem 3.7 If 2 lines are cut by a transversal so Alternate Exterior Angles transversal, then each pair Alternate Interior Angles/|| Lines that each pair of AI s is , then the Theorem (AE s Thm.) of AE s is . Theorem (AI s/|| Lines Thm.) lines are ||. Postulate 3.2 2 non-vertical lines have the same Postulate 3.5 If you have 1 line and 1 point NOT on that Slope of || Lines slope if and only if they are ||. || Postulate line, ONE and only ONE line goes through that point that’s || to the 1st line. Theorem 6.6 Theorem 6.4 A midsegment of a is || to one In ACE with ̅̅̅̅̅ 𝑩𝑫 || ̅̅̅̅ 𝑨𝑬 and Midsegment Thm. Proportionality Thm. intersecting the other 2 sides in distinct side of the , and its length is ½ ̅̅̅̅ 𝑩𝑨 ̅̅̅̅ 𝑫𝑬 the length of that side. points, = . ̅̅̅̅ 𝑪𝑩 ̅̅̅̅ 𝑪𝑫 Postulates and Theorems to IDENTIFY CONGRUENT TRIANGLES: SSS, ASA, SAS or AAS Postulates and Theorems to IDENTIFY SIMILAR TRIANGLES: AA, SSS or SAS Linear Equation in Slope-Intercept Form Linear Equation in Point-Slope Form y = mx + b m = slope, b = yintercept y – y1 = m(x – x1) m = slope, (x1, y1) = 1 point on the line Linear Equation in Standard Form Ax + By = C I – Numbers and coefficients can only be Integers. (No fractions or decimals.) P – The x coefficient must be Positive. (A > 0) O – Zero can only appear beside a variable Once. (If A = 0, then B ≠ 0) D – Numbers and coefficients can only be Divisible by 1. (GCF = 1) S – Variables can only be on the Same side of the equal sign. CI s: 2 inside || lines on SAME side of transversal. CO s: 1 inside || lines & 1 outside || lines, on OPPOSITE sides of transversal. AI s: 2 inside || lines on OPPOSITE sides of transversal. AE s: 2 outside || lines on OPPOSITE sides of transversal. AE CO AI CO CI AE AI/ CI