GRADE 8 Common Core MODULE 2 Name _______________________ Period _____ GRADE 8 Common Core MODULE 2 THE CONCEPT OF CONGRUENCE Topic A Name ______________ Period ______________ Dr. Basta, Mrs. Marotta, Mrs. Sirianni

Name __________________________ Period __________ Objectives Lesson 1: Why Things Move Around? a. Students are introduced to vocabulary and notation related to rigid motions (e.g. transformation, image, and map). b. Students are introduced to transformations of the plane and learn that a rigid motion is a transformation that is distance preserving. c. Students use transparencies to imitate a rigid motion that moves or maps one figure to another figure in the plane.   Lesson 2: Definition of Translation and Three Basic Properties a. Students perform translations to figures along a specific vector. Students label the image of the figure using appropriate notation. b. Students learn that a translation maps lines to lines, rays to rays, segments to segments, and angles to angles. Lesson 3: Translating Lines a. Students learn that when lines are translated they are either parallel to the given line, or the lines coincide. b. Students learn that translations map parallel lines to parallel lines. Name __________________________ Period __________

Objectives (continued)   Lesson 4: Definition of Reflection and Basic Properties a. Students know the definition of reflection and preform reflections across a line using a transparency. b. Students show that reflections share some of the same fundamental properties with translations (e.g. lines map to lines, angles and distance preserve motion, etc.) Students know that reflections map parallel line to parallel lines c. Students know that for the reflection across the line L, then every point P, not on L, L is the bisector of the segment joining P to its reflected image P’. Lesson 5: Definition of Rotation and Basic Properties. a. Students know how to rotate a figure a given degree around a given center. b. Students know that rotations move lines to lines, rays to rays, segments to segments, and angles to angles. Students know that rotations preserve lengths of segments and degrees of measures angles. Students know that rotations move parallel lines to parallel lines. Lesson 6: Rotations of 180° a. Students learn that a rotation of 180° moves a point on the coordinate plane (a, b) to (-a, -b) b. Students learn that a rotation of 180° around a point, not on a line, produces a line parallel to the given line. Name __________________________ Period __________

Name __________________________ Period __________ MODULE 2 VOCABULARY 1   Transformation- A rule that shows how a point, segment, ray, line or angle is moved (transformed) in some way. Basic Rigid Motion- Transformations that preserve size and shape. They include: translations, reflections, and rotations. Sequence of Transformations- one or more transformations that occur one after another if there is more than one transformation Plane- a flat surface that extends endlessly in all directions Map- to move a point to a new location Pre-image- the original figure of a how a point, segment, ray, line or angle Image- the new figure (transformed figure) of a point, segment, ray, line or angle Prime- the symbol used to identify the transformed image ie,P is transformed to P’ Inverse Transformation- a move that undoes the original transformation; an image is transformed from the image back to the pre-image Vector- is a directed segment (ray) that shows the direction and magnitude (how far) of the translation; the endpoint of the vector is the starting point and the arrow of the vector points toward the transformed image and is called the endpoint Name __________________________ Period __________

Name __________________________ Period __________ MODULE 2 VOCABULARY 2   Translation (slide) - when a figure is moved along a given vector; Translation symbol: T Reflection (flip) - when a figure is moved across a line (mirror image); Reflection symbol: r Line of Reflection- the line the pre-image is mapped over Bisector- the line of reflection is a bisector because it is equidistance from both the pre-image and the image Rotation (turn) - when a figure is moved around a point a specific number of degrees; Rotation symbol: R Clockwise- the direction the hands on the clock move; when rotating a figure clockwise (CW) the number of degrees is negative Counter-clockwise- the opposite direction the hands on the clock move; when rotating a figure counter-clockwise (CCW) the number of degrees is positive Center of Rotation(fixed point)- the point from which a figure is turned Name __________________________ Period __________

INTRODUCTION to BASIC RIGID MOTIONS Lesson 1: Why move things around? INTRODUCTION to BASIC RIGID MOTIONS Why move things? How do we move things about in a plane without changing the object’s size or shape? Basic Rigid Motion A transformation that is preserved with respect to distance. That is, the distance between any two points in the pre-image must be the same as the distance between the post-images of the two points. Name __________________________ Period __________

INTRODUCTION to BASIC RIGID MOTIONS Lesson 1: Why move things around? INTRODUCTION to BASIC RIGID MOTIONS Isometry is when the distance between any two points in the original figure is the same as the distance between their corresponding images in the transformed figure (image). Reflections, rotations, translations are isometries. Dilation is not an isometry. Name __________________________ Period __________

Name __________________________ Period __________ Lesson 1: Why move things around? How do things move? Name __________________________ Period __________

Labeling, is it really necessary? Lesson 1: Why move things around? Labeling, is it really necessary? A A’ B C B’ C’ Pre-Image Image Prime _____________________ Map ______________________ Name __________________________ Period __________

Name __________________________ Period __________ Lesson 1: Why move things around? Exploratory Challenge Describe, intuitively, what kind of movement (basic rigid motion) will be required to move the original figure on the left to each of the figures (1, 2 and 3) on the right. To help with this exercise, you may use a transparency to copy the figure on the left. Note that you are supposed to begin by moving the original figure to each of the locations in (1), (2), and (3). Original (2) (1) (3) _______________________________________________ (2) _______________________________________________ (3) _______________________________________________ Name __________________________ Period __________

Name _______________________ Period _____ Lesson 2: Definition of Translation and Three Basic Properties BASIC RIGID MOTION #1 Translation Think: Translate from English to Spanish Intuitively --- “slide” from one language to another VECTOR – a segment in a plane A vector has two endpoints: one “end” is the starting point and the other is the endpoint. The starting point is labeled with a single capitol letter, i.e, A. The ending point after the transformation is labeled with a single capitol letter with a prime symbol, i.e, A’ Name __________________________ Period __________

Name _______________________ Period _____ Lesson 2: Definition of Translation and Three Basic Properties Draw at least three different vectors, and show what a translation of the plane along each vector will look like. Describe what happens to the following figures under each translation using appropriate vocabulary and notation as needed. Name __________________________ Period __________

Name __________________________ Period __________ Lesson 3: Translating Lines Translating Lines Line has been translated along vector AB resulting in L’ . What do you know about lines L and L’? Line L is parallel to L’ Name __________________________ Period __________

Lesson 3: Translating Lines Translate angle XYZ, point A, point B, and rectangle HIJK along vector EF. Sketch the images and label all points using prime notation. What is the measure of the translated image of angle XYZ. How do you know? Name __________________________ Period __________

Name __________________________ Period __________ Lesson 3: Translating Lines Name __________________________ Period __________

Name __________________________ Period __________ Lesson 3: Translating Lines Name __________________________ Period __________

Name __________________________ Period __________ Lesson 3: Translating Lines TRANSLATION SUMMARY A translation maps line to lines, segments to segments, rays to rays and points to points (2) A translation preserves lengths of segments (3) A translation maps angles to angles (4) A translation preserves the degree of an angle Name __________________________ Period __________

Lesson 4 Definition of Reflection and Basic Properties BASIC RIGID MOTION #2 Reflection Think: Looking into the mirror Intuitively --- “flip” Tool: a mirror or mira mira = means “to see” Let’s do some mira activities … Name __________________________ Period __________

Name __________________________ Period __________ Lesson 4: Definition of Reflection and Basic Properties Reflect triangle ABC across line L. Notice that point B did not move. C and C’ are equidistant (the same distance) from Line L as are A and A’. The line of reflection is a BISECTOR because it is equidistant from both the pre-image and the image Give an example of a counterexample: Name __________________________ Period __________

Lesson 4: Definition of Reflection and Basic Properties Reflect Figure R and triangle EFG across line L. Label the reflected images. Name __________________________ Period __________

Lesson 4: Definition of Reflection and Basic Properties Reflecting over the X axis Name __________________________ Period __________

Lesson 4: Definition of Reflection and Basic Properties Reflecting over the Y axis Name __________________________ Period __________

Name __________________________ Period __________ Lesson 4: Definition of Reflection and Basic Properties Name __________________________ Period __________

Name __________________________ Period __________ Lesson 4: Definition of Reflection and Basic Properties Name __________________________ Period __________

Name __________________________ Period __________ Lesson 4: Definition of Reflection and Basic Properties Name __________________________ Period __________

Name __________________________ Period __________ Lesson 4: Definition of Reflection and Basic Properties REFLECTION SUMMARY A reflection maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle A reflection preserves lengths of segments (3) A reflection preserves degrees of angles Name __________________________ Period __________

Lesson 5 Definition of Rotation and Basic Properties BASIC RIGID MOTION #3 Rotation Think: A car on ice doing a 180 or a 360 Intuitively --- “turn” A positive degree of rotation moves the figure counterclockwise and a negative degree of rotation moves the figure clockwise. -360 < d < 360 Center of Rotation: The fixed point from which a figure is turned. Name __________________________ Period __________

Name __________________________ Period __________ Lesson 5: Definition of Rotation and Basic Properties A positive degree of rotation moves the figure counterclockwise and a negative degree of rotation moves the figure clockwise. -360 < d < 360 Name __________________________ Period __________

Name __________________________ Period __________ Lesson 5: Definition of Rotation and Basic Properties Let there be a rotation by around the center O. Explain why a rotation of degrees around any point never maps a line to a line parallel to itself.   A segment of length cm has been rotated degrees around a center . What is the length of the rotated segment? How do you know? An angle of size has been rotated degrees around a center . What is the size of the rotated angle? How do you know? Name __________________________ Period __________

ROTATIONS in the COORDINATE PLANE Lesson 5: Definition of Rotation and Basic Properties ROTATIONS in the COORDINATE PLANE Name __________________________ Period __________

Name __________________________ Period __________ Lesson 5: Definition of Rotation and Basic Properties Name __________________________ Period __________

Name __________________________ Period __________ Lesson 5: Definition of Rotation and Basic Properties Name __________________________ Period __________

Name __________________________ Period __________ Lesson 5: Definition of Rotation and Basic Properties Name __________________________ Period __________

Name __________________________ Period __________ Lesson 5: Definition of Rotation and Basic Properties ROTATION SUMMARY A rotation maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle A rotation preserves lengths of segments (3) A rotation preserves degrees of angles Name __________________________ Period __________

Lesson 6 Rotations of 180 degrees are special! The picture below shows what happens when there is a rotation of around center O? 180 degree rotation maps 1st angle onto 2nd angle Name __________________________ Period __________

Name __________________________ Period __________ Lesson 6: Rotations of 180 degrees What are the angles in the coordinate points? Name __________________________ Period __________

Name __________________________ Period __________ Lesson 6: Rotations of 180 degrees The figures below show triangles and A’B’C’, where the right angles are at B and B’. Given AB = A’B’ = 1, BC = B’C’ = 2, AB is not parallel to A’B’, is there a 180° rotation that would map ΔABC to ΔA”B”C”? Explain. No, because a 𝟏𝟖𝟎˚ rotation of a segment will map to a segment that is parallel to the given one. It is given that 𝑨𝑩 is not parallel to 𝑨’𝑩’, therefore a rotation of 𝟏𝟖𝟎˚ will not map ∆𝑨𝑩𝑪 onto ∆𝑨′𝑩′𝑪′ Name __________________________ Period __________

Name __________________________ Period __________ Lesson 6: Rotations of 180 degrees Name __________________________ Period __________

Name __________________________ Period __________ Lesson 6: Rotations of 180 degrees Name __________________________ Period __________

Name __________________________ Period __________ Lesson 6: Rotations of 180 degrees Name __________________________ Period __________

Name __________________________ Period __________ Lesson 6: Rotations of 180 degrees Name __________________________ Period __________

Name __________________________ Period __________ Lesson 6: Rotations of 180 degrees Name __________________________ Period __________

Name __________________________ Period __________ Lesson 6: Rotations of 180 degrees Name __________________________ Period __________

Name __________________________ Period __________ Lesson 6: Rotations of 180 degrees Name __________________________ Period __________