# Lesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations

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1 Math Buddies -Grade Lesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations Goal: Identify congruent and noncongruent figures Recognize the congruence of plane figures resulting from geometric transformations such as translation (slide), reflection (flip) and rotation (turn). Identify figures that are symmetric and lines of symmetry Vocabulary: Congruent figures have the same size and shape. The angles and line segments that make up the plane figure are exactly the same size and shape. Two shapes or solids are congruent if they are identical in every way except for their position; one can be turned into the other by rotation, reflection or translation. A figure or shape is Symmetrical when one-half of the figure is the mirror image of the other half A Line of Symmetry divides a symmetrical figure, object, or arrangement of objects into two parts that are congruent if one part is reflected (flipped) over the line of symmetry Transformation is an operation that creates an image from an original figure or pre-image. Translations, Reflections and Rotations are some of the transformations on the plane. Although there is a change in position for the original figure, there is no change to the shape or size of the original figure. Translation (Slide) is a transformation of an object that means to move the object without rotating or reflecting it. Every translation has a given direction and a given distance. Reflection (Flip) is a transformation of an object that means to produce its mirror image of the object on the opposite side of a line. Every reflection has a mirror line or a line of reflection. A reflection of an "R" is a backwards "R" Rotation (Turn) is a transformation of an object that means to turn it around a given point, called the center. Every rotation has a center of rotation, an angle of rotation, and a direction (counterclockwise and clockwise). Tessellations are patterns of shapes that cover a plane without gaps (holes) or overlaps are called tessellations. Related SOL: 4.17 The student will b) identify congruent and noncongruent shapes; and c) investigate congruence of plane figures after geometric transformations such as reflection (flip), translation (slide) and rotation (turn), using mirrors, paper folding, and tracing.

3 Math Buddies -Grade How do you know these are the same shapes? When we say two objects are congruent, does the color of the shape matter? (no) 8. Say: On paper, draw this symbol. Say: The mathematical symbol used to denote congruent is. The symbol is made up of two parts: ~ which means the same shape (similar) and = which means the same size (equal). Congruent Symbol Activity 1.2: Congruent or Not 1. Say: Open your book to Lesson #13: Student Activity Sheet #1:Congruent or Not. Look at the various shapes and determine whether they are congruent. Put a check under yes or no to indicate your answer. Then explain why they are or are not congruent. 2. Answers: 1. No, not the same size. 2. Yes, even though one is shaded. 3. No, different size and shaped triangles. 4. Yes, lines don t change the shape or size. 5. No, different size. 6. Yes, different position but the same shape and size. Activity 1.2: Tantalizing Triangles 1. Say: We can further refine our definition of congruent figure by saying that two shapes or solids are congruent if they are identical in every way except for their position; a figure can be moved by slides, flips or turns, and still be congruent. 2. Open the set of colored pencils for the Math Buddies to use and give each student one piece of patty paper to use as tracing paper. Say: Open your book to Lesson #13: Student Activity Sheet #2: Tantalizing Triangles. The objective of this activity is to find the tantalizing triangles that are congruent. To determine if they are congruent, carefully trace one of the triangles and then move the traced triangle around the page to find others that are congruent to it. Remember it does not matter what position the shape is in relative to another shape. Color any congruent triangles you find with the same color pencil. Then trace a second triangle and continue the same process. There are four different shaped triangles and all triangles should be colored. Good luck! Answers: Set #1: A, E, N, K are congruent Set #2: D, I, M, P are congruent Set #3: C, J, H, L are congruent Set #4: B, G, O, F are congruent Goal 2: Recognize the congruence of plane figures resulting from geometric transformations such as reflection (flip), translation (slide), and rotation (turn). Activity 2.1: Warm-Up: Transformations with Tangrams 1. Describing figures and visualizing what they look like when they are transformed through translations (slides), reflections (flips), and rotations (turns), or when they are put together or

4 Math Buddies -Grade taken apart in different ways are important aspects of the geometry program in elementary school. In this activity, students will use the seven tangram pieces to explore the transformation of shapes as they work to solve a few tangram puzzles. The potential for a high-quality spatial visualization experiences provided this activity that involves the use of manipulatives should enhance student understanding of transformations. The manipulative to be used is Tangrams, which are an ancient Chinese moving piece puzzle, consisting of 7 geometric shapes. 2. Give each student a set of tangrams and say: This is a set of seven tangram pieces from the ancient Chinese puzzle. The Tangram shapes were used for recreational activity in China thousands of years ago. The word Tangram is derived from tan, meaning Chinese, and gram, meaning diagram or arrangement. Spread them out on the table and point to the pieces as I say them: the square, two small triangles, one medium triangle, two large triangles, and one parallelogram. 3. Say: Let s examine each of the five different Tangram pieces, and determine the area of each piece, assuming that the small triangle has an area of one unit. Answers Small Triangle 1 square unit Square 2 square units Parallelogram 2 square units Medium Triangle 2 square unit Large Triangle 4 square unit 4. Say: You can use all seven pieces to make a figure or your can use a given number to make a figure. We are going to make a square of different sizes using a defined number of pieces. Let s try these tasks together. Select one or more based upon time constraints. Possible solutions follow. Can you make a square using one piece? (use the square piece) Can you make a square using two pieces? (two small triangles or two large triangles) Can you make a square using three pieces? (two small triangles and one medium triangle) Can you make a square using four pieces? Can you make a square using five pieces? Note: Using six pieces can t be done Can you make a square using seven pieces? (see below)

5 Math Buddies -Grade Say: Please use the seven tangram pieces to make one of the figures you select on Lesson#20: Student Activity Sheets #3A or #3B. You must use all seven pieces for each figure. I will check your answers once you inform me that you have completed a figure. 6. Answers: Activity 2.2: Transformations: Translations (Slides) 1. Say: You have been working with the Tangram pieces. While you worked to manipulate the shapes to create the different figures, often you were visualizing what they would look like once you had transformed them. You had a chance to move around the tangram pieces using a variety of transformations. 2. Say: Transformations is a word used to describe a category of movements that you can make with a shape. We will be studying three transformations: translations, rotations, and reflections. 3. Take out Lesson #13: Teacher Sheet #1. Refer to the top of the sheet as you describe translation transformations. Say: Translations are like slides, like sliding down a playground slide where you move from high to low but you are still sitting upright when you hit the bottom. A translation "slides" an object a fixed distance in a given direction. The original object and its translation have the same shape and size, and they face in the same direction. [Note: The word "translate" in Latin means "carried across".] When you are sliding down a water slide, you are experiencing a translation. Your body is moving a given distance (the length of the slide) in a given direction. You do not change your size, shape or the direction in which you are facing. Translations can be seen in wallpaper designs, textile patterns, mosaics, and artwork. 4. Say: Open your student books to Lesson #13: Student Activity Sheet #4. Look at the pentagons (five sided figures) at the top left hand side of the page. In mathematics, the translation of an object is called its image. If the original object was labeled with letters, such as ABCDE, the image may be labeled with the same letters followed by a prime symbol (like an apostrophe), A'B'C'D'E'.

6 Math Buddies -Grade Think of polygon ABCDE as sliding two inches to the right and one inch down. Its new position is labeled A'B'C'D'E'. 5. Say: A translation moves an object without changing its size or shape and without turning it or flipping it. Take out the Pattern Blocks and say: Here are some pattern blocks. Take out a blue parallelogram, a green triangle and a red trapezoid. On the pattern block grid paper, draw the translations of each shape by first placing it on the original figure and then sliding the pattern blocks the distance and the direction indicated by the arrow. To simplify this process we have only labeled one vertex of the shape with a letter. The image of the shape should have the same letter followed by the prime symbol in its new position as it had in it s original position. Check for accuracy of drawing. Ask: Did your shapes look different as a result of your translations? (no they do not change size or shape, just position) 6. Say: Now look at Part B on Activity Sheet #4. For each of the four problems, check yes or no to indicate whether one figure is the translation of the other. 7. Answers: 1. yes 2. no (change in size) 3. yes (doesn t need a slide line) 4. yes Activity 2.3: Transformations: Rotations (Turns) 1. Again take out Lesson #13: Teacher Sheet #1. Refer to the middle of the sheet as you describe the rotation transformation. Say: Rotations are turns, like when a basketball player pivots on one foot, or when a Ferris wheel turns around the center of the wheel. Look at this picture on the teacher page. To rotate a shape, you need to identify three things. First you must identify the point around which you are turning the shape, called the center of rotation. Second, you need to know the direction of the turn, clockwise or counterclockwise. Third, you need to know the angle, the number of degrees of the turn, or the fractional part of 1 whole turn (e.g. turn, or turn). Notice that the picture displays a clockwise rotation of the R around a center point, and where the angle of the turn is 90 degrees, or a one-quarter turn. 2. Say: Open your student books to Lesson #13: Student Activity Sheet #5: Discover Rotation. Notice the letter B being rotated four times around the center of the two intersecting lines. What is the direction of the rotation, clockwise or counterclockwise? (clockwise) What is the angle of the rotation for each turn? (90 degrees, or a onequarter turn) You might think of a rotation like putting an object on a plate or a Lazy

8 Math Buddies -Grade Say: Now open your student books to Lesson #13: Student Activity Sheet #6: Rotation and Reflection With Pattern Blocks. At the bottom of the page in Part B, I would like Math Buddy A to place a red trapezoid on the left side of the line, touching the line. Once the trapezoid is placed, say: Now, I would like Math Buddy B to place a red trapezoid on the right side of the line to show a reflection of this pattern block. Does everyone agree that this is the reflection image of the pattern block on the left. If not, make the corrections. This is one example; other placements of the trapezoid lead to other arrangements. Line of Reflection Reflection Image of Reflection 5. Say: Now let s try a more challenging task. At the bottom of the page in Part B, I would like Math Buddy A to make a pattern block design on the left side of the line so that the design touches the line. Once the pattern design is complete, I would like Math Buddy B to make the reflection of this pattern block design on the right side of the line to show the designs reflection. Once complete, say: Does everyone agree that this is the reflection of the pattern block design across the line of reflection? Do we need to make any corrections? 6. Say: Now, switch rolls, and Math Buddy B will create the design on the right side, and Math Buddy A will create it s reflection on the left side. Once complete, say: Does everyone agree that this is the reflection of the pattern block design across the line of reflection? Do we need to make any corrections? Activity 2.5: Rotation or Reflection? 1. Say: Now open your student books to Lesson #13: Student Activity Sheet #7: Rotation or Reflection? Here is a table of figures. Use your knowledge to decide whether the second figure, the image, is a rotation or a reflection of the first. Once you decide, check under the column heading of this transformation. Some images may represent a rotation and a reflection, so check both. 2. Answers: 1. Rotation (1/4 turn) 2. Reflection (across a horizontal line) or Rotation (1/2 turn) 3. Reflection (across a vertical line) 4. Rotation (3/4 turn clockwise; or turn counterclockwise) 5. Rotation (1/4 turn clockwise) 6. Reflection (across a horizontal line) 7. Reflection (across a horizontal line) 8. Reflection (across a vertical line) 9. Reflection (across a vertical line) 10. Rotation (3/4 turn clockwise; or turn counterclockwise) Activity 2.6: Is the Shape Reflection-Congruent? 1. Give each student a geo-reflector and introduce the students to its parts. Place the georeflector in front of the student so that the beveled edge is down (touching the desk) and the beveled edge is facing the student.

9 Math Buddies -Grade Point to the parts of the geo-reflector as you describe them to the students. Say: This is a geo-reflector. Feel the top edge of the geo-reflector. It has square corners for edges. Feel the Bottom edge of the geo-reflector. Is it the same as the top edge? (No) Notice that it is not as thick as any other edge on the geo-reflector. It has a beveled edge on the front face of the geo-reflector and a square corner edge on the back face of the georeflector. When you are working, always keep the beveled edge of the geo-reflector facing you so that you are looking into the front face of the geo-reflector. 3. Then review the parts of the geo-reflector by asking: a. How can you tell the top from the bottom? (The beveled edge is on the bottom.) b. How is the beveled edge different from all the other edges of the geo-reflector? (It is a different thickness.) c. How can you tell which face is the front? (By finding the beveled edge that is on the front face.) 4. Say: Now, go back to the bottom of Student Activity Sheet #6. Place the Geo-Reflector on the line of reflection and rotate your book around so that the Geo-Reflector is sitting horizontally, parallel to the table s edge. 5. Say: Now take out a yellow pattern block and place it anywhere between you and the Geo-Reflector. Using a pencil draw the perimeter of the yellow hexagon. Now, making sure the hexagon stays in this same spot, look through the Geo-Reflector and what do you see? (reflection of the hexagon) Yes, you see the reflection of the hexagon. Now I would like you to draw the perimeter of the reflection of the hexagon free hand. Once this is done, say: Remove the Geo-Reflector and pattern block leaving the drawing of the original figure, the line of reflection, and the drawing of the figure s reflection, called the image of reflection. 6. Say: Now take out a few pattern blocks and place them in front of the Geo-Reflector and look through the Geo-Reflector at their reflection. 7. Say: Now we are going to check to see whether two shapes are congruent as a result of a reflection. Take out Lesson #13: Student Activity Sheet #8: Is the Shape Reflective- Congruent. Place the Geo-Reflector between the two figures and move it around so that when you look into the Geo-Reflector you can see whether the one figure fits on top of the other. The figure between you and the Geo-Reflector, or what is in front of the Geo-Reflector is called the object. Notice that the object is outlined in black. The figure behind the Geo-Reflector is the image. What color is the image? (It is outlined in the color of the Geo-Reflector as a result of looking through the colored plastic.) 8. If the object and the image are congruent (e.g. same size and same shape), the pair of shapes are reflective-congruent. Use the Geo-Reflector to determine whether the other pairs of figures are reflective-congruent. Check Yes if they are and No if they are not. If they are congruent as a result of the reflection, draw the line of reflection by placing your pencil on the beveled edge and drawing along that edge when the object reflects onto the image.

10 Math Buddies -Grade Answers: A.) Yes B.) No C.) No D.) Yes E). No F.) Yes G.) No H.) Yes Goal 3: Identify and Draw Lines of Symmetry Activity 3.1: Lines of Symmetry 1. Ask students: What is a line of symmetry? (A line of symmetry divides a symmetrical figure, object, or arrangement of objects into two parts that are congruent if one part is reflected (flipped) over the line of symmetry.) Symmetry is everywhere in nature, art, music, mathematics, and beyond. Can you think of anything that is symmetrical? (Answers might include a butterfly, the letter H, a pair of pants, etc.) 2. In this activity, students will enhance their understanding of symmetry, particularly, reflectional symmetry, using the Geo-Reflector. Say: In our last activity, when shapes were congruent as a result of a reflection, we were able to draw a line of reflection. This line represented the line across which the objects were flipped. In this activity we will use the Geo-Reflector on individual shapes as a line of symmetry. The reflection will produce the other congruent half of the shape. Consequently, we will learn that a line of symmetry is a line that divides a figure in to congruent halves, each of which is the reflection image of the other. 3. Say: Take out Lesson #13: Student Activity Sheet #9: Line of Symmetry. The dotted line on each shape is the line of symmetry. Place your Geo-Reflector on the dotted line and draw the other side of the shape by tracing its reflection. 4. Answers: Line of Symmetry Activity 3.2: Polygons: How Many Lines of Symmetry? 1. Say: Take out Lesson #13: Student Activity #10: Polygons: How Many Lines of Symmetry? The polygons on this page are regular polygons. Regular polygons are polygons that have congruent sides and congruent angles; that is sides of the same lengths and angles of the same angle measure. 2. Say: You are going to determine how many lines of symmetry each of these polygons has using the Geo-Reflector. Move your Geo-Reflector around on the shape to find

11 Math Buddies -Grade lines of symmetry. When you find a line of symmetry, where one side can be reflected on the other, draw that line of symmetry by placing your pencil on the recessed (beveled) edge of the Geo-Reflector and drawing that line. As you complete each polygon, report the number of lines of symmetry for the identified shape in the table below. Work on this activity now and then we will summarize your findings in the table once you have finished. Lines of Symmetry: Triangle: 3 Square: 4 Pentagon: 5 Hexagon: 6 3. Say: Now, let s review the data you have collected in the table. How many lines of symmetry did you find for the equilateral triangle? (3) As you look back at these lines, notice that each line went through one vertex and through the midpoint of the side opposite the vertex. Now look at the five sided pentagon. How many lines of symmetry did you find for the pentagon? (5) How are these lines of symmetry similar to the lines of symmetry in the triangle? (Each line of symmetry went through one vertex and through the midpoint of the side opposite the vertex.) 4. Say: How many lines of symmetry did you find for the square? (4) As you look back at these lines, notice that two line went from one vertex through to the other vertex, and two line went from one midpoint through to the other midpoint on the opposite side. Now look at the six sided hexagon. How many lines of symmetry did you find for the hexagon? (6) How are these lines of symmetry similar in the hexagon similar to the lines of symmetry in the square? (Each line of symmetry went from one vertex to the opposite vertex, or from one midpoint to the opposite the midpoint.) 5. Say: Now, let s look at the numbers. Is there any relationship between the number of sides in a regular polygon and the number of lines of symmetry? (Yes, when finding lines of symmetry in regular polygons, the number of lines of symmetry equals the number of sides in the polygon.) Lesson #13: Assessment of Student Learning 1. Have students complete the thirteen multiple-choice assessment items independently by circling the correct answer. 2. Once complete, discuss the items that the students answered incorrectly, asking them to explain their thinking and reasoning about how they chose each answer. Answer Key: 1. B 2. C 3. A 4. C 5. B 6. A 7. C 8. B 9. C 10. J 11. G 12. A 13. J

12 Math Buddies -Grade Lesson #13: Student Activity Sheet #1 Congruent or Not? Look at these figures and see if you can pick congruent figures. Check yes if the figures are congruent and no if the figures are not congruent. Congruent or Not? Yes No Congruent or Not? Yes No

13 Math Buddies -Grade Lesson #13: Student Activity Sheet #2 Tantalizing Triangles Find out if the tantalizing triangles are congruent using tracing paper. Color any congruent triangles you find the same color. Hint: There are four congruent shapes for each of four different shapes! A B C D E H F G L I J K P M N O Four Congruent Triangles are: Four Congruent Triangles are: Four Congruent Triangles are: Four Congruent Triangles are:

14 Math Buddies -Grade Lesson #13: Student Activity Sheet #3A Tangram Puzzles

15 Math Buddies -Grade Lesson #13: Student Activity Sheet #3B Tangram Puzzles

16 Math Buddies -Grade Lesson #13: Student Activity Sheet #4 Translation With Pattern Blocks Translation "slides" an object a fixed distance in a given direction. The original object (A) and its translation (A ) have the same shape and size, and they face in the same direction. Part A: Translate the pattern blocks the distance and the direction indicated by the arrows and draw the image of the translation. Part B: Check yes or no to indicate whether one figure is a translation of the other. Translation or Not? Yes No Translation or Not? Yes No

17 Math Buddies -Grade Lesson #13: Student Activity Sheet #5 Discover Rotation Rotation B Center Center of Rotation One-Fourth Turn or Rotation of 90 o Check the yes or no box to indicate whether one figure is a rotation of another. Rotation or Not? Yes No Rotation or Not? Yes No

18 Math Buddies -Grade Lesson #13: Student Activity Sheet #6 Rotation and Reflection With Pattern Blocks Part A: Math Buddy A makes a pattern block figure on line A. Math Buddy B makes the one-quarter rotation of Math Buddy A s pattern block figures on line B. Line A Center of Rotation Line B Part B: Math Buddy A makes a pattern block figure on one side of the line. Math Buddy B makes its reflection on the other side of the line. Line of Reflection

19 Math Buddies -Grade Lesson #13: Student Activity Sheet #7 Rotation or Reflection? Check The Correct Transformation(s): Rotation Reflection

20 Math Buddies -Grade Lesson #13: Student Activity Sheet #8 Is the Shape Reflective-Congruent? Use your Geo-Reflector to check if the shapes are congruent as a result of a reflection. Check Yes if they are and draw the line of reflection; otherwise check No. A. Yes No E. Yes No B. Yes No F. Yes No C. Yes No G. Yes No D. Yes No H. Yes No

21 Math Buddies -Grade Lesson #13: Student Activity Sheet #9 Line Of Symmetry Using the Geo-Reflector

22 Math Buddies -Grade Lesson #13: Student Activity Sheet #10 Polygons: How Many Lines of Symmetry? Use the Geo-Reflector to draw as many lines of symmetry as you can find for each regular polygon. Complete the chart identifying the number of lines of symmetry. Shape Name of Shape Number of Sides Triangle 3 (Equilateral Triangle) Number of Lines of Symmetry Square 4 Pentagon (Regular Pentagon) Hexagon (Regular Hexagon) 5 6

23 Math Buddies -Grade Lesson #13: Student Assessments 1. The arrow below moved 90 degrees clockwise or turn. 4. The example below is a demonstration of what? This is an example of what? A. Translation B. Rotation C. Reflection A. Translation B. Rotation C. Reflection 2. The example below is a demonstration of a. 5. In the example below, the triangles going from left to right is an illustration of a. A. Translation B. Rotation C. Reflection A. Translation B. Rotation C. Reflection 3. The change in the position of the triangles in Set A to the position of the triangles in Set B is an illustration of a. Set A Set B 6. What is it called when the arrow in picture A is moved up to the position in picture B? Picture A Picture B A. Translation B. Rotation C. Reflection A. Translation B. Rotation C. Reflection

24 Math Buddies -Grade The arrow below in picture B is a mirror image of the arrow in picture A. This transformation is called a. 10. Picture A Picture B A. Translation B. Rotation C. Reflection 8. The example below is a demonstration of a. A. Translation B. Rotation C. Reflection 9. In which figure below is the line NOT a line of symmetry? Figure A Figure B Figure C 11. Which pair of figures does NOT show a translation? A. Figure A B. Figure B C. Figure C

26 Math Buddies -Grade Lesson #13: Teacher Sheet #1 Transformations: Translations, Rotations and Reflections Translation: To translate an object means to move it a given distance in a given direction without rotating or reflecting it. Rotation To rotate an object means to turn it around. Every rotation has a center of rotation and an angle of rotation. 90 o angle is of a turn. Reflection To reflect an object means to flip it to produce its mirror image. Every reflection has a line of reflection along which it is flipped. Translation Slides a given distance in a given direction Rotation Turns around a center point of rotation, for a given angle or identified turn (example of turn ) reflection Flips across a line of reflection

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