Materials: 2 (red) Miras 40 Assorted Pattern Blocks to make symmetric patterns Tracing Paper (Patty Paper) 2 Scissors

Transcription

1 Math Buddies Grade Lesson # 10 Congruence and Symmetry Goals: Understand that congruent shapes match exactly by laying one on top of the other, using tracing procedures, and using the Mira. Understand that symmetrical figures can be divided into two halves that are the mirror image of each other by using paper folding, the Mira, pattern blocks, and tracing paper. Identify and draw a line of symmetry in an object or an arrangement of objects. Vocabulary Congruent figures are figures having exactly the same size and shape. 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. Related SOL: 2.21 The student will identify and create figures, symmetric along a line, using various concrete materials The student, given appropriate drawings or models, will identify and describe congruent and symmetrical, two-dimensional (plane) figures, using tracing procedures. Materials: 2 (red) Miras 40 Assorted Pattern Blocks to make symmetric patterns Tracing Paper (Patty Paper) 2 Scissors Goal 1: Understand that congruent shapes match exactly (15 minutes) Activity 1.1: Congruence by Tracing (5 minutes) 1. Begin the lesson by discussing the meaning of congruent. Tell students: When one figure fits exactly over another, the two figures are said to be congruent. 2. Give students a sheet of tracing paper (Patty Paper). Tell students: Open to Lesson #10: Student Activity Warm-Up #1. Tell students: Your goal is to determine if one figure is congruent to the other. Use the tracing paper and a pencil to copy one of the figures. Then, move your copy of the first figure to see if it fits exactly over the second

2 Math Buddies Grade figure. If it does, the figures are congruent. If it doesn t, they are not congruent. Repeat this process for # 1 6 to determine the answers. Answers: 1.) Congruent 2.) Congruent 3.) Not Congruent 4.) Congruent 5.) Not Congruent 6.) Not Congruent 3. Ask students: What did you notice about congruent figures? How do their sizes compare? How do the shapes compare? (If two figures are congruent, then they are the same size and the same shape.) Activity 1.2: Testing For Congruence Using The Mira 1. Introduce the students to the parts of the Mira. Place the Mira in front of the student so that the beveled edge is down (touching the desk) and that the beveled edge is facing the student (not on the opposite side) as they face the Mira. 2. Point to the parts of the Mira as you describe them to the students. Say: This is a Mira. Feel the top edge of the Mira. It has square corners for edges. Feel the Bottom edge of the Mira. Notice that it is not as thick as any other edge on the Mira. It has a beveled edge on the front face of the Mira and a square corner edge on the back face of the Mira. Always keep the beveled edge facing you so that you are looking into the front face of the Mira. 3. Then review the parts of the Mira by asking: How can you tell the top from the bottom? (The beveled edge is on the bottom.) How is the beveled edge different from all the other edges of the Mira? (It is a different thickness.) How can you tell which face is the front? (By finding the beveled edge that is on the front face.) 4. Tell the students: Take out Lesson #10: Student Activity #1: Testing for Congruence with the Mira. Explore the properties of the Mira by asking the students to: Place the Mira between the two figures and move it around so that when you look into the Mira you can see that one figure fits on top of the other. The figure in front of the Mira is called the object. Notice that the object is outlined in black. The figure behind the Mira is the image. What color is the image? (It is outlined in red as a result of looking through the red plastic Mira.) Use the Mira to determine whether the other pairs of figures are congruent (by reflection). Write Yes or No. Answers: A.) Yes D.) Yes B.) Yes E.) No C.) No F.) Yes

3 Math Buddies Grade Activity 1.3: Finding the Missing Puzzle Piece 1. Ask students: Have you ever put together a puzzle? Some people look for the puzzle pieces by looking at the space that needs to be filled. What they are doing is looking for a puzzle piece that is congruent to the empty space. 2. Tell students: You are going to do a few problems that require you to find the piece that is congruent to the empty space like finding a puzzle piece. Open to Lesson #10: Student Activity #2 and identify the piece of the puzzle that is missing. Answers: 1. B 2. C Goal 2: Understand that symmetrical figures can be divided into two halves that are the mirror image of each other. (10-12 minutes) Activity 2.1: Testing for Symmetry by Paper Folding 1. Begin this part of the lesson by having the students watch you. Show students the clown face from Lesson #10: Student Activity Warm-Up #2: Clown Face. Draw a vertical line down the center of the clown s face. Tell students: I want you to examine both sides of the clown s face. Ask questions such as the following: Do both sides of the clown s face have an eye? (yes) Do both sides have an ear? (yes) Do both sides of the hat have the same design? (yes) Are both sides exactly the same? (yes) If I cut the face in half, could I slide one half over so that it would fit exactly on top of the other half? (yes) 2. Cut the face in half along the line. Tell students to: Slide one half of the face over and place it on top of the other half. Are both sides the same? (No) They should see that they are not the same. Say: Both halves of the clown s face have the same features, but their positions are reversed. 3. Ask: What would happen if you flipped one of the halves over on top of the other half? Would it then fit on top of the other half? (Yes) Have the students flip one half and place it on top of the other half to see that the two halves match exactly. 4. Explain to students: A shape, such as this clown face, is SYMMETRICAL if you can divide it in half and flip or fold one of the halves so that it fits exactly on top of the other half. 5. Explain to students that: Shapes are usually folded rather than flipped so that they can be kept whole. Tell students that: The process of folding a shape in half to see if it is symmetrical is called the fold test.

4 Math Buddies Grade Activity 2.2: Fold Test 1. Ask students to: Take out Lesson #10: Student Activity #3: Using the Fold Test for Symmetry. Give each student a pair of scissor and have one cut out two of the shapes on this activity page and the other person cut out the remaining two. 2. Tell students: Now that you have the shapes cut out, use the fold test to determine which of the shapes have symmetry. For each shape, ask students: Can you fold this shape so that the two halves fit together exactly? Answers: Square: Yes (four different ways) Parallelogram: No Rectangle: Yes (two different ways) Triangle: Yes (three different ways) 3. Tell students: Some shapes have more than one line of symmetry. If students notice this as they are working with the shapes, encourage them to investigate this idea further. Ask students: Can you determine how many different ways the square shape can be folded to show symmetry? (four different ways) How many ways can you find for the rectangle? (two different ways) How many ways can you find for the triangle? (three different ways) 4. Conclude this activity by asking: If a shape can fold in half so that one side fits on the other, what can we say about the shape? (it is symmetrical) Activity 2.3: Symmetry 1. Ask students to: Take out Lesson #10: Student Activity #4: Things That Are Symmetrical. Tell students: Symmetry is everywhere in nature, art, music, mathematics, and beyond. In this activity, students begin to develop an awareness of symmetry, particularly, reflectional symmetry, using the Mira. 2. Tell students: In addition to the fold test, we can use the Mira to check for symmetry in shapes. Say to students: You are going to determine whether the objects on this page are symmetrical by placing the Mira on each shape and moving it around to determine whether one side fits on the other. Answers: Things that are Symmetrical Star: yes Bee: yes Butterfly: yes Foot: no Leaf: yes Sail Boat: no 3. Tell students that: Because each half of the symmetric shapes forms a mirror image of the other half, this type of symmetry is often called mirror symmetry or reflectional symmetry. Then use the Mira to check for mirror symmetry with a

5 Math Buddies Grade few nonsymmetrical shapes. Students should see that the Mira does not reflect the other half of the nonsymmetrical shapes. 4. Tell students: Now take the Mira and go to one of the shapes that is symmetrical. Using the Mira properly (with the Mira edge facing you), find a place on the shape where one half of the shape fits on the other. Now holding the Mira firmly in place, draw a line along the Mira edge. 5. Say: This line is a line of symmetry. It 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. 6. Look at the other symmetrical shapes and draw their line of symmetry. If the students notice more than one line of symmetry, you may wish to say: Do any shapes have more than one line of symmetry? (yes the star it has five lines of symmetry) Goal 3: Identify and draw a line of symmetry in an object or an arrangement of objects (8-10 minutes) Activity 3.1: Lines of Symmetry 1. Ask students to: Take out Lesson #10: Student Activity # 5: Lines of Symmetry. 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.) 2. Ask students to: Look at the dotted lines on each shape. Tell whether the line is a line of symmetry. Circle Yes or No Answers: Lines of Symmetry 1. Yes 2. Yes 3. No 4. Yes (both are)

6 Math Buddies Grade Activity 3.2: Symmetrical Designs 1. Ask students to: Take out Lesson #10: Student Activity # 6: Symmetrical Design Challenge. Tell students: You will now use pattern blocks to create symmetrical designs. 2. Have students: Take turns making one-half of a symmetrical design. Tell student: One of you will create your design along one side of the line. When you have finished, the other student will take the Symmetrical Design Challenge by making the other half of the design symmetrical in both shape and color. You should check the final design, making sure that the design is symmetrical in shape and in color. Example: Activity 3.3: Lines of Symmetry 1. Ask students to: Take out Lesson #10: Student Activity # 7: How Many Lines of Symmetry? Ask the students to: Use your Mira to find lines of symmetry on each of the shapes. Draw each line of symmetry along the Mira edge. Fill in the chart for each shape to show the number of lines of symmetry found for each of the shapes. 2. Do you notice anything between the number of sides on the shape and the number of lines of symmetry? [Note: When using regular polygons, (where all sides are equal), the number of sides on a shape is equal to the number of lines of symmetry.] Lines of Symmetry: Not Shown Triangle: 3 Square: 4 Rectangle: 2 Pentagon: 5 Hexagon: 6

7 Math Buddies Grade Lesson #10:Assessing Student Learning: (5 minutes) 1. Have students: Complete the six problems in the Student Assessment section. Once completed, review each problem with the student. 2. If the student made errors, help them understand how to obtain the correct solution. Answers to the Student Assessments: 1. B 2. G 3. B 4. H 5. F 6. G

8 Math Buddies Grade Lesson #10: Student Activity Warm-Up Congruence by Tracing By tracing one shape, determine whether the pair of figures is congruent. Circle the answer. 1.) Congruent 2.) Congruent Not Congruent Not Congruent 3.) Congruent 4.) Congruent Not Congruent Not Congruent 5.) Congruent 6.) Congruent Not Congruent Not Congruent

9 Math Buddies Grade Lesson #10: Student Activity #1 Testing for Congruence with the Mira Use your Mira to check whether each pair of figures is congruent by reflection. Write, Yes or No. A. Are they congruent? B. Are they congruent? C. Are they congruent?

10 Math Buddies Grade D. Are they congruent? Lesson #10: Student Activity #1 (page 2) Testing for Congruence with the Mira E. Are they congruent? F. Are they congruent?

11 Math Buddies Grade Lesson #10: Student Activity #2 Finding the Missing Puzzle Piece For each shape below, determine the puzzle piece that is missing. 1.) 2.) A.) A.) B.) B.) C.) C.)

12 Math Buddies Grade Lesson #10: Student Activity Warm-Up #2 Clown Face

13 Math Buddies Grade Square Lesson #10: Student Activity #3 Using the Fold Test for Symmetry Parallelogram Rectangle Triangle

14 Math Buddies Grade Lesson #10: Student Activity #4 Things That Are Symmetrical Circle the objects that are symmetrical. Put an X on the objects that are not symmetrical.

15 Math Buddies Grade Lesson #10: Student Activity # 5 Lines of Symmetry Look at the shapes shown below. Tell whether the dotted line is a line of symmetry. Circle Yes or No. 1.) Line of symmetry? YES NO 2.) Line of symmetry? YES NO 3.) Line of symmetry? YES NO 4.) Lines of symmetry? YES NO

16 Math Buddies Grade Lesson #10: Student Activity # 6 Symmetrical Design Challenge One student creates a design along one side of the line. When he or she has finished, another student takes the Symmetrical Design Challenge by making the other half of the design symmetrical in both shape and color.

17 Math Buddies Grade Lesson #10: Student Activity # 7 How Many Lines of Symmetry? Look at the shapes shown below. Draw as many lines of symmetry as you can find. Fill in the chart below to show how many lines of symmetry each shape has. Shape Name of Shape Number of Sides Number of Lines of Symmetry Triangle 3 Square 4 Rectangle 4 Hexagon 6

18 Math Buddies Grade Lesson #10: Student Assessment 1. 2.

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