Symmetry
Resource Overview Quantile® Measure: Skill or Concept:
EM Identify and make figures with line symmetry. (QT‐G‐85)
Excerpted from:
Gourmet Learning 1937 IH 35 North Suite 105 New Braunfels, TX 78130 www.gourmetlearning.com © Gourmet Learning
This resource may be available in other Quantile utilities. For full access to these free utilities, visit www.quantiles.com/tools.aspx. The Quantile® Framework for Mathematics, developed by educational measurement and research organization MetaMetrics®, comprises more than 500 skills and concepts (called QTaxons) taught from kindergarten through high school. The Quantile Framework depicts the developmental nature of mathematics and the connections between mathematics content across the strands. By matching a student’s Quantile measure with the Quantile measure of a mathematical skill or concept, you can determine if the student is ready to learn that skill, needs to learn supporting concepts first, or has already learned it. For more information and to use free Quantile utilities, visit www.Quantiles.com.
1000 Park Forty Plaza Drive, Suite 120, Durham, North Carolina 27713 METAMETRICS®, the METAMETRICS® logo and tagline, QUANTILE®, QUANTILE FRAMEWORK® and the QUANTILE® logo are trademarks of MetaMetrics, Inc., and are registered in the United States and abroad. The names of other companies and products mentioned herein may be the trademarks of their respective owners.
3rd Grade
Geometry
Student Expectation: Students will recognize [congruence] and symmetry; Students are expected to create two-dimensional figures with lines of symmetry using concrete models and technology
Unit 2 – Lesson 2 The student recognizes congruence and symmetry. The student is expected to create twodimensional figures with lines of symmetry using concrete models and technology. Teacher note: The concept of symmetry is introduced in this lesson. However, the focus of this lesson is to have students create the lines of symmetry using a variety of concrete models. Lesson 3 will focus on students identifying the lines of symmetry.
Study the TEKS . . .
Prior Knowledge The primary focus in 2nd grade is to have students compare and contrast figures using attributes. Students also cut apart two-dimensional geometric figures and identify the new geometric figures formed. The concept of symmetry has not formally been introduced.
Next Steps
3rd
In 4th grade, students will connect transformations to symmetry. In addition, they will use reflections to verify that a shape has symmetry.
Grade In third grade . . . Although this is the first year 3rd graders are introduced formally to symmetry, mastery of this concept is crucial, since they will be asked to take it a step further in 4th grade when they are introduced to transformations and reflections. Gourmet Curriculum Press, Inc.©
1
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will create symmetry using their bodies
Focus Activity
K
Symmetry
C
Teacher note: The concept of symmetry can be challenging for students. In this introductory lesson, students will use their bodies to create “symmetry” with a partner. This kinesthetic action will help students begin to understand the concept of symmetry. Group size: pairs (two students of similar height, if possible) Materials: large, open space (can be done outside if weather permits); 1 hand-held mirror Before class: Locate an empty room or move desks aside; gather materials. Directions: • Hold up the mirror and prompt students to discuss its purpose (possible response: to see yourself). If necessary, explain “reflection” (the image seen in a mirror). If time permits, allow each student to see his/her reflection in the mirror. • Explain that in this activity, one student will “reflect” another student. Demonstrate this with two volunteers (it is helpful if they are about the same height). Have the two students stand shoulder-to-shoulder. The first student will extend his/her left arm to the side. The second student will extend his/her right arm. Ask the class if they think the students are “reflecting” each other. (Many will say “no” because the students aren’t extending the same arm.) • Explain that to check whether each is a “reflection” of the other, the students will move to face each other - like they are looking in a mirror - while maintaining their extended arm position. Once this has been done, ask the class again if they are “reflecting” each other. (Most will now say “yes.”) • Choose two different student volunteers of similar height for the next demonstration. Have each student raise his/her right arm into the air. Again, pose the question of reflection. Then, verify by having students face each other. Explain that even though the students both have an arm raised in the air, it is not on the same side in the reflection position, so we say one is NOT a reflection of the other. • Have students find a partner who is about the same height and spread out around the room. Students will follow the same procedure as demonstrated. They can move their arms, heads, or legs (take caution with extended legs when moving into the reflection position!), or they can bend from side-to-side at the waist. Students then can check for accuracy in reflection by moving face-to-face. • Allow ample time for partners to try several positions and for the students to grasp the concept of “reflection.”
2
Gourmet Curriculum Press, Inc.©
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will be introduced to vocabulary relating to symmetry
Initial Instruction—Part I—Vocabulary Symmetry
K
symmetry: an exact matching of parts on opposite sides of a dividing line line of symmetry: figure
the line that divides a symmetrical
mirror image: an image that has its parts arranged with the right and left reversed or opposite of each other, as they would appear if seen in a mirror reflect: to show an image of something on a shiny surface such as a mirror reflection: the image seen on a shiny surface such as a mirror reflection symmetry: one half of an image is a reflection of the other half
Gourmet Curriculum Press, Inc.©
3(T)
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will identify real-life symmetrical objects
Initial Instruction—Part II Symmetry
K
C
Teacher note: Before students can begin to create symmetrical figures, they must first grasp the concept of symmetry. What better way to learn about symmetry than with symmetrical objects found in real life! In this Initial Instruction, students will be exposed to real-life symmetry and then look for symmetrical objects in the classroom. The purpose of this activity is simply to determine if the objects are symmetrical, NOT to identify the line of symmetry. That skill will be introduced in Lesson 3. Group size: whole class, then pairs Materials: Instructional Strategy, pages 4-5; pictures, transparency pages 6-7; raw uncooked spaghetti, pipe cleaner, or straw Before class: Gather materials. Directions (Part I): • Review with students what they learned about symmetry through the Focus Activity and Vocabulary lesson. If necessary, review the definition of “symmetry.” A quick demonstration of folding a paper in half, cutting out 12 of a heart along the fold, and revealing a symmetrical heart is one suggestion. • Follow the Instructional Strategy to guide students in the first part of this lesson.
Questioning Technique Instructional Strategy Say: Symmetry is all around us. Later we will look for things in our classroom that are symmetrical. Now let’s look at some pictures of real-life symmetrical objects. Place transparency page 6 on the overhead. Use sheets of paper to cover all pictures except the ice-cream cone. Hold up the spaghetti, pipe cleaner, or straw. Say: Let’s use this (spaghetti) as our line of symmetry and see if this ice-cream cone is symmetrical. Place the spaghetti vertically down the center of the ice-cream cone. Ask: Is it symmetrical? (Yes.) Ask: What does that mean? (The ice-cream cone is the same on both sides of the line of symmetry.) Place the spaghetti horizontally across the ice-cream cone. Ask: Is it symmetrical now? (No.) Ask: Can someone explain why it’s not symmetrical? (It is not a line of symmetry because the ice-cream cone is not the same on both sides of the dividing line.)
4
Gourmet Curriculum Press, Inc.©
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will identify real-life symmetrical objects
Initial Instruction—Part II Symmetry Questioning Technique Instructional Strategy Now uncover the picture of the butterfly. Say: Let’s check a butterfly. Raise your hand if you think a butterfly is symmetrical. Place the spaghetti on the line of symmetry. Ask: Did we prove it? (Yes.) How do we know? (The butterfly is the same on both sides of the line of symmetry.) Now place the spaghetti slightly to the left or right of the line of symmetry. Ask: Is the butterfly still symmetrical if I move the spaghetti? (No.) Ask: Why not? (The spaghetti is not dividing the butterfly on the line of symmetry, so one side is bigger than the other side. The butterfly is not symmetrical when divided like this.) Ask: What is important to remember when determining if an object or figure is symmetrical? (The line of symmetry must separate the object or figure into exact matching parts.) Continue with the remaining pictures on pages 6-7. Be sure to show vertical, horizontal, and diagonal symmetry when applicable. When finished, move on to Part II of this lesson. Directions (Part II): • Now that students understand real-life symmetry, have them look for examples within the classroom. Students can work in pairs for this activity. It may be necessary to establish boundary rules; e.g., students can’t stand on desks to reach objects. • Determine the amount of time students will have to find an object. Once they have found an object, students take it back to their seats until everyone is ready to share. • When students have found objects and/or time runs out, allow students to share what they found with the class one at a time. They can use the spaghetti to show their object’s line of symmetry. Extension: Have the class visually search for symmetrical objects while walking down the hallway, playing outside at recess, or eating lunch in the cafeteria. They can record or draw these objects in a “Log of Symmetry.”
Gourmet Curriculum Press, Inc.©
5
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will identify real-life symmetrical objects
Initial Instruction—Part II—Pictures Symmetry
6(T)
Gourmet Curriculum Press, Inc.©
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will identify real-life symmetrical objects
Initial Instruction—Part II—Pictures Symmetry
Gourmet Curriculum Press, Inc.©
7(T)
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will create two-dimensional figures with lines of symmetry using concrete models
Initial Instruction—Part III Symmetry
K C Ap
Teacher note: Now that students feel more confident with symmetry, they are ready to begin creating symmetrical two-dimensional figures. In this activity, students use pattern blocks to create a symmetrical design. Group size: whole class, then pairs Materials: Instructional Strategy, pages 8-9; line of symmetry, page 10; standard pattern blocks or those provided in the Resource Section, pages 1-4 Before class: Gather a set of pattern blocks for each pair of students, or copy pages 1-4 from the Resource Section for each pair using the following color guide. Cut the blocks apart. Copy the line of symmetry, page 10, for each student. Prepare an overhead set of pattern blocks by coloring the blocks provided on pages 1-4 in the Resource Section according to the color guide. • hexagons - on yellow paper • trapezoids - on red paper • rhombuses - on blue paper • triangles - on green paper Directions: • Place the line of symmetry on the overhead, and follow the Instructional Strategy below to explain the activity to the whole group. • When the class is comfortable creating reflections, distribute a set of pattern blocks to each pair and a line of symmetry to each student. • Each student will create a design on one side of his/her line of symmetry. Then the students will carefully switch papers (or chairs) and duplicate the design on the other side of the line of symmetry. • Allow enough time for students to repeat this process several times, creating different designs. Move around the room as the students work to monitor their understanding and accuracy.
Questioning Technique Instructional Strategy Say: This line represents the line of symmetry. Ask: Who can tell me what that means? (the line that divides a symmetrical figure) Say: I am going to make a design using pattern blocks on one side of the line of symmetry. I want to use several different blocks, but not too many. I also want to be sure that my design lines up along the line of symmetry.
8
Gourmet Curriculum Press, Inc.©
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will create two-dimensional figures with lines of symmetry using concrete models
Initial Instruction—Part III Symmetry Questioning Technique Instructional Strategy Make a simple design (see the example) on the overhead with the pattern blocks being careful to place the edge of the design on the line of symmetry. Example:
Ask: Now that I have my design, I want to create a symmetrical design on the other side of the line. What does that mean? (You want to make a design on the other side of the line of symmetry that is a mirror image of the first one.) Ask: Remember the activity we did when we “reflected” each other’s body position? What did we learn about things that are “mirror images” of each other? (Mirror images are reflections of each other, which means that things are reversed.) Say: Let’s look at this design and see if we can make the same design on the other side of the line of symmetry. I’ll place the first block. (Choose a prominent block in the design, and place it in the appropriate position on the other side of the line. This will help students to identify the position of the other blocks.) Ask: Who can come place another block on the overhead so that it is symmetrical to this design? What does that mean? (The block will go in the opposite position on the other side of the line of symmetry.) Why? (Symmetrical designs are reflections of each other.) Allow a student volunteer to choose a pattern block and place it in the appropriate spot on the overhead. Ask: Does everyone agree that this is the correct spot? (If student does not place the block in correct symmetrical position, then discuss why it is incorrect. Then have him/her place it in the correct location.) Continue this process, allowing different volunteers to place other blocks on the overhead. Be sure to have the class check for accuracy after each block is placed into the design. When the design is complete: Ask: Does this look like a symmetrical design? (Yes.) Ask: How do we know that it is symmetrical? (The design on one side of the line is a reflection of the design on the other side of the line.) If the class experiences difficulty with this demonstration, create another design on the overhead, and follow the same procedure. If not, distribute the pattern blocks and line of symmetry pages to students. Gourmet Curriculum Press, Inc.©
9
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will create two-dimensional figures with lines of symmetry using concrete models
Initial Instruction—Part III—Line of Symmetry Symmetry
10 ( T )
Gourmet Curriculum Press, Inc.©
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will create symmetrical designs using M&M candies
Initial Instruction—Part IV Symmetry Optional Reading Activity
C Ap An
Teacher note: This book activity is similar to the pattern block activity from Initial Instruction–Part III. However, the use of M&M candy makes this a great symmetry review activity that the students are sure to enjoy! Group size: whole class, then pairs Materials: M&M Graph Paper, page 12; The M&M’s Brand Chocolate Candies Counting Book by Barbara Barbieri McGrath; large bag of M&M candies (or similar multicolored, round candy such as Skittles or Sweet Tarts); small plastic bags; paper plate to hold extra candy Before class: Copy the M&M Graph Paper for each student. Make extra copies, laminate them, and place them at a center. Obtain a copy of the book; purchase candy and separate it into small plastic bags with equal numbers of each color M&M in each bag (10-12 M&Ms per bag per student). Pour the remaining M&Ms onto the paper plate to use as “extras.” Directions: • Read the story aloud to the class. Discuss how the book uses M&Ms to teach counting and how students will use M&Ms to practice symmetry. • Distribute the bags of M&Ms and the M&M Graph Paper to students. Allow them to choose a partner. • The first partner will use his/her M&Ms to create a design on one half of his/her M&M Graph Paper. Emphasize that students should not use all their candies in the design. • When the design is completed, the second partner will use his/her M&Ms to make a symmetrical design on the other side of the line of symmetry. If the student does not have the correct color of M&Ms in his/her bag, he/she may get the appropriate colored one from the “extra” paper plate. • Designs must be checked for accuracy before partners can begin a new design. • When finished, students can keep their bags of M&Ms as a treat.
Gourmet Curriculum Press, Inc.©
11
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will create symmetrical designs using M&M candies
Initial Instruction—Part IV—M&M Graph Paper Symmetry Optional Reading Activity
12
Gourmet Curriculum Press, Inc.©
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will create two-dimensional figures with lines of symmetry
Initial Instruction—Guided Practice Symmetry
C Ap An
Teacher note: The class will create symmetrical “quilt squares” on the overhead using color tiles in a “follow the leader” type activity. If overhead tiles are not available, this can still be done as a whole group by having students circle around you while you create the design with standard color tiles. Group size: whole class Materials: quilt samples, transparency page 14 (optional); quilt square, transparency page 15; overhead or standard color tiles or those provided on transparency page 16; pictures or samples of symmetrical quilt squares (optional) Before class: If using the overhead: Gather overhead color tiles, or use those provided on page 16. Use permanent markers to color the squares, and then cut out all the tiles. If not using the overhead: Make 1 copy of the quilt square, page 15. Gather color tiles. Directions: • If available, display quilts or quilt squares. If not, use the quilt samples, transparency page 14. Discuss how many quilt squares are symmetrical. • Explain that the class will be creating symmetrical “quilt squares” using color tiles. Display the quilt square, transparency page 15. Point out the line of symmetry, and review how the design made on one side of the line must reflect the design made on the other side of the line in order for the quilt square to be symmetrical. • Place one color tile anywhere on the left side of the symmetry line. Be sure to place the tile so it touches the line. Have a student volunteer select an identical color tile and place it in the correct position on the right side of the symmetry line. Ask students if the tile is symmetrical to the first tile. • Select another color tile and place it on the left side of the symmetry line. Again have a student place an identical tile in the correct, symmetrical position on the right side of the line. Continue to have the class check for accuracy. • Challenge students’ understanding of symmetry by placing a color tile so it does not touch a previously placed tile. • Continue in this manner until each square has been covered with a color tile. Be sure to check for correct placement of each color tile before proceeding to the next one. • If necessary, remove color tiles and begin a new quilt square, so all students have the opportunity to place a tile on the quilt square. Extension: Students can follow this same procedure with a partner to create their own quilt squares. Simply make enough copies of the quilt square page for each pair of students, and gather enough color tiles for students to use.
Gourmet Curriculum Press, Inc.©
13
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will create two-dimensional figures with lines of symmetry
Initial Instruction—Guided Practice—Quilt Samples Symmetry
14 ( T )
Gourmet Curriculum Press, Inc.©
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will create two-dimensional figures with lines of symmetry
Initial Instruction—Guided Practice—Quilt Square Symmetry
Gourmet Curriculum Press, Inc.©
15 ( T )
Unit 2 – Lesson 2
Geometry
Student Expectation: Students will create two-dimensional figures with lines of symmetry
Initial Instruction—Guided Practice—Color Tiles Symmetry
16 ( T )
Gourmet Curriculum Press, Inc.©
Unit 2 – Practice #2
Geometry
Directions: Read each problem carefully. Decide which answer best completes the question. Show your work.
Problem #1
Problem #2
Students were asked to create a symmetrical building. Which of the following is not symmetrical?
Which of the following is true about this bracelet?
A F
B
G There is not a line of symmetry for this bracelet. It is not symmetrical. H All the figures on the bracelet are quadrilaterals (four-sided figures). J
C
The string holding the bracelet together is the line of symmetry.
There are more rhombuses than ovals.
Problem #3 The class was asked to draw the line of symmetry to show a heart is symmetrical. Rachel’s heart looked like this:
D
Did Rachel answer correctly? Why or why not?
Gourmet Curriculum Press, Inc.©
25 ( T )
Unit 2 – Practice #2
Geometry
Directions: Read each problem carefully. Decide which answer best completes the question. Show your work.
Problem #4
Problem #5
Carlos created the following design using pattern blocks on his paper.
Use crayons or colored pencils to make a design on the left side of the line. Then color the mirror image on the right side of the line.
Margo wants to make a mirror image of it. Which of the following shows Margo’s design? F
G
H
J Is your design symmetrical? How can you tell?
26 ( T )
Gourmet Curriculum Press, Inc.©
Unit 2 – Practice #2
Geometry
Directions: Read each problem carefully. Decide which answer best completes the question. Show your work.
Problem #6
Problem #7
Doron built half of a symmetric skyscraper with the blocks shown below. Which of the answer options could NOT be the completed building?
Use circles and squares to design a building that has a vertical line of symmetry. Draw the building in the space provided.
F
G Problem #8 Look at this object.
H
Is this a symmetrical object? Why or why not?
J
Gourmet Curriculum Press, Inc.©
27 ( T )
Unit 2 – Practice #2
Geometry
Directions: Read each problem carefully. Decide which answer best completes the question. Show your work.
Problem #9 Aaron laid some checkers on one half of a checker board. Which of the lettered squares on the other half would have a checker on it, if his design used the fold of the board as a line of symmetry?
Problem #10 Draw the other half of this symmetrical leaf.
B
D
A
C
Bonus: Why would it be important to understand symmetry if you were a dentist?
28 ( T )
Gourmet Curriculum Press, Inc.©
Unit 2 – Application #2
Geometry
Directions: Read each problem carefully. Decide which answer best completes the question. Show your work. Problem #1
Problem #3
Which shape below has only one line of symmetry?
Students crafted space ships using building blocks. Which of the following is not symmetrical?
Problem #4 Destiny has strings of beads hanging down from the door frame to her closet instead of a door. Each string looks like the one shown. Which of the following is true about this string of beads? Problem #2 Draw a symmetrical face in this circle.
Only the circles have more than 1 line of symmetry. The string is a horizontal line of symmetry. The string is a vertical line of symmetry. The circles and stars have only one line of symmetry.
Gourmet Curriculum Press, Inc.©
29
Unit 2 – Application #2
Geometry
Directions: Read each problem carefully. Decide which answer best completes the question. Show your work. Problem #5
Problem #7
Look at this card from a deck of cards.
Dr. Doolittle often referred to a special animal called the “Push me, pull you.” It was a symmetrical animal that had a head at each end. Carl used blocks to create the left side of this imaginary creature. How would the pattern blocks be displayed on the right side of the line of symmetry to complete the animal?
Is it symmetrical?
How do you know?
Problem #6 Which of the following is NOT true about symmetry? All squares are symmetrical. Symmetry is when one half of an object is a mirror image of the other half. A football is symmetrical. The line of symmetry is always horizontal.
30
Gourmet Curriculum Press, Inc.©
Unit 2 – Application #2
Geometry
Directions: Read each problem carefully. Decide which answer best completes the question. Show your work. Problem #8 Raquele and Ali are playing a game with 2-colored discs. Raquele creates a design with the tiles, and Ali has to create the other half of her symmetric design.
If you placed a mirror on the dark line in this design, which of the following would you see in the mirror?
Raquele’s design looks like this:
Color in the circles of their final design. Problem #9
Problem #10 Draw the other half of this ice cream cone so it is symmetrical.
Gourmet Curriculum Press, Inc.©
31
