Symmetry has bothmathematical significance and visual appeal, and

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1 SHOW 116 PROGRAM SYNOPSIS Segment 1 (1:36) MATHMAN: SYMMETRY In this video game, Mathman confronts a variety of polygons and must select only those that have a line of symmetry. Flip and Fold: Seeing Symmetry Segment 2 (1:03) SYMMETRY PATTERNS An animated short piece illustrates pleasing patterns that can be generated from simple shapes through the use of symmetry. Segment 3 (4:17) GENERAL MATHPITAL: ASYMMETRIOSIS This episode of GENERAL MATHPITAL finds a patient (an irregular shape) suffering from asymmetriosis a lack of symmetry. The doctors suggest ways of chopping up the shape and reassembling it so that it will be symmetric. The cure they finally agree on needs only one cut and one reattachment. Segment 4 (2:22) SQUARE ONE CHALLENGE: CUTTING OUT F S In this game show excerpt, the challenge question involves a rectangular piece of paper that has been folded twice; contestants must imagine two symmetries simultaneously to figure out what the unfolded paper will look like. Maria and Buster review the answers that were offered for the CUTTING OUT F S question to demonstrate that two of them are not symmetrical around the horizontal fold. INTRODUCTION Symmetry has bothmathematical significance and visual appeal, and has been used in art and architecture throughout history. This module concentrates on symmetries that are reflections. A figure is symmetric if it looks exactly the same when it is flipped over a line. Students are asked to imagine changing geometric figures to make them symmetrical. This kind of spatial and geometric reasoning is emphasized in the NCTM s Curriculum and Evaluation Standards for School Mathematics (K 4, 48ff and 5 8, 112ff). BEFORE VIEWING Review with your students what it means for an object or picture to be symmetric around a line of symmetry: A shape is symmetric if there is a line around which it can be flipped and still look exactly the same, in exactly the same position. (Such a line is called a line of symmetry.) Discuss some examples that can be found in the classroom. For example, an ordinary rectangular (non-square) piece of paper has two lines of symmetry because if you flip the paper as shown, it looks exactly the same; you can t tell that it s been flipped. Stress that an object has to look exactly the same to qualify as symmetric. This letter A is not symmetric. If you flipped it around a vertical line drawn through the point on the top, the thicker diagonal line would be on the left instead of the right. Use an example like this to introduce the word asymmetric, which means not symmetric

2 MathTalk TUNING IN GENERAL MATHPITAL STOP THE TAPE after the first doctor explains his proposal for making the patient symmetric. Ask if anyone sees another way to do it. There will be an opportunity to explore this further later on. AFTER VIEWING SQUARE ONE CHALLENGE STOP THE TAPE after the question is posed so that your students can have some time to think about, discuss, and/or model their answers. STOP THE TAPE on each of the unfolded answers that Maria and Buster review so that students can study and discuss what lines of symmetry they have. Point out that the correct answer has two lines of symmetry, and the incorrect answers have only one. SYMMETRY PATTERNS Return to this animation and STOP THE TAPE at various points. Discuss whether the pattern on the screen is symmetric, and if so, where the line or lines of symmetry are. Note that sometimes flipping the pattern around a line reverses the black and white pattern, which means that it is not symmetric around that line. On this pattern from the segment, for example, ask students to find the one line of symmetry the diagonal from top left to bottom right. The other diagonal is not a line of symmetry. activity SYMMETRY SURGERY: EXPLORING LINES OF SYMMETRY This activity is based on the situation described in the GENERAL MATHPITAL segment. Students explore and test their own ideas about how to develop symmetrical shapes by designing shapes and rearranging them on a grid. 1. Use the 12 squares to review what happened with the patient before the operation. 2. Recall that the cure proposed at the end of the segment required one square to be cut and reattached, as shown below on the right. MATERIALS FOR THE TEACHER: Some way of recreating the patient in the sketch either 12 squares of construction paper (all the same color) and masking tape, OR square sticky-backed note paper, OR 12 small transparent squares that can be used on an overhead projector FOR THE STUDENTS: graph paper OR square sticky-backed note paper Before operation Proposed cure 100

3 GEOMETRY 3. Ask if anyone can see other ways to create a symmetric figure. Tell students that all 12 squares must be used, and they must be reattached so that complete edges are connected. Discuss students ideas and suggestions. 4. Display the latest fee schedule from General Mathpital: What would the doctors charge the patient for the operation that they ended up doing? (They made one cut and created one vertical line of symmetry, so the charge would have been $100 $30, or $70.) What about for the operation that was first proposed in the segment? (They made six cuts, resulting in one vertical line of symmetry, so the total bill would have been $100+(6x 30), or $80. That is, the hospital would have had to pay the patient $80. No wonder they abandoned that method!) 5. Work with the entire class to calculate the bill for several different operations that students suggest. Or ask volunteers to propose operations on the patient, while the other students calculate the bill individually or in pairs. GENERAL MATHPITAL FEES FOR SURGERY One line of symmetry, horizontal or vertical: $ 100. An additional line of symmetry, horizontal or vertical: 50. One diagonal line of symmetry 150. each Reattachment (suturing) FREE Any cut, per edge 30. each This activity can also be presented as a game in which points are scored for making shapes with various lines of symmetry. Use the numbers from the fee schedule above, calling them points instead of dollar amounts. The object of this game is to get the largest number of points. 6. Get students to work in pairs to figure out who can do an operation that results in the highest fee. Give them plenty of time to explore this problem. To get students started, suggest an operation like this, which gives the patient one diagonal line of symmetry: The fee for this operation would be $150+(3x $30), or $60. cut cut cut This is the most expensive operation I came up with $210 but there MIGHT be one that costs even more! 7. Encourage your students to make up their own asymmetrical patients and challenge their classmates to find clever ways to transform them into symmetrical objects. 101

4 keep thinking figure A MathTalk SYMMETRY THAT WORKS LIKE MAGIC Harry Blackstone, a famous magician, does a magic trick that is based on symmetry (or, actually, a lack of symmetry). This trick is easy to do with your class. Put a row of about 10 or 12 ordinary paper clips on the overhead projector, like figure A. The trick depends on the fact that paper clips are not symmetric around the vertical midline. When you first put down the paper clips, make sure the one on the right end is oriented differently from all the rest. Ask a student to think of a number between one and five. Turn your back and have the student slide that number of paper clips from the left side of the row to the right side, moving them all over to the left so that you can t tell how many have been moved. The paper clips might then look like figure B. When you spot that paper clip after the student has moved some, it s easy to count how many were moved. After some mind-reading mumbo-jumbo, you announce the number that the student was thinking of! In figure B, it s three, because three paper clips have been moved to the right of the turned clip. figure B As students catch on to the trick, ask them not to reveal the secret to the rest of the class. They can do the trick too! MENTAL UNFOLDING You and your students can create challenging variants of the paper-folding question from the SQUARE ONE CHALLENGE segment. Start with one similar to the one in the show. Ask students to draw a sketch of what the paper will look like when it is unfolded. Then unfold it and see. (It will have two lines of symmetry.) Next fold a piece of paper into sixths. Point out that this one has only one line of symmetry when it is unfolded. Discuss with students what they think will happen if they keep increasing the number of folds. 102 Look at odd versus even numbers of times you folded! FOLD IN FOURTHS 1. right edge to left edge 2. bottom up to top edge 3. cut out a UNFOLDED: FOLD IN SIXTHS 1. left side first 2. then right side 3. then bottom edge up to top 4. cut out a

5 GEOMETRY A REGULAR HEXAGON is a hexagon whose sides are all the same length and whose angles are all the same size. FOR THE PORTFOLIO Students can explore the different kinds of symmetries that can result from gluing together regular hexagons (as opposed to squares). Ask students to extend the GENERAL MATHPITAL idea to asymmetrical patients who are composed of regular hexagons. Devise some examples of patients who look terribly asymmetrical, but who can be cured by making some clever cuts and reattachments. For example, a block of three hexagons in this patient can be cut off along the heavy lines, and reattached as indicated. The new figure has a hexagonal hole in the middle and two lines of symmetry. Ask students to explain how figures made of regular hexagons are different from ones made of squares. BEFORE AFTER Notice that more lines of symmetry are now possible, since a regular hexagon has six lines of symmetry. (A square has only four lines of symmetry.) CURRICULUM CONNECTIONS Symmetry plays a role in areas of mathematics besides geometry. For example, the fact that multiplication is commutative means that the numbers in the ordinary multiplication table are symmetric around the main diagonal. Symmetry has application in probability too for example, the numbers on a number cube are equally likely to occur when the cube is tossed because of the cube s symmetries. X Math Talk CONNECTIONS FAMILY INVOLVEMENT Get students families involved in math! Send copies of reproducible page 104 home with your students so they can play it with their families. NUMBER SENSE Both Sides of Zero: Playing with Positive and Negative Numbers Calculations using positve and negative numbers 103

6 NAME MathTalk Family Page 1995 Children s Television Workshop FLIP AND FOLD: SEEING SYMMETRY 104 HOW TO PLAY Cut out the Symmetri-scope. Then carefully cut out the small square inside it to make a window. Each player in turn places the Symmetri-scope on the design to reveal a portion of the design that has one or more lines of symmetry. Players get one point for each line of symmetry they find. When the players have agreed on the score for that turn, the player traces the outline of the square onto the design. Nothing inside that square can be used again by any player. When no player can find any more squares with symmetries, the game is over and the player with the most points wins. SYMMETRI-SCOPE Cut out this center square to make your scope.