For many years, geometry in the elementary schools was confined to

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1 SHOW 118 PROGRAM SYNOPSIS Segment 1 (1:39) MATHMAN: RECTANGLES Mathman is told to eat only the rectangles that appear on his video game board. He notes that rectangles must have four right angles (and that therefore all squares are rectangles). Getting into Shapes: Playing with Polygons INTRODUCTION Segment 2 (2:12) SQUARE ONE CHALLENGE: SLICING A TETRAHEDRON In this excerpt from the game show, contestants visualize the result of slicing a regular tetrahedron (a solid shape with four faces, all of which are equilateral triangles). One way of slicing creates a new triangular face. Slicing through the midpoints of four edges creates a square. Segment 3 (6:40) DIRK NIBLICK: BANANZA Fearless leader of the Math Brigade Dirk Niblick once again comes to the rescue in a mathematical situation. To resolve a family feud, he must figure out how to divide a triangular field into four (and later three) parts of the same size. For many years, geometry in the elementary schools was confined to learning the names for various kinds of shapes, along with formulas for finding the perimeters and areas of certain figures. Recently there has been more emphasis on exploring different aspects of shapes how they fit together to form new shapes, how they can be subdivided into other shapes, symmetries of shapes, patterns involving shapes, and how shapes in two and three dimensions relate to each other. This module provides a range of ways to explore shapes. BEFORE VIEWING Many students have the mistaken idea that a rectangle must have two sides of different lengths; the MATHMAN segment reminds them that a square is also a rectangle. Many students also think that squares and rectangles must be drawn so that their edges are parallel to the edges of the paper or TV screen. That s why all the rectangles in this Mathman game are drawn so the edges aren't parallel to the edges of the screen. Introduce the words face, edge, and midpoint (of a line segment) if your students are not already face midpoint familiar with them. Rather than a formal definition, give some examples, using a rectangular box. } edge 111

2 MathTalk DURING VIEWING AFTER VIEWING SQUARE ONE CHALLENGE STOP THE TAPE after the question is asked so that your students can have a moment to think about and discuss the problem. Ask students if they can imagine other ways of dividing up the triangular plot into three or four pieces of equal area. There are many ways to do this, some of which are shown below. Notice that the pieces don t have to have the same shape to have the same area. activity SLICING SOLIDS The SQUARE ONE CHALLENGE question, in which a slice is made through the midpoints of four edges of a regular tetrahedron, is an example of how a two-dimensional shape can be created from a threedimensional one. The kind of mental imagery that is needed to see what the new shape will look like is difficult for many students. It can be developed by manipulating real three-dimensional shapes (not just pictures of them), predicting what the outcome of various slices will be, and then comparing the actual results with one's predictions. 1. If you have made a demonstration tetrahedron, you can use it to be sure that everyone understands the question. Mark three edges and show how a slice would result in a triangle. Then show how a slice through the midpoints of four edges would create a new square face. 2. Pass out a copy of the reproducible for each person. Ask what shape could be made by cutting out the figure and folding it on the solid lines. (a cube) Everyone should then cut out the figure, fold it up into a cube, and hold it together without taping the edges. MATERIALS FOR EACH GROUP OF TWO OR THREE STUDENTS: copies of reproducible page 116 (if possible, copy this onto construction paper, lightweight card stock, or oaktag) scissors two or three pieces of paper transparent tape rulers FOR TEACHING DEMONSTRATION: helpful but not essential: a model of the tetrahedron that was used in the SQUARE ONE CHALLENGE question. You can make one by taping together four equilateral triangles of the same size. 3. Ask students what would happen if you took a knife and made a straight slice through the dashed line. (One corner of the cube will come off, creating a hole. If this were a solid cube, a new face would be created.) Ask students what shape they think the new face or the new hole will be. (a triangle) Some students find it helpful to imagine putting a roof over the hole. 112

3 GEOMETRY What size and shape would the roof have to be? (It s an equilateral triangle with each side about 1.5 inches long.) You can ask your students to sketch (on a separate piece of paper) what the shape would look like. 4. Now discuss what would happen if you started with the original cube and cut along the tire-track line. Ask students to sketch what they think the roof would look like. (This time it will be a trapezoid. If your students don t know that word, they may describe the shape as a four-sided polygon with two parallel sides and two non-parallel sides. The concept is more important than the vocabulary!) 5. Ask them to imagine the shape that would be created by slicing on the dotted line. This is harder partly because no matter how you turn the cube, you can see only part of the shape. How many sides will it have? (six) Again students can sketch what they think it will look like. The shape is a hexagon (six-sided polygon). 6. Now assign one person in each group to cut along one of the three different kinds of lines, and then tape together the edges of what remains of the cube. Each person then should cut another piece of paper that will form the roof for the resulting hole. (One way to do this is to place the hole face down on a sheet of paper. Mark the vertices of the shape [a triangle, trapezoid, or hexagon], and complete the shape by connecting the vertices with a ruler. Then cut out the roof and tape it over the hole.) Students will see that cutting along the tire-track line creates a trapezoid and cutting along the dotted line creates a regular hexagon. Have them compare their predictions to the results and discuss. How did students know what the shape would be? Can they explain any surprises they found? 7. As an extension, you can ask if there is a way of making a slice that will result in a five-sided face. (Yes, there is. One way is shown here.) 113

4 keep thinking MAPPING THE RANGE As a group project, it can be fun to make a large drawing of the Cartwrong ranch and see how many equal subdivisions can be carried out. 80 cm Start with a very large piece of paper and draw an equilateral triangle that s 80 cm on a side. Draw one line segment that is 80 cm long, and then use two meter sticks to locate the third vertex of the triangle. (We chose 80 cm because 80 is easy to divide repeatedly by 2, getting 40, 20, 10, 5, ) For the sake of simplicity, call the area of the triangle one unit. (It s actually about 2771 square cm.) Now draw an inner triangle by connecting the midpoints of each side the points that are 40 cm from each vertex like this: This splits the triangle into four equal-sized or congruent pieces. Look at the land that each of the three brothers gets. At first, he gets one of the four triangles ( of a unit) and the middle triangle is left over to be divided again. Each brother gets (in addition to what he already has) of the new inner triangle. That s of, or, and again a middle triangle is left over. Now draw a still smaller inner triangle, with sides that are 10 cm long. At this stage each brother gets an even smaller triangle, which is of of,or of a unit. As you add more and more of these tinier and tinier triangles, what fraction of the entire area is each brother getting closer and closer to? ( of the whole area) 80 cm cm MathTalk Did someone in that sketch say something about a RECURSION? A recursion is simply an endlessly repeated mathematical process in which each step depends on previous steps. In this case, the process is splitting up the middle triangle into four smaller triangles, giving the three outer ones to the three brothers, and repeating the process on the even smaller middle triangle. Students who are familiar with decimals might want to verify this with a calculator, like this: =.25 + = = = = and so on 114

5 GEOMETRY FOR THE PORTFOLIO Students may want to write up this investigation to include in their math portfolios. The same sort of slicing that we did with a cube in the main activity can be done with any solid. A good one to experiment with is the regular octahedron, which has eight faces, all equilateral triangles. Individually or in small groups, students can build regular octahedrons, imagine what the new face would look like if various slices were made, and then construct the resulting solid shapes. (It is possible to create hexagons, pentagons, and different kinds of quadrilaterals by slicing an octahedron, as the drawings here suggest.) hexagon You can make a regular octahedron by folding up something that looks like this: CURRICULUM CONNECTIONS The ideas in this module are widely applicable in other parts of the geometry curriculum. As you and your students encounter various shapes, look for ways in which they could be combined or split up into other shapes. As you build students spatial understanding in this way, the focus should be on the ideas rather than on the vocabulary. Use technical terms (like trapezoid or tetrahedron ) only when they are needed to communicate an idea clearly. quadrilateral pentagon MathTalk CONNECTIONS GEOMETRY Measured Steps: Measuring Length Dividing shapes into smaller pieces 115

6 NAME MathTalk GETTING INTO SHAPES: PLAYING WITH POLYGONS Cut along the solid outer line and fold into a cube. Imagine what shape would be created if you cut along the dotted line, the dashed line and the tire-track line. Then try it and see. Tire-track line Dashed line Dotted line Children s Television Workshop

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