THE MORE, THE BETTER. Getting Ready. The Activity. Overview. Introducing

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1 THE MORE, THE BETTER Getting Ready What Yo ll Need Tangrams, set per child 7-piece Tangram Sqares, page 99, per grop (optional) Overview Children search for all the convex Tangram shapes that can be made with different nmbers of pieces that represent the same fractional part of the seven-piece Tangram sqare. In this activity, children have the opportnity to: create and compare a variety of polygons recognize convex and concave polygons assign fractional amonts to each Tangram piece se eqivalence work with fractions as an area model GEOMETRY NUMBER Comparing Classifying Polygons Fractional eqivalence Spatial visalization The Activity Introdcing Show children a sqare made from the seven Tangram pieces. Tell children that the 7-piece sqare has a vale eqal to. Ask them to find the fractional vale of each piece. Trace the sqare, showing the otline of each piece. /6 /8 /8 As volnteers give the vale of each piece and explain their reasoning, record the fractional amont in the appropriate place on the tracing. /4 /6 /8 /4

2 On Their Own How can yo make a Tangram shape that has a certain area? Work with a grop. Each of yo make a shape that has no indents and whose area is /2 the area of the sqare made with all 7 Tangram pieces. Okay Not okay Compare yor shapes. Record only the ones that are different. Cont the nmber of pieces sed. Make as many more shapes as yo can that are also /2 the area of the 7-piece sqare, bt each time se a different nmber of pieces. Find the least nmber of pieces possible, the most nmber of pieces, and all the possibilities in-between. Do this activity several more times. Each time, instead of /2, select one of these fractional amonts to be the area of yor shape: /8, /4, 3/8, 5/8, 3/4. Look at yor grops soltions. Make a list of what yor grop observes. The Bigger Pictre Thinking and Sharing Discss one fractional amont at a time. Have volnteers share the shapes they fond and what they noticed. Use prompts sch as these to promote class discssion: How did yo decide which Tangram pieces to se for yor shapes? How did yo know yor shape was the correct fractional amont of the 7-piece sqare? Were some fractional amonts easier to create than others? Which fractional amont cold be made in the most ways? Why? Which fractional amont cold be made in the fewest ways? Why? Did yo ever find it impossible to bild a shape with no indents that represented the fractional amont yo needed? Explain. Extending the Activity. Give children the labeled Tangram sqare on p. 99 or have them make their own. Show them how to represent the sqare with a nmber sentence sch this: =. Now have children write nmber sentences for each of the shapes they made when they did the activity. 2. Have children repeat the activity bt allow concave shapes.

3 Teacher Talk Where s the Mathematics? As they bild shapes whose areas are a specific fractional amont of the area of the 7-piece sqare, children deepen their nderstanding of eqivalence. They also focs on a geometric attribte convex vs. concave often sed to describe, identify, and classify polygons. One-half of the area can be shown in more ways than any other fractional amont. And, while there is only one way to show one-half with two pieces, there are many ways to show it with five pieces. Children might start with the two large Tangram triangles, then sbstitte two small triangles for one of them, thereby making a shape with three pieces. Or, they may sbstitte instead one medim triangle and two small triangles to prodce a forpiece shape that is either a triangle or one of the qadrilaterals shown below. Some children think in terms of shapes only, sch as medim triangle eqals 2 small triangles or large triangle eqals the 3 smaller triangles. Others se the fractional names of the pieces. For example, they will refer to the large triangles as the 4-pieces, the medim triangle, the sqare, and the parallelogram as the -pieces, and the two small triangles as the 6-pieces. It is not necessary for every grop to have investigated every fraction before starting a class discssion. Here are some soltions for each of the other fractional amonts. Shapes with the area of the 7-piece Tangram sqare

4 Shapes with 4 the area of the 7-piece Tangram sqare Shapes with 3 the area of the 7-piece Tangram sqare Shapes with 5 the area of the 7-piece Tangram sqare Shapes with 3 4 the area of the 7-piece Tangram sqare Another way to focs on the data is to have the class organize the data in a chart like this. Nmber of Pieces Fractional Amont This format helps children find missing shapes or discover why some are impossible. Althogh the parallelogram, the sqare, and the medim triangle together represent 3 of the area of the 7-piece sqare, all the shapes that can be made with them are concave. Therefore, in this activity, 3 can only be represented with for pieces. Similar reasoning can explain why 3 4 cannot be represented with fewer than five pieces. Considering 5 first as , then as , gives the minimm nmber of pieces that works. Sbstitting the small triangles for one medim triangle confirms that = 2 6 at the same time as it prodces an acceptable five-piece shape.

5 7-PIECE TANGRAM SQUARES