The Elementary School Math Project. Struts n Stuff

Transcription

1 The Elementary School Math Project Struts n Stuff Math Grows Up (Patterns/Relationships) Objective Students will identify the relationship between the number of sides in a regular polygon and the number of struts (diagonals) needed to make each polygon rigid. Overview of the Lesson Students build polygons using strips of paper and paper fasteners to explore the attributes of triangles, rectangles, pentagons, and hexagons. From their explorations they discern that triangles form the only rigid polygon. They are challenged to determine the least number of struts or supports they would need to add to the rectangles, pentagons, and hexagons to make them rigid. Students then record their findings in a chart and look for patterns. They use the pattern to state a rule that shows the relationship between the number of sides in a polygon and the number of struts needed to make it rigid. To further reinforce these concepts, students make a graph showing this relationship. Students use the graph to predict the number of struts needed to make different polygons rigid. Finally, the class discusses the relationship between the number of sides of a polygon and the number of triangles formed by the struts. Students look for a pattern in order to make a generalization. Page 1

2 PBS MATHLINE Materials Teacher: Large summary chart Large sheet of graph paper Polygon models made from paper strips and fasteners (Activity Sheet: Template Page - Strips ) Each Student: 25 strips of paper, 1 cm x 7 cm, with one hole punched close to each end (using a heavier weight, i.e.: manila folders will be easier for students to manipulate) One plastic bag for paper strips 10 strips of paper, 1 cm x 30 cm, with holes punched on one end of each strip (see Activity Sheet: Template Page - Struts ) 25 paper fasteners (1/2 inch) Scissors 2 sheets of graph paper (Activity Sheet: Graph Grids ) Several hole punchers (Students can share) Student journal Procedure Distribute plastic bags containing the 25 1cm x 7cm strips of paper and 25 paper fasteners to each student. (See Activity Sheet: Template Page - Strips ). Link the strips of paper together with the paper fasteners to demonstrate how to construct polygons using these materials. Have the students make a triangle, rectangle, pentagon, and hexagon in this manner. Allow the students ample time to explore these shapes. Encourage students to share their observations with the class. Be sure students notice that the triangle is the only shape that is rigid while the other polygons collapse and do not hold their shape. Challenge students to find ways to make the rectangle rigid like the triangle using struts or supports. Demonstrate how a strut could be added by cutting the paper strip to the necessary length, punching a hole in the end of the strut, and securing it with a paper fastener. Distribute scissors, paper hole punchers, and 10 1cm x 30cm paper strips to the students. (See Activity Sheet: Template Page - Strips ). Tell the students that there are four rules for the challenge. ESMP Struts n Stuff Lesson Guide Page 2

3 PBS MATHLINE RULE 1. RULE 2. RULE 3. RULE 4. The struts must go from vertex to vertex. The struts may not cross each other. The polygon must keep its original number of sides. Use the fewest number of struts as possible to make the shape rigid. After students have been given enough time to experiment with the shapes and struts, encourage them to share their findings. Students should discover that one strut can be placed diagonally across the rectangle to make it rigid. Have students make a chart with two columns in their journal. Students should label one column, Number of Sides in the Polygon, and the other should be labeled, Number of Struts Needed to Make It Rigid. Using chart paper, demonstrate how to fill the chart using the rectangle as an example. Instruct students to follow the same procedure they did with the rectangle to determine the fewest number of struts necessary to make the pentagon and hexagon rigid. (see Activity Sheet: Possible Solutions ). As students explore with the paper manipulatives, have them record their findings in the chart and write observations about the discoveries. Those students who finish early may explore polygons with a larger number of sides. When students have completed the tasks, have them discuss their findings. Focus the students attention on patterns in their data. Ask the students to predict the number of struts that would be needed for a heptagon, an octagon, etc. Challenge students to find a generalization or rule they could use for determining the number of struts needed to make any regular polygon rigid. (If n equals the number of sides in a regular polygon, then the number of struts needed to make the shape rigid is n-3). Ask students to determine the number of struts that would be necessary for a regular polygon with 100 sides. (Ans: 97). Distribute graph paper to each student. (See Activity Sheet: Graph Grids ). Guide the class as needed in constructing a graph depicting the number of sides in a polygon and the number of struts needed to make it rigid. Have the class analyze the data graphed. Students should use the graph and the rule to find the number of struts for a given number of sides and find the number of sides given the number of struts. Have students make a second chart in their journals recording the Number of Sides in the Polygon, and the Number of Triangles Formed by the Addition of the Struts. They may use their constructions to gather this data. After the students have recorded their findings, discuss any patterns they may have noticed. Lead students to formulate the generalization or rule that if n equals the number of sides of the regular polygon, the number of triangles formed is n-2. A second graph can be made and compared to the first graph. ESMP Struts n Stuff Lesson Guide Page 3

4 PBS MATHLINE Mathematically Speaking... Though this lesson is an interesting investigation for students to explore the attributes of polygons, it also provides students experiences with beginning algebra. As students look for patterns, form generalizations, and plot coordinate pairs, they are beginning to think algebraically. Having students express the rules or generalizations concerning the relationship between the number of sides in a polygon and the number of struts needed to make it rigid in algebraic terms further reinforces this concept. Note that the struts are diagonals, thus students are discovering some of the relationships between polygons and the number of diagonals. Extensions & Connections Have students search for examples of triangles used in construction for support at home and school. They can refer to books and magazines that show architecture and engineering, particularly bridge building. Children can also experiment with tinker toys, straw and toothpick building, or other construction projects. Invite an architect or engineer to class to discuss the importance of the triangle in architecture. The Internet has some examples of triangles as used in construction. Have students build a structure out of newspaper rolls as introduced in the video. Resources Educational Development Center, Structures: Teacher s Guide for Structures Delta Education, Hudson, New York. Shape and Size 2, The Nuffield Project John Wiley and Sons, ESMP Struts n Stuff Lesson Guide Page 4

5 PBS MATHLINE Ideas for Online Discussion (Some ideas may apply to more than one standard of the NCTM Professional Standards for Teaching Mathematics.) Standard 1: Worthwhile Mathematical Tasks Students are encouraged to use patterns to make generalizations in the form of rules or formulas, and then display and interpret their findings in the form of graphs. Share some of the ideas you have used to promote sound, significant mathematics. Standard 2: Teacher s Role in Discourse When students were in groups, how did the teacher probe students to explain their thinking to others? What techniques do you use? Standard 3: Students Role in Discourse Were students focused on making sense of the problem in terms of mathematical ideas? How did students use different ways of representing and explaining their findings? Standard 6: Analysis of Teaching and Learning When you are taking a risk by designing activities to teach a mathematical concept that will be unusually challenging to students, how do you assess the effectiveness of your teaching? In order to teach in a way that has greater benefits and potentially greater risks, who, if anyone, should you involve? ESMP Struts n Stuff Lesson Guide Page 5

6 PBS MATHLINE ESMP: Struts n Stuff Template Page - Strips

7 PBS MATHLINE ESMP: Struts n Stuff Template Page - Struts

8 PBS MATHLINE ESMP: Struts n Stuff GRAPH GRIDS

9 Struts n Stuff Possible Solutions Square 1 strut (diagonal) Pentagon 2 struts Pentagon 2 struts Hexagon 3 struts Hexagon 3 struts PBS MATHLINE ESMP: Struts n Stuff