District: Whitehall Central School District

District: Whitehall Central School District

School District Representative : Mac Petrequin

SLO Project Name: Product Design: Soccer Ball

Author: Mac Petrequin

E-mail:

Grade Levels: 10, 11

Subject Area: Mathematics

Learning Context: What must be true of the polygon panels in a soccer ball in the above graphic? In this project, students will investigate this design and find out why it seems to work so well. Compass and straightedge constructions of polygons inscribed in circles, Law of Sines, Ratio and Proportion, Use of Protractor.

Learning Standards: The following strands from the Geometry and Algebra II and Trigonometry curricula (Revised by NYS Board of Regents March 2005) are addressed in this lesson:

G.G.20 Construct an equilateral triangle (hexagon, pentagon) using a straightedge and compass, and justify the construction.

G.G.51 Investigate, justify, and apply theorems about arcs determined by the rays of angles formed by 2 lines intersecting a circle when the vertex is on the circle (2 chords).

G.G.52 Investigate, justify, and apply theorems about arcs of a circle cut by 2 parallel lines.

A2.A.57 Sketch and use reference angles for angles in standard position.

A2.A73 Solve for an unknown side or angle using the Law of Sines and Law of Cosines.

Assessments: Project for whole class or small groups to discover, plan the production of a paper soccer ball based upon measurements of a commercial standard.

Student Outcomes: Through the use of the above mathematic skills, students will produce the proper sized paper circles needed to inscribe polygons that will become the faces of the ball, and then stitch them together with stapler to produce the ball.

Procedures:

1) Students will need to discover the properties of the standard ball: the types of polygons, and number of each kind that make up the whole ball. The dimensions of the polygons should not be measured. It is the particular ratios that we want to discover that would apply to any size ball.

2)Using a 5” circle for the inscribed hexagons, students will determine central angle measure (60 degrees). Carefully find the center of each circle by drawing 2 or more diameters with light pencil. Use protractor to draw 6 such angles to form the vertices of the hexagon. Draw the hexagons (20), but do not cut off the remaining circle arcs; they will be needed to staple through. Students will determine that from a 5-inch circle will come an inscribed hexagon with 2.5 inch side lengths. Take care to measure the side lengths for congruence.

3)Students will now have to determine the necessary radius length for black pentagons as follows. Determine the central angle (72 degrees) and using this as the vertex angle of an isosceles triangle, find the base angle measures (54 degrees). Since the side of the triangle opposite the 72 degrees must be 5 inches, students can use the Law of Sines to find the other side measures, r (radii) as follows:

Sin 72 = Sin 54

5 rr = 4.25 (rounded)

And the ratio of pentagon radius to hexagon radius is 4.25 to 5, or 17/20, or 85%.

4)Students will make the 4.25 inch circles for the pentagons (12). Same procedure as hexagons but with 5 carefully measured 72 degree angles and carefully found circle centers.

5)Polygons can now be made by folding “tabs” along the inscription lines. Begin the stitching of the ball with a group of 5 hexagons and one pentagon. The pentagon will be at the center, with the 5 hexagons clustered around. Match the tabs and staple as close to the actual fold lines as possible. Add 5 new pentagons to the edge of the cluster, and continue the addition of the new “panels” of the ball. Some flipping and folding back and forth is necessary to get good fits. The last panel can be glued in.

Instructional Modifications: Note: ERROR IN MEASUREMENT IS INEVITABLE! But can be minimized with systems. Introduce students to the concept of quality control through precise measurements, acceptable tolerances, and people checking other people’s work. Don’t be afraid to re-fold or even throw out bad panels. The use of glue instead of staples will yield a “tighter” ball, at the cost of wait time as the glue sets.

When we did the soccer ball project in my kids, we used carboard 9-inch baker’s cake bottom forms, which made for a nice large rigid ball. (An afterthought was that we could have used a flexible silicon calk to seal the seams.) Other such commercial paper and cardboard circles may be available and they save on having to cut out circles drawn from a compass. If the centers are marked, so much the better. Different colors, paint, autographs, and decorations limited only to the imagination. My kids painted our ball, signed it and presented it to their soccer coach on fall awards night.

Time required: Depends on how many are working and how many balls you want to make. Ours was made during free periods and after school. I would guess 2-4 class periods.

Resources: (materials) Circles to be cut from paper or cardboard: 20 of radius = r, 12 of radius = .85 r, compasses, protractors, scissors, rulers, staples, glue, perhaps tape.