Logo type; soccer, anyone? Donna Bearden.
It seems as if everyone has been bitten by the soccer bug, even the Logo turtle!
It all started with a fourth grade boy from Venezuela who lives and breathes soccer. He is in the United States temporarily while his dad goes to school at the University of Dallas. Carlos belongs to the computer club at Forrest Park elementary school in Irving, TX.
Looking for a way to combine his love for soccer with his involvement with a computer, we began looking at soccer balls instead of just kicking them. What we discovered was an intriguing pattern of pentagons and hexagons. Would the students, who had already learned to teach the turtle to draw hexagons and pentagons, be able to put them together in the soccer ball pattern?
The computer club was divided into teams. Each team was challenged to teach the turtle to draw the pattern of a pentagon surrounded by five hexagons-- not as easy a task as it first seems.
One team defined a procedure for a pentagon and another for a hexagon and then started trying to figure out how to put them together. The other team figured they could define a hexagon, then make a ring of hexagons, and end up with a pentagon in the center. After all, that is the way it looks on the ball.
Their looks of triumph quickly changed to puzzled ones when they realized the ring of hexagons had, of all things, a hexagon in the center.
How could that possibly be when they could look at the soccer ball and see a ring of hexagons with a pentagon in the center? Something very strange was going on. The difference, of course, was that the soccer ball was a three-dimensional object that they were attempting to draw in two dimensions.
The first week ended in a tie--0 to 0, with everyone puzzled but not discouraged. They knew there must be a way to do it and were eager to try again.
The second week we brought an old soccer ball with us. It had long since seen its last game, but we gave it a chance to score one more goal. By cutting it apart, the kids could see how the pattern had to be split to enable it to lie flat. (If you don't have an old soccer ball, try using the peel of an orange or grapefruit.)
Once they saw the splits that allowed the curve of the ball to flatten out on the table, they returned to the computers more determined than ever. Most of them had to leave at the end of the hour, but Carrie Simms, a fourth grader, stayed long after everyone else had left and figured out one way to draw the pattern. Here is her solution:
REPEAT 5 [FD 20 RT 72]
REPEAT 6[FD 20 RT 60]
REPEAT 4[LT 132 FD 20 RT 60 HEX]
The Teachers Try
At the Microcomputers in Education Conference at Arizona State University in March, we presented the soccer ball puzzle to a group of teachers, most of whom were first time Logo users. Most of the adults were as puzzled but as determined as the students had been. John Onacki, an Arizona math teacher, came up with this solution almost by accident:
TO DESIGN :R :D
REPEAT 5[POLY :R :D FD :D LT 72]
TO POLY :R :D
REPEAT :R[FD :D RT 360/ :R]
By giving the command DESIGN with an input of 6 for :R and any number for :D, you create the soccer ball pattern.
Once the kids in the computer club were able to draw the pattern on the screen, we decided to take the two-dimensional drawing and turn it back into a three-dimensional soccer ball. Because there are 12 pentagons on a soccer ball, we printed out 12 copies of the pattern. The "players' then cut out each pattern. Rather than cut out the long, narrow triangle between each hexagon, they cut only one side of the triangle and brought the sides of the two hexagons together, securing them with tape.
They could then experience the curve of the ball beginning to take place. By overlapping the 12 pieces of the ball, they taped together one fantastic paper soccer ball. They had completed the circle, going from three dimensions to two dimensions and back to three.
But that wasn't the end. It had started as a cultural experience and through some creative imagination, ended as quite another. Just before we taped the last piece of the ball in place, one of the kids said, "Hey, let's fill it with candy and turn it into a pi nata!' And that is exactly what we did.