By Harold; diSessa, Andrea Abelson
Turtle Geometry provides an leading edge application of mathematical discovery that demonstrates how the powerful use of non-public desktops can profoundly switch the character of a student's touch with arithmetic. utilizing this booklet and some uncomplicated laptop courses, scholars can discover the houses of house by way of following an imaginary turtle around the monitor. the idea that of turtle geometry grew out of the brand staff at MIT. Directed through Seymour Papert, writer of Mindstorms, this staff has performed broad paintings with preschool little ones, highschool scholars and college undergraduates.
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Additional info for Turtle Geometry: The Computer as a Medium for Exploring Mathematics
4. Consider: How could a turtle, while walking along drawing a POLY, know when the figure has been completed? ) Thinking locally, the turtle knows only two things, position and heading. Neither of these is truly local, for to measure them usually involves a coordinate system. But the one locally computable quantity we know about-total turning-can do the trick. The closed-path theorem says that if the path closes, then total turning must be a multiple of 360°. How about the converse: If the total turning reaches a multiple of 360°, will the path be closed?
8. [DD] Show that a simple spirolateral can never be unexpectedly closed. [H] 9. [DD] Can INSPI produce unexpectedly closed figures? 10. [P] Invent some disguised looping programs like INSP!. Give a formula for the total turning of the basic loop in terms of the inputs to the procedure. Find inputs that draw figures with simple symmetry. Find inputs that draw unbounded figures. Determine whether any of these figures are unexpectedly closed. 43 Symmetry of Looping Programs 11. When we summarize turtle paths as state-change operators, the closed paths are precisely those operators that leave the turtle's state unchanged.
5. Rewrite the POLYSTOP program recursively, so that it doesn't use the REPEAT command. [A] 6. What is the sum of the interior angles of an n-gon? What is the interior angle of a regular n-gon? Show how these formulas can be easily derived by using the simple-closed-path theorem. [HA] 7. Suppose we have a simple arc (an arc that does not cross itself) and that we join the endpoints of the arc by a straight line. 17). 18 Solve for A in terms of e. total turning over the arc to the (interior) angles that the arc makes with the line.
Turtle Geometry: The Computer as a Medium for Exploring Mathematics by Harold; diSessa, Andrea Abelson