The Maze
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Introduction

Look at the maze in the figure to the left. (Click on it to see it full-sized). The path from the bottom of the picture to the top is built from two sets of half circles. Each set consists of 8 half circles with radii 1 to 8 cm. Circles on the left side have centers in the upper part of the two central points, and circles on the right have centers in the lower part. Carefully study the path from right to left. Run your finger over it.

Activity
Supplies: pencils, scissors, paper
Optional Tools: Wicki Stix, measuring tape, meter sticks, string

1. Show children the picture and explain how to draw it. It is harder than it seems, so give them plenty of time and lots of help.

One way to draw it:

  • Put 18 tick marks one cm apart along an imaginary straight line from the top of the page to the bottom.
  • Number the tick marks from one to 18.
  • With the point of your compass on point number 9, draw half-circles on the right with radii
    1, 2, 3, 4, 5, 6, 7, and 8 cm.
  • With the point of your compass on point number 10, draw half-circles on the left with radii
    1, 2, 3, 4, 5, 6, 7, and 8 cm.
    • Another way to draw it:

  • Put 18 tick marks one cm apart along an imaginary straight line from the top of the page to the bottom.
  • Number the tick marks from 1 to 18.
  • With the point of your compass on point number 10, draw concentric circles with radii
    1, 2, 3, 4, 5, 6, 7, and 8 cm. (See Figure 1)
  • Cut the maze in half along the imaginary straight line through the center of the circles (See Figure 2) . Shift one of the halves one cm up or down to create the maze (See Figure 3).
    • 2. Let children draw ants or spiders and decorate their drawing. Have them run their finger over the path.

    3. How long is the path through the maze? How far does the ant have to travel to get through the maze in the picture below? (The class may make estimates.)

    click on me

    First Solution
    The total area of the maze is one circle; the area of a circle with radius 8 cm is

    * 8² = 201 sq. cm

    But the path is 1 cm wide, so the length of the path is

    201 sq. cm / 1 cm = 201 cm,

    approximately 2 meters.

    Second Solution
    If you travel in the middle of the path you are covering the distance of a perimeter of 8 circles with radii

    .5 cm, 1.5 cm, ..., 7.5 cm

    Thus you travel

    2 * * (.5 + 1.5 + ... + 7.5) = * (1 + 3 + ... + 15) = * 64 = 201 cm


    Third Solution
    With tape, fasten a string at one end of the maze. Follow the maze with the string to the center. Measure the string, and double the length.

    Fourth Solution
    Using Wicki Stix, make a path through the maze to the center. Measure the Wicki Stix and double the length. After the solution is found, show this distance using meter sticks or a measuring tape.

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