Jossey-Bass Teacher - Math Wise Phần 10 ppsx

3. 1/2-Phase Mobius Strip: ¨ (a) The line will meet itself; there is just

1 line, and therefore this Mobius strip has only side. (b) Cutting ¨

down the middle results in a single ‘‘new’’ strip twice as long and

1/2 as wide; further, the new strip is non-Mobius. (c) When cut ¨

down the middle, the new non-Mobius strip splits into 2 strips ¨

that are linked together.

4. 1/3-Phase Mobius Strip: ¨ (a) The continuous line will miss itself on

the ‘‘first pass,’’ which is when you have gone all the way around

the paper. But it will meet itself on the ‘‘second pass.’’ (b) When cut

1/3 of the way in, the result will be a small, ‘‘fat’’ loop interlinked

with a longer, ‘‘narrow’’ loop. The narrow loop is non-Mobius and ¨

the fat loop is Mobius. Further, the fat loop is the center of the ¨

original Mobius strip, and the narrow one is its outside edge. ¨

5. Extension 1: Mobius strips are in common use as conveyor (and ¨

other) belts, because they will, theoretically, last twice as long as

regular belts. The reasoning for this is that the wear is distributed

evenly to all portions of a Mobius belt, whereas a regular belt ¨

wears only on one side.

Puzzlers with Paper 387

Chapter 100

Create a Tessellation

Grades 4–8

× Total group activity

× Cooperative activity

× Independent activity

× Concrete/manipulative activity

× Visual/pictorial activity

× Abstract procedure

Why Do It:

This project allows students to explore regular tessellations

and then create M. C. Escher–type tessellations of their own.

M. C. Escher, a Dutch artist who lived from 1898 to 1972,

created drawings of interlocking geometric patterns (or tes￾sellations).

You Will Need:

Each student will require a large sheet of light-colored draw￾ing or construction paper that is fairly stiff; a small square

of tagboard (about file-folder weight) measuring 2-1/2 inches

on a side; tape; pencils; scissors; rulers; and colored markers.

Some examples of Escher-type tessellations or reproductions

of Escher’s work may also prove to be helpful. (You can find

examples by going online to http://images.google.com and

typing in M. C. Escher).

How To Do It:

1. A tessellation of a geometric plane is the filling of that

plane with repetitions of figures in such a way that no

figures overlap and there are no gaps. With this infor￾mation, students are to search out and explore the

many regular tessellations that are found in everyday

388

locations. Such everyday tessellations are most often composed of

regular polygons, including squares, triangles, and hexagons (for

example, ceramic tile patterns on bathroom floors, brick walls, or

chain-link fences).

2. Next, explore some examples of the Escher-type tessellations and

tell the students that they will be learning the logical procedures

for developing similar tessellations of their own. When it is time

to construct the tessellations, it is suggested that the class work

together as they create their first tessellations; that is, the students,

even though they will probably use different designs, should

complete together the steps outlined in the Example.

3. After students have completed their first tessellations, engage them

in a discussion of the ‘‘motion’’ geometry they accomplished—in

this instance, the cutting out of segments and the subsequent

‘‘slide’’ motion to move these to their new locations. (In other

instances of motion geometry, such cut-out segments might be

‘‘flipped,’’ ‘‘turned,’’ ‘‘stretched,’’ or ‘‘shrunk.’’) This discussion,

involving the logic of creating tessellations, should include such

questions as What happens when you ? and

What might happen if ? Finally, allow the par￾ticipants to try out some of their ideas as they attempt the creation

of more tessellations.

Create a Tessellation 389

Example:

Each student should follow the steps below to create his or her first

tessellation.

C

B

D

A

C

B

D

A

Step 1: Label your tagboard square

with the vertices A, B, C, and D as

shown above.

Step 2: Draw a continuous line that

connects vertex B with vertex C

and cut along that line to get a

cut-out piece.

C

B

D

A

C

B

D

A

Step 3: Slide the cut cut-out piece

around to the opposite side, place

the straight edge BC against AD,

and tape them together.

Step 4: Draw a continuous line that

connects vertex D with vertex C

and cut along that line.

C

B

D

A

Step 5: Slide this cut-out piece

around from the bottom. Place it

on top, with the straight edge DC

against AB, and tape them

together. Your tessellation pattern

is now complete.

Step 6: Place the pattern on your

drawing paper and trace it. Then

slide the pattern (up, down, left, or

right) until it is against a matching

edge and trace again. Continue

until the entire drawing paper is

filled with repeating patterns. You

may use colored markers to

emphasize your tessellation

pattern.

(Note: See the completed bird-like

tessellation on the prior page.)

390 Logical Thinking