## Cutting Up in Class, Fractal Style

### Richard Angelini Sr.

#### Description

A simple to make, hands-on manipulative, three dimensional, model of fractions, mirror images, tiling, fractals, tessellations, multiplying fractions, dividing fractions, and exponents; created from a single sheet of paper.

#### Objectives

The student compares and orders fractions and decimals using graphic models, number lines, and symbols.

The student given a two-dimensional figure, creates a new figure by increasing or decreasing the original dimensions.

The student uses manipulatives and drawings to solve problems requiring spatial visualization.

#### Materials

Each student receives:
-1 Pair of scissors
-2 Sheets of colored paper
-1 Bottle of white glue
-1 Ruler
-Diagrams of how to fold and cut the paper (see the associated file)
-Pre and Post tests (see associated file)

#### Preparations

Use a paper cutter to cut each standard sheet of colored paper, (8 ˝ X 11), 2 sheets given to each student, to exactly 8 in width. Plan on two sheets per student, one the sheet to be cut and the other the mounting sheet.

#### Procedures

Introduction:
Tell the students: Today we will be experimenting with mirror images, fractions, and repeating shapes. You will learn how to multiply fractions in your head, using pictures. You will learn how to simply reproduce very complicated designs the way computers do. But, before we get to the math, lets create a structure. At this time, administer the pretest found in the associated file. This is optional. It may be helpful to define for the students circumference and area. Write on the board: circumference is the distance around the outside of any shape and area is the measurement of the surface enclosed within the boundaries of the sides of the shape. Example: the circumference of the classroom is the distance around all the outside walls. Area is the surface of the floor of the classroom.

Procedure:
(It may be helpful if the teacher builds the structure as the students are building it to demonstrate the correct procedure. Also, you will find organizing the students into teams of three or four will aid them in understanding the procedures. They can tutor and the slower students will be helped by observing the quicker students. Diagrams are included in the associated file to help with constructing the model.)

First, students will build the structure. Tell the students:
1. Look at the diagrams. Follow along as I demonstrate how to make the model. Take two sheets of colored paper. Choose two different colors, that you think look good together. Fold them in the middle, the short way, (8 inches). You are going to make an object that is similar to a book or birthday card. Hold the papers so the folded edge is on the left side and the open edges are on the right side each time you measure.

2. Choose one paper to be the inside color. On this paper measure accurately to the middle of the fold, (on the left side), and mark.

3. Next, mark half way from the edge and the middle mark on each end, creating a ź way mark on each side of the middle mark. Now make a cut using the scissors on the ź way marks ˝ way to the open side of the paper.

4. Fold the cut up so that the folded edge lines up with the top edge. (Open the paper and
push the original fold of the cut area up and in the opposite direction of the fold so that it leaves an empty area along the left, or folded side. Refold the paper. This creates a large folded-in portion centered in the paper.

5. Repeat the process inside the fold, creating a second fold ˝ half the width of the first fold and folding to ˝ the remaining distance to the right side of the paper.

6. Again fold the cut up so that the folded edge lines up with the top edge.

7. Repeat the process one more time.

8. Now fold the other colored paper the same way as the first.

9. Glue the two papers together making a form similar to a greeting card. When the glue is dry, open it up 90 degrees and observe.

Next, explain to the students the structures uses. Choose one or all of the uses to teach.

Multiplying and Dividing Fractions:
1. Tell the students: The model you have made clearly shows what happens when you multiply irrational numbers, (fractions). If you multiply one-fourth by one-fourth and again one-fourth you will get one-sixty-fourth;
ź X ź X ź= 1/64
Which is really saying that if you divide ź into 4 parts and then again divide all those parts into 4 parts each and then again divide all those parts into 4 parts; the result will be 64 parts. Now, when you began to fold and then measure to the middle, you were dividing 1 by a half and then each part by ˝ again to equal ź. You set a pattern in motion. (This is most important in understanding fractals).

2. Tell the students: Look at the smallest cube. Can you see how many of the smallest cubes it will take to fill the middle cube? If you place one on top of another in your mind, then you will use 8 of the smallest cubes. Next, look at the middle cube. Can you see it will take the same amount of middle cubes to fill the large cube as it took smallest cubes to fill the middle cube? Ask them if anyone knows how many cubes will it take to fill the big cube. Ask them if anyone knows how many cubes will it take to fill the big cube.

3. Next, tell the students: If it takes 8 small cubes to fill the middle cube and if it takes 8 middle cubes to fill the large cube then 8 X 8 = 64 small cubes in the one large cube.

Exponents:
Write on the board: 4 X 4 = 16 4 to the 2nd
16 X 4= 64 4 to the 3rd
Ask if any student can offer a formula for what is happening.
Explain to the students that 4 to the 3rd is 64 and 4 to the 2nd is 16. You can see this in our paper model. The smallest cube is 4 X 4 and the next cube is 16 X 4.

Tessellations:
Tell the students: This is a model of a repeating pattern called a tessellation. A tessellation is a complex pattern made with simple rules that will repeat forever. In this paper cutting, the rule is simple: Keep dividing by 4 by first taking one half out of the middle and then folding to one half of the remainder. Ask them if they know what ˝ X ˝ equals? (1/4)

Fractals:
Tell the students: A fractal is a repeating pattern that repeats with symmetry infinitely in all directions. In this simple fractal, the pattern can be imagined to repeat to the infinite large or the infinite small, (if it was possible to keep cutting and folding paper). In each repeating step you will find a self-similar pattern. That means little identical versions of the pattern and bigger identical versions of the same pattern. Tessellations cannot get infinately small or big and still be self-similar. That is why fractals are different than tessellations. Fractals also must have a place of beginning for the rules, called an identity that establishes the original conditions. The identity is considered the initial condition from which it is generated to the infinite. It holds the rule in a mathematical formula. This mathematical formula is to keep dividing by ˝ in each of the two two-dimensional directions. This will create a three dimensional model; one with three measurable distances: height, width, and depth. In this model each of the three dimensions are the same length, so a cube is formed.

Mirror Image;
Tell the students to observe the bilateral symmetry of the paper structure. The left side matches the right. The mirror image was created by the rules. The fold is a good physical representation of duplication of rules to create a mirror image. You may want to hold your model up to a mirror and see if you can identify any differences.

Tiling:
Show the students how the mathematical rule, or formula has created a consistent and repeating form when viewed from the top. The rule will continue to tile into the infinite and will cover all the area.

#### Assessments

Pre-test: Use the results as an introduction to the lesson. A KWL chart on the board may be helpful. After the lesson, have the students compare their answers on the pre-test to the post-test and complete the chart using the post-test.
Use this as a formative:
1. Assign teams of 4 students for this exercise.
2. At each instruction, give 4 points for the team first completing that step.
3. Give 8 points for the team that can first state the number of smallest cubes it will take to fill the middle tube and 8 points for the statement of how many middle cubes it will take to fill the largest cube.
4. Give 16 points for the first team to offer the correct formula: 4 X 4 = 16 X 4 = 64.
The posttest can be used as a summative only if students demonstrate that they are ready to be summatively assessed on these benchmarks.

#### Extensions

This lesson lends itself well to degree of study. After the cutting and pasting are completed, the teacher may decide what level of depth should be studied.
Extensions: Continue to study fractals, (non-Euclidian geometry) with web URLs.

#### Web Links

A Fractals Unit for Elementary and Middle School Students
Fractals

Chaffey High School's Fractal Links of the Web
Fractals