Fractal Fraction Fun, by Michael Naylor

By combining a study of fractions and some simple geometry, students can make art of infinite beauty

You may have seen calendars or t-shirts with pictures of fractals – complicated and colorful images made by computers. Fractals are figures with infinite complexity, but in reality they are not really complicated at all. They are made by repeating very simple rules. The beautiful patterns created by these rules make fractals an ideal activity for any grade level and allow children to study fractions and similarity in a fascinating new way.

Fractal tree (Grades K-3)
Materials: For each child, one piece of construction paper, four strips of different colored construction paper, glue stick and scissors. The strips of paper should be about two cm wide and cut widthwise from standard-sized paper.

1. Select one strip to be the tree trunk and glue it in the lower center of the page. Fold the remaining three strips in half and cut along the crease to make six half-size strips.

2. Select two strips of one color and make two branches coming up from the top of the tree trunk. Glue them in place. Cut the remaining four pieces in half to make eight quarters.

3. Select four strips of one color and make two branches coming up from each of the two branches added before. Cut the remaining four quarters in half to make eight eighths.

4. Place two of these small pieces branching out of each of the smaller branches you added in the last step and glue in place.

   Your tree is complete!

Questions for discussion: What would happen if you kept going? What happens to the length of the branches with each step? If the tree trunk is length one, what are the lengths of each of the other colored pieces? What is the total length of the pieces of each color?

Does each branch look like the entire tree? If you could continue a fractal forever, each part, no matter how small, would look exactly the same as the entire figure.

Fractal triangles (Grades 3-8)
Both of the following fractals are built on triangles. They may be cut out of paper, drawn with pencil and paper or created with a computer.

To make a "Sierpinski Triangle," have your students start with an equilateral triangle and use the following rule: connect the midpoints of each triangle and remove the center. Use the pictures below to help you see how this works.

When your students mark the midpoints of each side of the triangle and connect, they'll have divided the triangle into quarters. Ask them to shade in the center triangle and consider it removed from the figure. This leaves the other three smaller triangles. Now the rule is repeated on each of these three triangles. This should leave nine tiny triangles…have your students keep going!

Your students can complete a chart to examine the mathematical properties of this shape. They'll be thinking about fractions and hunting for algebraic patterns.

A "Koch Snowflake" (see Figure 1 above) is also made by first drawing an equilateral triangle. Triangles are then connected to the center third of each side. How many sides does this figure have? (12) What is the name of a twelve-sided figure? (a dodecagon).

The rule is repeated — a triangle is drawn on the center third of each of the 12 sides, making the 48-sided figure that follows.)

Fractal art (Grades 4-8)
Students can create their own fractals, making beautiful mathematical designs and exploring the patterns they find.

As we've seen, fractals are created by repeating a rule over and over on a smaller and smaller scale. Here's one made with a square and the rule: add a smaller square to the center of each side. After only a few steps, it becomes extremely complicated and beautiful.

To get students started, it's useful to post this plan:

1. Choose a basic shape and rule.

2. Apply your rule on the shapes until you have smaller copies of the base shape.

3. Repeat step two many times.

A classroom fractal gallery
Have each of your students make an exhibition piece, drawing their fractals using rich colors or other media and displaying their work with a chart and a description of the mathematical patterns in their drawings. Using their new creations, your students will find mathematical connections between their fractals and other classmates' fractals. Have a discussion about what these fractals have in common, and why?

For a sample of the Koch Snowflake worksheet click here. PDF 189KB

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.