## A Fractal Coast

This activity can be introduced and used in conjunction with social studies lessons on geography, particularly coastlines and mapping. Students will need a basic understanding of fractals before starting the lesson. Plan on one 45-60 minute class for this activity.

#### Materials

- Pencils
- Paper
- Rulers
- Maps

#### Instructions

- Share a map with students and view as a whole class.
- Zoom in on a coastline (choose one in your area to connect and engage students) and ask:
- How can we measure this coastline?

- Allow students to discuss answers in partner-pairs, then lead a conversation about methods.
- Set aside maps for later and distribute paper and rulers.
- Have students turn their papers sideways and draw a nine-inch line horizontally on their papers. Ask them to divide the line into three equal parts and erasing the middle section, leaving the two side sections intact.
- Next, instruct students to connect their two line segments with an upwards 'v' shape taking up the center section. Discuss how the pattern is now broken into four sections.
- Make more fractals by having students divide each of the four sections into a new upwards 'v' shapes. How many sections are there now?
- Instruct students to repeat this exercise two more times independently. How many sections do they now have?
- Ask students to return to the map and set their fractal side-by-side. How are the coastline and fractal similar?
- Now ask students to measure the coastline using rulers by breaking into sections as they did with the fractal activity.
- Finish by having students explain to a partner how fractals and coastlines are the same.

## Three-Dimensional Fractals

Students love to get busy and to eat. This activity combines these two loves into one fun way to explore fractals. They'll be using their understanding of fractals in cooperative problem-solving and critical-thinking groups. Allow one typical 45-60 minute class period for this activity.

#### Materials

- Mini-marshmallows (dry out for a day before the lesson)
- Toothpicks
- Protractors
- Colored pencils

#### Instructions

- Divide students into partner-pairs.
- Distribute toothpicks, protractors, and mini-marshmallows to each group.
- Have students label their math notebooks 'Fractals' and ask:
- How are fractal patterns tetrahedrons?

- Share answers, then instruct students to use their materials to make a basic tetrahedron. Four marshmellows and six toothpicks can be used to build the simplest version of this three-dimensional shape.
- Have groups use their protractors to measure angles and sides and record in notebooks, along with a sketch.
- Next have students create a chart for the following information:
- Number of tetrahedrons
- Number of marshmallows used
- Number of toothpicks used
- Length of sides

- Have students input information for their tetrahedron.
- Now create foursomes and have them combine their tetrahedrons, the fill in their charts.
- Tell students they will combine with another group and have them predict what their new data will be, then have them combine, assemble and measure.
- Continue on in this way until the whole class has assembled and measured.
- When finished, ask students:
- Where have you seen examples of this shape in real life?
- How can tetrahedrons be used?

## Fractals in Nature

This lesson is literally right outside your window. Students will be using concepts of fractals to determine the relationship between fractals and trees! We'll have them measure the ratios of a tree's branches to show how math is all around us. Plan on 60-90 minutes for the core lesson.

#### Materials

- Protractors
- Millimeter rulers
- Calculators

#### Instructions

- Take students outside and identify a tree. Make sure they will be able to reach branches to measure.
- Have students observe the tree and lead a discussion:
- What patterns do you notice about the tree's branches?
- How is a tree fractal?
- What shapes do you see in the tree's branches?

- Now have students draw a sketch of the tree in their notebooks and label as follows:
- Tree base - A
- First branch point - B
- Second branch point - C
- Third branch point - D

- Label any subsequent branch points as needed.
- Instruct students to create a chart with the following column headers:
- Section/name
- Distance between sections
- Quotient of adjacent sections
- Ratio of adjacent sections

- Instruct students to input rows for each distance in the first column. For example, AB, AC, AD, BC, etc.
- When this data is complete, have students compare the branch sizes by finding the quotient by dividing the length of one branch by the length of another. For example, the first equation is AB/BC.
- After all results are recorded, have students finish the chart by writing the ratios.
- Now ask:
- What patterns do you notice in this data?
- How are fractals visible in this tree? How are they visible in other nature?