### MATH: Fractal Trees

Time Required: Two 45-minute sessions
Skill Level: Intermediate

In this FRACtivity, students will examine the natural ecology of tree branching, while exploring mathematical ratios and quotients to help interpret data. Students will work in pairs to create their own Doodled fractal tree as a tangible example of fractals. Within this activity, students will measure length and angles of branching, using rulers and protractors. For younger grade levels, more simplified mathematical concepts can be explored like patterns and symmetry.

Note: Any links outside of the3doodler.com are optional resources. We can’t ensure their upkeep or accuracy. ### Lesson Plan

#### Instructions

Step 1 - PREPARATION

Take a nature walk. Suggest that mathematics can be found in nature. Direct students to explore patterns in tree branches. Ask students to suggest other naturally occurring elements with similarly repeated branching. (Possible responses: rivers, cracks in ice or glass, blood vessels, lightning, neurons in the brain, creases on the palm of your hand.)

Step 2

Define a fractal: A repeating pattern. Branching fractals are one type of fractal found in trees. Project your tablet or computer screen for students to view and explore this link on fractal trees.

Step 3

Demonstrate how to draw a fractal tree stencil on your whiteboard or poster size paper.

a. Draw the trunk.
b. At the end of the trunk, split by some angle and draw two more branches.
c. Repeat at the end of each branch about 6-8 times.

Step 4

Model how to use a 3Doodler to doodle over the stencil. Do NOT peel the tree off of the paper.

Step 5

On stencil paper, label each branch segment and measure using a ruler. Example: AB
Record measurements on theFractal Tree Worksheet.

Step 6

Demonstrate how to compare the measurement of each branch segment using division and a calculator.

*Divide the length of branch AB (6 ml.) by branch BC (3 ml.) to get a quotient of 2 ml. Calculate the successive branches, two at a time, in this same manner and record the quotients on the Fractal Tree Worksheet.

Step 7

Demonstrate how to state the relationship between the length of branches as a ratio. Example: 6:3.

Step 8

Demonstrate how to measure the angle of each successive branching. Label each angle.
*See Appendix. Measure with a protractor and record on Fractal Tree Worksheet.

Step 9

Divide students into pairs. Instruct students to complete this same activity and record related observations on the Fractal Tree Worksheet.

Step 10

Hand out paper, pencils, 3Doodlers, rulers, protractors and the Fractal Tree Worksheet.

Step 11

Circle to assist and assess.

#### Wrap Up

Students will share their fractal trees and discuss their findings based on recordings in Fractal Tree Worksheet. What is the relationship of successive branch lengths and angles?

#### Assessment

The teacher will assess students’ work based on their fractal trees, Fractal Tree Worksheets, and feedback during a discussion.

#### Possible Extensions

Students create a fractal forest with related poetry about their tree. Create a 3Doodled base for each tree and assemble all of the student's poetry and trees as a 3D display. Explore more fractals with Sierpinski Triangles and fractions.

#### Vocabulary

• acute angle - An angle that measures less than ninety degrees but more than zero degrees.

• angle - the (rotational) space between two intersecting lines.

• collaboration - to work jointly with others or together especially in an intellectual endeavor.

• drawing - the art or technique of representing an object or outlining a figure, plan, or sketch by means of lines.

• fractal - any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size.

• line - an infinite extent which is one-dimensional and straight.

• measurement - the act or process of measuring; a figure, extent, or amount obtained by measuring.

• nature - the physical world and everything in it (such as plants, animals, mountains, oceans, stars, etc.) that is not made by people.

• obtuse angles - Angles larger than 90 degrees.

• pattern - an artistic, musical, literary, or mechanical design or form.

• perpendicular - being at right angles to a given line or plane.

• problem-solving - the process or act of finding a solution to a problem.

• ratios - the relationship in quantity, amount, or size between two or more things.

• repeat - renewed or recurring again and again.

• symmetry - the correspondence in size, form, and arrangement of parts on opposite sides of a plane, line, or point; regularity of form or arrangement in terms of like, reciprocal, or corresponding parts.

• tree - a woody perennial plant having a single usually elongate main stem generally with few or no branches on its lower part.

#### Educational Standards

Common Core
CCSS.MATH.CONTENT.4.OA.C.5

Generate and analyze a pattern.

In This Lesson

Students will construct a stencil and Doodle a tree as a fractal pattern, then analyze the relationship between the length of branch segments and angles.

Common Core
CCSS.MATH.CONTENT.4.MD.B.4

Represent and interpret data.

In This Lesson

Students will record their measurements as data and interpret their findings.

Common Core
CCSS.MATH.CONTENT.4.MD.C.5

Geometric measurement: understand concepts of angle and measure angles.

In This Lesson

Students will measure length and angles of tree branches.

Common Core
CCSS.MATH.CONTENT.4.MD.C.6

Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

In This Lesson

Students will use a protractor to measure the branching angles on the Doodled fractal tree.

Common Core
CCSS.MATH.CONTENT.4.G.A.1

Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

In This Lesson

Students will draw a stencil of a fractal tree composed of line segments and angles.

Common Core
CCSS.MATH.CONTENT.4.G.A.3

Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

In This Lesson

Students will recognize the symmetry within branching.

CS Teachers
1A-A-3-5

Decompose (break down) a larger problem into smaller sub-problems with teacher guidance or independently.

In This Lesson

Students will break down the process of analyzing a fractal tree through a multistep process of a designing a fractal tree stencil, Doodling it, recording measurements and interpreting data.

ISTE
1C

Use technology to seek feedback that informs and improves their practice and to demonstrate their learning in a variety of ways.

In This Lesson

Students will use the 3Doodler to create a fractal tree.

ISTE
6B

Create original works or responsibly repurpose or remix digital resources into new creations.

In This Lesson

Students will use a 3Doodler to create a fractal tree.

ISTE
7A

Use collaborative technologies to work with others, including peers, experts or community members, to examine issues and problems from multiple viewpoints.

In This Lesson

Students will seek feedback from partner to interpret data on measurement of angles and lines on Doodled fractal tree.

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