### Math: Parabolic Art and Geometry

Time Required: One or Two 50-minute sessions
Skill Level: Beginner

Students will work in pairs to graph Parabolic Curves over 4 quadrants of an x- and y- axis, and then create a series of conjectures about Parabolic Curves based on their measurement of lines, angles, symmetry and patterns. The resulting product will be similar in appearance to string art.

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### Lesson Plan

#### Instructions

Step 1

Whole group: Challenge the class to create a curve from a series of straight lines using bamboo or any type of wooden skewers.

Step 2

After giving students some time to practice, introduce them to the concept of a Parabolic curve.

Step 3

Share the goal: Students will create a curve from a series of straight lines using a pencil, ruler and graph paper.

Step 4

Model how to do it. Project your computer or tablet screen on the board and display Parabolic Curve Template A. Use a ruler to draw straight lines to connect like numbers on each pair of axes. Show students how to draw a line from number 1 on the x- axis to number 1 on the y- axis. Do the same for each pair of numbers.

Step 5

Demonstrate how to use the 3Doodler to connect point to point. When creating a new line, anchor it to the graph paper, then lift the line gently and extend it to the next point anchoring it before ending it. Use a 2-3 color pattern for aesthetics. Allow plastic to harden. Peel off graph paper and adhere to contrasting color construction paper background with double-sided tape or by welding it to the paper with 3Doodled beads of plastic.

Step 6

Instruct students to observe patterns, angles and lines of intersection. Record their observations on the board.

Step 7

Divide students into pairs. Hand out one copy of Parabolic Curve Template A to each pair. Hand out protractors and rulers. Instruct students to measure lines and angles as they work. Tell students to record their measurements and observations.

Step 8

Ask students to come up with a conjecture or rule to explain how to form a Parabolic Curve. Ask: "What do we know to be true?" "Do you need a maximum or minimum number of lines?"Help students understand the following: All lines must be straight. All lines must intersect each other at a 90º angle.

Step 9

Rather than share assertions about parabolic curves, lead students to conjectures through an effective line of questioning.

#### Wrap Up

Instruct students to tackle Parabolic Curve Template B and Parabolic Curve Template C. Discuss whether their original conjectures still hold true. Meet in a group to discuss.

#### Assessment

The teacher will assess students’ work based on their final products, as well as each student's participation in the group discussion.

#### Possible Extensions

Create a Parabolic Curve inside of an obtuse angle and an acute angle. Analyze the difference and develop a conjecture or rule for each. Create Parabolic Curves inside of different polygons. Create a conjecture. Code a Parabolic Curve using programs on a website such as Code or Scratch.

#### Vocabulary

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

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

• conjecture - a proposition (as in mathematics) before it has been proved or disproved.

• geometry - a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids; broadly : the study of properties of given elements that remain invariant under specified transformations.

• line art - graphic material that consists of lines or (areas of pure black and pure white).

• math - the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.

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

• parabolic curves - in mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits any of several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

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

• proofs - in mathematics, a proof is an inferential argument for a mathematical statement.

• protractor - an instrument for laying down and measuring angles in drawing and plotting.

• rule - a determinate method for performing a mathematical operation and obtaining a certain result.

• ruler - a smooth-edged strip (as of wood or metal) that is usually marked off in units (such as inches) and is used as a straightedge or for measuring.

• straight lines - an object in geometry that is characterized as a straight, thin, one-dimensional, zero width object that extends on both sides to infinity.

• string art - art that is made with strings that is often mathematical in nature, in that it is regularly crafted on an x and y axis.

• 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.

#### Educational Standards

Common Core
CCSS.MATH.PRACTICE.MP2

Reason abstractly and quantitatively.

In This Lesson

Students will form conjectures about Parabolic Curves based on observations and measurements.

Common Core
CCSS.MATH.PRACTICE.MP4

Model with mathematics.

In This Lesson

Students will analyze and evaluate by creating a concrete Parabolic Curve with a 3Doodler.

Common Core
CCSS.MATH.PRACTICE.MP7

Look for and make use of structures and patterns.

In This Lesson

Students will note the structures and patterns of Parabolic Curves through measurement of lines, angles and observations.

Next Gen Science
3-5-ETS1-2

Generate and compare multiple possible solutions to a problem based on how well each is likely to meet the criteria and constraints of the problem.

In This Lesson

Students will generate and compare multiple 3Doodled Parabolic Curves in order to draft a series of rules and conjectures.

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 larger problem of drafting a series of rules and conjectures for Parabolic Curves by 3Doodling, measuring and observing their concrete models.

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 visually demonstrate the the relationship between lines, angles, symmetry and patterns in Parabolic Curves.

ISTE
4D

Exhibit a tolerance for ambiguity, perseverance and the capacity to work with open-ended problems.

In This Lesson

Students will demonstrate willingness and competency within an open-ended task that has more than one possible outcome.

ISTE
6B

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

In This Lesson

Students will use a 3Doodler to design a series of Parabolic Curves.

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 use a 3Doodler with a partner to complete a series of Parabolic Curves.

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