Using Geometry to Create and Appreciate Art

Title

Using Geometry to Create and Appreciate Art

Target Audience

Structured for 9-10th grade geometry but adapting it for algebra II class is possible

Project Description and Overall Goals

In this project-based unit, high school geometry students will discover how geometry can be used to create and appreciate art. By closely studying works of art, students will gain a new appreciation for the artistic field and have an increased awareness of the functionality of mathematics outside the classroom. In developing this project based approach, we intend for our future students to explore cross disciplinary applications of mathematics.

Our unit will take place over three weeks of class time. The three weeks are broken into five sub units for the students. The first sub unit is the introduction and includes: watching a video, breaking into project groups, and a field trip. The second is an in depth study of the Pythagorean Theorem. The third is a sub unit focused on triangle similarity and congruence. In the fourth sub unit, the class will do a geometric analysis of two artworks. The fifth is lab time for the students to create their project and then to present their work.

First Sub Unit: Introduction

To start off the unit, our students will watch an anchor video. The video includes interviews with artists in which they discuss the different triangle properties that they used to create sculptures. The videos shows a time lapse segment of a painting being created step-by-step so that the students can see how fundamental geometry is to planning a piece of art. The video ends with the assignment: create and analyze a piece of art using geometry. After the students watch the video, they will break into their project groups and begin to assign roles within the group.

The next class period the students will take a field trip. They will either go to a local art museum or they will go to a library. In either setting the students will find examples of different influential artists whose work demonstrates triangle properties. Both of these experiences will help the students understand how geometry is fundamental in many types of art.

Second Sub Unit: The Pythagorean Theorem

The next lesson will be focused on the Pythagorean Theorem. First the students will prove that the theorem holds true for all right triangles using algebra and properties of area. Then the students will learn about the different ways ancient cultures applied the Pythagorean Theorem. Finally the class will do practice word problems in which they have to apply the theorem.

Third Sub Unit: Similar and Congruent Triangles

Next, we begin a two-part lesson dealing with similar and congruent triangles and their applications. Students will partake in the exploration of shadows as it relates to proportionality and systems of similar and congruent triangles. Students will be shown examples of artist Shigeo Fukuda’s work, which demonstrate his mastery of using shadow properties to create his artwork.

After the introduction to similar and congruent triangles, students will discover new and different applications of these triangle properties. Students are given a task to find the height of a flagpole without directly measuring it. In this application, students will be able to use mirrors and yard sticks, among other supplies to create and carry out a method of calculating the height of the flag pole with indirect measurements. Students will be able apply their understanding of the Pythagorean Theorem and other triangle properties learned in previous lessons in carrying out the activity. To conclude this section of the unit, we will discuss how mirror reflections could be used by artists in the field.

Fourth Sub Unit: Geometric Analysis of Artwork

The next topic will be an in-depth geometric analysis of two pieces of art. In the first exploration, the project groups will be given a bowl. The bowl will be set on a piece of paper that includes a drawn in shadow of the bowl. The project groups will use triangle properties to determine the precise position of the light that cast the shadow. Then the groups will outline their step by step procedure of problem solving strategies and present them to the class. Once the class has reached a consensus on how to solve the problem, they will analyze La Vigilia. In their analysis, the project groups will determine how many light sources are casting the shadows, and the position of each light source.

The second day of this sub unit, students will analyze the painting MayanTemple. In their project groups, the students will explore three different triangle properties that are demonstrated in this painting. Our intention is for the students to notice similar and congruent triangles and how they are related to this painting. The groups will also analyze distance, ratio of area, and applications of the Pythagorean Theorem. After the students have finished their explorations, each group will present their different findings and turn in an application card. The application card asks that each group explain the main concept covered in class that day, and at least one idea of how they can apply this concept to their final project.

Fifth Sub Unit: Creating Projects and Presenting

The next phase of the unit will be focused on the students creating their artwork. The project groups will explore different design possibilities on Geometer’s Sketchpad. They will also make up a detailed plan for what materials they need, responsibilities of each group member, and a plan of how they will complete their project before the due date.

The final product that the students will turn in consists of three components. The project groups are required to make a piece of art that demonstrates two triangle properties learned in this unit. The groups will be required to give a five-minute presentation explaining their process in making the art and the different geometry concepts that they used. Each student will also turn in an individual write-up in which they refer to the artists that influenced the design of the artwork and explain which triangle properties their group art piece demonstrates.Once the students have turn in their artwork, papers, and made their presentations, we will conclude the unit with a final test.

Driving Question

How is math used to create and appreciate art?

Project Objectives

Each of the lessons in this unit has a set of objectives for each of the students to achieve. In the Discovering the Pythagorean Theorem benchmark lesson, students will be able to state and prove the Pythagorean Theorem in addition to applying the theorem to real world examples. In the Triangle Similarity benchmark lesson, students will be able to state what it means for two triangles to be proportional. Also, students will be able to model a light-object-shadow system using a system of similar right and non-right triangles.

In the Triangle Similarity and Some Applications investigation lesson, students will be able to describe and carry out a method of calculating the height of a flagpole without actually measuring the height itself and use mirrors to create a system of similar triangles that could be used to indirectly calculate the height of an object. In leading up to the actual creation of there are work, students will be able to use these triangle properties to analyze actual art pieces in the Is La Vigila Geometrically Sound investigation lesson. For the final product, students will use the concepts covered throughout the lesson in the creation of their artwork.

Rationale

The lessons in this unit are developed around the Texas Essential Knowledge and Skills (TEKS), which are a set of statewide standards created by the Texas Education Agency. Our project is designed to cover the skills that students are meant to learn over a 3-week period in Geometry class. The lessons focus on triangle properties such as the Pythagorean Theorem, triangle similarity and triangle congruence

In carrying out our project, we wish to blur the lines between art and mathematics and show students that there are very real connections between what’s taught in their geometry classroom and what professionals use in the field. The idea of using art to teach mathematics has become quite popular in math and museum education. For example, The Blanton Museum of Art at the University of Texas has recently added new tours that are geared towards math classes. Students participating in these tours are exposed to math concepts in the context of the works of art at the Museum.

Background:

Understanding various triangle properties is essentialto create a quality product for this project. Each property or concept that students discover will serve as a tool that students could use in the planning and creation their artwork.

Students are first introduced to the Pythagorean Theorem through a benchmark lesson involving the discovery of the theorem’s proof and its applications. By knowing the mathematical relationship between the lengths of a right triangle, students could calculate the length of the unknown side given the length of two sides. This, as we found out in creating a sample of the product, comes in handy while taking measurements in trying to align certain elements of the art piece.

Our other benchmark lesson involves the investigation of shadows as it relates to the concept of similar triangles. Being able to model a light source-object-shadow system with a system of similar triangles allows students to utilize similar triangle properties in manipulating shadows and lighting in their art piece.

Other than with shadows, similar triangles can be demonstrated using a person’s perspective. Holding out an object in front of you until it completely covers your vision of another object creates a system of similar triangles involving your eye, the object held, and the object that was covered. The various concepts explored in this unit such as the Pythagorean Theorem and similar triangles allow for greater range of possibilities in the students’ final products.

Standards:

These are the various TEKS in which our lessons revolve around:

(G.1) The student is expected to recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes.

(G.2) The student analyzes geometric relationships in order to make and verify conjectures.

(G.4) The student uses a variety of representations to describe geometric relationships and solve problems.

(G.5) The student uses a variety of representations to describe geometric relationships and solve problems.

(G.8) The student is expected to derive, extend, and use the Pythagorean Theorem.

(G.11) The student applies the concepts of similarity to justify properties of figures and solve problems.

Final Product

The final product of this unit has two primary components: the presentation of the artwork created and the report analyzing their art piece. Attached is a rubric detailing the criteria in which each student is to be graded.