Case Studies
Beautiful Triangles

Beautiful Triangles: Exploring Angles and Lines

GENERATIVE TOPIC:

Form Conveys Meaning  

כמה דכייף איניש דעתיה טפי מעלי

"Bend your mind upward"

(Talmud Sanhedrin 26b)

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The Beautiful Triangle: geometric figures convey thoughts, feelings and values differently for each of us
The Beautiful Triangles unit is the math component of the larger generative topic or overarching theme, "Form conveys meaning." Grade 7 students examine triangles of different shapes, sizes and forms, and come to discover that they can express different thoughts and emotions which can vary from one person to another. For example, an equilateral triangle, which has three equal angles of 60 degrees and side lengths of equal measures, can express the idea of equality or perfection. At the same time, it can also be viewed as something impossible or out of one’s reach since it is the most difficult triangle to construct.
Specifically, throughout this unit, students learn how to classify triangles and master concepts of congruence, similarity, ratio, angle and triangle theorems. Students demonstrate their understanding by discovering angles in the world around them with a focus on kinesthetic activities. The unit culminates in the Beautiful Triangle project in which students learn how to create precise geometric constructions to convey themes, ideas, feelings, and experiences similar to the abstract expressionist style of Wassily Kandinsky.
Visual Arts and Mathematics
Abstract art relies on form (shape, line, and colour) to convey meaning to the viewer. The decisions made by the artist impact the viewer, and the viewer uses their own ability to interpret form to internalize their own understanding of the work. The subjectivity of meaning conveyed through form becomes clear in this unit. As students look at different triangles and decide which is beautiful to them, they use their judgement and critical thinking  skills to derive meaning from form.
 
As artists, they reverse their thinking and develop forms that will convey meaning back to a viewer. Students look to Artist-in-Residence Wassily Kandinsky, the pioneer of abstract art, to learn the art of using form to convey meaning. We take inspiration from Kandinsky precisely because he eschews retinal (representational) art, and favours shapes, lines, and colours to express ideas and feelings.
Essential Questions
1. What is an angle?
2. How do we name angles and triangles?
3. What makes a shape beautiful, meaningful, and interesting? And, How can angles, lines, and triangles express ideas and emotions?
4. How can triangles serve as tools for accessing information that is beyond our reach?
5. How can we use information that we have to find information that is missing?
Math Learning Goals
1. Students understand how to measure and construct angles precisely by using rulers and protractors.
2. Students understand that mathematicians divide angles into five categories, each with its own name: acute, right, obtuse, straight, and reflex.
3. Students understand how to precisely construct and classify triangles using a compass, ruler, and protractor.
4. Students understand that the sum of a triangle’s interior angles is 180 degrees, and they apply this property to determine the interior angles of all regular polygons.
5. Students learn about the concepts of congruent and similar triangles, specifically in relation to Thales’ method for calculating unknown heights.
6. Students master the mathematical concepts of congruency, similarity, ratio, angles, and triangle theorems.
The Beautiful Triangle
Asking students to identify which triangle is the most beautiful to them allows them to take a step back, and see the beauty inherent in shapes, but also the way a shape can evoke emotions. The triangle becomes a metaphor for our emotions – at times a balanced equilateral triangle, at other times a sharply pointed scalene, other times an isosceles triangle standing tall.
 
The triangle comes to represent the diversity that can be found within a single shape that contains very clear rules (3 sides, 3 angles, the sum of the angles equals 180 degrees). Much like the makeup of each student in the class, triangles can be very different, yet each one contains its own beauty, and its qualities can be interpreted subjectively by others. Each triangle is unique, each triangle is beautiful.
The Project
As artists and mathematicians, the students then apply their understanding of geometry to create a geometric artwork, similar to those created by Kandinsky. The piece must centre around triangles, but they may use other polygons as well. Students are then tasked with writing an artist statement discussing what message they want to convey and what theme underlies their artwork.
Artist Statement
“This piece of artwork, made using a simple protractor, compass and ruler, conveys my feelings about the problems in our world. I started at a single point, and surrounded it by many geometric shapes, demonstrating that from one single action, many come after. To quote Isaac Newton, “For every action, there is a reaction.” All the different colours represent different religions and races, and together they work to create this picture. The colours change from frame to frame, showing the evolving views and actions of the people in our world. I left four triangles incomplete to show the altering world and how there is space for growth, improvement and expansion. There is no true background to show that the people and races make up the world, and aren’t simply a part of it. Each and every point in the picture contributes to the entire experience. Keep your mind to what you want to do and you will…make a difference.”
– Jonah B.

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Concepts:

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Application of Skills:

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Mental Math: