Geometric Designs
You are working for a stained-glass company and need to analyze and design geometric patterns. You will use your ability to work with geometric shapes to calculate dimensions of shapes including angle measurements, side measurements, and area. In addition, you will need to be creative in order to make your own stained-glass design.
Task
In this project you will play the part of an artist working for a stained-glass company. When considering different glass designs there are several factors to consider: 1) area of shapes, 2) dimensions of shapes including angle and side measurements, and 3) aesthetic appeal (color, design, shapes utilized.). Working together with your group, you will analyze several designs and finally design your own stained glass window.
Instructions
Solve each problem in order and save your work along the way, as you will create a professional report at the conclusion of the project.
· First problem: Area of Shapes
Most stained-glass windows are a combination of many shapes. The artists come up with sketches and designs. Then the glassmaker needs to determine the side lengths of each individual shape and total area. Look carefully at each glass below and determine the shapes, their side lengths, and area.
Stained-Glass #1
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Name of Shape |
Sketch of shape with side lengths in centimeters |
Area of shape in square centimeters |
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Total area in square centimeters |
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Stained-Glass #2
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Name of Shape |
Sketch of shape with side lengths in centimeters |
Area of shape in square centimeters |
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Total area in square centimeters |
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· Second Problem: Area of Shaded Regions
Along with combining various shapes, stained-glass makers use a variety of different colors to enhance the beauty of their designs. Look carefully at each glass below and determine the areas of each different color in the stained-glass design.
Stained-Glass #3 Rectangles
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Color |
Area of shape in square centimeters |
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Orange |
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Red |
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Yellow
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Total area in square centimeters |
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Stained-Glass #4 Trapezoid
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Color |
Area of shape in square centimeters |
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Aqua-Blue |
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Purple |
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Total area in square centimeters |
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· Third Problem: Dimensions and Area of Pattern Blocks
In order to make intricate designs, it is nice to have shapes with known area and dimensions (side lengths and angles). In this problem, you will be determining the area (to the nearest hundredth) and dimensions of each pattern block. The area of the gray rhombus is given. Each side of the square, gray rhombus and equilateral triangle has a length of 2.5 cm. These pattern blocks will be used in the fourth problem to design your own stained glass window.
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Pattern Block |
Number of Equilateral Triangles in Shape |
Sketch with dimensions of sides and angles |
Area of Shape (to the nearest hundredth) |
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Equilateral Triangle
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1 |
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Blue Rhombus
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2 |
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Isosceles Trapezoid
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3 |
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Regular Hexagon
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6 |
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Square
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Does not apply |
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Gray Rhombus
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Does not apply |
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Area = 3.125 square cm |
· Fourth Problem: Make your own Stained-Glass Window
Using the pattern blocks from the template, you need to design your own stained-glass window. You should try to make it appealing in design and color. After you have your design, frame it with a square so it could be used in a window pattern. It does not have to be a “fitted” square meaning that it could be larger than the shape without touching each of the sides of the design. You should color the space between the inside design and the square. Measure the length of the square and determine the angle measurements of the glass needed to fill the square. Make a sketch of the entire design. Fill in the chart below for each shape and determine the total area of the inside shape and the area of the fill between the inside shape and the square.
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Pattern Block |
Number in Design |
Area of Shape |
Total Area |
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Equilateral Triangle
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Blue Rhombus
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Isosceles Trapezoid
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Regular Hexagon
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Square
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Gray Rhombus
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Area = 3.125 square cm |
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Total Area of Pattern Blocks (in square cm) |
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Area Square = ______________ square cm
Area of Fill Needed (Between Inside Shape and Square) = ________________ square cm
Collaboration
Get together with another group to compare your answers to each of the four problems. Discuss how you might analyze the shapes differently or combine your answers to make a more complete and convincing analysis of the situation.
Conclusions
Present your solution in a way that makes it easy for someone beginning geometry to be able to understand your results. Be sure to clearly explain your reasoning at each stage and conclude with recommendations about combining shapes, areas of different shaded shapes, and the design and making of a stained-glass window. You should look at different designs of windows and stained-glass windows to see how your design compares. Explain how your results can transfer to these different situations by scaling your window design.
Grade
Your project will be given a score of 1 to 4, with 4 being the highest score possible. Your project should correctly identify and justify mathematically: (1) calculating area of combined shapes, (2) calculating areas of shaded regions, (3) analyzing and finding dimensions and areas for each of the pattern blocks, and (4) designing your own stained-glass window with a sketch. You will be evaluated based on the following:
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Score |
Content |
Presentation/Communication |
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4 |
• The solution shows a deep understanding of the problem including the ability to identify the appropriate mathematical concepts and the information necessary for its solution. • The solution completely addresses all mathematical components presented in the task. • The solution puts to use the underlying mathematical concepts upon which the task is designed and applies procedures accurately to correctly solve the problem and verify the results. • Mathematically relevant observations and/or connections are made. |
• There is a clear, effective explanation detailing how the problem is solved. All of the steps are included so that the reader does not need to infer how and why decisions were made. • Mathematical representation is actively used as a means of communicating ideas related to the solution of the problem. • There is precise and appropriate use of mathematical terminology and notation. • Your project is professional looking with graphics and effective use of color. |
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3 |
• The solution shows that the student has a broad understanding of the problem and the major concepts necessary for its solution. • The solution addresses all of the mathematical components presented in the task. • The student uses a strategy that includes mathematical procedures and some mathematical reasoning that leads to a solution of the problem. • Most parts of the project are correct with only minor mathematical errors. |
• There is a clear explanation. • There is appropriate use of accurate mathematical representation. • There is effective use of mathematical terminology and notation. • Your project is neat with graphics and effective use of color. |
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2 |
• The solution is not complete indicating that parts of the problem are not understood. • The solution addresses some, but not all of the mathematical components presented in the task. • The student uses a strategy that is partially useful, and demonstrates some evidence of mathematical reasoning. • Some parts of the project may be correct, but major errors are noted and the student could not completely carry out mathematical procedures. |
• Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics. • There is some use of appropriate mathematical representation. • There is some use of mathematical terminology and notation appropriate of the problem. • Your project contains low quality graphics and colors that do not add interest to the project. |
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1 |
• There is no solution, or the solution has no relationship to the task. • No evidence of a strategy, procedure, or mathematical reasoning and/or uses a strategy that does not help solve the problem. • The solution addresses none of the mathematical components presented in the task. • There were so many errors in mathematical procedures that the problem could not be solved. |
• There is no explanation of the solution, the explanation cannot be understood or it is unrelated to the problem. • There is no use or inappropriate use of mathematical representations (e.g. figures, diagrams, graphs, tables, etc.). • There is no use, or mostly inappropriate use, of mathematical terminology and notation. • Your project is missing graphics and uses little to no color. |
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Monterey Institute for Technology and Education 2011 |