Shapes & Designs
Shapes and Designs Objectives and Standards | |
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Goals for Shapes & Designs
Students should...
- Understand the properties of polygons that affect their shape
- Understand special relationships among angles
- Understand the properties needed to construct polygons
Investigation 1
Vocabulary | |
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The objectives of the Investigation are to: develop student understanding of critical polygon properties (especially degree measure for angles); ability to estimate angle and rotation measures using benchmark angles; skill in using standard tools for measuring and drawing angles; intuition about ways that triangular shapes can be characterized by minimal side and angle information
In Shapes & Designs 1.3 students will be estimating angle measures. Students can you the button to the right to practice their estimations of angle measure using the Bee Dance Game. Also located on the right is a pdf file of all vocabulary from Investigation 1.
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Investigation 2
Vocabulary | |
File Size: | 293 kb |
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The central objectives of Investigation 2 are to develop student understanding of several basic properties of polygons and their applications—the size and sums of interior and exterior angles and the explanation of tessellations in designs like the surfaces of honeycombs. Students frequently use variables to represent quantities such as the number of angles/sides and the total angle sum and construct simple equations to reason about the quantities in question.
In 2.3, students examine properties of tessellations. Students can use the button on the right to build their own tessellations on the computer.
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Investigation 3
Vocabulary | |
File Size: | 196 kb |
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The central objective in this Investigation is to develop student understanding of several basic properties of triangles and quadrilaterals. Students learn to apply these properties to building structures, making mechanical devices, and creating works of art. The Investigation explicitly addresses CCSSM objective 7.G.2, which calls for drawing geometric figures with given conditions, especially, “Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.” In the prior Investigations, students constructed some shapes, which depended more on angles. This Investigation uses both angle and side conditions as constraints.
The Investigation has five Problems. The first engages students in exploring the conditions for constructing triangles when given three potential side lengths, leading to the discovery of the SSS congruence criterion. The second Problem returns to the question of conditions that do and do not uniquely define triangles. The third explores the same question about quadrilaterals, revealing the structural instability of those figures. The fourth Problem uses results from work on parallelograms to explore the angle relationships when parallel lines are cut by a transversal. The fifth Problem explores reflectional and rotational symmetry in polygons and designs made from those figures as part of the introduction to the Quadrilateral Game. This game engages students in a review of the many important properties of common polygons including side-angle relationships and symmetry. In the game, students draw polygons on a grid from given specific conditions. They are then challenged to design a Triangle Game.
The Investigation has five Problems. The first engages students in exploring the conditions for constructing triangles when given three potential side lengths, leading to the discovery of the SSS congruence criterion. The second Problem returns to the question of conditions that do and do not uniquely define triangles. The third explores the same question about quadrilaterals, revealing the structural instability of those figures. The fourth Problem uses results from work on parallelograms to explore the angle relationships when parallel lines are cut by a transversal. The fifth Problem explores reflectional and rotational symmetry in polygons and designs made from those figures as part of the introduction to the Quadrilateral Game. This game engages students in a review of the many important properties of common polygons including side-angle relationships and symmetry. In the game, students draw polygons on a grid from given specific conditions. They are then challenged to design a Triangle Game.
Problem 3.1 develops student understanding of the basic facts that in a triangle the sum of any two sides must be greater than the third; and that once three acceptable side lengths have been chosen, there is only one triangular shape with those side lengths. Students can use Virtual Polystrips to explore this problem.
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The objective of Problem 3.3 is to extend the Triangle Inequality Property into a somewhat parallel result for quadrilaterals. Also, the Problem reveals the fact that quadrilaterals are not rigid figures. This is why, when they are used in construction, it is common to see bracing, turning quadrilateral frames into linked triangles. The sum of any three side lengths of a quadrilateral is greater than the fourth side. If a quadrilateral can be built from four side lengths, different shapes of quadrilaterals can also be built from them. Additionally, a quadrilateral’s shape will change in response to pressure on the vertex. This is how different shapes can be produced with the same arrangement of side lengths.
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