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Course 1, Unit 6 - Patterns in Shape

Overview
The intent of this unit is to review, deepen, and extend students' understanding of two- and three-dimensional shapes, their representations, their properties, and their uses. The fundamental idea of this unit is one of shape—what gives shapes their form and how the shape of an object often influences its function. The unit provides an introduction to mathematical reasoning as a way to discover or establish new facts as consequences of known or assumed facts. As such, the unit lays the groundwork for ideas of mathematical argument or proof that will be developed formally in Courses 2, 3, and 4. The focus here is on careful visual reasoning, not on formal proof.

Key Ideas from Course 1, Unit 6

  • The Triangle Inequality: This relationship among the lengths of the sides of a triangle is developed on pages 363-364. (The quadrilateral analog to this inequality is developed on page 365.)

  • Conditions that are sufficient for testing congruence of triangles: side-side-side, side-angle-side, and angle-side-angle are developed on pages 369-371.

  • The Pythagorean Theorem: If the lengths of the sides of a right triangle are a, b, and c, with the side length c opposite the right angle, then a2 + b2 = c2. For example:

    (This relationship is taught in most middle school classes. It was reviewed in Course 1 Unit 1 on page 50 and used for various review tasks in this course.)

  • The converse of a statement: The converse of an if-then statement reverses the order of the two parts of the statement. For example, the converse of the Pythagorean Theorem is: If the sum of the squares of the lengths of two sides of a triangle equals the square of the length of the third side, then the triangle is a right triangle. The converse of the Pythagorean Theorem has a very practical use - that of ensuring that a right angle is formed. (See page 380 Problem 3.)

  • Visualize and represent two- and three-dimensional shapes: Students build models of polyhedra and consider properties of polygons and polyhedra such as symmetry and rigidity.

  • Polygon: closed figures in a plane formed by connecting line segments endpoint-to-endpoint with each segment meeting exactly two other segments. For example, three sides would make a triangle, four a quadrilateral, five a pentagon, as below.

  • Polyhedron (plural - polyhedra): A three-dimensional counterpart of a polygon, made up of a set of polygons that encloses a single region of space. Exactly two polygons (faces) meet at each edge and three or more edges meet at each vertex. (See the examples below and page 429.)

  • Name, analyze, and apply properties of polygons and polyhedra: Polygons are frequently classified by the number of sides they have (page 400). For example, a 10-sided polygon is called a decagon. Characteristics of pyramids, prisms, cylinders, and cones are developed in Lesson 3 (pages 424-431).

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