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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).

See the student text book (page 460) for the Summarize the Mathematics questions. Solutions follow. Encourage students to look at their Math Toolkits or investigation notes to help refresh their ideas about these questions if they need help.

Summarize the Mathematics

	i.	Every polygon is made up of pairs of connected edges (segments) joined at vertices (endpoints). Polygons enclose a single region of a plane. The sum of the measures of the interior angles of any (convex) polygon with n sides is (n - 2)180°. The sum of the measures of the exterior angles of any (convex) polygon is 360°. See the "Design a Linkage" custom tool in the Course 1 CPMP-Tools interactive geometry software.
	ii.	Every quadrilateral is formed from four connected edges (sides). The sum of the lengths of three of the sides must be greater than the fourth side length in order to create a quadrilateral. The sum of the measures of the interior angles of every quadrilateral is 360°. Every quadrilateral tiles the plane. Quadrilaterals are not rigid figures. If the sum of the lengths of the shortest and longest sides is less than or equal to the sum of the lengths of the remaining two sides, then the shortest side (crank) can rotate completely. When the shortest side is the frame, both cranks will rotate 360°. (See page 367.)
	iii.	The sum of the measures of the interior angles of every triangle is 180°. Also, the sum of the lengths of any two sides will always be greater than the length of the third side. This property is called the Triangle Inequality. In addition, every triangle is a rigid shape. Every triangle tiles the plane.
	Two polygons are congruent if the corresponding angles and sides are congruent. This means that the two polygons will have the same shape and size.
	i.	Each of the following sets of corresponding parts will be sufficient to test for congruence of two triangles: three sides (SSS) two sides and an included angle (SAS) two angles and an included side (ASA) or two angles and a non-included side (AAS)
	ii.	In order to test whether two parallelograms are congruent, you would need a pair of corresponding adjacent sides congruent and any one pair of corresponding angles congruent. So, of the above tests, SAS could be used for a parallelogram.
	iii.	If a polygon can be subdivided into triangles that can be shown to be congruent, then you can reason to properties of the polygon such as congruent sides, angles, or diagonals. For parallelograms, opposite angles are congruent and the measures of consecutive angles sum to 180°.
	The converse of the Pythagorean Theorem is: If a triangle has side lengths a, b, c satisifying a² + b² = c², then the triangle is a right triangle. The general idea behind the justification of the converse of the Pythagorean Theorem involves recognizing that if you have a triangle with side lengths a, b, and c that satisfy a² + b² = c², then any right triangle that you would make with sides of length a and b would have a hypotenuse of length c (Pythagorean Theorem) and so would be congruent to the given triangle with side lengths a, b, and c. Thus, that original triangle must also be a right triangle. The converse of the Pythagorean Theorem provides a way to test if a convex is square, that is, if it makes a right angle.
	i.	Polyhedra are three-dimensional shapes formed from polygon shapes. A polygon encloses a region of the plane with edges that meet at vertices, while a polyhedron encloses a region of three-dimensional space with polygonal faces that meet in pairs at edges and edges that meet at vertices.
	ii.	Polyhedra can be represented with three-dimensional models such as the straw and pipe-cleaner models or with clay or foam. They can also be represented in two dimensions with different views as in orthographic drawings, with oblique drawings, or with nets.
	iii.	The angle measures of congruent copies of polygons must add to 360° in order to tile the plane. In order to form a polyhedron, the sum of the angle measures must be less than 360° at each vertex.
	iv.	Tests for line and plane symmetry for two-dimensional and three-dimensional shapes respectively involves looking for mirror images across a line or a plane. Tests for rotational symmetries in two- and three-dimensions involve looking for rotations (about a point or a line) less than 360° so that the figure and its turn image coincide.
	v.	The key idea in bracing shapes is to stabilize the vertices of the shape. This is accomplished by adding braces to the shape to form triangles (triangulation). This works because triangles themselves are rigid shapes.