Course 1, Unit 5 - Patterns in Space and Visualization

Summary
In the "Patterns in Space and Visualization" unit, students continue their development of geometric concepts and skills. Students define, identify, and sketch prisms, pyramids, cones, and cylinders. They learn to apply the concepts and computational formulas for perimeter and area of some two-dimensional shapes and surface area and volume of some three-dimensional shapes. The hands-on activities are designed to assist students in developing visualization skills, essential in understanding different kinds of symmetry. Students also begin to classify polygons and analyze their properties, the study of which becomes more formal in Courses 2 and 3.

Key Ideas from Course 1, Unit 5

• Polygon: a two-dimensional shape made of non-intersecting straight line segments that enclose a space. For example, three sides would make a triangle, four a quadrilateral, five a pentagon, as below.

• Pyramid: a three-dimensional shape which has a polygonal base, triangular faces, and comes to a point at a vertex above the base.

• Right prism: a three-dimensional shape which has two congruent polygonal bases and lateral rectangular faces which are perpendicular to the bases. (In this unit, prisms and cylinders are assumed to be right unless otherwise indicated.)

• Right circular cylinder: has two congruent circular bases, perpendicular to the lateral base.

• Cone: has a circular base and comes to a point above the base.
• Lateral faces: the wraparound rectangular faces, perpendicular to the base of a prism, or the triangular faces of a pyramid.
• The volume of a prism or cylinder is found by multiplying the area of the base times the height.
• Special quadrilaterals - squares, rectangles, rhombi, trapezoids, kites, and parallelograms: A square has four equal sides and right angles at each vertex; a rectangle has two pair of opposite equal sides and right angles at each vertex; a rhombus has four equal sides; a trapezoid has one pair of parallel sides; a kite has two pairs of adjacent equal sides; a parallelogram has two pairs of opposite sides parallel. (Squares, rectangles, and rhombi are all particular types of parallelograms.)

• Properties of special quadrilaterals: As well as the above definitions, each quadrilateral also has other properties. For example, a parallelogram has opposite sides equal, opposite angles equal, and the diagonals bisect each other; a square has all of the properties of a parallelogram and also equal and perpendicular diagonals.
• Line of symmetry: a line that cuts a shape into two symmetric halves, or mirror images. Squares have four lines of symmetry, rectangles have two, parallelograms have none, and rhombi have two. (See dashed lines on above diagram.)
• Isosceles: A trapezoid or triangle is isosceles if it has two equal length legs.

• Rotational symmetry: the property of a shape which can be rotated around a specific point through an angle less than 360° to fit back into its own outline. For example, a regular hexagon would rotate 60° around its center and fit back into its own outline.
• Rigid: Triangles are rigid, that is, with three given sides the angles are fixed. (This is not true of other polygons.)
• Space-shapes: three-dimensional shapes
• Net: shows all the lateral faces of a polygon and how they connect to each other. For example:

• Pythagorean Theorem: applies to right-angled triangles only and says that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. For example:

• Tiling of a plane, or tessellation: Some shapes can be repeatedly fit together to cover a plane, leaving no gaps. (See student book page 390.)