Checkpoint - Course 1, Unit 5
Patterns in Space and Visualization

In each unit, there is a final lesson and Checkpoint that helps students summarize the key ideas in the unit. The final Checkpoint will generally be discussed in class, with the teacher facilitating the summarizing, and students making notes in their Math Toolkits (teachers may just refer to this as "notes") of important points they need to remember, adding illustrative examples as needed. If your student is having difficulty with any investigation in this unit, this Checkpoint and the accompanying answers may help you recall the concepts involved, and give you the big picture of what the entire unit is about. If your student has completed the unit, then a version of this Checkpoint summary should be in his or her notes or toolkit.

Possible Responses to Unit Summary Checkpoint
The words in bold represent vocabulary and concepts that students should have mastered in this unit.
a. Compare and contrast prisms, pyramids, cylinders, and cones.
  • Pyramids and cones have one base, whereas cylinders and prisms have two identical parallel faces, each of which may be considered a base. (Students develop a working definition of right prism in this unit. They know that a right prism must have rectangular lateral faces that are perpendicular to the bases. The bases are polygons but are not necessarily regular. A pyramid must have triangular lateral faces.)
  • Cylinders and cones have circles for bases; pyramids and prisms have polygons for bases.
  • The volume of a prism or cylinder is found by multiplying the area of the base times the height.
  • Bases of pyramids and prisms may be any polygon. Faces of pyramids are triangles; faces of right prisms are rectangles.
  • Cones and cylinders have a curved lateral surface, whereas prisms and pyramids have a lateral face for each edge of the base.
b. Compare and contrast the six special quadrilaterals.
  • The six special quadrilaterals considered in this unit are square, rectangle, parallelogram, rhombus, trapezoid, and kite.
  • Squares have 4 lines of symmetry, rectangles and rhombi have 2, kites have one, and parallelograms and trapezoids, in general, have none.
  • For squares, rectangles, parallelograms, and rhombi, opposite sides are parallel and have the same length, and the diagonals bisect each other.
  • All four sides are the same length in squares and rhombi.
  • The diagonals are the same length in rectangles and squares.
  • All, except kites and trapezoids, have rotational symmetry.
These properties and others are proved in Course 3.
c. For each shape in Parts a and b, describe a real life application of that shape. What properties of the shape contribute to its usefulness?
Examples of applications can be found in fields from art to engineering. Useful properties may include:
Triangles and, therefore, tetrahedrons, are rigid. Because of this, they are used in construction. Points on the curved surface of cylinders are the same distance from a line connecting the centers of the circular bases. Thus, cylinders are useful for rolling on a flat surface. Squares enclose a greater area for a fixed perimeter, than any other quadrilateral. Likewise, cubes enclose a greater volume, for a fixed surface area, than any other rectangular prism.
d. Describe a variety of ways you can represent (draw or construct) space-shapes.
Space-shapes (the term used for three-dimensional shapes) can be represented using face-views, isometric drawings, or geometric sketches according to which representation is more helpful. They could also be drawn as two-dimensional nets. (A net shows all the lateral faces and how they connect to each other.) Space shapes can be constructed out of solid material such as clay.
e. Pose a problem situation which requires use of cubic, square, and linear units of measure. Describe how you would solve it.
One situation in which all three types of measures would be required is related to a swimming pool. The amount of water needed to fill the pool (volume) would require cubic units of measure. The amount of material needed for a pool cover (area) would be reported in square units. The distance around the pool (length) would use linear units of measure.
f. Describe how to make a strip pattern and how to make a tiling of the plane.
A strip pattern or tiling of a plane typically has a specific unit of design that is repeated. The repeat may be accomplished by translating the design unit repeatedly, or by reflecting and translating, or by rotating and translating. (Students used these ideas to create tessellations.)

If you would like to see specific problems from Course 1, Unit 5, a link is provided to Examples of Tasks from Course 1, Unit 5. If you are interested in following up on the Geometry strand in general, then the Scope and Sequence will help you see where different concepts are introduced. On the Geometry page, you can read an explanation of the main geometry concepts as they are developed in all four courses.

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