9 Cubes Everywhere
In this lesson, students use cubes to develop spatial thinking and review basic geometric
principles through real-life applications. Students are given the opportunity to build and
take apart structures based on cubes.
Learning Objectives
Students will:
Construct and visualize two- and three-dimensional shapes
Engage in explorations designed to enhance spatial visualization skills
Observe and identify characteristics of two- and three-dimensional objects
Develop geometry vocabulary and converse about geometric ideas
Sort, classify, and make and test conjectures
Materials
Assorted boxes and models of rectangular prisms (including cubes)
Toothpicks
Mini marshmallows (for best results, allow marshmallows to dry out a day or two)
Square grid or dot paper
Isometric dot paper (optional)
Tag board
Cellophane tape
Exploring Cubes Activity Sheet
Historical Photos Activity Sheet
Instructional Plan
The following pictorial puzzle is a challenge to the visual thinking of the students and
serves as a good introduction to the lesson.
On a trip to a city surrounded by many waterways, a series of photographs of some
historical landmarks (a lighthouse, a water tower, and a steeple) was taken from the deck
of a cruising ferryboat. Unfortunately, the resulting pictures were dropped and got mixed
up. Can you find in which order the pictures were taken? The map is a view of the
waterway region.
Project the following images onto the overhead projector or television screen.
Students can work in groups of 2 or 3 to solve the problem, "Can you find in which order
the pictures were taken?"
Students should discover the following order: C, B, F, A, D, E. Students can use various
reference points to help them (e.g. C is the first picture because the steeple is the first
thing seen from the boat, it has the greatest height, and you can see the other two
landmarks from that point.) Students may also use the compass to help them determine
the direction of one landmark with respect to another
He Exploring Cubes activity sheet provides the questions to be asked in this lesson.
Cooperative groups work well in this activity, namely small groups of 2 or 3 students.
Throughout the lesson, ask students questions which assist students to think of basic
geometric concepts and to develop vocabulary. Concepts such as point, vertex, line, edge,
parallel and perpendicular lines, skew lines, plane, angle, right angle, parallel and
perpendicular planes, square, rectangle, diagonal, and so on, naturally arise in student
discussions and in response to questions.
These activities give students an opportunity to see, touch, and manipulate tangible
embodiments of the concepts. They allow students to build visualization skills as they
study relationships among attributes of geometric objects and figures. Students might
draw some of the networks on tag board and construct cubes and open boxes. Students
might make up their own versions of creating a "building" with given front, side, and top
views to challenge their classmates.
Solutions to the Exploring Cubes activity sheet:
The number cube and box of cookies are examples of rectangular prisms. Students will
give various examples of objects they have seen. The milk container and rectangular
prism are not examples of cubes, because not all 6 faces are congruent.
Students should have used 12 toothpicks and 8 marshmallows. Students should see
parallel lines (e.g. the toothpicks on the top of the cube in the front and back), and they
should see perpendicular lines (e.g. a horizontal toothpick meets at a right angle with a
vertical toothpick). Students may view the top face and the bottom face as parallel planes,
and they may consider the top face and the front face as perpendicular planes (for
example.)
Student answers will vary depending on their drawings and the way they labeled the
vertices.
The first, fourth, and fifth pictures could be a view made from the number cube net
shown. 6 squares are needed to form a net for a cube.
There are twelve different pentominoes. Pentominoes are not considered "different" if the
same figure can be obtained by slide, flip, or turn motions. Eight of the twelve
pentominoes can be folded into open boxes (i.e., a cube without a lid).
Student answers will vary, depending on the building they draw.
95 cubes were needed to build the design in the figure. Students may use several methods
for solving this problem, including actually building the figure and counting the number
of cubes used
NCTM Standards and Expectations
Precisely describe, classify, and understand relationships among types of two- and threedimensional objects using their defining properties.
Describe sizes, positions, and orientations of shapes under informal transformations such
as flips, turns, slides, and scaling.
Recognize and apply geometric ideas and relationships in areas outside the mathematics
classroom, such as art, science, and everyday life.
Use two-dimensional representations of three-dimensional objects to visualize and solve
problems such as those involving surface area and volume.
Use visual tools such as networks to represent and solve problems.
Draw geometric objects with specified properties, such as side lengths or angle measures