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.
The above images are also available for individual student use. Distribute the Historical Photos activity sheet.
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.
The 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 tagboard 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
- 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.