An
Exploration in Connections between

Pyramids
and Prisms

Facilitators Guide

**Required materials: **(for each working group)

· Cards that
are 5x7 inches or bigger

· Scissors

· Tape

· Rice

·
Ruler with metric

·
Pencil

**Getting started:**

· Hand out
activity guide to students, so they can follow along.

· Introduce the topic and show
the students examples of what they will be constructing.

· Lead a discussion about some of
the vocabulary before you start or about finding midpoints, height of the
pyramid, ratio of the volumes. Talk
about differences between faces, edges, and bases. Discuss why the term side is avoided.

· Ask if they have any questions
before starting

· Break students into groups of
four, and let them start the activity.

**Introduction:** In this activity you will be using an
ancient Egyptian method for finding the volume of a pyramid.

· Gather needed materials: 1 5x8 card, scissors, tape, ruler, and a pencil.

**Activity 1: **Constructing a Pyramid

- You will construct a four-sided pyramid, containing only the lateral faces.

· Cut out two 5x6 cm rectangles, be sure use the corner of the card as a reference point so that you obtain 90 degree angles.

· Find and mark the midpoint of the 6 cm edge.

· Then draw a line from the midpoint to each opposite corner; cut along these lines. Be sure to label the 6cm edge, you will need it as a reference point.

· Repeat process for the second card.

· You should now have 4 congruent right triangles and 2 congruent isosceles triangles.

· Tape two right triangles together to form an isosceles triangle that is congruent to the other isosceles triangle. Then tape together the remaining two right triangle in a similar fashion.

· Tape the four triangles together so that you obtain a pyramid with an open base.

· Calculate its height.

**Activity 2:** Constructing a Rectangular Prism

· You will construct an open rectangular prism (so only one base is necessary).

· Construct a base for the prism that will have the same dimensions as the base of the pyramid.

· Construct the four lateral faces of the prism. Each face should have a width equal to that of the base and a height equal to the height of the pyramid.

· Tape the four sides together and to the base.

**Activity 3:** Comparing Volumes

· Turn the pyramid upside down and fill it completely with rice. Make sure that the rice is level and fills the inside of the pyramid

· Then pour the rice from the pyramid into the rectangular prism.

· If this does not fill the prism, repeat the last two steps until prism is full.

· How many pyramids did it take to fill the prism?

· What is the ratio of the volume of the pyramid to the volume of the prism?

**Conclusion:**

· Lead
students in discussion of findings.

·
Why is this important?

·
Give the students the
formulas for the volumes of a prism and a pyramid.

·
Then have the
students calculate the volumes of their prism and pyramid.

**Some
Useful Definitions:**

__Congruent:__ two figures that can be transformed
into each other by a rotation, reflection, or translation are congruent.
(Basically the two figures are the same shape and size.)

__Isosceles
triangle:__ a triangle with at least two congruent edges.

__Rectangular
prism:__ a closed box consisting of six rectangular faces each connected at
90 degree angles. The bases must be congruent and parallel. (If all faces are
square, it is known as a cube.)

__Right
Triangle:__ a triangle with a 90 degree angle

__Lateral
Faces__ of a Pyramid: the triangular faces that meet at one vertex.

__Pythagorean
Theorem__: For a right triangle with legs *a
*and *b* and hypotenuse *c*, _{}. See picture: