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

 

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: