Geometry Manual Volume 3

Home , Parallelepiped

CHAPTER 3: VOLUME

I. Introduction

II. Volume as a Measurement of Space

III. Volume of Solids

A. Right-angled Parallelepiped

1. Passing from One Solid to a Succeeding One

2. Passing from One Solid to a Non-Successive One

B. Volume of Solids: Not Parallelepipeds

1. Volume of a Prism with a Triangular Base

2. Volume of a Prism with a Rhomboid Base

3. Volume of a Prism with a Hexagonal Base

C. Volume of Solids: Not Prisms

1. Volume of a Pyramid with a Square Base

2. Volume of a Pyramid with a Triangular Base

3. Solids of Rotation

a. Volume of the Cone

b. Volume of the Cylinder

c. Volume of the Sphere, Ellipsoid, and Ovoid

D. Volume of Regular Polyhedrons

I. INTRODUCTION

The child has had contact with solids in the Children’s House. Now we

view solids from two viewpoints:

1. With what material and experience has the child met the concept of

“body” before?

Body can have three states: solid, liquid, and gas. In this study we consider

only the solid state.

The child has met solids in his own environment above and beyond the

Children’s House. In the Children’s House he has met many materials regarding

the visual sense:

Visual discrimination of color, size, shape, and form. All these materials:

the Red Rods, Pink Cubes, Brown Prisms, Cylinders, and Geometric Solids, have

put him in contact with solids.

2. When has the child met measurement of solids before?

Volume is number. The child met this in the first presentation of the Golden

Bead material. Later he met the Cubes of the powers.

Materials for the Study of Volume:

The Yellow Volume material

Box of neutral wood cubes, 2 cm edge

The red rods

The broad stair

The solid cylindrical insets

Thousand cube

Cubes of the powers (colored cubes)

Series of geometric solids from Children’s House

Box of hundred squares

First and fourth rectangle from the area material

II. VOLUME AS A MEASUREMENT OF SPACE (SOLIDS)

Material:

The box of neutral cubes

Presentation:

“There are a certain number of cubes in this box. Let’s take out a few and

see what we can do with them.”

Example: Take 12 cubes.

1. Line them up in a row. I have placed one cube twelve times.

2. Form a rectangle. I have 2 cubes repeated 6 times.

3. Form a second rectangle. I have 3 cubes repeated 4 times. (This exercise also

shows an equivalence of solids.)

Example: With 6 cubes make all the figures that you can.

Aim: Equivalence among solids working with the material only.

Age: 8 years approximately

Material:

Yellow volume material

It contains a right-angled parallelepiped with a square base. (This is the

same dimension as the thickest and largest brown prism.)

There are 5 right-angled parallelepipeds with rectangular bases. Together

these equal the larger figure.

Presentation:

One of the first solids the child has met is the cube (Pink Cubes). With the

cube, we have a constant difference in the edges, from the smallest to the largest.

The difference in size with the cubes are more easily discernible.

1. “What is this?” It is a cube. (Smallest cube)

2. “What is this?” It is a cube (Largest cube) We use the cube to measure

volume.

We calculate the volume of any solid by determining how many like cubes

we can fit in the solid; the size of the cube is not important.

3. Take the large parallelepiped. How many cubes does it contain?

4. Imagine we cut this in 5 thick slices. Show the other pieces and how they are

all the same size.

5. “Let’s see how many of these small cubes would fit in one of these.” Show

they have one dimension the same.

6. Take the squared figure. Count the squares by counting the two sides:

10 x 5 = 50 cubes.

7. We have 5 pieces, each with 50 cubes. The entire figure must contain 250

cubes.

8. Take the box of neutral cubes. This contains exactly 250 cubes. “Let’s count

them.” Count 10 across and 5 cubes across. Show that there are 5 layers,

10 x 5 x 5. We obtained 250 by adding 50 + 50 + 50 + 50 + 50 = 250.

9. Take small labels. We multiply 5 x 5 x 10 to obtain the number of cubes.

Place labels at the edges.

10. Let’s give a name to the edges.

5 = a

5 = b

10 = c

Volume is: a x b x c

Finding the Volume

Volume = Abh

To reinforce the concept:

1. Take 18 small cubes (from the edges of the box - 3 edges). Build these same

edges with the small cubes. ( 4 x 4 x 10 with 10 being the height). Check the

resulting figure with the large parallelepiped.

2. Take enough squares to build the base. What touches the plane of the table

equals the base. Superimpose the large figure on this base.

a. 52

b. Construct on this base the edge that is the height. Each cube of the

height represents another layer like the one of the base.

Comparison: Work with the golden bead cube, a construction of a cube

with ten squares.

Volume is the area of the base x the height. V = Abh

This is the formula Dr. Montessori preferred.

Exercise:

Calculate the volume of the brown prisms, red rods, and pink cubes. This

may be done as either a x b x c or Abh.

Work out a table of volumes.

1. Passing from One Solid to a Succeeding One

Brown Prisms

Material:

The first and second of the brown prisms (smallest).

Separate box of 19 wooden prisms the same as the smallest brown prism.

Presentation:

1. Take the smallest brown prism

2. Add one neutral prism.

3. Compare with the second brown prism. One dimension is the same.

4. To make the brown prisms equal, we must add three neutral wooden prisms.

first —— second: 1 + 1+ 1 = 3 prisms

From prism 7 to 8:

1. Add 7 prisms along 2 edges.

2. Add one at the corner

7 —— 8: ( 2 x 7 ) + 1 = 15 prisms

See Mathematics Manual: Passing from one square to a succeeding one.

2. Passing from One Solid to a Non-Successive One

The material here is not sufficient for this work but is could be done on

paper.

See the Mathematics Manual: Passing form one square to a non-successive

one.