CHAPTER 3: VOLUME. 1. Passing from One Solid to a Succeeding One. 2. Passing from One Solid to a Non-Successive One

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1 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 Montessori Research and Development

2 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. Montessori Research and Development

3 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 Montessori Research and Development

4 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.) Montessori Research and Development

5 Example: With 6 cubes make all the figures that you can. Aim: Equivalence among solids working with the material only. Age: 8 years approximately Montessori Research and Development

6 III. THE VOLUME OF SOLIDS: A. THE RIGHT-ANGLED PARALLELEPIPED 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. Montessori Research and Development

7 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 = 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 = A b h 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. Montessori Research and Development

8 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 = A b h 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 A b h. Work out a table of volumes. Montessori Research and Development

9 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: = 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. Montessori Research and Development

10 2. Passing from One Solid to a Non-Successive One paper. The material here is not sufficient for this work but is could be done on one. See the Mathematics Manual: Passing form one square to a non-successive Montessori Research and Development