## Tuesday, September 21, 2010

M6M3. Students will determine the volume of fundamental solid figures (right rectangular prisms, cylinders, pyramids and cones).
a. Determine the formula for finding the volume of fundamental solid figures.
b. Compute the volumes of fundamental solid figures, using appropriate units of measure.

There is actually a standard in Grade 5 that discusses the volume:
M5M4. Students will understand and compute the volume of a simple geometric solid.
c. Derive the formula for finding the volume of a cube and a rectangular prism using manipulatives.
d. Compute the volume of a cube and a rectangular prism using formulae.
So, what are the difference between these two standards? There are two obvious differences in these two standards. First, the Grade 5 standard involves the volume of "simple geometric solids," while the Grade 6 standard deals with "fundamental solid figures." Specifically, in Grade 6, students are expected to determine the volume of cylinders, pyramids, and cones in addition to cubes and rectangular prisms learned in Grade 5. So, the Grade 6 standard deals with a wider range of solids than the Grade 5 standard does.

Another difference is that, in Grade 5, students are to derive the formula using manipulatives while the Grade 6 standard does not mention the use of manipulatives. So, how do we expect Grade 6 students to derive the formula?

In Grade 5, students may determine the volume of cubes and rectangular prisms by filling them with unit cubes. Those experiences parallel what students might have done as they determine the area of squares and rectangles using unit squares. From these experiences, students learn that the dimensions of cubes and rectangular prisms can tell us the number of unit cubes that fit in each dimension. Thus, they can conclude that the volume of a rectangular prism can be calculated by multiplying its length, width and height.

The solids students explore in Grade 6 cannot be filled with unit cubes because of their shapes. So, how can students determine the formula for those solids? One important step is to re-visit the formula for the volume of cubes and rectangular prisms. When we determine the number of unit cubes inside a rectangular prism, we typically figure out the number of unit cubes in one layer, then multiply the result with the height, which signifies the number of layers. However, the first product, the number of unit squares in a single layer is equal to the area of the rectangular base. Thus, we can express the formula for calculating the volume of a rectangular prism as (Area of Base) x height, instead of length x width x height.

When we consider the volume formula for a rectangular prism as (Area of Base) x height, a natural question is whether or not this formula can be applied to prisms whose bases are something other than rectangles. Students can explore this question with triangular prisms and other prisms. Through such an exploration, they will find that the formula applies to any prism - and cylinders.

The volume formula for pyramids (and cone) is slightly different. It may be difficult to derive the volume formula for pyramids/cones directly. In fact, what we need to do is to relate the volume of a pyramid/cone to the related prism/cylinder, which has the congruent base and the same height as the pyramid/cone. A common way to establish this relationship is to have students actually fill up both a pyramid and the related prism (there are commercially made sets available for this purpose) with water or rice grains. Through such experimentations, students can establish the relationship that the volume of a pyramid/cone is a third of the volume of the related prism/cylinder. Thus, the volume formula for a pyramid is simply (Area of the base) x height ÷ 3 - if students have already learned multiplication of fractions before this unit, the formula can be written as (1/3) x (Area of the base) x height.

It may be useful to have students actually cut out (or the teacher demonstrate cutting) a cube into 3 congruent square pyramids like this - I apologize the poor quality of my 3-D drawing, and I hope you get the idea from this picture.

Note that these pyramids are different from most pyramids students seen in K-8 curriculum. Pyramids students study typically has the vertex that is not on the base to be directly above the center of the base. These pyramids, in contrast, has the vertex directly above one of the vertices of the base.

Clearly, such a demonstration does not establish the 1:3 relationship of the volume of any pyramid to the volume of the related prism. However, it may still be a worthwhile experience for students to have. There is, I believe, a commercially made puzzle that asks you to make a cube out of 3 congruent pyramids.