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.

Saturday, September 4, 2010

M7G3 - Proportional Relationships (7)

M7G3. Students will use the properties of similarity and apply these concepts to geometric figures.
b. Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures.

This standard is another example of how proportional relationships play an important role in the middle school mathematics curriculum. We say two figures are similar if one can be made to overlap the other exactly through a combination of translation (slide), rotation (turn), reflection (flip), and dilation (magnification). The parts of two similar figures that match up are called corresponding angles, sides, etc. In a pair of similar figures, we know that corresponding angles are congruent and the ratios of corresponding segments are constant - and the value of this ratio is the scale factor. For example, the two quadrilaterals shown below are similar.

Therefore, angles A and E, B and F, C and G and D and H are congruent, respectively. Moreover, the ratios of the lengths of sides, AB: EF, BC:FG, CD:GH, and DA:HE, are constant, and in this case the ratio is 1:2. The scale factor depends on which of the two figure we consider as the base of the comparison. So, if we consider quadrilateral ABCD as the base, the scale factor, in this case, is 2. On the other hand, if we consider quadrilateral EFGH as the base, the scale factor is 1/2.

Suppose AB = 4 cm, BC = 2 cm, CD = 5 cm, and DA = 6 cm. Then, EF = 8 cm, FG = 4cm, GH = 10 cm, and HE = 12 cm. Let's organize these lengths in a table.

Now, you see that as the length in ABCD doubles and triples (from 2 cm to 4 cm or 6 cm), the length of EFGH also doubles and triples. Even when the length becomes 2.5 times as long, from 2 cm to 5 cm, in ABCD, the corresponding length also becomes 2.5 times as long, 4 cm to 10 cm. Thus, the lengths in these two figures are in a proportional relationship. In general, if two figures, X and Y, are similar, the lengths in these two figures are in a proportional relationship. Thus, we can apply all the tools we discussed previously in representing this relationship. So, if we use a double number line, the relationship can be represented something like this:

Thus, if we know a side in Figure X is 15 cm and the corresponding side in Figure Y is 6 cm, we can use that relationship to determine the length of any side can be determined if we know the length of the corresponding sides. Suppose, we know another side in Figure X is 20 cm, the relationship can be represented in a double number line like this:

On the other hand, if you know the length of a side in Figure Y is 4 cm, the relationship will be represented like this:

Another feature of a proportional relationship is that the quotients of corresponding quantities are constant. So, if we divide the lengths in Figure X by the corresponding lengths in Figure Y, the quotients are constant. We can also use this relationship to represent the two situations above like this:

These tables basically show the four values (including the missing value represented by a ?) from the double number line representations above. Note that in the second table, the columns are in the reverse order. A number line has a particular direction, i.e., as you move to the right, the numbers become larger, However, a table does not have such an inherent directionality. So, for students, it might be more natural if we place the relationship as they are presented.

In any event, since 15 x 0.4 = 6, ? = 20 x 0.4 -- 0.4 is the scale factor if we consider Figure X as the base. For the second problem, we can say that since 6 x 2.5 = 15, ? = 4 x 2.5 -- 2.5 is the scale factor if we consider Figure Y as the base.

As is the case with the conversion of measurements from one unit to another, what is important is to help students develop an understanding that mathematics is a web of relationships. The focus of this standard is not just for students to find the missing lengths in similar figures. We also want them to understand that what they have learned previously, namely proportional relationships, can be used to represent, interpret, and investigate new situations.