6th Grade - Math

LESSON PROBLEM: Measurement: You know how to calculate the area of a surface, say a floor, a field, a lake. But how would you find the volume of a book, or a box of rice cookies?


STUDY GUIDE

Different things are measured in different units. Money is measured in dollars and cents, time in hours and seconds, weight in grams and kilograms, length or distance in meters and kilometers and area in square meters and square centimeters.

Volume is the space something occupies. When calculating the area of a surface, we consider only the floor area, the surface it covers. For example, we calculate the area of the floor in a room to find how much it would cost to be carpeted. But most things have their body as well; they have a third dimension - the height, as well as the length and width.

If you have a big container, and you want to know the number of rice chip packets that can be stored in the container, how do you do it? You consider the third dimension of the chip packet as well as that of the container.

You have to store the year's rice crop, and need to know the size of the storage space required for storing it.

We need to know the volume for all of these.

When calculating the area of a surface, we consider its length and width only. We multiply the two lengths, and get the area which is expressed in square units.

When calculating the volume of an object, we consider its three dimensions; we multiply length, width and height, and express the volume in cubic units.

Remember the concept of a cube. A cube is an object where all three dimensions are the same size. If each dimension of an object is 1 cm, then its volume would be 1 cm³ (1 cm x 1 cm x 1 cm = 1 cm³).

If you place two cubes side by side, they will occupy a space 2 cm long, 1 cm wide and 1 cm high. The volume is 2 cm x 1 cm x 1 cm = 2cm³
If you arrange 8 cubes in two rows of 4 cubes each, then the surface area would be 4 cm in length, and 2 cm in width, and the height would remain 1 cm only. The volume of this block of 8 cubes would be 4 cm x 2 cm x 1 cm = 8 cm³.
If you place another layer of 8 cubes on top of the first layer of 8 cubes, then the block of cubes would occupy a space 4 cm long, 2 cm wide, and 2 cm high. The volume of the block would be 4 cm x 2 cm x 2 cm = 16 cm³

In the three examples you see how to get the volume of an object by multiplying its three dimensions. The formula for finding the volume of a solid is :

Length x Width x Height = Volume

Find the volume of these packages of rice products:

1.A box of rice cookies measuring 25 cm in length, 12 cm is width and 8 cm in height
2.A bag of rice measuring 52 cm x 30 cm x 10 cm
3.A box of rice noodles measuring 30 cm x 8 cm x 6 cm
4.A box of rice pilaf measuring 24 cm x 16 cm x 12 cm

In the examples above, we have used centimeter units. But we can use all of the other big and small units for measuring length. It can be meters, or even millimeters. To convert one unit into another, we only need to divide or multiply by ten. See the example below:

Let's look at one of the examples that we have already done. A box measures 25 cm x 12 cm x 8 cm. If we have to calculate its volume in centimeters, it is simple. Multiply the three numbers and express the answer in cubic centimeters. If we multiply 25, 12 and 8, we get 2400 as our answer. So it is 2,400 cm³.

If we want to express this answer in cubic meters, then divide by 1 followed by three zeros, because we are moving from centimeters to meters. Remember there are 100 centimeters in a meter. If you want to convert a cubed unit to one step above or below it, then you divide or multiply by 1000. In this case we are moving two steps, from centimeters to meters. Divide by 1,000,000 (1000 X 1000). Move six decimal places to the left. Let's try another one.

Assume that you want to express the dimensions of the box in cubic millimeters, the next unit smaller than the centimeter. 1 centimeter = 10 millimeters. First convert all the centimeter dimensions of the box into millimeters, and then calculate its volume. When coverting centimeters into millimeters, the dimensions of the box will be 250 mm x 120 mm x 80 mm. (Multiply each dimension by 10.)

To find the volume, multiply the three dimensions of the box:

250 mm x 120 mm x 80 mm = 2,400,000 mm³

In centimeters the volume of the box is 2,400 cm³

So 2,400 cm³ = 2,400,000 mm³

Compare the two units. There are three zeros added to the number. To convert one cubic unit, multiply or divide the number by 1 followed by three times the zeros in the steps between the two units. Centimeter and millimeter are next to each other. So they have a difference of one step. Therefore, multiply by 1 followed by three zeros. If the units are two steps apart, then multiply or divide by 1 followed by six zeros.

Convert answers for the three questions given above to cubic millimeters and to cubic decimeters. Let's do the first one together:

The box measures 25 cm x 12 cm x 8 cm. We want to find its volume in cubic decimeters. Decimeter is a unit one step higher, so divide its volume by 1 followed by three zeros. The volume of the box in cubic centimeters is 2400 cm³. Divide it by 1000, to get 2.400 or 2.4 cubic decimeters as the volume. (Remember not to put zeros at the end of a number, which is to the right of a decimal point.)

Conversion Table for Cubic Units.

1 cu. meter = 1000 cu. decimeter
1 cu. decimeter = 1000 cu. centimeter
1 cu. centimeter = 1000 cu. millimeter

Between millimeters and meters, there are three steps. To convert cubic meters to millimeters, multiply by 1 followed by nine zeros (three times the number of zeros for the number of steps between the two units).

You can also convert cubic measurements between the metric and U.S. systems.




ACTIVITIES

Measure different boxes of rice snacks, and find their volume.




EXTENDED LEARNING

You know the dimensions of rice snack boxes. Now you have to send them by a hired delivery truck to a store. Since you have to pay a fixed amount for the truck, you do not want to leave any empty space. How would you find the number of boxes that can be loaded into the truck?




VOCABULARY

  • volume
  • space
  • surface
  • dimension
  • cube
  • layer
  • convert


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