Volume: teacher workshop activity (written in 2003 by Don Yost)

Teacher notes follow, for a student lab

Each group will need 24 plastic cubes. Come to the front and check out your cubes; count them.

1. How many are there?

2. What color are they?

Make a large block 1 block by 3 blocks by 4 blocks.

If you wanted to tell someone about the big block, you could count the number of small blocks in the big block.

3. How many small blocks are in the big block?

You should have counted 24 blocks. Each block is called a cubic centimeter, since it is a cube and each side is one centimeter long. The number of these cubes that could fit into an object is called the “volume”. Therefore, the volume of the big block is 24 cubic centimeters.

4. If each edge of the cubes is one centimeter (cm) long, how long are each of

the edges of the big square?

You should have answered 3 cm, 4 cm, and 2 cm. In some figures, some of the edges are the same length and it becomes difficult to tell them apart. One trick to keep track of all the edges is to pick one corner of the big block, and use the three edges that lead from this one corner. Try several corners, and you will notice that it doesn’t matter what corner you pick, you will always get the same three sides. These sides are sometimes called width, height, and length, and which is which just depends on how the block is sitting.

5. What is 2 X 3 X 4?

Now arrange the blocks in a large block 1 by 2 by 12. When you take the blocks apart, do not twist them, as this will shear off the small square pegs. Simply pull straight out when you take them apart.

6. How many blocks are in this large block?

7. What is 1 X 2 X 12?

8. What is the volume of this large figure?

Now arrange the blocks in a large block 3 X 1 X 8.

9. How many blocks are in this large block?

10. What is 3 X 1 X 8?

11. What is the volume of this large figure?

You should notice that there are several ways to find the volume of a figure.

a. You can count the number of one-cm. blocks that are in the figure.

b. You can multiply the width by the length by the height.

12. What is the volume of a block with sides 6 cm by 1 cm by 2 cm?

VOLUME MEASUREMENT AND MODELING: teacher notes

(written in 2008 by Don Yost)

Apparatus

24 interlocking plastic cubes

One-inch cube

Pre-lab discussion

Length tells you how far. Area tells you how much cover. Volume concerns amount. Students build some different shapes out of cubes. In order to identify each, pick a corner of the block and identify how many cems are in each of three directions from that corner. If students pick a corner and there are 2 cems in one direction, 3 in another, and 4 in the remaining direction, they could call this a 2 X 3 X 4 block. Students should conclude that this is the same as a 4 X 2 X 3 block. Ask students if there are any other blocks related to these two. They should identify 2 X 4 X 3 for example.

These are hard questions to answer without actually building these blocks.

Activity:

Build the 2 X 3 X 4 block and count and record the number of cubes used. On a data table mark 2 X 3 X 4 block and next to it count and fill in the number of small cubes to make this block.

Next build a 3 X 4 X 1 block. Record the block and how many cubes it has. Then build a 1 X 6 X 4 block and a 2 X 2 X 6 block recording the block and how many small cubes it has in your data table. Also try and record.

Lab performance notes

Students should quickly realize that each block has the same number of cubes and therefore the same volume. They should also realize that changing the orientation doesn’t change the shape or volume of the block.

Post-activity discussion

The number of cubes in a block is called its volume. Since it takes 3 different cem measurements for volume, volume is measured in “cem cem cems” or “cubic cems”.

As in the case of area, there is a shortcut for counting the number of cubes. Students should come up with length times width times length. Students should also conclude that the choice of width, length, and height depends on orientation and is somewhat arbitrary.

While we have been working with cems, someone in a different town may not understand how long a cem is. This might be serious if two towns built railroad tracks toward each other and didn’t use the same measurement. This actually happened in Europe in some countries. Each country has a different width of train track, and when you take the train from one country to the other, you have to change trains at the border.

In order not to have railroad and other measuring problems, most of the world has agreed not to use the term “cem”, but instead uses the term “centimeter”... The cubes, then, are 1 cm. on an edge, the area of one face is 1 sq. cm. , and the volume is 1 cubic cm. (often called 1 cc.) If you have ever had a shot, you know about cc’s. When the doctor calls for a 3 cc. injection, you can expect three “cubes” of serum.

Deployment

Using the cubes, determine the volume of a one-inch cube.

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