Volume with Fractional Edges

What is volume?

Volume is how much space a box takes up. You measure it in cubic units. A cube that is 1 unit on each side holds one cubic unit.

A box with a side that ends in a half is the new part. A whole-number cube will not fit that side neatly.

Why fractions are not a problem

The fix is to use a smaller cube. A cube with an edge of 1/2 fits any side that is a whole number or a half.

Then you count the cubes in one layer, and count how many layers stack up.

A 2 1/2 by 1 1/2 by 3 box holds 15 cubes in a layer and 6 layers, so 90 cubes in all. Each cube is 1/2 by 1/2 by 1/2, so the 90 cubes fill 11.25 cubic units.

What is this module?

The module shows a box you can spin. Three sliders set the length, the width, and the height, in steps of one half.

The box is packed with small cubes. The Add layer button fills it one layer at a time, so you can count.

Two readouts sit below the box. One counts the cubes. The other multiplies length times width times height. They always agree.

What they're learning

• A fractional edge does not stop you from finding volume.
• You can pack the box with small cubes whose edge is the unit fraction that fits.
• Volume is cubes per layer times number of layers.
• Counting the cubes gives the same answer as V = l × w × h.
• Volume is measured in cubic units.

Aliases

• Volume
• Fractional edge lengths
• Unit cubes
• Packing a box

Refresher

Volume is the space inside a box, measured in cubic units. When an edge length is a fraction like 1/2, pack the box with smaller cubes whose edge is that fraction, then count them.

Try this together

You don't have to teach a whole lesson. A few minutes with the sliders is enough.

1. Set length 2 1/2, width 1 1/2, height 3. Ask how many cubes fit in the bottom layer before pressing anything.
2. Press Add layer once. Count the cubes together. There are 15.
3. Press Add layer a few more times. Each layer adds 15 cubes. Six layers make 90.
4. Compare the two readouts. The cube count times 1/2 × 1/2 × 1/2 is 11.25. Multiply 2.5 × 1.5 × 3 and you get 11.25 too.

If the numbers feel slippery, start with a whole-number box first. Set 2, 2, 2 and count the 8 cubes. Then change one edge to 2 1/2 and watch the smaller cubes appear.

Words you might hear

• Volume: how much space a shape takes up.
• Rectangular prism: a box with flat rectangular faces.
• Edge: one of the sides of the box.
• Unit cube: a cube used to fill the box. Here its edge is 1/2, so it holds 1/2 × 1/2 × 1/2 of a cubic unit.
• Unit fraction: a fraction with 1 on top, like 1/2.
• Cubic units: the unit for volume. A 1 × 1 × 1 cube is one cubic unit.
• Layer: one flat row of cubes stacked on the floor of the box.

Deeper dive

Take a 2 1/2 by 1 1/2 by 3 box. A half-unit cube fits the edges neatly. The bottom layer holds 5 cubes across and 3 cubes deep, which is 15 cubes. The box is 6 half-units tall, so 6 layers stack up.

Fifteen cubes per layer times six layers is 90 cubes. Each cube holds 1/2 × 1/2 × 1/2 = 1/8 of a cubic unit, so 90 × 1/8 is 11.25 cubic units. Multiply 2.5 × 1.5 × 3 and you get 11.25 too.