Volume with Fractional Edges: Packing with Smaller Cubes

In this lesson:

• Pack a fractional-edge prism with unit-fraction cubes
• Compute the volume from a cube count
• Show why gives the same answer

What You Will Be Able to Do

By the end, you will:

1. Pick the right unit-fraction cube for a fractional-edge prism
2. Count how many cubes fit and compute total volume
3. Multiply by the cube's volume — not just the count
4. Show that is packing, regrouped

A Jewelry Box: Does a Whole Cube Fit?

A jewelry box measures in by in by in.

A cube leaves a half-inch gap — and we can't slice the cube.

Recap: Whole Cubes Inside a Whole Prism

A prism packs with unit cubes:

Each cube has volume , so count equals volume — a coincidence we're about to lose.

A Smaller Cube: The Half-Inch Cube

A unit-fraction cube has edge . Try a -inch cube on the jewelry box:

• Length: cubes
• Width: cubes
• Height: cubes

Half-Inch Cubes Fill the Prism Cleanly

Three rows along the length, two across, two high — no gaps.

Total Cube Count: Twelve Half-Inch Cubes

The half-inch cubes pack as a array.

So is the volume cubic inches?

Predict before the next slide.

Predict: Is Cubic Inches?

Twelve cubes fit. Two answers on offer:

• A. cubic inches (one count = one cubic inch each)
• B. — each cube isn't a cubic inch

Pick A or B. Justify in one sentence.

One Half-Inch Cube Has Volume

A cube with edge inch has volume:

Each cube takes up cubic inch — not 1.

Total Volume: Count Times Cube Volume

Check by formula: . They match.

Twelve cubes, but volume — not 12.

Halves Don't Always Work — Try Thirds

• A -cube can't fill a -edge cleanly
• Use a -cube when any edge is in thirds

Your Turn: Pack a Thirds-Edge Prism

Use -inch cubes.

1. Cubes along each direction:
2. Total cubes:
3. Volume of one cube:
4. Total volume: cubic inches

Quick Check: Prism

Use -inch cubes.

• Cubes along each direction:
• Total cubes:
• One cube's volume:
• Total volume: cubic inch

If you wrote , you used the count as the volume.

Bridge: Does Multiplying Edges Give the Same?

For our jewelry box:

• Packing-and-counting:
• Multiplying the edges:

Both give . Coincidence — or is it the same calculation?

The Regrouping: Same Numbers, Grouped Differently

The same multiplication, regrouped to show the prism edges.

Pair Them Up: Edges Reappear

Each "count cube edge" recovers one prism edge.

The Theorem: Is Packing, Regrouped

For any right rectangular prism with fractional edges:

This is what the standard means by "show the volume is the same as multiplying the edge lengths."

Two-Way Check: Prism

By packing (use -cubes):

• Counts: cubes
• One cube: cu in
• Total:

By formula:

Your Turn: All Four Steps Unscaffolded

Solve fully on your own:

1. Pick the cube — what fraction of an inch?
2. Count along each direction
3. Total volume by packing
4. Verify with

No scaffolding — produce the whole chain.

Answers and Common Errors to Avoid

1. Cube edge: inch
2. Counts:
3. Total: cu in
4. Formula: ✓

Common error: (forgot to multiply by )

Key Takeaways and Warnings to Remember

✓ Pick a unit-fraction cube whose edge divides every prism edge

✓ Total volume = (cube count) (volume of one cube)

✓ is packing, regrouped — same calculation

Cube count is NOT the volume

, not

Next Lesson: Applying the Volume Formulas

You've earned the formula. The next deck:

• Apply to mixed-number prisms — convert first
• Use — same volume, computed in pieces
• Solve real-world problems with fractional dimensions

Storage boxes, planters, fish tanks, moving boxes.