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Volume with Fractional Edges | Lesson 2 of 2

Applying and : Mixed Numbers and Real-World Problems

In this lesson:

  • Apply to mixed-number prisms
  • Use as the same calculation
  • Solve real-world fractional-volume problems
Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

What You Will Be Able to Do

By the end, you will:

  1. Convert mixed numbers to improper fractions before multiplying
  2. Apply to fractional-edge prisms cleanly
  3. Use and pick any face as the base
  4. Solve real-world volume problems with unit conversion
Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

A Storage Box with a Mixed-Number Edge

How many cubic feet does this box hold?

One edge is a mixed number. How do we multiply cleanly?

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Bridge: We Earned the Formula in Deck 1

Deck 1 proved: is packing, regrouped.

So we can apply the formula directly — no need to pack every time.

The cube count and one-cube volume hide inside l, w, and h.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

The Procedural Rule: Convert First

Mixed numbers must become improper fractions before multiplying:

Don't multiply whole and fractional parts separately — it doesn't generalize.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Worked Example:

Step 1 — Convert:

Step 2 — Multiply numerators and denominators:

Volume = 30 cubic feet.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Worked Example:

Convert: ,

Volume = cubic units.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Worked Example:

The expression two-thirds times three-fourths times six written horizontally with the 3 in the numerator of three-fourths and the 3 in the denominator of two-thirds connected by colored arrows showing them cancel; below, the simplified expression two-fourths times six equals one-half times six equals 3

Cancel common factors first — the answer is much simpler than it looks.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Quick Check:

Convert and multiply.

If you got 12, you've got the rhythm.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

A Second Formula:

A 4 by 3 by 2.5 rectangular prism on the left; on the right, three smaller versions of the same prism each with a different face shaded as the base — first with the 4 by 3 base shaded and h equals 2.5 labeled, second with the 4 by 2.5 base shaded and h equals 3 labeled, third with the 3 by 2.5 base shaded and h equals 4 labeled; all three labeled V equals 30

  • = area of one base face
  • = height (perpendicular to the base)
Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Worked Example: Compute V = Bh Directly

Take the face as the base.

  • square units
  • units

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Try a Different Base — Same Volume

Same prism, different base choice:

  • Base : , ,
  • Base : , ,

Three different bases, three different heights, one volume.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Why Have Two Formulas for One Shape

is the more general statement.

  • Rectangular prism: , so
  • Triangular prism (Grade 7): is a triangle
  • Cylinder (Grade 8):

is the special case; is the general rule.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Mixed Practice — Two Items, Two Formulas

  1. lwh: — find
  2. Bh: Base area sq ft, height ft — find

Pick the formula that matches what's given.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Real Prisms Bring New Complications

A rectangular fish tank with three labeled dimensions: 2.5 ft along the length, 1.25 ft along the width, 1.25 ft along the height; annotations on the right read volume in cubic feet then convert to gallons; gallons are the actual answer

  • Mixed units (feet and inches)
  • Capacity questions (gallons, not cubic feet)
  • Downstream decisions (does it fit?)
Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Fish Tank: ft

Capacity (1 cu ft 7.48 gallons):

The volume is the input. "How many gallons" is the answer.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Planter: ft

Notice the cancellation: .

Cancel the threes and the twos — the answer is much simpler than it looks.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

The Unit-Conversion Trap: 12 vs 1728

A 1-foot cube outlined on the left; on the right, the same cube subdivided into a 12 by 12 by 12 grid of small inch-cubes with the count 1728 labeled prominently underneath

  • foot inches (linear)
  • cubic foot cubic inches

Volume conversion is the linear factor cubed — not 12.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Moving Box: Convert Inches to Cubic Feet

Convert dimensions first: in ft, in ft

Convert dimensions, then multiply — safer than converting at the end.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Find the Error in This Conversion

A student computed cu in.

Then converted: cu ft.

Where did they go wrong?

Hint: cubic feet of moving box is the size of a small house.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Your Turn — Soil for a Planter

A planter is ft.

Soil comes in -cu-ft bags.

How many bags do you need?

Compute the volume, then divide. Round up.

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Answers and Common Errors to Avoid

  • Mixed practice: ;
  • Soil bags: cu ft, bags

⚠️ Common errors: count = volume; instead of ; missed cancellation

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Key Takeaways and Warnings to Remember

✓ Convert mixed numbers improper fractions before multiplying

✓ and — same calculation, regrouped

✓ Real-world: convert units first; cancel before multiplying

⚠️ cu ft cu in (not 12)

⚠️ Volume is often the input — answer the actual question

Grade 6 Mathematics | Standard 6.G.A.2
Volume with Fractional Edges | Lesson 2 of 2

Next: Surface Area of the Same Prism (6.G.A.4)

You can now find the volume of any rectangular prism.

In Standard 6.G.A.4, you'll find its surface area using a net:

  • Unfold the prism into a flat figure
  • Add the areas of all six faces

Same prism, different measurement.

Grade 6 Mathematics | Standard 6.G.A.2