Volume with Fractional Edge Lengths - Sixth Grade Mathematics
Calculating the volume of rectangular prisms with fractional edge lengths reinforces multiplying fractions and establishes how fractional unit cubes pack within spatial boundaries.
Packing with Fractional Unit Cubes
Consider a rectangular prism whose dimensions are not whole integers, such as a prism measuring 1 1/2 inches by 1/2 inch by 3/4 inch. How do we count unit cubes when whole unit cubes do not fit neatly?
Conceptualizing Fractional Unit Cubes:
Imagine small unit cubes whose edge length is a unit fraction, such as 1/4 inch.
1 whole unit cube (1 in. x 1 in. x 1 in.):
How many (1/4 in. x 1/4 in. x 1/4 in.) cubes fit inside a standard 1 in.^3 cube?
- Length: 4 fractional cubes of 1/4 in.
- Width: 4 fractional cubes of 1/4 in.
- Height: 4 fractional cubes of 1/4 in.
Total = 4 * 4 * 4 = 64 small cubes fit in 1 whole cubic inch.
Each small cube has a volume of (1/4)^3 = 1/64 cubic inch!
To find prism volume by packing fractional cubes: Total Volume = (Number of small fractional cubes) * (Volume of one fractional cube).
Example: Prism measuring 3/4 in. by 1/2 in. by 1/4 in.
Convert all fractions to a common denominator of 4:
Length = 3/4 in. --> 3 small cubes of 1/4 in.
Width = 1/2 in. = 2/4 in. --> 2 small cubes of 1/4 in.
Height = 1/4 in. --> 1 small cube of 1/4 in.
Total small cubes = 3 * 2 * 1 = 6 cubes.
Since each cube has volume 1/64 in.^3:
Total Volume = 6 * (1/64) = 6/64 = 3/32 cubic inches!
The Standard Formula Applied to Fractions
The standard volume formula V = l * w * h applies identically to fractions and mixed numbers. Mixed numbers must always be converted to improper fractions prior to multiplication.
The Fractional Volume Procedure:
Step 1: Convert all mixed numbers into improper fractions.
Step 2: Multiply numerators across: Numerator = l_num * w_num * h_num.
Step 3: Multiply denominators across: Denominator = l_den * w_den * h_den.
Step 4: Simplify or convert back to a mixed number in lowest terms.
Worked Problem:
Find the volume of a rectangular wooden block measuring:
l = 2 1/2 cm, w = 1 1/4 cm, h = 3/5 cm
Step 1: Convert to improper fractions:
l = 2 1/2 = 5/2 cm
w = 1 1/4 = 5/4 cm
h = 3/5 cm
Step 2: Multiply using V = l * w * h:
V = (5 / 2) * (5 / 4) * (3 / 5)
Step 3: Simplify by canceling the common factor of 5:
V = (1 / 2) * (5 / 4) * (3 / 1)
V = (1 * 5 * 3) / (2 * 4 * 1)
V = 15 / 8 cm^3
Step 4: Convert to mixed number:
15 / 8 = 1 7/8 cubic centimeters.
Finding a Missing Fractional Dimension
If the volume and two fractional dimensions are given, divide the volume by the product of the known dimensions.
Problem: A prism has volume V = 3/8 cubic foot.
Its length is 3/4 foot and its width is 1/2 foot. What is its height h?
Step 1: Calculate base area B:
B = l * w = (3/4) * (1/2) = 3/8 square foot.
Step 2: Solve for height:
V = B * h
3/8 = (3/8) * h
h = (3/8) / (3/8) = 1 foot.
Exercises
Exercise 1. Calculate the volume of each rectangular prism, expressing answers as simplified fractions or mixed numbers: (a) l = 3/4 m, w = 2/3 m, h = 1/2 m; (b) l = 1 1/2 ft, w = 2/3 ft, h = 3/4 ft; (c) A cube with edge length 2/5 cm; (d) l = 3 1/3 in., w = 1 1/5 in., h = 2 1/2 in.
Exercise 2. How many small cubes with edge length 1/3 inch are required to completely fill a rectangular prism measuring 1 inch by 2/3 inch by 1 1/3 inches?
Exercise 3. A jewelry box has a volume of 4 1/2 cubic inches. Its length is 2 1/4 inches and its width is 2 inches. What is the height of the box?
Exercise 4. A rectangular planter box has interior dimensions of 3 1/2 feet long, 1 1/2 feet wide, and 1 1/4 feet deep. How many cubic feet of potting soil are needed to fill the box to the brim?
Solutions
Solution to Exercise 1. For (a), V = (3/4) * (2/3) * (1/2) = (6 / 24) = 1/4 m^3. For (b), convert 1 1/2 to 3/2: V = (3/2) * (2/3) * (3/4) = 1 * (3/4) = 3/4 ft^3. For (c), V = (2/5)^3 = (2/5) * (2/5) * (2/5) = 8/125 cm^3. For (d), convert to improper fractions: 3 1/3 = 10/3, 1 1/5 = 6/5, 2 1/2 = 5/2. V = (10/3) * (6/5) * (5/2). Cancel common factors: (10/3) * (6/2) = (10/3) * 3 = 10 in.^3.
Solution to Exercise 2. Convert each dimension to multiples of 1/3 inch: Length = 1 in. = 3/3 in. (3 cubes); Width = 2/3 in. (2 cubes); Height = 1 1/3 in. = 4/3 in. (4 cubes). Total cubes required = 3 * 2 * 4 = 24 cubes. Check volume: 24 * (1/3)^3 = 24 * (1/27) = 24/27 = 8/9 in.^3; Formula: 1 * (2/3) * (4/3) = 8/9 in.^3 (Confirmed!).
Solution to Exercise 3. Convert numbers to improper fractions: Volume = 4 1/2 = 9/2 in.^3; Length = 2 1/4 = 9/4 in.; Width = 2 in. Base area B = l * w = (9/4) * 2 = 18/4 = 9/2 in.^2. Using V = B * h: (9/2) = (9/2) * h. Divide both sides by 9/2: h = 1 inch.
Solution to Exercise 4. Convert all dimensions to improper fractions: l = 3 1/2 = 7/2 ft, w = 1 1/2 = 3/2 ft, h = 1 1/4 = 5/4 ft. V = (7/2) * (3/2) * (5/4) = (7 * 3 * 5) / (2 * 2 * 4) = 105 / 16 = 6 9/16 cubic feet of potting soil.