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Find the Volume of a Right Rectangular Prism with Fractional Edge Lengths

Show all work. Convert any mixed numbers to improper fractions before multiplying. Include cubic units in all answers.

Grade 6·21 problems·~35 min·Common Core Math - Grade 6·standard·6-g-a-2

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Recall / Warm-Up

What is the volume of a rectangular prism with length 5 cm, width 3 cm, and height 4 cm?

12 cm³

24 cm³

60 cm³

47 cm³

What is $\frac{3}{4} \times \frac{2}{3}$ ?

$\frac{5}{7}$

$\frac{6}{12} = \frac{1}{2}$

$\frac{5}{12}$

$\frac{3}{8}$

Convert $2\frac{1}{2}$ to an improper fraction.

Fluency Practice

$Rectangular prism with dimensions 2 ft by 1/2 ft by 1/3 ft, showing unit fraction cubes of edge 1/6 ft being packed along the 2 ft length.$

A rectangular prism has dimensions $2 \times \frac{1}{2} \times \frac{1}{3}$ feet. You pack it with cubes of edge length $\frac{1}{6}$ ft. How many small cubes fit along the length (2 ft)?

Find the volume of a rectangular prism with $l = \frac{3}{4}$ m, $w = 2$ m, $h = \frac{5}{2}$ m.

Find the volume: $l = \frac{2}{3}$ ft, $w = \frac{3}{4}$ ft, $h = \frac{1}{2}$ ft.

A prism has a base area of $B = 7$ cm² and a height of $h = \frac{3}{2}$ cm. Find its volume using $V = Bh$ .

Find the volume: $l = 1\frac{1}{2}$ in, $w = 2$ in, $h = \frac{3}{4}$ in. (Convert the mixed number first.)

Varied Practice

A prism has dimensions $\frac{1}{2} \times \frac{1}{3} \times \frac{1}{4}$ cm. You pack it with cubes of edge $\frac{1}{12}$ cm. Each small cube has volume $\frac{1}{1728}$ cm³. The formula $V = lwh$ gives $\frac{1}{2} \times \frac{1}{3} \times \frac{1}{4} = \frac{1}{24}$ cm³. Which statement explains why the packing method and the formula agree?

They give the same answer by coincidence.

The formula V = lwh counts all the unit-fraction cubes because each dimension tells how many cubes fit in that direction.

The packing method only works for whole-number dimensions.

The formula gives an approximation, not the exact answer.

A rectangular prism has $l = 2.5$ cm, $w = 4$ cm, $h = \frac{1}{2}$ cm. Find the volume in cm³.

A prism has $l = \frac{5}{4}$ ft, $w = \frac{2}{3}$ ft, $h = 3$ ft. Which expression gives the correct volume?

$\frac{5}{4} + \frac{2}{3} + 3$

$\frac{5}{4} \times \frac{2}{3} \times 3$

$2\left(\frac{5}{4} \times \frac{2}{3}\right) + 2\left(\frac{5}{4} \times 3\right) + 2\left(\frac{2}{3} \times 3\right)$

$\frac{5}{4} \times \frac{2}{3}$

Find the volume of a prism with $l = 2\frac{1}{3}$ m, $w = \frac{3}{4}$ m, $h = 2$ m. Convert the mixed number first.

Word Problems

A fish tank has length $1\frac{1}{2}$ ft, width $\frac{2}{3}$ ft, and height $\frac{3}{4}$ ft.

What is the volume of the fish tank in cubic feet? (Show the conversion of any mixed numbers.)

If 1 cubic foot of water weighs about 62 pounds, approximately how many pounds of water can the tank hold?

A sandbox is $2\frac{1}{2}$ m long, $1\frac{1}{2}$ m wide, and $\frac{1}{4}$ m deep.

Write a multiplication expression for the volume, then evaluate.
Expression = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
Volume = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ m³

expression:

volume:

A gift box has a base area of $\frac{5}{8}$ ft² and a volume of $\frac{5}{16}$ ft³.

Use $V = Bh$ and solve for the missing height.
Division expression for $h$ = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
Height = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ft

division expression:

height:

Small boxes with dimensions $\frac{1}{2}$ ft by $\frac{1}{3}$ ft by $\frac{1}{4}$ ft are stacked inside a large box that is 2 ft by 1 ft by 1 ft.

How many small boxes fit inside the large box? (Divide each dimension.)

Error Analysis

Raj finds the volume of a prism with $l = 2$ cm, $w = \frac{1}{2}$ cm, $h = 3$ cm. He writes:
$V = 2 + \frac{1}{2} + 3 = 5\frac{1}{2}$ cm³

What error did Raj make? What is the correct volume?

Raj added instead of multiplied. Correct volume: $V = 2 \times \frac{1}{2} \times 3 = 3$ cm³.

Raj used the wrong formula. He should have used $V = Bh$ , giving the same answer.

Raj's computation is correct; 5.5 cm³ is the right volume.

Raj should have converted $\frac{1}{2}$ to a whole number before computing.

Priya finds the volume of a prism with $l = 1\frac{1}{2}$ ft, $w = 2$ ft, $h = \frac{3}{4}$ ft. She writes:
$V = 1 \times 2 \times \frac{3}{4} = \frac{3}{2} = 1\frac{1}{2}$ ft³

What error did Priya make? What is the correct volume?

Priya dropped the $\frac{1}{2}$ from the mixed number $1\frac{1}{2}$ . Correct: convert first to $\frac{3}{2}$ , then $V = \frac{3}{2} \times 2 \times \frac{3}{4} = \frac{9}{4} = 2\frac{1}{4}$ ft³.

Priya used the wrong formula. She should have computed $V = Bh$ instead.

Priya's answer is correct.

Priya should have only used the fractional part $\frac{1}{2}$ , not the whole part 1.

Challenge / Extension

A prism has volume $\frac{3}{4}$ ft³. Its base dimensions are $\frac{3}{2}$ ft and $\frac{1}{2}$ ft. What is the height of the prism?

A rectangular prism has edge lengths $l$ , $w$ , and $h$ . If you double all three dimensions, what happens to the volume? Explain using the formula, and give a specific numerical example with fractional dimensions.

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