Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
In fifth grade students found the volume of boxes with whole-number edges by counting unit cubes and multiplying. Sixth grade extends this to edges like 2 1/2 inches. The approach starts concretely: pack the prism with small cubes whose edges are a unit fraction, such as 1/2 inch. A box 2 1/2 by 1 1/2 by 2 inches holds 5 by 3 by 4 = 60 half-inch cubes, and since each half-inch cube has volume 1/8 cubic inch, the box holds 60/8 = 7 1/2 cubic inches.
Students then confirm that multiplying the edge lengths directly, 2 1/2 × 1 1/2 × 2, gives the same 7 1/2. That match is the point: it shows why V = lwh still works when the numbers are fractions. The second formula, V = bh, multiplies the area of the base by the height, which highlights that volume is layers of the base stacked up. Applications include filling boxes, aquariums and planters, where students must choose units carefully and express answers in cubic units.
A cube with edges of 1/2 inch has volume 1/2 × 1/2 × 1/2 = 1/8 cubic inch. Students often assume halving the edge halves the volume.
Some students add length, width and height. Building a small prism from cubes shows volume grows by multiplying.
Multiplying whole parts and fraction parts separately, so 2 1/2 × 3 1/2 becomes 6 1/4, is a common slip. Converting to improper fractions first avoids it.
A box is 2 1/2 in long, 1 1/2 in wide and 2 in tall. Find its volume by counting half-inch cubes and by using V = lwh.
Answer: The volume is 7.5 cubic inches (7 1/2 in³).
Use real cubes and boxes where possible, or interactive drawings that show layers. Asking how many 1/2-inch cubes make one cubic inch (8) gives students a concrete reason the fractional cube volume is 1/8.
Test questions often provide fractional dimensions in a context and ask for the volume, or give the volume and two dimensions and ask for the third, which links to one-step equations.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 30
V = 4 × 5/2 × 3 = 60/2 = 30 cubic centimeters.
Answer: D) 1/8 ft³
V = 1/2 × 1/2 × 1/2 = 1/8 cubic foot.
Answer: 50
V = bh = 12.5 × 4 = 50 cubic inches.
Answer: B) 54
Each inch holds 3 small cubes along it, so the box holds 3 × 3 × 6 = 54 cubes.
Answer: 9/8 (also accepted: 1.125, 1 1/8)
V = 3/2 × 3/4 × 1 = 9/8 cubic feet, which is 1 1/8 or 1.125 ft³.
A full lesson with slides, activities and an exit ticket on volume with fractional edge lengths, pitched to grade 6 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 6.G.A.2 with an answer key, ready in about a minute.
Make a worksheet →Turn volume with fractional edge lengths into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Packing with cubes like 1/2-inch cubes shows why multiplying fractional edge lengths gives the volume, rather than asking students to trust the formula.
They are the same calculation. In V = bh, b is the area of the base (l × w), so the volume is that area times the height.