Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
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
Volume is the amount of space a solid figure takes up. Fifth graders begin by meeting the unit used to measure it: a cube whose edges are each 1 unit long, called a unit cube, which has a volume of one cubic unit. Just as area is measured by covering a flat shape with unit squares, volume is measured by filling a solid with unit cubes.
The key condition is packing without gaps or overlaps. A solid that can be filled exactly by n unit cubes has a volume of n cubic units. Students build rectangular prisms from cubes, count them and describe the result in cubic units, which lays the foundation for the volume formulas in 5.MD.C.5. They also distinguish volume from area and from perimeter: a box can have a large surface yet small volume, and two prisms with different shapes can have the same volume. Getting this idea right now prevents years of mixed-up units later.
In a picture of a prism, hidden cubes inside or at the back are easy to miss. Building the prism with real cubes, then drawing it, helps students account for every cube.
Counting the square faces on the outside of a cube structure gives surface area, not volume. Volume counts the cubes, not their faces.
Writing '24 square units' for a volume shows confusion between two and three dimensions. Volume is always in cubic units.
If cubes do not fill a space completely, the count is not the volume. The definition requires no gaps and no overlaps.
A box is filled with unit cubes in 2 layers. Each layer has 3 rows of 4 cubes. What is the volume of the box?
Answer: The volume is 24 cubic units.
Hands-on building is essential. Give students a small box and unit cubes and ask them to predict, then check, how many cubes fill it. Comparing two different boxes that hold the same number of cubes shows that shape and volume are separate ideas.
Assessment items show cube structures and ask for the volume, ask which unit measures volume, or ask whether two figures have the same volume.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: A) Cubic inches
Volume measures three-dimensional space, so it uses cubic units such as cubic inches.
Answer: 18 (also accepted: 18 cubic units)
Each unit cube is 1 cubic unit, so 18 cubes make a volume of 18 cubic units.
Answer: 30 (also accepted: 30 cubic units)
One layer has 2 × 5 = 10 cubes, and 3 layers have 30 cubes, so the volume is 30 cubic units.
Answer: B) They have the same volume
Volume counts the unit cubes, and both use 12, so both have a volume of 12 cubic units.
Answer: A) The surface area
Counting outside faces measures the area of the surface. Volume counts the cubes that fill the solid.
A full lesson with slides, activities and an exit ticket on understanding volume, pitched to grade 5 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 5.MD.C.3 with an answer key, ready in about a minute.
Make a worksheet →Turn understanding volume into a quiz students answer online that marks itself, with a class summary for you.
Build a test →A cube with every edge 1 unit long. It has a volume of one cubic unit and is the basic tool for measuring volume.
Because volume is only measured correctly when the cubes fill the space exactly. Gaps undercount and overlaps overcount.