Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
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
After counting cubes, students connect volume to multiplication. Packing a right rectangular prism with unit cubes shows that the number of cubes in one layer is the length times the width, which is the area of the base, and the number of layers is the height. So the volume can be found by multiplying the three edge lengths, V = l × w × h, or by multiplying the base area by the height, V = b × h. Both formulas describe the same layered counting.
Students then apply the formulas to real and mathematical problems with whole-number edge lengths, such as finding how much sand a sandbox holds or which shipping box has more room. The standard also introduces the idea that volume is additive. An L-shaped solid made of two non-overlapping rectangular prisms has a volume equal to the sum of the two prisms' volumes. Splitting a composite shape in different ways and getting the same total is a good check. Representing a product such as 2 × 3 × 4 as a prism also illustrates the associative property, since the cubes can be grouped in any order.
Students sometimes compute 3 + 4 + 5 instead of 3 × 4 × 5. Linking each factor to the cube-layer picture shows why multiplication is needed.
In V = b × h, b is the area of the base, not the length of a base edge. Writing B for base area can help separate the two ideas.
When splitting an L-shape, students may include the corner region in both prisms. The two parts must not overlap, or the volume is overcounted.
After multiplying, students sometimes write square units. Three dimensions multiplied together give cubic units.
A planter is made from two rectangular prisms that do not overlap. Prism A is 6 ft by 2 ft by 3 ft. Prism B is 4 ft by 2 ft by 3 ft. What is the total volume?
Answer: The planter's volume is 60 cubic feet.
Have students build a prism, record the number of cubes in the bottom layer and the number of layers, and fill in a table. After several prisms they usually state the formula themselves. Then give composite shapes made from real cubes and ask for two different ways to split them.
Tests include formula problems, missing-dimension problems (a 120 cubic inch box with base 4 by 5 inches: how tall?), and composite solids shown in a drawing with labeled dimensions.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 120 (also accepted: 120 cubic cm)
V = l × w × h = 8 × 5 × 3 = 120 cubic centimeters.
Answer: 168 (also accepted: 168 cubic inches)
V = b × h = 24 × 7 = 168 cubic inches.
Answer: A) 6
The base area is 4 × 5 = 20 square inches. Height = 120 ÷ 20 = 6 inches.
Answer: 48 (also accepted: 48 cubic meters)
3 × 3 × 2 = 18 and 5 × 3 × 2 = 30. Adding gives 18 + 30 = 48 cubic meters.
Answer: C) 2 + 6 + 4
2 × 6 × 4, 12 × 4 and 6 × 8 all equal 48. Adding the edges gives 12, which is not a volume.
Answer: 24 (also accepted: 24 cubic feet)
V = 6 × 4 × 1 = 24 cubic feet.
A full lesson with slides, activities and an exit ticket on volume formulas and composite figures, pitched to grade 5 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 5.MD.C.5 with an answer key, ready in about a minute.
Make a worksheet →Turn volume formulas and composite figures into a quiz students answer online that marks itself, with a class summary for you.
Build a test →They are the same calculation. l × w gives the area of the base, b, so b × h is just a shorter way to write l × w × h.
No. Fifth grade uses whole-number edge lengths. Volumes of prisms with fractional edges are a sixth grade topic (6.G.A.2).