Primary Standards:

MAFS.5.MD.3.3: Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

1. A cube with side length 1 unit, called a “unit cube”, is said to have “one cubic unit” of volume, and can be used to measure volume.
2. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

MAFS.5.MD.3.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

MAFS.5.MD.3.5: Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

1. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base.  Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
2. Apply the formulas V = l x w x h and V = B x h for rectangular prisms to find the volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems.
3. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Content Knowledge:

In 5th grade, students develop an understanding of volume, as well as explore ways to measure it.  At this grade level, volume is measure in whole number cubic units.  Initial explorations should involve students filling prisms with cubes and counting the volume.  As they explore the concept further, they will decide on appropriate strategies, such as counting each cube, or using multiplication or addition methods for finding the total.  They will use their experiences to estimate volume and solve problems related to volume.

Although students can count to find the number of cubic units, through exploration they discover there are more efficient methods.  Rather than telling students the formula for finding the volume of a right rectangular prism, providing opportunities for students to gather data, observe those data, and gain insights of their own allows them to make sense of the formulas they discover (multiplying length × width × height or multiplying base × height).

As students explore further, they realize they can decompose complex 3-D figures into separate prisms to determine the volume.  Through this work they recognize volume is additive. As important as understanding how to calculate volume is understanding when and why we calculate volume.  Students will experience real world problems involving the application of finding missing volumes.

GCG 3 – Learning Goal:  As a mathematician, I will be able to explore and define the concepts of volume

• Step 1: Identify and explain that volume is measured in cubic units that are packed without gaps or overlaps
• Step 2: Determine the volume of a rectangle by counting the number of cubes it takes to fill a rectangular prism

GCG 2 – Learning Goal:  As a mathematician, I will be able to use dimensions to solve problems involving volume

• Step 1: Relate multiplication and multiplicative reasoning to use dimensions of prism to determine volume
• Step 2: Explain and use a formula to determine the volume of rectangular prisms
• Step 3: Solve problems involving volume

GCG 3 – Learning Goal:  As a mathematician, I will be able to recognize volume as additive and determine the volume of complex figures

• Step 1: Use concrete models to recognize volume as additive. Find the volume of complex figures (two non-overlapping right rectangular prisms)
• Step 2: Apply understanding of addition and multiplication to solve real world problems involving volume of complex figures