Getting Started

The student is unable to identify the dimensions of a unit prism that can be used to pack the right rectangular prism.

The student cannot identify the unit fraction dimensions of the small prism. The student:

• Provides incorrect unit fractions (e.g., , , and ).
• Provides dimensions that are not unit fractions.
• Attempts to find the least common denominator of the given dimensions. 

What is a unit fraction?

What unit fraction divides evenly into ?

If the length of the large prism is cm, and the length of the small prism is a unit fraction, what length might the small prism have so it could fit evenly into the length of the large prism?

Ensure that the student is familiar with rectangular prisms and the terms used to describe their parts and dimensions such as base, length, width, and height.

Review the meaning of unit fraction. Guide the student to decompose fractions into sums of unit fractions (e.g., = + + ). Assist the student in determining the unit fraction measurement for each dimension of the small prism. Then ask the student to complete the task worksheet.

Moving Forward

The student is unable to identify the number of unit prisms needed to fill a right rectangular prism.

The student is able to determine the dimensions of the unit prism but cannot determine the number of unit prisms needed to pack the large prism. The student:

• Adds the numerators of the dimensions providing an answer of 14.
  
  
  
  
• Rewrites each dimension with a common denominator and then multiplies numerators and divides by the common denominator.
  
  
  
  
• Multiplies the denominators of the dimensions.
• Provides an estimate or guess.

How many small prisms would fit into the large prism lengthwise? What about width or height?

How many small prisms could fit onto the base of the large prism? How many layers would be needed to fill the large prism?

Review the concept of volume of a rectangular prism as the number of unit cubes needed to fill the prism. Construct a prism with linking cubes and, using one cube as the unit of measure, show the student that the volume can be determined by counting the number of cubes needed to construct the prism. Guide the student to determine the number of cubes in one layer (which is numerically equivalent to the area of the base). Next have the student determine how many layers are needed to complete the prism (which is equivalent to the height). Guide the student to multiply the number of cubes in the bottom layer by the number of layers needed to fill the prism. Relate this strategy to the formula for finding the volume of a rectangular prism (e.g., V = l x w x h or V = B x h where B is the area of the base).

Guide the student to determine the number of unit prisms that can span the length, width, and height of the larger rectangular prism. For example, explain that if the length of the base of the unit prism is cm and the length of the larger prism is cm, then three unit prisms can span the length of the base of the larger prism (since 3 x = ). Then assist the student in observing that if the unit prism is used as the unit of measure of volume, the larger prism can be described as a 3 x 2 x 9 prism. Using a counting approach or the volume formula, guide the student to determine that 54 unit prisms are needed to fill the larger prism.

Provide additional opportunities to determine appropriate dimensions of a unit prism and the number of units needed to fill a rectangular prism with fractional dimensions.

Consider administering the MFAS tasks for standard 5.MD.3.3 to assess the student’s understanding of volume.

Almost There

The student cannot explain how the volume of the larger prism is related to the number of small prisms needed to fill it.

The student is able to determine the number of unit prisms needed to pack the large prism but does not relate the volume of the unit prism to the volume of the large prism. The student:

• Explains the number of smaller prisms that fill the larger prism is the volume.
  
  
  
  
• Provides a general statement such as, “Volume is how much something can hold.”
  
  
  
  
• Writes an unclear statement.
  
  

What is volume? How is it measured?

If the unit prism is the unit of measure of volume, what is the volume of the larger prism?

Does the unit prism have volume? What is its volume?

Guide the student to consider the unit prism as the unit of volume measurement and explain that the volume of the larger prism is 54 unit prisms. Ask the student to determine the volume of the larger prism in cubic centimeters, given that the volume of the unit prism is cubic centimeter (by multiplying 54 x ). Then ask the student to determine the volume of the larger prism by using the volume formula (by multiplying x x ). Guide the student to understand why these two strategies result in the same volume.

Consider implementing the MFAS task Clay Blocks (6.G.1.2) to further assess the student’s understanding of the volume of a rectangular prism.

Got It

The student provides complete and correct responses to all components of the task.

The student determines the dimensions of the unit prism are ¼ cm, 1/5 cm, and 1/8 cm and that 54 unit prisms are needed to completely fill the larger prism. The student explains that the volume of the large prism is 54 times the volume of the small prism or 54 unit prisms.

Can you find the volume of the small prism?

Could you find the volume of the larger prism without first calculating the volume of the smaller prism?

Ask the student to find the volume of the larger prism in two ways:

1. By using the formula V = lwh.
2. By finding the volume of the unit prism and multiplying it by 54.

Then ask the student to explain why these two strategies result in the same volume.

Consider implementing the MFAS task Clay Blocks (6.G.1.2) to further assess the student’s understanding of the volume of a rectangular prism.