5th Grade Mathematics
Instructional Week 22
Develop and use the formula for finding the volume of right, rectangular prisms
Paced Standards:
5.M.5: Apply the formulas V = l × w × h and V = B × h for right rectangular prisms to find volumes of
right rectangular prisms with whole-number edge lengths to solve real-world problems and other
mathematical problems.  +
5.M.6: 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 and other mathematical problems. 
Connections to other 5th Grade Standards:
5.M.3: Develop and use formulas for the area of triangles, parallelograms, and trapezoids. Solve realworld and other mathematical problems that involve perimeter and area of triangles, parallelograms
and trapezoids, using appropriate units for measures.
5.M.4: 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
or multiplying the height by the area of the base.
Prerequisite/Foundational Standards:
4.M.4: Apply the area and perimeter formulas for rectangles to solve real-world problems and other
mathematical problems. Recognize area as additive and find the area of complex shapes composed of
rectangles by decomposing them into non-overlapping rectangles and adding the areas of the nonoverlapping parts; apply this technique to solve real-world problems and other mathematical
problems.
3.M.5: Find the area of a rectangle with whole-number side lengths by modeling with unit squares,
and show that the area is the same as would be found by multiplying the side lengths. Identify and
draw rectangles with the same perimeter and different areas or with the same area and different
perimeters.
3.M.6: Multiply side lengths to find areas of rectangles with whole-number side lengths to solve realworld problems and other mathematical problems, and represent whole-number products as
rectangular areas in mathematical reasoning.
3.M.7: Find perimeters of polygons given the side lengths or by finding an unknown side length.
3.AT.5: Determine the unknown whole number in a multiplication or division equation relating three
whole numbers.
5th Grade ISTEP+ Toolkit
Indianapolis Public Schools
Curriculum and Instruction
Teacher Background Information
Standard: 5.M.5: Apply the formulas V = l × w × h and V = B × h for right rectangular prisms to find volumes of right
rectangular prisms with whole-number edge lengths to solve real-world problems and other mathematical problems.
Teacher Background Information:
After students fully understand 5.M.4, then they can learn the formulas V =l x w x h and V = B x h for right
rectangular prisms as efficient methods for computing volume, maintaining the connection between these
methods and their previous work with computing the number of unit cubes that pack a right rectangular prism.
They use these competencies to find the volumes of right rectangular prisms with edges whose lengths are whole
numbers and solve real-world and mathematical problems involving such prisms.
Students also recognize that volume is additive and they find the total volume of solid figures composed of two right
rectangular prisms. For example, students might design a science station for the ocean floor that is composed of
several rooms that are right rectangular prisms and
that meet a set criterion specifying the total
volume of the station. They draw their station and
justify how their design meets the criterion.
These standards involve finding the volume of right
rectangular prisms (see picture to the
left).Students should have experiences to describe
and reason about why the formula is true.
Specifically, that they are covering the bottom of a
right rectangular prism (length x width) with
multiple layers (height). Therefore, the formula
(length x width x height) is an extension of the
formula for the area of a rectangle.
Indianapolis Public Schools
Curriculum and Instruction
5.M.6
Standard: 5.M.6 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 and other mathematical
problems.
Teacher Background Information:
This standard calls for students to extend their work with the area of composite figures into the context of volume.
Students should be given concrete experiences of breaking apart (decomposing) 3-dimensional figures into right
rectangular prisms in order to find the volume of the entire 3-dimensional figure.
Students need multiple opportunities to measure volume by filling rectangular prisms with cubes and looking at
the relationship between the total volume and the area of the base. They derive the volume formula (volume
equals the area of the base times the height) and explore how this idea would apply to other prisms. Students
use the associative property of multiplication and decomposition of numbers using factors to investigate
rectangular prisms with a given number of cubic units.
Indianapolis Public Schools
Curriculum and Instruction
5.M.6
Standard: 5.M.6 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 and other mathematical
problems.
Teacher Background Information:
Example:
When given 24 cubes, students make as many rectangular prisms as possible with a volume of 24 cubic units.
Students build the prisms and record possible dimensions.
Example:
Students determine the volume of concrete needed to build the
steps in the diagram:
Process Standards to Emphasize with Instruction of 5.M.6:
5.PS.1: Make sense of problems and persevere in solving them.
5.PS.2: Reason abstractly and quantitatively.
5.PS.4: Model with mathematics.
5.PS.8: Look for and express regularity in repeated reasoning.
Indianapolis Public Schools
Curriculum and Instruction
Instructional Week 22
5 Grade Mathematics Assessment
th
Name: _____________________________
1.
The right rectangular prism shown below has a length of 9 inches, width of 4 inches, and a height of 7 inches.
7 in.
4 in.
9 in.
Which of the following equations could be used to find the volume of the prism?
I.
II.
III.
IV.
A)
B)
C)
D)
2.
V = 36 x 7
V = (9 + 4) x 7
V=9x4x7
V = 36 + 36 + 36 + 36 + 36 + 36 + 36
I. and II. only
II. and III. only
I. III. and IV. only
I. and III. only
Roger’s Burger Shop needs to buy a new freezer to keep all of his meat cold. Which one of the freezer
options below has the greatest volume?
Fry Freezer: 5 ft. by 6 ft. by 8 ft.
Frigid Freezer: 4 ft. by 3 ft. by 12 ft.
February Freezer: 6 ft. by 4 ft. by 9 ft.
Funky Freezer: 8 ft. by 3 ft. by 8 ft.
A)
B)
C)
D)
Fry Freezer
Frigid Freezer
February Freezer
Funky Freezer
Indianapolis Public Schools
Curriculum and Instruction
3a)
The truck below is used to transport ovens. The trailer has interior measurements of 8 feet wide, 40
feet long, and 12 feet high.
How many oven boxes, with the measurements of 4 feet wide, 4 feet long, and 4 feet high, will it take to
fill the entire trailer?
Show all work
Answer _________________
3b)
The same truck is used to transport refrigerator boxes with the dimensions of 4 feet wide, 4 feet long,
and 9 feet high. The refrigerator boxes may NOT be laid on their sides. If the trailer is filled with the
greatest number of refrigerator boxes, how many cubic feet of space will NOT be used?
Show all work
Answer _______________
Indianapolis Public Schools
Curriculum and Instruction
4)
Box A is 12 feet long, 4 feet wide and 4 feet high. What is the combined volumes of both boxes?
8 ft.
4 ft.
5 ft.
B
A
A)
B)
C)
D)
5)
192 ft3
288 ft3
352 ft3
320 ft3
Peggy will fill the tank below with 756 cubic inches of dirt.
14 in.
18 in.
6 in.
How high will Peggy fill the tank with dirt?
A)
7 inches
B)
14 inches
C)
38 inches
D) 1512 inches
Indianapolis Public Schools
Curriculum and Instruction
6a)
Lisa is building a dollhouse out of cardboard boxes in the shape of right rectangular prisms. She will
attach Box A to Box B. A model is shown below.
8 in.
18 in.
A
10 in.
B
8 in.
30 in.
Write an expression that can be used to find the total volume, in cubic inches, of the dollhouse.
Volume of the dollhouse = _______________________________________________
6b)
What is the total volume, in cubic inches, of the dollhouse?
Show all work
Answer _______________________
Indianapolis Public Schools
Curriculum and Instruction
Question
1
2
3a
3b
4
5
6a
6b
2
1
0
Grade 5
Instructional Week 22 Assessment
Answer Key
Correct Answer
C
A
60 oven boxes
960 cubic feet
C
A
22 x 8 x 10 + 18 x 8 x 8
or other equivalent expression
2912 cubic inches
Standard(s)
5.M.5
5.M.5
5.M.5
5.M.5
5.M.6
5.M.5
5.M.6
5.M.6
Content Rubric
A score of two indicates a thorough understanding of the mathematical concepts embodied in the
task. The response
 shows content related work executed correctly and completely.
A score of one indicates a partial understanding of the mathematical concepts embodied in the
task. The response
 contains errors in the content related work
A score of zero indicates limited or no understanding of the mathematical concepts embodied in
the task.
Indianapolis Public Schools
Curriculum and Instruction