6.G.A.2: Volume of Rectangular Prisms with Fractional Edges
In plain English: 6.G.A.2 is the Common Core grade 6 math standard that asks students to find the volume of a box shape (a right rectangular prism) whose edges are fractions. Students pack the box with small cubes, such as 1/2-inch cubes, and show that the count gives the same volume as multiplying the edges. Then they use V = l w h and V = b h to solve real problems.
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Solve real-world and mathematical problems involving area, surface area, and volume. Also written as 6.G.2 · Official standard
Students find the volume of right rectangular prisms (box shapes with six rectangular faces, or flat sides, and square corners) whose edges (the segments where two faces meet) are fractions, such as 2 1/2 inches. Volume is the amount of space inside a solid, measured in cubic units: a cube with edges of 1 inch holds 1 cubic inch. In grade 5, students packed boxes with whole unit cubes. Here they pack boxes with smaller cubes whose edge is a unit fraction (a fraction with 1 on top, such as 1/2 or 1/3). Eight cubes with 1/2-inch edges fill 1 cubic inch, so each one is 1/8 cubic inch.
Students count the small cubes, turn the count into cubic units, and show that the result matches the product of the edge lengths. Then they use the two formulas the standard names: V = l w h (length times width times height) and V = b h, where b is the area of the base. They apply both to real problems, such as how many bags of soil fill a garden bed or how deep the water in an aquarium is. The prisms in this standard are always rectangular, and the numbers stay friendly: halves, thirds, quarters and simple mixed numbers.
Learning Objectives
By the end of this lesson, students will be able to:
Explain why a cube with 1/2-unit edges has a volume of 1/8 cubic unit, and a cube with 1/3-unit edges has a volume of 1/27 cubic unit
Find the volume of a prism with fractional edges by packing it with unit fraction cubes and converting the count to cubic units
Show that the packing count gives the same volume as multiplying the edge lengths
Use V = l w h and V = b h to find volumes, and a missing edge, in real-world and mathematical problems
Prior Knowledge Required
Students should already be comfortable with:
Understanding volume as the number of unit cubes that fill a solid 5.MD.C.3
Finding volumes of prisms with whole-number edges with V = l × w × h and V = b × h 5.MD.C.5
Multiplying fractions and mixed numbers 5.NF.B.45.NF.B.6
Show a small box that is 4 inches long, 3 inches wide and 2 inches tall. Ask students to answer without a formula at first:
Warm-Up Prompt
"How many 1-inch cubes fill this box? Now suppose we fill it with smaller cubes whose edges are 1/2 inch. How many of those fit along each edge? How many fill the box?"
With 1-inch cubes: 4 × 3 × 2 = 24 cubes, so the volume is 24 cubic inches. With 1/2-inch cubes: 8 fit along the length, 6 along the width and 4 up the height, so 8 × 6 × 4 = 192 small cubes. Ask: why are there 8 times as many? Each 1-inch cube splits into 2 × 2 × 2 = 8 small cubes. So each small cube is 1/8 cubic inch, and 192 × 1/8 = 24 cubic inches, the same volume. Some students will say there are only twice as many small cubes; build one 1-inch cube from eight smaller cubes to settle it.
Direct Instruction20-25 minutes
Part 1: Packing with unit fraction cubes. When an edge is a fraction, such as 2 1/2 inches, choose small cubes whose edge fits it exactly: 1/2-inch cubes fit 2 1/2 inches 5 times. Find how many small cubes make 1 cubic unit (8 for 1/2, 27 for 1/3, 64 for 1/4). Then count the small cubes in the box and divide by that number. Diagram 1 shows the first example.
Packing with 1/2-inch cubes
A small box is 2 1/2 in long, 1 1/2 in wide and 2 in tall. Pack it with 1/2-inch cubes. What is its volume?
A raised garden bed is 6 ft long and 2 1/2 ft wide, and it will be filled with soil 3/4 ft deep. Soil comes in bags of 1 1/2 cubic feet. How many bags are needed?
A vase with a square base, 4 in by 4 in, holds 60 cubic inches of water. How deep is the water?
Equation: b = 4 × 4 = 16 sq in; 60 = 16 × h, so h = 60 ÷ 16 = 3 3/4 in
Part 2: Why multiplying the edges works. In the first example, the number of small cubes is 5 × 3 × 4. Each edge in inches is that number of halves: 2 1/2 = 5 × 1/2, 1 1/2 = 3 × 1/2 and 2 = 4 × 1/2. Multiplying the edges multiplies the counts and the 1/2s: (5 × 1/2) × (3 × 1/2) × (4 × 1/2) = (5 × 3 × 4) × 1/8. That is exactly the number of cubes times the volume of one cube. So packing and multiplying always give the same volume.
V = l w h: multiply the length, the width and the height. Use the same unit for all three edges.
V = b h:b is the area of the base, the face the prism stands on, and h is the height. The base area tells how many cubes cover the bottom layer, and the height tells how many layers there are. Diagram 2 shows the garden bed from the fourth example.
Missing edge: if you know the volume and the base area, divide: h = V ÷ b, as in the fifth example.
Units: lengths are in inches or feet, base areas in square units, and volumes in cubic units.
Guided Practice15 minutes
Pairs solve four problems, one at a time. For the packing problems, they sketch the prism on isometric dot paper (paper with dots in a triangle pattern, used to draw boxes in 3D) and mark the small cubes along each edge. After each problem, one pair explains its work.
Guided practice problems (answers are for the teacher)
Problem
Task
Answer
1
A prism is 3/4 in by 1 in by 1 1/2 in. Pack it with 1/4-inch cubes and find its volume.
A paperback book is 8 in by 5 1/2 in by 3/4 in. What is its volume?
8 × 5 1/2 × 3/4 = 33 cubic inches
4
A cooler has an inside base of 1 1/2 ft by 1 ft and is 1 1/4 ft deep. What is its volume?
b = 1 1/2 sq ft; V = 1 1/2 × 1 1/4 = 1 7/8 cubic feet
Listen for students who count the small cubes and stop there, reporting 72 cubic inches in Problem 1. Ask them how many small cubes make 1 cubic inch.
Independent Practice10-15 minutes
Students solve five problems on their own. (1) A box is 2 1/2 in by 1 in by 1/2 in. How many 1/2-inch cubes fill it, and what is its volume? (10 cubes, so 1 1/4 cubic inches.) (2) How many cubes with 1/3-ft edges fill a 1-foot cube? How many fill a box 2 ft by 1 ft by 2/3 ft, and what is its volume? (27; then 6 × 3 × 2 = 36 cubes, so 36/27 = 1 1/3 cubic feet.) (3) A tissue box is 4 1/2 in by 4 1/2 in by 5 in. Find its volume. (101 1/4 cubic inches.) (4) A prism has a base area of 10 1/2 sq cm and a height of 3 cm. Find its volume. (31 1/2 cubic centimeters.) (5) A drawer is 1 1/2 ft by 2 ft at the bottom and holds 1 1/2 cubic feet. How deep is it? (b = 3 sq ft, so h = 1 1/2 ÷ 3 = 1/2 ft.)
Closure5 minutes
Exit ticket: (1) Explain in one or two sentences why a cube with 1/2-inch edges has a volume of 1/8 cubic inch. (Eight of them, 2 × 2 × 2, fill a 1-inch cube.) (2) Find the volume of a box 3 1/2 in by 2 in by 1 1/2 in in two ways: with V = l w h, and by counting 1/2-inch cubes. (7 × 4 × 3 = 84 cubes, 84 ÷ 8 = 10 1/2 cubic inches, and 3 1/2 × 2 × 1 1/2 = 10 1/2.)
Differentiation Strategies
For Struggling Students
Let students build every packing problem with centimeter cubes first, calling each cube 1/2 unit, and group 8 cubes into one 2 × 2 × 2 block to see 1 cubic unit
Give a three-column chart (cubes along the length, width and height) before students multiply
Start with prisms where only one edge is a fraction, such as 3 by 2 by 1 1/2, before using two or three fractional edges
For Advanced Students
Ask for three different boxes with fractional edges that all have a volume of 6 cubic inches, and the number of 1/2-inch cubes in each
Ask how the volume changes when every edge of a box is cut in half, and test it with a box 3 by 2 by 1 1/2
Ask why 1/4-inch cubes could not exactly pack a box with an edge of 1 1/3 inches, and which unit fraction cubes would
Assessment Guidance
What to Look For
Check that students choose a cube size that fits every edge exactly, count the cubes along each edge correctly, and divide the count by the number of small cubes in 1 cubic unit (8, 27 or 64). They should be able to explain why the count and the product of the edges agree. With the formulas, look for correct multiplication of mixed numbers (convert to fractions or decimals first) and for b being an area, not a length. Every answer should use cubic units, and real-world answers should answer the question asked, such as the number of bags to buy.
02
Classroom Activities
3 Activities
1
Build It with Half-Unit Cubes
20 minPairs
Pairs build prisms from centimeter cubes. For this activity, we invent a bigger length unit: 1 unit is 2 cm. So each centimeter cube has an edge of 1/2 unit, and a 2 × 2 × 2 block of 8 cubes is 1 cubic unit. Record every volume in cubic units, not cubic centimeters. Pairs build each prism on the card, count the cubes, find the volume in cubic units and check it by multiplying the edges.
Build Cards (4 cards)
Card A: 1 1/2 by 1 by 1 unit (3 × 2 × 2 = 12 cubes, 1 1/2 cubic units)
Card B: 2 by 1/2 by 1 1/2 units (4 × 1 × 3 = 12 cubes, 1 1/2 cubic units)
Card C: 2 1/2 by 1 1/2 by 1 unit (5 × 3 × 2 = 30 cubes, 3 3/4 cubic units)
Card D: 1 by 1 by 1 unit (2 × 2 × 2 = 8 cubes, 1 cubic unit)
Procedure
Build Card D first and keep it as your 1 cubic unit
Build each other card, count the cubes along each edge, and multiply to get the total number of cubes
Divide the count by 8 to get the volume in cubic units, then multiply the edge lengths on the card and compare
Record both results in a table and sketch each prism on isometric dot paper
Discussion Questions
Which two cards have the same volume but different shapes? (Cards A and B)
Which card has the largest volume? (Card C)
Why do we divide the number of cubes by 8 and not by 2?
Modification for Distance Learning
Use a free online 3D cube-building tool or drawings on isometric dot paper. Students share a screenshot of each prism with its cube count and volume.
2
Measure Real Boxes
20 minGroups of 3-4
Groups measure 3 real boxes (for example a cereal box, a tissue box and a shoebox) to the nearest 1/4 inch. They find each volume with V = l w h and with V = b h, and put the boxes in order from least to greatest volume.
Procedure
One student measures, one records, and one computes; roles rotate for each box
Measure the length, width and height of each box to the nearest 1/4 inch
Compute V = l w h. Then compute the base area b = l w and V = b h, and check that the answers agree
Estimate first: which box do you think holds the most? Then compare with your results
Sample Measurement
A cereal box measured 7 3/4 in long, 2 1/2 in wide and 11 in tall. Base area: 7 3/4 × 2 1/2 = 19 3/8 sq in. Volume: 19 3/8 × 11 = 213 1/8 cubic inches. Your boxes will give different numbers.
Discussion Questions
Did your estimate match the order of the volumes? Which box surprised you?
Which face did you use as the base? Would a different face give a different volume?
Why might two groups measuring the same box get slightly different volumes?
Challenge Variation
Groups design a box that holds the same volume as their cereal box but has a square base, and give its edge lengths to the nearest 1/4 inch.
3
Shipping Carton Packing Problem
15 minPairs
A company ships small gift boxes that are cubes with 1/2-foot edges. Pairs figure out how many gift boxes fit in a shipping carton and connect the count to the carton's volume.
The Problem
The carton is 2 ft long, 1 1/2 ft wide and 1 ft tall
How many gift boxes fit along each edge? (4, 3 and 2)
How many fill the carton? (4 × 3 × 2 = 24 gift boxes)
What is the carton's volume? (2 × 1 1/2 × 1 = 3 cubic feet, and 24 × 1/8 = 3)
Procedure
Draw the carton on isometric dot paper, one dot spacing for each 1/2 ft
Mark the gift boxes along each edge and find the total
Write the volume two ways and explain why they agree
Challenge Variation
Design a carton that holds exactly 48 gift boxes. Give its edge lengths in feet and its volume. (One answer: 2 ft by 2 ft by 1 1/2 ft, which is 4 × 4 × 3 = 48 boxes and 6 cubic feet.)
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Box Packed with 1/2-Inch Cubes
A box 2 1/2 in long, 1 1/2 in wide and 2 in tall holds 5 × 3 × 4 = 60 cubes with 1/2-inch edges. Eight such cubes make 1 cubic inch (right), so the volume is 60 ÷ 8 = 7 1/2 cubic inches, the same as 2 1/2 × 1 1/2 × 2. Both drawings are isometric: every edge is drawn at the same scale.
Diagram 2: V = b × h for a Raised Garden Bed
The garden bed is 6 ft long, 2 1/2 ft wide and filled 3/4 ft deep, drawn to scale. Its base area is 15 sq ft, so the soil fills 15 × 3/4 = 11 1/4 cubic feet, and 8 bags of 1 1/2 cubic feet are needed. Dashed lines show hidden edges.
04
Homework Assignment
~30 min
6.G.A.2 Homework: Volume with Fractional Edges
Directions: Show your work for every problem. For packing problems, give the number of small cubes along each edge, the total number of cubes, and the volume. Write cubic units with every volume.
Part 1: Packing with Unit Fraction Cubes (Problems 1-2)
A box is 3 in long, 1 1/2 in wide and 1 in tall. (a) How many 1/2-inch cubes fit along each edge, and how many fill the box? (b) What is the volume in cubic inches? (c) Show that multiplying the edge lengths gives the same volume.
A prism is 1 1/4 units long, 1/2 unit wide and 3/4 unit tall. (a) How many cubes with 1/4-unit edges fit along each edge, and how many fill the prism? (b) What is its volume in cubic units? (c) Check your answer by multiplying the edges.
Part 2: V = l w h and V = b h (Problems 3-4)
A pencil box is 8 1/2 in long, 3 in wide and 1 1/2 in tall. Find its volume with V = l w h.
An aquarium has a base 2 ft long and 1 ft wide. The water is 1 1/4 ft deep. Find the base area b and the volume of the water with V = b h. Then check with V = l w h.
Part 3: Real-World Problems (Problems 5-6)
A sandbox is 5 ft long and 4 ft wide. It will be filled with sand 3/4 ft deep. Sand is sold in bags of 1/2 cubic foot. How many bags are needed?
A box has a volume of 22 1/2 cubic inches. It is 5 in long and 3 in wide. (a) How tall is it? (b) How many 1/2-inch cubes would fill it?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Packing with Cubes
Correct cubes along each edge, total count, and volume in cubic units
Correct count but not converted to cubic units, or one error
Count and volume missing or incorrect
Connecting Packing and Multiplying
Shows that the product of the edges equals the packing volume
Computes both but does not compare them
No comparison
Using the Formulas
V = l w h and V = b h used correctly with fractions, including the missing height
One computation error with fractions
Formulas not used or used with the wrong measurements
Real-World Answers
Answers the question asked (bags, depth) with cubic units and sensible rounding
Correct volume but the final question is not answered
No real-world answer
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order, and keep scrap paper for sketches and fraction work. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
How many cubes with 1/2-inch edges fill a box that is 1 1/2 in long, 1 1/2 in wide and 1 in tall?
Answer: B
Along the edges, 3, 3 and 2 small cubes fit (each inch holds two 1/2-inch cubes), so 3 × 3 × 2 = 18 cubes fill the box. Choice A is the volume, 1 1/2 × 1 1/2 × 1 = 2 1/4 cubic inches, not the number of small cubes. Choice C adds the counts, 3 + 3 + 2. Choice D doubles the volume, as if each small cube were 1/2 cubic inch: 2 1/4 ÷ 1/2 = 4 1/2.
Question 2 of 20 · Multiple Choice
A wooden block is a cube with edges of 1 1/2 inches. What is its volume?
Answer: D
Pack it with 1/2-inch cubes: 3 fit along each edge, so 3 × 3 × 3 = 27 small cubes, each 1/8 cubic inch, and 27 ÷ 8 = 3 3/8 cubic inches. Multiplying gives the same: 3/2 × 3/2 × 3/2 = 27/8 = 3 3/8. Choice A adds the three edges, 1 1/2 + 1 1/2 + 1 1/2. Choice B is 1 1/2 × 1 1/2 = 2 1/4, the area of one face in square inches. Choice C reports the number of small cubes as the volume.
Question 3 of 20 · Multiple Choice
A prism is packed with 45 cubes whose edges are 1/3 ft: 5 along the length, 3 along the width and 3 up the height. What is its volume?
Answer: A
27 cubes with 1/3-ft edges make 1 cubic foot (3 × 3 × 3), so the volume is 45/27 = 1 2/3 cubic feet. Check: the prism is 5/3 ft by 1 ft by 1 ft, and 5/3 × 1 × 1 = 1 2/3. Choice B divides by 3, as if 3 small cubes made 1 cubic foot. Choice C divides by 9, which is only one face of 3 × 3 cubes. Choice D reports the number of cubes as the volume.
Question 4 of 20 · Multiple Choice
Maya packed a box 2 1/2 in by 2 in by 1 in with 1/2-inch cubes and counted 40 cubes. Which calculation gives the volume and matches her count?
Answer: C
Each 1/2-inch cube is 1/8 cubic inch, so 40 cubes make 40 ÷ 8 = 5 cubic inches, and 2 1/2 × 2 × 1 = 5 as well. Choice A treats each small cube as 1/2 cubic inch. Choice B adds the edges. Choice D counts the cubes correctly but reports the count as cubic inches.
Question 5 of 20 · Multiple Choice
What is the volume of a prism that is 4 1/2 cm long, 2 cm wide and 3 1/3 cm tall?
Answer: B
V = l w h = 4 1/2 × 2 × 3 1/3 = 9 × 10/3 = 30 cubic centimeters. Choice A adds the three edges. Choice C drops both fractions and multiplies 4 × 2 × 3. Choice D drops the 1/2 in 4 1/2 and multiplies 4 × 2 × 3 1/3.
Question 6 of 20 · Multiple Choice
A shipping box is 2 1/2 ft long, 1 1/4 ft wide and 2 ft tall. What is its volume?
Answer: D
V = 2 1/2 × 1 1/4 × 2 = 5/2 × 5/4 × 2 = 50/8 = 6 1/4 cubic feet. Choice A adds the edges, 2 1/2 + 1 1/4 + 2. Choice B keeps only the whole-number parts, 2 × 1 × 2. Choice C multiplies only two edges, 2 1/2 × 1 1/4, which is the area of the base, not the volume.
Question 7 of 20 · Multiple Choice
A prism has a base area of 12 3/4 sq in and a height of 4 in. What is its volume?
Answer: A
V = b h = 12 3/4 × 4 = 48 + 3 = 51 cubic inches. Choice B adds the base area and the height. Choice C divides the base area by the height. Choice D multiplies only the 12 by 4 and forgets the 3/4.
Question 8 of 20 · Multiple Choice
In the formula V = b h for a rectangular prism, what does b stand for?
Answer: C
b is the area of the base, found by multiplying the length and the width of the face the prism stands on. It tells how many unit cubes cover the bottom layer, and h tells how many layers there are. Choice A is a length, not an area. Choice B counts cubes along an edge, not the whole layer. Choice D is the perimeter of the base.
Question 9 of 20 · Multiple Choice
A square baking pan is 8 in by 8 in. Batter fills it to a depth of 1 1/4 in. What is the volume of the batter?
Answer: B
The base area is 8 × 8 = 64 sq in, so V = b h = 64 × 1 1/4 = 64 + 16 = 80 cubic inches. Choice A adds 8 + 8 + 1 1/4. Choice C is the base area only, with no height. Choice D multiplies only one side of the pan by the depth, 8 × 1 1/4.
Question 10 of 20 · Multiple Choice
A box has a volume of 18 cubic feet. Its base is 4 ft by 3 ft. How tall is the box?
Answer: D
The base area is 4 × 3 = 12 sq ft. Since V = b h, 18 = 12 × h, so h = 18 ÷ 12 = 1 1/2 ft. Check: 12 × 1 1/2 = 18. Choice A multiplies 18 by 12 instead of dividing. Choice B subtracts 4 + 3 = 7 from 18. Choice C divides 18 by only one side of the base, 3.
Question 11 of 20 · Multiple Choice
How many cubes with 1/4-inch edges fill a box that is 1 in long, 1/2 in wide and 1/2 in tall?
Answer: A
Along the edges, 4, 2 and 2 small cubes fit (each inch holds four 1/4-inch cubes), so 4 × 2 × 2 = 16 cubes fill the box. Check: 16 ÷ 64 = 1/4 cubic inch, and 1 × 1/2 × 1/2 = 1/4. Choice B counts only one row, the 4 cubes along the length. Choice C is the number of these cubes in a whole cubic inch, but the box is smaller than that. Choice D adds the counts, 4 + 2 + 2.
Question 12 of 20 · Multiple Choice
A box is 4 in long, 2 in wide and 1 1/2 in tall. It is packed full of sugar cubes with 1/2-inch edges. How many sugar cubes are in the box?
Answer: C
8, 4 and 3 sugar cubes fit along the edges, so 8 × 4 × 3 = 96. Check: the volume is 4 × 2 × 1 1/2 = 12 cubic inches, and each cubic inch holds 8 sugar cubes, so 12 × 8 = 96. Choice A is the volume in cubic inches, not the number of cubes. Choice B multiplies the volume by 2 instead of 8. Choice D multiplies the volume by 4, which is the number of 1/2-inch squares on one square inch of a face.
Question 13 of 20 · Multiple Choice
Container A is 3 in by 2 1/2 in by 2 in. Container B is 4 in by 1 1/2 in by 2 1/2 in. Which container holds more?
Answer: B
Container A: 3 × 2 1/2 × 2 = 15 cubic inches. Container B: 4 × 1 1/2 × 2 1/2 = 15 cubic inches. They hold the same amount. Choice A compares the sums of the edges, which is not volume. Choice C compares base areas (7 1/2 and 6 sq in) and ignores the heights. Choice D looks at only one edge.
Question 14 of 20 · Multiple Choice
A box is 1 1/2 in by 1 in by 3 1/2 in. Which explanation shows why its volume is 5 1/4 cubic inches?
Answer: A
3 × 2 × 7 = 42 half-inch cubes fill the box, and each is 1/2 × 1/2 × 1/2 = 1/8 cubic inch, so the volume is 42 ÷ 8 = 5 1/4 cubic inches, which matches 1 1/2 × 1 × 3 1/2. Choice B divides by 8 for the wrong reason: the number of corners has nothing to do with volume; 8 small cubes fill 1 cubic inch because 2 × 2 × 2 = 8. Choice C reaches the right number with a wrong count (21, half the real count) and a wrong cube volume (1/4 cubic inch). Choice D also divides by 8 for the wrong reason: 8 cubes lining up along 4 inches is a fact about length, not about how many cubes fill 1 cubic inch.
Question 15 of 20 · Short Answer
A prism is 1 unit long, 3/4 unit wide and 1/2 unit tall. How many cubes with 1/4-unit edges fill it? What is its volume? Show that multiplying the edges gives the same answer.
4, 3 and 2 small cubes fit along the edges, so 24 cubes fill it. 64 of these cubes make 1 cubic unit, so the volume is 24/64 = 3/8 cubic unit. Multiplying: 1 × 3/4 × 1/2 = 3/8, the same.
Question 16 of 20 · Short Answer
A toy chest is 3 ft long, 1 1/2 ft wide and 1 1/4 ft tall. Find its volume with V = l w h.
A planter box has a base area of 3 3/4 sq ft. It is filled with soil 2/3 ft deep. How many cubic feet of soil are in it?
V = b h = 3 3/4 × 2/3 = 15/4 × 2/3 = 30/12 = 2 1/2 cubic feet.
Question 18 of 20 · Short Answer
A box is 5 in long, 2 1/2 in wide and 1 1/2 in tall. Find its volume with V = b h, and then with V = l w h. Do you get the same answer?
Base area: b = 5 × 2 1/2 = 12 1/2 sq in. V = b h = 12 1/2 × 1 1/2 = 18 3/4 cubic inches. V = l w h = 5 × 2 1/2 × 1 1/2 = 18 3/4 cubic inches. Yes, the same: b is just l × w done first.
Question 19 of 20 · Short Answer
An aquarium is 3 ft long and 1 1/2 ft wide. The water is 1 1/3 ft deep. One cubic foot of water is about 7 1/2 gallons. About how many gallons of water are in the tank?
Base area: 3 × 1 1/2 = 4 1/2 sq ft. Volume of water: 4 1/2 × 1 1/3 = 9/2 × 4/3 = 6 cubic feet. Gallons: 6 × 7 1/2 = about 45 gallons.
Question 20 of 20 · Short Answer
A carton is packed with 64 gift boxes that are cubes with 1/2-ft edges, arranged 4 by 4 by 4. What are the carton's edge lengths and its volume?
Each edge holds 4 boxes of 1/2 ft, so each edge is 4 × 1/2 = 2 ft. Volume: 2 × 2 × 2 = 8 cubic feet. Check: 64 × 1/8 = 8.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.G.A.2 mean?
6.G.A.2 means students find the volume of box-shaped prisms whose edges are fractions. First they pack the box with small cubes, such as cubes with 1/2-inch edges, and show that the count matches the product of the edges. Then they use V = l w h and V = b h to solve real problems.
How is 6.G.A.2 different from grade 5 volume?
The difference is fractional edge lengths. In grade 5 (5.MD.C.5), students packed prisms with whole unit cubes and used V = l × w × h with whole numbers. In 6.G.A.2, the edges can be fractions or mixed numbers, so students pack with smaller cubes and multiply fractions.
Why do students pack boxes with fraction-sized cubes?
Packing shows why the formula still works with fractions. When an edge is 2 1/2 inches, whole 1-inch cubes do not fit exactly, but 1/2-inch cubes do. Counting them and converting to cubic inches gives the volume, and students see that it equals the product of the edges.
Why is a 1/2-inch cube only 1/8 of a cubic inch?
Because 8 of them fill a 1-inch cube: 2 along the length, 2 along the width and 2 up the height. The same idea gives 27 cubes with 1/3-inch edges in a cubic inch (3 × 3 × 3) and 64 cubes with 1/4-inch edges (4 × 4 × 4). Building a 2 × 2 × 2 block from small cubes is a quick way to show it.
What is the difference between V = l w h and V = b h?
They are the same calculation in a different order. In V = b h, b is the area of the base (length times width), and h is the height. V = b h is handy when the base area is given, or when you think of the prism as layers stacked h high. For a box, b h and l w h always give the same volume.
What are common mistakes with 6.G.A.2?
A common mistake is reporting the number of small cubes as the volume, without dividing by 8, 27 or 64. Others are adding the edges instead of multiplying, using b as a length instead of an area, dropping the fraction part of a mixed number, and writing square units instead of cubic units.
Does 6.G.A.2 include other prisms, like triangular prisms?
No, 6.G.A.2 is only about right rectangular prisms. Volumes of other right prisms and composite solids come in grade 7 (7.G.B.6), and cylinders, cones and spheres come in grade 8 (8.G.C.9).
How do you find a missing edge from the volume?
Divide the volume by the area you know. If a box holds 10 cubic feet and its base is 5 ft by 1 1/3 ft, the base area is 6 2/3 sq ft, so the height is 10 ÷ 6 2/3 = 1 1/2 ft. This uses division of fractions from 6.NS.A.1.
How does 6.G.A.2 show up on tests?
Test items usually give a box with fractional edges and ask for its volume, the number of small cubes that fill it, or a missing edge. Some show a drawing of a box packed with small cubes and ask students to connect the count to the formula. Word problems often involve boxes, tanks, planters or shipping cartons.
How can parents help with volume at home?
Measure boxes together with a ruler marked in quarter inches, such as cereal or shoe boxes, and find their volumes. Ask which box holds more before you compute. Filling a small box with sugar cubes or dice and counting them is a good hands-on check.
07
Related Standards
6 standards
These standards connect to 6.G.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.MD.C.5Prerequisite
Find volumes of rectangular prisms with whole-number edges by packing and by formulas
Lesson coming soon
5.NF.B.4Prerequisite
Multiply a fraction or whole number by a fraction, including areas with fraction sides
Lesson coming soon
Alongside
6.G.A.4Parallel
Use nets of rectangles and triangles to find surface area
