6.G.2 What does it mean to use fractional edges? (Day 1)
Domain:
Geometry
Big Idea (Cluster):
Solve real-world and mathematical problems involving area, surface area, and volume.
Common Core Standards:
6.G.2 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=lwh and V = Bh to find volumes of right
rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical
problems.
Mathematical Practice(s):
MP 1 Make sense of problems and persevere in solving them
MP 2 Reason abstractly and quantitatively
MP 3 Construct viable arguments and critique the reasoning of others
MP 5 Use appropriate tools strategically
MP 6 Attend to precision
MP 8 Look for and express regularity in repeated reasoning
Content Objectives:
Language Objectives:
Students will be able to find the volume of a cube
using fractional edge lengths.
Write, justify and explain your strategy for solving for
volume with rectangular prism using fractional edge
lengths.
Students will be able to recognize that the size of the
packing cube will not change the volume of the
given rectangular prism.
Vocabulary:
Area of the Base
Cube
Denominator
Dimensions
Edge Lengths
Edges
Prior Knowledge: Concepts students need to know
Fractional
Height
Length
Numerator
Rectangular Prism
Unit Cube
Width
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Right Rectangular Prism
Fraction multiplication
Volume of Right Rectangular Prism formula with
whole number edges (Grade 5 standard beginning
2014-2015)
Cross-sections of rectangular prisms (Grade 5
standard beginning 2014-2015)
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6.G.2 What does it mean to use fractional edges? (Day 1)
Questions to Develop Mathematical Thinking:
Common Misconceptions:
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Students may not recognize that the area of the base
face has the same numerical value as the volume of the
base layer. Emphasize that the base layer always has a
height of 1.
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When would we use fractional edges?
How come we get the same volume regardless of
the size cubes we use?
What is important to know about improper
fractions and fractional edges?
What strategies or formulas help you find the
volume of a rectangular prism?
How does your strategy for solving volume with
whole number cubes help when packing with
fractional edge cubes?
Students may struggle to divide up the fractional edges
into the fractional unit. Continuously encourage
students to draw or build a model to understand
packing with fractional edge cubes.
Students need to be able to use both the standard
algorithm for finding volume of a rectangular prism
with any edge length and finding volume by packing
with a set fractional length cube.
ASSESSMENT:
Observe student work and listen to student discussions for:
 Accurate solving of volume with whole number edges
 Accurate re-drawing of cube with ½-inch lengths and solving for volume with fractional edge
 Explanations of both strategies to solve for volume (V=lwh and V = Bh)
 Connections between volume with whole numbers and volume made by fractional edge cubes
Use student responses on B1 as a formative assessment.
District EOU Assessment
MATERIALS:
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1-2 Task Instructions per table
Minimum of 55 Blocks per table
INSTRUCTIONAL PLAN:
Launch: (10 minutes) Warm - up
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On the board, draw two identical 2 inch x 2 inch x 2 inch rectangular prisms (cube). Label one with whole
numbers. Ask students to copy both onto their paper. Find the volume of the whole number prism.
Solution: V=8 in3
 Ask students to consider what the prism would look like if you only had ½-inch blocks. Give students 2
minutes with their team to sketch out and determine the volume using ½-cubes. Have a whole class
discussion. Look for teams that are able to show how many ½-inch blocks would fit along the length,
width, and height.
Solution: Four ½-inch blocks fit along the length, width, and height. This means 64 ½-inch cubes would fill
the rectangular prism (cube). Therefore, V=4/2 x 4/2 x 4/2, V= 64/8 in3, V= 8in3)
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6.G.2 What does it mean to use fractional edges? (Day 1)
Explore: (30 minutes)
Give students the task card and have them read directions as a team. Ask students what student actions both
teacher and student should see and hear during the task. For example, we should see all students writing
responses on their labsheet, drawing new box, and showing their work. We should hear all students checking
their work, asking questions, and justifying their reasoning.
Check in with tables throughout the activity. Move students towards formalizing their understanding with the
goal of students summarizing different strategies to their classmates.
When I observe students:
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Listen for student engagement in the task
Check student work for accurate solving of volume with whole number edges (MP 6)
Listen for students identifying multiple strategies to solving volume (MP 2 and MP 3)
Listen and read student justifications of Leslie and Deb both solving for volume accurately (MP 3)
Listen and read student explanations of both strategies to solve for volume (V=lwh and V = Bh) (MP 2
and MP 8)
Watch for teams sticking together
Listen for students making sense of packing with ½-inch cubes (MP 1)
When students struggle to re-draw the box with fractional edges, ask students how fractions and whole
numbers are the same if you have a specific fraction denominator like 1/2.
Accurate re-drawing of cube with ½-inch lengths and solving for volume with fractional edge
Questions to Develop Mathematical Thinking as you observe:
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Use “because” statements to explain how Leslie and Deborah could both be right?
Tell me why the volume remains the same?
How come it works to use fractional edges?
What if I only had a 1 inch ruler and I had to find the volume of a dice? Would fractional measures
make sense?
When might we use fractional edges?
How come we get the same volume regardless of the size cubes we use?
What is important to know about improper fractions and fractional edges?
What strategies or formulas help you find the volume of a rectangular prism?
Solutions:
A. Both Deb & Leslie are correct because they both each have an accurate strategy for solving volume of a
rectangular prism. Deb’s strategy is more efficient and shows a solid understanding of layering the base area
by height. Volume is 64 in3.
B.
1. See student drawings.
2. 8/2 inch x 8/2 inch x 8/2 inch
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6.G.2 What does it mean to use fractional edges? (Day 1)
3. 8x8x8 is 512 so, 512 ½-inch cubes would pack into the box.
4. The volume of one ½ - inch cube would be ½ x ½ x ½ which equals 1/8 in3.
5. The volume of the box is 512/8 in3 or 64 in3. The volume is staying the same since 8/2 x 8/2 x 8/2 =
512/8. The fraction 512/8 reduces to 64.
Summarize: (10 minutes)
Have students share out their evidence that Leslie or Deb’s strategy for solving for volume was accurate. Ask
students, “Do you agree or disagree?” and “Why?”
Primary focus of the summary is on B1. Have students share how they drew the ½-inch cubes into the 4x4x4
inch rectangular prism (cube). Then, have students share out their justification for the volumes being the
same or different. Students need to recognize that the volume does not change when the unit used to measure
or pack the figure changes. Students also need to recognize when using fractional length cubes that the
numerator identifies how many of that fractional edge cubes will fit along the length, width and height. The
denominator identifies the part of the whole.
The future connections for students is the understanding that when they re-draw the original figure using ½inch cubes (8/2 x 8/2 x 8/2) that they can find the number of ½-inch cubes that will pack inside by multiplying
8x8x8. The volume of each fractional cube is ½ x ½ x ½ . Therefore, number of fractional cubes packed in
figure multiplied by volume of one fractional cube equals volume of the whole figure. (8x8x8) x (1/2 x ½
x1/2) = 64 in3. Packing a rectangular prism with a fractional edge cube is an alternate strategy for finding
volume of a rectangular prism besides V=lwh or V=Bh.
Feedback for lesson improvement:
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