Making common fractional edges when presented with unlike denominators-Day 3 (6.G.2)
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 4 Model with mathematics
MP 6 Attend to precision
MP 7 Look for and make use of structure
Content Objectives:
Language Objectives:
Students will be able to find the volume of a right
rectangular prism using fractional edge lengths.
Explain the relationship between the fractional or
decimal edge unit cube, total number of cubes that
pack into a prism, and the volume of the prism.
Students will be able to solve volume using unlike
denominators.
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
Fractional
Prior Knowledge:
Height
Length
Numerator
Rectangular Prism
Unit Cube
Unlike Denominators
Width




Finding the volume of rectangular prisms with
fractional edge lengths and packing with fractional
edge cubes
Identifying fractional edge cubes based on
dimensions
Multiplying of fractions
Volume of right rectangular prism formula with
whole number edges (Grade 5 standard beginning
2014-2015)
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Making common fractional edges when presented with unlike denominators-Day 3 (6.G.2)
Questions to Develop Mathematical Thinking:
Common Misconceptions/Challenges:

Challenge: Students may struggle to divide up the
fractional edges into the fractional unit.



What does an improper fraction tell us when
finding volume?
When might we need to know how to pack with
fractional edges into a rectangular prism?
What strategies or formulas help you find the
volume of a rectangular prism?
If you were to pack a prism with fractional edge
cubes, how would you determine the
arrangement of the cubes inside the prism
without guessing and checking?
Strategy: Continuously encourage students to draw or
build a model to understand packing with fractional
edge cubes.
ASSESSMENT:
Observe student work and listen to student discussions for:
 Accurate identification of fractional edge unit cube based on making common denominators
 Accurate use of volume formula with fractional edge lengths
 Accurate labeling of volume with unit3
 Connections between the volume of one fractional edge cube and the volume of the rectangular prism
Question 4 could be used as a formative assessment.
MATERIALS:


1-2 task cards per table
Unit cubes
INSTRUCTIONAL PLAN:
Launch: (10 minutes)
On the board, draw a 1 ½ in. by 2 ¼ in. by 3 in. right rectangular prism. Ask students for strategies they are
using to solve for volume. Students might identify the standard algorithm, packing with fractional edge unit
cubes, or using a drawing. Give students a few minutes to solve for the volume. Go to tables and prompt stuck
tables to revisit their work from prior volume lessons. Students will likely either solve for volume by formula
or filling with ¼-inch cubes. If students are solving for volume by using cubes, ask them how they know the
edge length of the unit cube based on the rectangular prism dimensions.
81
1
Solution: If students used the standard algorithm they would get for volume either 8 in3 or10 8 in3. If
648
1
students used ¼-inch cubes, they would get a volume of 64 in3 which reduces to 10 8 in3.
 As a class, have students share out their strategies. If students used cubes to determine volume, ask the
group how they decided which size unit cube would relate to the dimensions of the rectangular prism.
Have students begin to make connections between the standard algorithm and packing with fractional
edge unit cubes.
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Making common fractional edges when presented with unlike denominators-Day 3 (6.G.2)
Explore: (30 minutes)
Give students the task card and have the students read the directions as a group. When students finish reading
directions, have them share out to the class the student actions both teacher and student should see or hear
during the task. For example, we should see all students writing all work on their paper, labeling volume
accurately, and leaning in to check work of group members. We should hear all students describing
relationship between unit cubes and total volume, answering the same question, and explaining solutions.
When I observe students:








Listen for students discussing the meaning of the task question and sticking together (MP 1)
Check student work for accurate volume solutions and labels (MP 6)
When students struggle to label accurately, ask students how the cubes used to pack the rectangular prism
connect with how we label volume.
Listen and read student responses about the relationship between the fractional edge cube volume and
total volume of the rectangular prism (MP 2)
When students struggle to determine the relationship, have the students look at the numerical values of the
fractional edge unit cube volume, the number of cubes that pack inside the rectangular prism, and total
volume before the solution has been reduced or changed to a mixed number.
Check student work for accurate choice of fractional edge cube based on rectangular prism dimensions
(MP 7)
When students struggle to determine the fractional edge unit cube, ask students how we might re-write
fractions to help us identify the unit cube edge length.
Encourage students to find more than one possible set of dimensions for the shoe box based on the unit
cube size and number of unit cubes that pack into the shoe box (MP 4 and 7)
Questions to Develop Mathematical Thinking as you observe:






Do you always need to turn a mixed number into an improper fraction when finding volume? Why?
What does an improper fraction tell us when finding volume?
When might we need to know how to pack with fractional edges into a rectangular prism?
How come we get the same volume regardless of the size cubes we use?
What strategies or formulas help you find the volume of a rectangular prism?
If you were to pack a prism with fractional edge cubes, how would you determine the arrangement of the
cubes inside the prism without guessing and checking?
Summarize:
Have students share their strategies for identifying the unit cube edge length when denominators are unlike.
Then, check student solutions for volume of unit cube and total volume of rectangular prism. Solutions are
identified below. Students should begin to move away from models and formalize the formula for finding the
volume of right rectangular prisms with rational edge lengths.
The next connection students will need to make relates to the arrangement of the unit cubes within the
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Making common fractional edges when presented with unlike denominators-Day 3 (6.G.2)
rectangular prism once you know the unit cube edge length. This connects the volume formula for
rectangular prisms with packing with fractional edge lengths cubes that occurs in real world contexts. When
students provide their solutions for 1b and 2b, this is an opportunity to make connections between the
numerators of the fractions when improper and the arrangement of the unit cubes within the rectangular
1
prism. For example, in 1b, there are 24- 5 unit cubes that are arranged in a 2 x 3 x 4 model in order to pack
inside the rectangular prism.
Solutions:
1a.
1b.
1c.
2a.
2b.
2c.
1
1
-unit cube with a volume of 125 unit3
2 x 3 x 4 = 24
1
24 5-unit cubes will fit inside the prism.
5
2
5
3
4
24
𝑥 5 𝑥 5 = 125 unit3
1
1
-unit cube with a volume of 64 unit3
18 x 10 x 3 = 540
1
540 4-unit cubes will fit inside the prism.
4
3
4
2
1
𝑥2 4 𝑥4 2 =
540
64
unit3
28
7
Volume could also equal 8 64 unit3 or 8 16unit3.
1
338
169
3.
4.
You would use a 8-inch cube to pack the prism. The volume would be 512 inch3 or 256 inch3.
The fractional edge unit cube volume multiplied by the number of unit cubes that fit inside a prism
equal the total volume of the rectangular prism. The fractional edge unit cube volume is the total
volume denominator and the number of fractional edge cubes that fit inside the prism is the numerator
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑐𝑢𝑏𝑒𝑠 𝑖𝑛𝑠𝑖𝑑𝑒 𝑝𝑟𝑖𝑠𝑚
of the total volume.
𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑐𝑢𝑏𝑒
5a.
See student responses. Some dimensions of her shoe box could be:
2 7 9
2 21 3
18 7 1
14 3 3
6 7 3
𝑥 𝑥 or 3 𝑥 3 𝑥 3 or 3 𝑥 3 𝑥 3 or 3 𝑥 3 𝑥 3 or 3 𝑥 3 𝑥 3
3 3 3
5b.
5c.
The volume would be 27 inch3 or 4 3 inch3.
See student responses.
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2
Feedback for lesson improvement:
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