TEACHER GUIDE TO CLARIFICATION
Finding volume with fractional edge lengths
This standard goes beyond just using fractional edge lengths and the volume formula.
Problem from ....
How many cubes with edge lengths 1/3" would be needed to fill the prism?
This type of questioning is different than how we have asked about volume with fractional edge lengths. It does take the concept of volume and fractions deeper. Students will have to apply visual spatial reasoning and their knowledge of greatest common factor.
Students can easily figure out the volume of cubic inches, but how can students see 120 cubes inside this
rectangular prism. A drawing or model will help to illustrate this.
Students can see an expression many different ways, allow them to explore and share why they are equivalent when representing different expressions
Previously students calculated the volume of right rectangular prisms (boxes) using whole number edges. The unit cube was 1 x 1 x 1. In 6th grade the unit cube will have fractional edge lengths. (i.e. ½ • ½ • ½ ) Students find the volume of the right rectangular prism with these unit cubes. For example, the right rectangular prism below has edges of 1¼”, 1” and 1½”. The volume can be found by recognizing that the unit cube would be ¼” on all edges, changing the dimensions to 5/4”, 4/4” and 6/4”. The volume is the number of unit cubes making up the prism (5 x 4 x 6), which is 120 unit cubes each with a volume of 1/64 (¼” x ¼” x ¼”). This can also be expressed as 5/4 x 6/4 x 4/4 or 120/64.
“Know the formula” does not mean memorization of the formula. To “know” means to have an understanding of why the formula works and how the formula relates to the measure (volume) and the figure. This understanding should be for ALL students.
Students need multiple opportunities to measure volume by filling rectangular prisms with blocks and looking at the relationship between the total volume and the area of the base. Through these experiences, students derive the volume formula (volume equals the area of the base times the height). Students can explore the connection between filling a box with unit cubes and the volume formula using interactive applets such as the Cubes Tool on NCTM’s Illuminations
In addition to filling boxes, students can draw diagrams to represent fractional side lengths, connecting with multiplication of fractions. This process is similar to composing and decomposing two dimensional shapes.
Students will be able to see the shape in its entirety. Students will also be able to see, for example, 2 ½ inch cubes, 4 ½ times and 3 inches tall. This multiplicative thinking, in fractional form, is important to develop.
The model shows a cubic foot filled with cubic inches. The cubic inches can also be labeled as a fractional cubic unit with dimensions
The models show a rectangular prism with dimensions 3/2 in., 5/2 in., and 5/2 in. Each of the cubic units in the model is 1/2 in3. Students work with the model to illustrate 3/2 x 5/2 x 5/2 = (3 x 5 x 5) x 1/8. Students reason that a small cube has volume 1/8 because 8 of them fit in a unit cube.
It is very important for students to continue to physically manipulate materials and make connections to the symbolic and more abstract aspects of geometry. Exploring possible nets should be done by taking apart (unfolding) three-dimensional objects. This process is also foundational for the study of surface area of prisms. Building upon the understanding that a net is the two-dimensional representation of the object, students can apply the concept of area to find surface area. The surface area of a prism is a composition of the areas for each face.
Multiple strategies can be used to aid in the skill of determining the area of simple two- dimensional composite shapes. A beginning strategy should be using rectangles and triangles, building upon shapes for which they can already determine area to create composite shapes. This process will reinforce the concept that composite shapes are created by joining together other shapes, and that the total area of the two-dimensional composite shape is the sum of the areas of all the parts.
Fill prisms with cubes of different edge lengths (including fractional lengths) to explore the relationship between the length of the repeated measure and the number of units needed. An essential understanding to this strategy is the volume of a rectangular prism does not change when the units used to measure the volume changes.
Example, A rectangular prism with a volume of 300 inches cubed can be filled with different cubes.
This could be filled with 300 - 1 inch cubes or 600 ½ inch cubes.
Since the focus in Grade 6 is to use fractional lengths to measure, if the same object is measured using one centimeter cubes and then measured using half centimeter cubes, the volume will appear to be eight times greater with the smaller unit. However, students need to understand that the value or the number of cubes is greater but the volume is the same.
Solve real-world and mathematical problems involving area, surface area, and volume.
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 edge lengths of the prism. Apply the formulas and to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
Coherence and Connections: Need to Know
In grade 5 students work with volume in terms of v = lwh
In sixth grade students should have experience exploring why the formula for volume could look two different ways.
lwh = Bh
Sixth grade students use fractional side lengths when solving for volume.
Examples of Opportunities for Connections among Standards, Clusters or Domains
Writing, reading, evaluating, and transforming variable expressions (6.EE.1-4) and solving equations and inequalities (6.EE.7-8) can be combined with use of the volume formulas V = lwh and V = Bh (6.G.2).
Examples of Opportunities for Connecting Mathematical Content and Mathematical Practices
Area, surface area, and volume present modeling opportunities (MP.4) and require students to attend to precision with the types of units involved (MP.6).
PARCC Model Content Frameworks: Mathematics Grades 3-11 (version 4.0). (2014, December 1).
It is interesting for students to explore the relationship between area, surface area and volume. An example problem might be for student to explore two different sized rectangular prisms with the same volume. We can ask, if the shapes have the same volume, will they have the same surface area? This problem would address MP 4 and MP 6 and connects to 6.G.4 (surface area).
Volume = 60 units cubed
Surface Area = 104 units squared
2 (5x2 + 2 x 6 + 6 x 5)
Volume = 60 units cubed
Surface Area = 128 units squared
2 (2 x 2 + 15 x 2 + 15 x 2)
Students also analyze and compose and decompose polyhedral solids.
They describe the shapes of the faces, as well as the number of faces, edges, and vertices.
They make and use drawings of solid shapes and learn that solid shapes have an outer surface as well as an interior.
They develop visualization skills connected to their mathematical concepts as they recognize the existence of, and visualize, component of three-dimensional shapes that are not visible from a given viewpoint (MP.1).
They measure the attributes of these shapes, allowing them to apply area formulas to solve surface area problems (MP.7).
They solve problems that require them to distinguish between units used to measure volume and units used to measure area (or length).
They learn to plan the construction of complex three-dimensional compositions through the creation of corresponding two-dimensional nets (e.g., through a process of digital fabrication and/or graph paper) 6.G.4.
For example, they may design a living quarters (e.g., a space station) consistent with given specifications for surface area and volume (MP.2, MP.7). In this and many other contexts, students learn to apply these strategies and formulas for areas and volumes to the solution of real-world and mathematical problems (6.G.1, 6.G.2). These problems include those in which areas and volumes are to be found from lengths or lengths are to be found from volumes or areas and lengths.
Common Core Standards Writing Team. (2012, June 23). Progressions for the Common Core State Standards in
Mathematics (draft). K-6 Geometry. Tucson, AZ: Institute for Mathematics and Educations, University of Arizona.
Connecting 6.G.2 to MP 7
In grade 6, students build on their understanding of volumes of right rectangular prisms that do not have measurements given in whole numbers. Students rely on their understanding that area and volume measures are additive and can be decomposed. When determining the volume of a 3 by 2 by 5 feet box students might see the structure of the volume of the box as 5 layers of 3 by 2 . Students apply multiplicative reasoning to determine volumes, looking for and making use of structure (MP7) they realize the height of the prism tells how many layers. Students show that the volume is the same as would be found by multiplying the edge lengths of the prism. They realize they do not have to count all the cubes to find the volume and they relate volume to the operations of addition, multiplication and equivalent formulas to see the structure that volume is both length x width x height as well as base x height.
PARCC Evidence Statements
Evidence Statement Text
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.
i) Tasks do not have a context. ii) Tasks require focusing on the connection between packing the solid figure and computing the volume
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
i) Tasks focus on using the formulas in problem-solving contexts.
1. A right rectangular prism has edge lengths , 2", and . How many cubes with edge lengths would be needed to
fill the prism? What is the volume of the prism?
To find the number of cubes: one might think how many cubes are in , the answer would be 6, 5 cubes in the
one inch and then another , 10 cubes in 2 inches and 7 cubes in 1
6 x 10 x 7 makes 420
2. Cube-shaped boxes will be loaded into the cargo hold of a truck. The cargo hold of the truck is in the shape of a
rectangular prism. The edges of each box measure 2.50 feet and the dimensions of the cargo hold are 7.50 feet by 15.00
feet by 7.50 feet, as shown below.
What is the volume, in cubic feet, of each box?
Determine the number of boxes that will completely fill the cargo hold of the truck. Use words and/or numbers to
show how you determined your answer.
ANSWER: The volume of each box is 15.625 cubic feet.
54 boxes completely fill the cargo hold of the truck. The length of the cargo hold is 15 feet, so 15 divided by
2.50 equals 6. The width and height of the cargo hold are each 7.5 feet, so 7.5 divided by 2.5 equals 3. So the 6 boxes
times 3 boxes times 3 boxes equals 54 total boxes that fit in the cargo hold.
Problems 1 and 2 adapted from...
3. I am moving to a new house. We have boxes that are 2 ft, by 1.5 ft by 2.75 foot
I have 30 boxes to move. What dimensions should the truck have? Draw a picture if needed to help.
I could stack the boxes like this.
2ft + 2ft + 2ft + 2ft +2ft long
2.75 + 2.75 deep
1.5 + 1.5 + 1.5 high
2 x 1.5 x 2.75 = 8.25 per box
8.25 x 30 = 247.5 cu. ft.
total volume of my boxes and the truck
(2 x 5) x (2.75 x 2 ) x (1.5 x 3)
10 x 5.5 x 4.5 Truck Dimensions
247.5 cu. ft.
4. Ann is moving and she needs to rent a truck that can carry 30 crates that are ft wide, ft long, and ft tall.
Which is the volume of the smallest truck that Ann can rent?
5. Each small cube has a length of cm.
Explain TWO different ways to calculate the volume of the rectangular prism that is NOT filled.
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