Topic Twelve: Volume of Solids

Desired Results

Transfer:

1. Makes sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Established Goals:

  • 5.MD.3: Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
    • a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.
    • b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
  • 5.MD.4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
  • 5.MD.5. Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume.
    • a. 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, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
    • b. Apply the formulas V

      l´w´h and V

      b ´h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems.
    • c. Recognize volume as additive. 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.

Prerequisite Standards:
  • 4.MD.1: Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …
  • 4.MD.2: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
  • 4.MD.3.: Apply the area and perimeter formulas for rectangles in real-world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
  • 5.NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm.

I Can Statements:
  • I can understand volume.
  • I can measure volume by counting unit cubes.
  • I can solve real world problems involving volume.
  • I can find the volume of an object using the formulas V = l x w x h and V = b x h.
Big Ideas:
  • Geometric Figures: Two and three-dimensional objects with or without curved surfaces can be described, classified, and analyzed by their attributes. An object's location in space can be described quantitatively.
  • Practices, Processes, and Proficiencies: Mathematics content and practices can be applied to solve problems.
Essential Questions:
  • How can three-dimensional shapes be represented and analyzed?
  • What does the volume of a rectangular prism mean and how can it be found?
Students will know...
  • Three-dimensional or solid figures have length, width, and height. Many can be described, classified, and analyzed by their faces, edges, and vertices. Many everyday objects closely approximate standard geometric solids.
  • The shape of a solid can sometimes be determined by analyzing different views of the solid.
  • Some problems can be solved by breaking apart or changing the problem into simper ones, solving the simpler ones, and using those solutions to solve the original problem.
  • Volume is a measure of the amount of space inside a solid figure. Volume can be measured by counting the number of cubic units needed to fill a three-dimensional object.
  • The volume of some objects can be found by breaking apart the object into other objects for which the volume of each can be found.
  • Some problems can be solved by using objects to act out the action in the problem. Some problems can be solved by reasoning about conditions in the problem.

Vocabulary:three-dimensional shape, cube, edge, face, vertex, vertices, cone, cylinder, prism, pyramid, volume, cubic unit
Students will be skilled at...
  • Identifying three-dimensional shapes according to faces, edges, and vertices
  • Identifying different views of a solid.
  • Using objects to act out and break apart problems into simpler ones in order to reach a solution.
  • Determining the volume of rectangular solids.
  • Counting the cubic units and using formulas to find the volume of rectangular prisms.
  • Finding volumes of irregular solids.
  • Using objects and reasoning to find the volumes of solid figures.

Assessment Evidence

Performance Assessment:
Other Evidence:

Learning Plan

Learning Activities:
12-1:Three-dimensional or solid figures have length, width, and height. Many can be described, classified, and analyzed by their faces, edges, and vertices. Many everyday objects closely approximate standard geometric solids.
12-2: The shape of a solid can sometimes be determined by analyzing different views of the solid.
12-3: Some problems can be solved by breaking apart or changing the problem into simper ones, solving the simpler ones, and using those solutions to solve the original problem.
12-4: Volume is a measure of the amount of space inside a solid figure. Volume can be measured by counting the number of cubic units needed to fill a three-dimensional object.
12-5:Volume is a measure of the amount of space inside a solid figure. Volume can be measured by counting the number of cubic units needed to fill a three-dimensional object.
12-6: The volume of some objects can be found by breaking apart the object into other objects for which the volume of each can be found.
12-7: Some problems can be solved by using objects to act out the action in the problem. Some problems can be solved by reasoning about conditions in the problem.
Resources:

Problem of the Month:
Cubism

Centers:
Build a Cubic Meter
Exploring Volume
Building Rectangular Prisms with a Given Volume
3D Structures
Roll a Rectangular Prism
Four Open Boxes
What's the Volume?
Ordering Rectangular Prisms by Volume
Designing a Toy Box
Designing a Cereal Box
Create a Sculpture
Comparing Buildings
Find the Volume
Joe's Buildings