Shapes and Volume

Students explore the volume of three-dimensional shapes, connecting it to the operations of multiplication and addition, as well as classify two-dimensional shapes hierarchically.


Unit 3

5th Grade

Unit Summary

In Unit 3, 5th grade students will explore volume of three-dimensional shapes (5.MD.3—5), connecting it to the operations of multiplication and addition (5.NBT.5, 4.NBT.4). They also use their understanding that they gradually built in prior grade levels to classify shapes in a hierarchy, seeing that attributes of shapes in one category belong to shapes in all subcategories of that category (5.G.3—4). 

In prior grade levels, students explored the idea of volume informally, comparing the capacity of various containers as being able to “hold more” or “hold less” (K.MD.2). Students have also explored one-dimensional and two-dimensional measurements of figures, developing a deep understanding of length in 2nd grade and of area in 3rd grade. In their exploration of area in 3rd Grade Math, students come to understand area as an attribute of plane figures (3.MD.5) and measure it by counting unit squares (3.MD.6), and they connect area to the operations of multiplication and addition (3.MD.7). 

Students have also explored two-dimensional shapes and their attributes extensively in previous grades. “From Kindergarten on, students experience all of the properties of shapes that they will study in Grades K–7, recognizing and working with these properties in increasingly sophisticated ways” (Geometry Progression, p. 3). In Kindergarten through 2nd grade, students focused on building understanding of shapes and their properties. In 3rd grade, students started to conceptualize shape categories, in particular quadrilaterals. In 4th grade, work with angle measure (4.MD.5—7) lent itself to classifying figures based on the presence or absence of parallel and perpendicular sides. 

Thus, this unit builds off of students’ well-established understanding of geometry and geometric measurement. Similar to students’ work with area, students develop an understanding of volume as an attribute of solid figures (5.MD.3) and measure it by counting unit cubes (5.MD.4). Students then connect volume to the operation of multiplication of length, width, and height or of the area of the base and the height; they also connect it to the operation of addition to find composite area (5.MD.5). Throughout Topic A, students have an opportunity to use appropriate tools strategically (MP.5) and make use of structure of three-dimensional figures (MP.7) to draw conclusions about how to find the volume of a figure. 

Students then move on to classifying flat shapes into categories and see that attributes belonging to shapes in one category are shared by all subcategories of that category (5.G.3). This allows students to create a hierarchy of shapes over the course of many days (5.G.4). Throughout this topic, students use appropriate tools strategically (MP.5) to verify various attributes of shapes including their angle measure and presence of parallel or perpendicular lines, as well as attend to precision in their use of language when referring to geometric figures (MP.6). They also look for and make use of structure to construct a hierarchy based on properties (MP.7).

In 6th Grade Math, students will explore concepts of length, area, and volume with more complex figures, such as finding the area of right triangles or finding the volume of right rectangular prisms with non-whole-number measurements (6.G.1, 6.G.2). Students will even rely on their understanding of shapes and their attributes to prove various geometric theorems in high school (GEO.G-CO.9—11). Thus, this unit provides a nice foundation for connections in many grades to come.

Pacing: 17 instructional days (15 lessons, 1 flex day, 1 assessment day)

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The following assessments accompany Unit 3.


Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.


Have students complete the Mid-Unit Assessment after lesson 10.


Use the resources below to assess student understanding of the unit content and action plan for future units.

Expanded Assessment Package

Use student data to drive instruction with an expanded suite of assessments. Unlock Pre-Unit and Mid-Unit Assessments, and detailed Assessment Analysis Guides to help assess foundational skills, progress with unit content, and help inform your planning.

Unit Prep

Intellectual Prep

Unit Launch

Before you teach this unit, unpack the standards, big ideas, and connections to prior and future content through our guided intellectual preparation process. Each Unit Launch includes a series of short videos, targeted readings, and opportunities for action planning to ensure you're prepared to support every student.

Intellectual Prep for All Units

  • Read and annotate “Unit Summary” and “Essential Understandings” portion of the unit plan. 
  • Do all the Target Tasks and annotate them with the “Unit Summary” and “Essential Understandings” in mind. 
  • Take the Post-Unit Assessment.

Essential Understandings

  • Volume refers to the amount of space a three-dimensional figure takes up. Two-dimensional figures have no volume. 
  • You can find the volume of a rectangular prism by counting individual cubic units; counting the number of cubic units in a “layer” and multiplying by the number of layers; or multiplying the length, width, and height of the figure. The latter two strategies correspond to the formulas $$V = b \times h$$ and $$V = l \times w \times h$$
  • You can calculate the volume of a rectangular prism by multiplying edge lengths in any order because of the associative property. 
  • One can “find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts” (CCSS, standard 5.MD.5c). This is possible because volume is additive.
  • Two-dimensional figures are classified by their properties into categories but can fit into more than one category at the same time. For example, a four-sided closed flat shape with one pair of parallel sides can be simultaneously classified as a polygon, a quadrilateral, and a trapezoid. 



cubic units



regular polygon

rectangular prism

unit cube


To see all the vocabulary for Unit 3, view our 5th Grade Vocabulary Glossary.


  • Centimeter cubes (Maximum of about 70 per student or small group)
  • Cardstock (Total of 5 sheets per student or small group) — Printing the Nets A-E templates on cardstock will make them sturdier, but plain paper can be used if materials are limited
  • Tape (1 per teacher)
  • Nets A-C (1 per student or small group)
  • Net D (1 per student or small group)
  • Net E (1 per student or small group)
  • Build My Prism Template (1 per pair of students)
  • Rectangular Prisms Template (1 per pair of students)
  • Optional: Inch cube (1 per teacher) — See Lesson 6 for more information.
  • Markers or crayons (2 per student)
  • Right-angle tool (1 per student) — This can be any tool used to verify right angle measures, e.g., a protractor, the corner of a piece of paper, etc.
  • Ruler (1 per student) — This can be any tool used to draw straight lines and verify equal lengths, e.g., the edge of a piece of paper, etc.
  • Polygons Template (1 per student or small group)
  • Parallelograms Template (1 per student or small group)
  • Triangles Template (1 per student or small group)

Unit Practice

Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Lesson Map

Topic A: Volume of Three-Dimensional Figures

Topic B: Classification of Two-Dimensional Shapes

Common Core Standards


Major Cluster

Supporting Cluster

Additional Cluster

Core Standards


  • 5.G.B.3 — Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
  • 5.G.B.4 — Classify two-dimensional figures in a hierarchy based on properties.

Measurement and Data

  • 5.MD.C.3 — Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
  • 5.MD.C.3.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.
  • 5.MD.C.3.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.C.4 — Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
  • 5.MD.C.5 — Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
  • 5.MD.C.5.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.
  • 5.MD.C.5.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.
  • 5.MD.C.5.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.

Foundational Standards


  • 3.G.A.1
  • 4.G.A.2

Measurement and Data

  • 3.MD.C.5
  • 3.MD.C.6
  • 3.MD.C.7
  • 4.MD.A.3

Number and Operations in Base Ten

  • 5.NBT.B.5
  • 5.NBT.B.6

Operations and Algebraic Thinking

  • 3.OA.B.5

Future Standards


  • 6.G.A.1
  • 6.G.A.2

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

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Unit 2

Multiplication and Division of Whole Numbers


Unit 4

Addition and Subtraction of Fractions/Decimals

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