Unit 2 — Understanding Volume and Operations on Fractions
Description
This unit develops understanding of volume as an attribute of three-dimensional space. Students recognize that volume can be measured by finding the total number of same-size unit cubes needed to fill a space. They understand that a unit cube is the standard unit for measuring volume and learn to use formulas V = l × w × h and V = B × h. Students decompose three-dimensional shapes to find volumes of composite rectangular prisms. Simultaneously, students develop fluency with addition and subtraction of fractions with unlike denominators by using equivalent fractions. They interpret fractions as division and solve word problems involving division of whole numbers that result in fractions. Students begin multiplying fractions and whole numbers, using visual models and area representations to understand why procedures work.
Essential Questions
- How do we measure volume?
- How are area and volume alike and different?
- How can you find the volume of cubes and rectangular prisms?
- How are equivalent fractions helpful when solving problems?
- How can fractions with different denominators be added together?
Learning Objectives
- Measure volumes by counting unit cubes and using standard formulas
- Recognize volume as additive and find volumes of composite rectangular prisms
- Add and subtract fractions with unlike denominators by converting to equivalent fractions
- Solve word problems involving addition and subtraction of fractions
- Use benchmark fractions to estimate and assess reasonableness of answers
- Interpret fractions as division and solve problems leading to fraction or mixed number answers
- Multiply whole numbers by fractions and fractions by fractions using visual models
- Find areas of rectangles with fractional side lengths
Supplemental Resources
- Unit cubes or base-ten blocks for hands-on volume measurement activities
- Graphic organizers showing fraction equivalence and operation strategies
- Sentence strips with fraction vocabulary for building mathematical language
- Chart paper for recording student strategies during volume and fraction investigations
- Printed images of rectangular prisms for volume problem-solving
Measurement
Number and Operations in Base Ten
Number and Operations—Fractions
Standards for Mathematical Practice
Students read and comprehend informational and literary texts to understand mathematical concepts and solve word problems. Students write explanations of mathematical thinking using precise vocabulary and demonstrate command of conventions including grammar and punctuation. Students engage in collaborative discussions about mathematical strategies and justify their reasoning with evidence.
Students collect, organize, and analyze data to identify patterns and make predictions in scientific investigations. Students use measurement tools and develop understanding of volume and capacity. Students engage in scientific practices including asking questions, developing models, conducting fair tests, and constructing explanations based on evidence.
Formative Assessments
- Classwork involving building rectangular prisms and calculating volumes
- Exit tickets showing understanding of equivalent fractions and fraction operations
- Individual work adding and subtracting fractions with visual models
- Group tasks exploring multiplication of fractions with area models
- Math journals recording strategies for solving fraction word problems
Summative Assessment
Unit 2 test assessing volume calculation, fraction operations, and problem-solving with fractions and volumes
Benchmark Assessment
Benchmark assessment tracking progress on fraction standards and volume understanding
Alternative Assessment
Students may demonstrate understanding of volume by physically building rectangular prisms with unit cubes and describing the dimensions orally, or by using visual models and grid paper instead of formula-based written tasks. For fraction operations, students may use manipulatives such as fraction strips or area models, with responses recorded through drawings, oral explanation, or teacher-guided written work with sentence frames provided.
IEP (Individualized Education Program)
Students receiving IEP services may benefit from the use of physical unit cubes and grid paper to build and visualize rectangular prisms before applying volume formulas, supporting both conceptual understanding and procedural fluency. Fraction work should be scaffolded with visual models such as fraction strips or area diagrams, and word problems may be broken into smaller steps with key information highlighted. Teachers should allow alternative output modes—such as verbal explanation or pointing to a visual—when written responses create a barrier to demonstrating understanding of volume or fraction concepts. Providing a reference card with the volume formulas and steps for finding equivalent fractions can reduce cognitive load and help students focus on reasoning rather than recall.
Section 504
Students with 504 plans should be given extended time on classwork and assessments involving multi-step fraction operations and volume calculations, as the density of procedural steps in this unit can require additional processing time. Preferential seating near the teacher during direct instruction on formulas and fraction models supports focus, and a low-distraction environment is especially helpful during independent problem-solving tasks. Providing printed copies of any board work—particularly visual models for fraction multiplication and composite prism diagrams—ensures students can reference key information without losing their place.
ELL / MLL
Multilingual learners should have access to a visual vocabulary reference that includes illustrated definitions of unit-specific terms such as 'unit cube,' 'volume,' 'denominator,' 'equivalent fraction,' and 'composite figure,' ideally supported in the student's home language where possible. Teachers should use physical models and diagrams consistently when introducing volume and fraction concepts, pairing verbal instruction with visual representations to make abstract ideas more concrete and accessible. Directions for tasks involving fraction operations or volume problems should be simplified and given in short steps, with an opportunity for students to restate the task before beginning.
At Risk (RTI)
Students who need additional support should be connected to prior knowledge of area and multiplication before volume concepts are introduced, and fraction work should begin with models using familiar benchmark fractions before moving to unlike denominators. Reducing the number of problems on a task while maintaining a focus on key concepts—such as finding volume with a formula or adding fractions with unlike denominators—allows students to build confidence and accuracy without becoming overwhelmed. Frequent check-ins during independent work, along with access to manipulatives like fraction tiles and unit cubes, provide important entry points for students who need to ground abstract concepts in hands-on experience.
Gifted & Talented
Advanced learners should be encouraged to explore volume beyond rectangular prisms by investigating how volume changes when dimensions are scaled, connecting this to multiplicative reasoning in ways that foreshadow proportional relationships. In fraction work, students can be challenged to analyze and explain why the procedures for multiplying or adding fractions work, using mathematical argumentation rather than simply applying steps. Opportunities to pose and solve original word problems that combine volume and fraction concepts—such as finding the volume of a prism with fractional dimensions—can push students toward deeper integration of the unit's two major content strands.