Unit 3 — More Operations on Fractions
Description
This unit extends fraction operations to multiplication and division in more complex contexts. Students interpret multiplication as scaling and compare products to factor sizes to make predictions about results. They multiply fractions by fractions and solve real-world problems involving fraction multiplication. Students divide unit fractions by whole numbers and whole numbers by unit fractions, using visual models and relating to the inverse relationship with multiplication. The unit also addresses decimal operations, extending place value understanding to multiplication and division of decimals to hundredths. Students convert measurement units within the same system and use conversions to solve multi-step problems. Throughout, students use concrete models, drawings, and strategies based on place value and properties of operations.
Essential Questions
- What does dividing a unit fraction by a whole number look like?
- What does dividing a whole number by a unit fraction look like?
- How can comparing factor size to 1 help us predict what will happen to the product?
- How can we use models to help us multiply and divide decimals?
- What happens to a measurement when you change its unit of measure to a related unit?
Learning Objectives
- Interpret multiplication as scaling and explain effects of factors on products
- Multiply fractions by fractions and solve real-world multiplication problems
- Divide unit fractions by whole numbers and whole numbers by unit fractions
- Solve real-world problems involving division of fractions
- Add, subtract, multiply, and divide decimals to hundredths using models and strategies
- Explain patterns in decimal point placement when multiplying or dividing by powers of 10
- Convert measurement units within the same system
- Solve multi-step problems requiring measurement conversions
Supplemental Resources
- Fraction bars or strips for visualizing multiplication and division operations
- Graph paper for recording decimal and fraction operations
- Printed problem cards for measurement conversion practice
- Sticky notes for recording student strategies during investigations
- Rulers and measuring tools for hands-on conversion experiences
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 solving multiplication and division fraction problems with visual models
- Exit tickets demonstrating understanding of scaling and fraction operations
- Individual problem-solving with decimals using concrete representations
- Group work on measurement conversion strategies and applications
- Math journals recording reasoning about fraction division and decimal operations
Summative Assessment
Unit 3 test assessing multiplication and division of fractions, decimal operations, and measurement conversions
Benchmark Assessment
— not configured —
Alternative Assessment
Students may demonstrate understanding through visual models, number lines, or manipulatives in place of written computation, with teacher guidance on interpreting multiplication as scaling and solving fraction operations. Reduced-complexity problems focusing on single operations (such as multiplying fractions or dividing unit fractions) and sentence frames for explaining reasoning may be provided as needed.
IEP (Individualized Education Program)
Students with IEPs may benefit from visual models such as fraction bars, area models, and number lines to support understanding of multiplication as scaling and fraction division concepts. Providing a reference sheet with key vocabulary, operation steps, and example models can reduce cognitive load while students focus on reasoning through problems. Teachers should consider allowing students to demonstrate understanding through oral explanation or the use of manipulatives rather than written work alone, particularly for multi-step fraction and decimal problems. Breaking complex problems into smaller, labeled steps and offering frequent check-ins during classwork can help students build accuracy and confidence across the unit.
Section 504
Students with 504 plans should be given extended time on classwork, exit tickets, and the unit assessment to allow full demonstration of their understanding of fraction operations and decimal concepts. Preferential seating near direct instruction and reduced-distraction environments support focus during multi-step problem solving involving measurement conversions and decimal operations. Providing a printed copy of any board work or models ensures students can reference examples without losing their place in multi-step tasks.
ELL / MLL
Multilingual learners benefit from visual representations—such as labeled area models, fraction strips, and conversion charts—that make abstract relationships in fraction multiplication and division more concrete and accessible across language levels. Key terms such as 'scaling,' 'unit fraction,' 'convert,' and 'product' should be pre-taught with visual support and, when possible, connected to students' home languages using bilingual glossaries or peer support. Directions for problem-solving tasks should be given in short, clear steps, and students should be encouraged to explain their reasoning using drawings or numbers when verbal or written English is a barrier.
At Risk (RTI)
Students who need additional support should be connected to prior knowledge of basic fraction concepts and whole-number multiplication before engaging with more complex fraction-by-fraction or decimal operations. Teachers can reduce the number of problems assigned while maintaining focus on the most essential concepts—such as understanding what happens to a product when multiplying by a fraction less than one—so students experience success and build momentum. Concrete manipulatives and visual models should remain available throughout the unit as entry points into abstract reasoning, and small-group instruction can provide targeted reteaching of scaling, division of unit fractions, or decimal placement as needed.
Gifted & Talented
Students who demonstrate early mastery of fraction multiplication and division should be encouraged to explore the mathematical reasoning behind why multiplying by a fraction less than one produces a smaller product, moving beyond procedural fluency toward conceptual justification. Extensions might involve applying fraction and decimal operations in multi-step, real-world contexts that require students to determine which operations and conversions are needed without being prompted—emphasizing mathematical modeling and interpretation. Students can also investigate patterns in decimal operations across powers of 10 more broadly, connecting to scientific notation or ratio reasoning as a bridge to more advanced mathematical thinking.