Unit 1 — Operations and Reasoning About Ratios
Description
Unit 1 establishes fluency with operations on fractions and multi-digit numbers while introducing ratio and rate concepts. Students interpret and compute quotients of fractions, solve division problems with visual models, and fluently divide and multiply multi-digit numbers and decimals using standard algorithms. The unit then builds ratio reasoning by having students understand ratio language, determine unit rates, and use ratio and rate to solve real-world problems involving unit pricing, constant speed, percentages, and measurement conversions. Students create tables of equivalent ratios, plot coordinate pairs, and solve problems through multiple representations including tape diagrams, double number lines, and equations.
Essential Questions
- Why does the process of invert and multiply work when dividing fractions?
- When I divide one number by another, do I always get a quotient smaller than the original number?
- What are equivalent ratios, and how can I use them to solve problems?
- How are ratios and rates similar and different?
- How are unit rates helpful in solving real-world problems?
Learning Objectives
- Interpret and compute quotients of fractions and solve word problems involving division of fractions.
- Represent division of fractions using visual models and explain using multiplication-division relationships.
- Fluently divide multi-digit numbers using the standard algorithm.
- Understand ratio concepts and use ratio language to describe relationships between quantities.
- Determine and use unit rates in the context of ratio relationships.
- Use ratio and rate reasoning to create tables of equivalent ratios and solve real-world problems.
- Solve unit rate problems including unit pricing and constant speed.
- Find percents as a rate per 100 and solve percentage problems.
- Fluently add, subtract, multiply, and divide multi-digit decimals.
- Find greatest common factors and least common multiples of whole numbers.
Supplemental Resources
- Printed word lists and vocabulary cards for ratio, rate, and unit rate terminology for building mathematical vocabulary
- Graphic organizers for comparing ratios and organizing equivalent ratio tables
- Printed passage sets and problem cards for context-based ratio and rate problems
- Sticky notes for exit tickets and formative feedback on ratio reasoning
- Chart paper for displaying and comparing different strategies for solving unit rate problems
The Number System
Ratios and Proportional Relationships
Standards for Mathematical Practice
Students use close-reading skills to understand and solve complex word problems and write mathematical reflections after each unit. Students utilize reading comprehension skills by acting out or drawing the order of important events in story problems. Students read and write stories to represent mathematical concepts.
Students apply mathematical reasoning to understand scientific phenomena including temperatures, data analysis, and quantitative relationships in science investigations.
Students understand how to read dates properly, interpret geographic and economic data, and use quantitative evidence to support historical and civic arguments.
Formative Assessments
- Homework practice on fraction division, multi-digit operations, and ratio problems
- Exit tickets assessing understanding of unit rates and equivalent ratios
- Journal writing reflecting on strategies for solving ratio and rate problems
- Task cards for practicing ratio table completion and unit rate calculation
- Self-assessments on procedural fluency with algorithms
Summative Assessment
Chapter tests covering operations with fractions and decimals, and ratio and rate reasoning; performance tasks requiring students to solve multi-step real-world problems involving ratios, rates, and unit pricing; extended projects applying ratio reasoning to plan events or analyze community problems
Benchmark Assessment
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Alternative Assessment
Students may demonstrate understanding of fraction division and ratio concepts through a combination of oral explanation with visual supports (models, diagrams, or manipulatives), teacher-led problem solving with reduced complexity or fewer steps, or recorded explanations paired with simplified written responses. Number lines, area models, and partially completed tables may be provided as scaffolds.
IEP (Individualized Education Program)
Students benefit from visual models such as tape diagrams, double number lines, and fraction bars to anchor abstract ratio and fraction division concepts to concrete representations. Providing graphic organizers that break multi-step ratio and rate problems into sequenced steps supports both processing and output. Allow students to demonstrate understanding through oral explanation or use of a calculator for computation-heavy tasks so that procedural barriers do not obscure conceptual understanding. Reducing the number of problems on practice assignments while maintaining coverage of key concepts—such as unit rate and fraction division—helps students build mastery without cognitive overload.
Section 504
Extended time on chapter tests and performance tasks is especially important in this unit given the multi-step nature of ratio, rate, and decimal computation problems. Preferential seating that minimizes distraction supports sustained attention during algorithm practice and problem solving. Providing a reference card with ratio and fraction operation procedures allows students to focus on reasoning rather than memorization during assessments.
ELL / MLL
Pre-teaching key vocabulary—such as ratio, rate, unit rate, quotient, and equivalent—with visual supports, examples, and home-language connections helps students access both the language and the mathematics of this unit. Using visual representations like double number lines, ratio tables, and diagrams as primary instructional tools reduces language load while keeping mathematical content accessible. Directions for multi-step problems should be simplified and chunked, and students should be encouraged to retell problem situations in their own words before computing.
At Risk (RTI)
Connecting ratio and rate concepts to familiar real-world contexts—such as comparing prices or calculating speed—provides meaningful entry points and activates prior knowledge of multiplicative thinking. Beginning with whole-number ratio relationships before introducing fraction and decimal contexts allows students to build confidence and fluency progressively. Frequent check-ins during the first few minutes of practice, along with structured problem formats that reduce extraneous complexity, help students experience early success and maintain engagement across the unit.
Gifted & Talented
Students ready for greater depth can explore the relationship between ratios, proportional reasoning, and algebraic thinking by investigating how ratio tables connect to linear equations and graphing on the coordinate plane. Encouraging students to analyze real-world data sets—such as unit pricing across different markets or speed data from athletics or engineering—challenges them to apply ratio and rate reasoning critically and draw evidence-based conclusions. Open-ended performance tasks that require students to design a solution to a community or economic problem using ratio and percentage reasoning provide meaningful complexity beyond procedural fluency.