Unit 1 — Operations on Rational Numbers & Expressions
Description
This unit extends students' understanding of operations with fractions to include all rational numbers, including positive and negative integers and decimals. Students represent addition and subtraction on both horizontal and vertical number lines, understanding additive inverses and absolute value as distance. They apply properties of operations to perform all four operations with rational numbers, convert between decimal and fraction forms, and interpret products and quotients in real-world contexts. The unit culminates with work on linear expressions with rational coefficients, where students add, subtract, factor, and expand expressions using properties of operations. Students also introduce irrational numbers and rational approximations, laying groundwork for later study of the real number system.
Essential Questions
- How do operations with fractions extend to all rational numbers, including negative numbers?
- How can we use properties of operations to solve problems and simplify expressions with rational numbers?
- What is the relationship between rational and irrational numbers, and how do we work with each?
Learning Objectives
- Represent addition and subtraction of rational numbers on number lines and interpret sums in real-world contexts.
- Add and subtract rational numbers, showing that distance between points on a number line is the absolute value of their difference.
- Multiply and divide rational numbers, including signed numbers, and interpret products and quotients in context.
- Convert rational numbers to decimal form using long division and determine whether decimals terminate or repeat.
- Apply properties of operations to add, subtract, multiply, and divide rational numbers fluently.
- Solve real-world problems involving all four operations with rational numbers.
- Add, subtract, factor, and expand linear expressions with rational coefficients.
- Rewrite expressions in equivalent forms to highlight relationships between quantities.
- Identify irrational numbers and use rational approximations to compare and locate irrational numbers on a number line.
Supplemental Resources
- Printed word lists for integer and fraction terminology for vocabulary building
- Number line templates on page protectors for dry-erase practice with addition and subtraction
- Index cards and sentence strips for sorting rational numbers by type (positive, negative, fractional, decimal)
- Graphic organizers for comparing equivalent fractions, decimals, and percents
- Chart paper for displaying properties of operations and anchor charts for multi-step problems
Expressions and Equations
The Number System
Ratios and Proportional Relationships
Students write in science notebooks, construct viable arguments, and critique reasoning of others using mathematical evidence and precise language to communicate mathematical thinking.
Students apply mathematical reasoning to analyze scientific data, represent relationships using equations and graphs, and solve real-world problems involving physical phenomena such as temperature, distance, and population dynamics.
Formative Assessments
- Exit and entrance tickets assessing understanding of rational number operations and properties.
- CPM checkups on specific standards (7.NS.A.1, 7.NS.A.2, 7.EE.A.1, 7.EE.A.2).
- Quizzes on fraction-to-decimal conversion and order of operations.
- Pair-and-share discussions comparing arithmetic and algebraic approaches to problems.
- Observations of student work on number line representations and expression simplification.
Summative Assessment
Unit 1 test covering all content standards; performance assessments on real-world problems involving rational number operations and expression rewriting.
Benchmark Assessment
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Alternative Assessment
Students may demonstrate understanding through oral explanation of rational number operations and number line representations, with a teacher or aide recording responses. Visual number lines, manipulatives, or partially completed work samples may be provided as scaffolds, with reduced problem sets focused on key concepts such as addition/subtraction or multiplication/division of rational numbers.
IEP (Individualized Education Program)
Students may benefit from visual scaffolds such as pre-labeled number lines and graphic organizers that break down the steps for operations with signed rational numbers. Providing a reference sheet with rules for integer operations, fraction-to-decimal conversion steps, and properties of operations supports working memory during multi-step tasks. Teachers should offer flexible output options—such as oral explanations or partially completed problem frames—for tasks involving expression writing and real-world problem interpretation. Assignments may be shortened to focus on mastery of core operations before extending to linear expressions with rational coefficients.
Section 504
Students should be given extended time on quizzes and the unit test, particularly for multi-step problems involving fraction-to-decimal conversion and expression simplification. Preferential seating that minimizes distractions is especially helpful during number line work and problem-solving tasks that require sustained attention. A reference card listing operation rules for rational numbers may be provided unless the assessment is specifically measuring recall of those rules.
ELL / MLL
Teachers should use visual representations—such as number line diagrams, color-coded sign rules, and annotated examples—to make the meaning of operations with rational numbers accessible across language levels. Key vocabulary for this unit, including terms like rational number, absolute value, additive inverse, coefficient, and irrational, should be introduced with visual support and practiced in context before students are expected to use them independently. Simplified, step-by-step directions for multi-part problems help students focus on the mathematical reasoning rather than language load, and connections to familiar real-world contexts such as temperature, money, or elevation can bridge understanding.
At Risk (RTI)
Entry points such as review of fraction and integer concepts from prior grades can help students connect new operations with rational numbers to what they already know. Teachers can reduce the complexity of initial tasks—for example, beginning with operations on integers or simple fractions before introducing mixed signs and decimal forms—so students experience early success and build confidence. Frequent, brief check-ins during work time allow teachers to catch and address misconceptions about sign rules or expression structure before they become ingrained, and structured practice routines with consistent formats reduce cognitive load for students who struggle with organization.
Gifted & Talented
Students who demonstrate early mastery of rational number operations can explore the algebraic structure underlying the properties they are applying, such as investigating why the product of two negatives is positive through formal arguments or number patterns. Extending work on linear expressions to include multi-variable expressions or those with more complex rational coefficients offers appropriate depth, as does exploring the distinction between rational and irrational numbers more rigorously—for example, investigating the density of rationals or constructing proofs of irrationality for simple cases. Encouraging students to create and solve their own real-world problems that require a combination of operations and expression rewriting develops both creativity and deeper mathematical reasoning.