Unit 3 — Equations, The Rational Number System and 2-D Geometry
Description
Unit 3 develops understanding of the rational number system and algebraic problem-solving. Students understand positive and negative numbers as representing quantities with opposite directions or values, such as temperature above/below zero or elevation changes. They locate rational numbers on number lines and coordinate planes, recognizing opposites and understanding absolute value as distance from zero. Students reason about order and inequalities with rational numbers. The unit also focuses on solving one-variable equations and inequalities using substitution and properties of operations. Students write equations to represent real-world situations and solve them. In two-dimensional geometry, students find areas of triangles, special quadrilaterals, and composite polygons by decomposing or composing them into shapes with known area formulas. They also draw polygons in the coordinate plane and find distances between points.
Essential Questions
- When are negative numbers used, and why are they important?
- Why is it useful to know the absolute value of a number?
- How do I use positive and negative numbers to represent quantities in real-world contexts?
- How can I use coordinates to find distances between points?
- How can we find the area of figures by decomposing or composing them?
Learning Objectives
- Represent quantities with positive and negative numbers in real-world contexts and explain the meaning of zero in context.
- Locate rational numbers and their opposites on horizontal and vertical number lines.
- Plot pairs of positive and negative rational numbers in the coordinate plane.
- Describe ordered pairs that differ only by signs as reflections across one or both axes.
- Interpret inequalities and determine relative positions of rational numbers on a number line.
- Explain the meaning of absolute value as distance from zero and as magnitude for positive or negative quantities.
- Use substitution to determine whether given numbers make equations or inequalities true.
- Solve real-world problems by writing and solving equations of the form x + p = q and px = q.
- Write inequalities of the form x > c or x < c to represent constraints or conditions in real-world problems.
- Solve real-world problems by graphing points in all four quadrants of the coordinate plane.
- Find areas of right triangles, other triangles, special quadrilaterals, and polygons by decomposing or composing into known shapes.
Supplemental Resources
- Number line diagrams (printed) for representing positive and negative rational numbers
- Coordinate plane grids for plotting points and finding distances
- Graphic organizers for decomposing composite figures into triangles and rectangles
- Sticky notes for recording equations and inequalities
- Chart paper for displaying strategies for solving one-step equations
Expressions and Equations
Geometry
The Number System
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 with number line representations of positive and negative numbers
- Exit tickets assessing understanding of absolute value and inequalities
- Journal writing explaining reasoning about opposite numbers and rational number order
- Task cards for plotting points and finding distances in the coordinate plane
- Self-assessments on solving one-step equations and writing equations from word problems
Summative Assessment
Chapter tests on rational numbers, equations, and area; performance tasks requiring students to solve multi-step equations and find areas of composite figures; extended projects using coordinate geometry to design figures or solve navigation problems
Benchmark Assessment
— not configured —
Alternative Assessment
Students may demonstrate understanding of rational numbers and equations through a combination of concrete manipulatives, visual number lines, and oral explanations in place of written work. Number sentences may be presented with partial fill-in options, visual supports such as pre-labeled coordinate planes, or simplified problem sets focused on one concept at a time.
IEP (Individualized Education Program)
Students with IEPs may benefit from visual supports such as number lines taped to their desks and graphic organizers that connect positive and negative numbers to real-world contexts like temperature or elevation. For equation-solving tasks, breaking multi-step problems into sequenced steps with worked examples nearby can reduce cognitive load and support independent problem-solving. Teachers should allow alternative output modes—such as oral explanation or guided written frames—when students demonstrate understanding verbally but struggle to produce written responses. Providing a reference sheet with area formulas and coordinate plane vocabulary ensures students can access content without working memory barriers interfering with reasoning.
Section 504
Students with 504 plans should receive extended time on assessments involving rational number operations, equation solving, and area calculations, as these tasks require sustained attention and multi-step processing. Preferential seating near instructional modeling and reduced-distraction environments support focus during complex tasks such as plotting coordinate pairs or decomposing composite figures. Providing printed copies of any coordinate grids, number lines, or problem sets displayed on the board ensures students can reference materials at their own pace without losing their place in a problem.
ELL / MLL
Multilingual learners benefit from a unit-specific visual vocabulary resource that pairs key terms—such as rational number, absolute value, inequality, quadrant, and decompose—with diagrams or illustrations that clarify meaning without relying solely on English text. Directions for tasks involving number lines, coordinate planes, or area problems should be given in short, clear steps, and teachers should ask students to restate directions in their own words before beginning. Where possible, connecting real-world contexts used in the unit (temperature, elevation, navigation) to students' home cultures or geographies can build meaningful background knowledge and language access simultaneously.
At Risk (RTI)
Students who need additional support should begin rational number concepts with familiar, concrete real-world situations—such as comparing temperatures or tracking money—before moving to abstract number line and coordinate plane representations. For equation-solving, entry points should emphasize substitution and checking answers rather than procedural fluency with multiple steps, allowing students to experience success and build confidence with algebraic reasoning. Area tasks can be scaffolded by starting with single, clearly labeled shapes and providing pre-drawn decomposition lines on composite figures so students can focus their reasoning on applying known formulas rather than on how to partition the shape.
Gifted & Talented
Advanced learners can extend their understanding of the rational number system by exploring how absolute value functions behave graphically in the coordinate plane or by investigating real-world applications of inequalities in contexts such as budgeting constraints or engineering tolerances. In equation work, students can be challenged to write and solve multi-variable or multi-step equations derived from self-generated real-world scenarios, justifying each step with properties of operations. For geometry, students can design coordinate-based figures that meet given area or perimeter constraints, then analyze how transformations across axes connect to the reflection properties of ordered pairs they have studied—integrating geometric reasoning with algebraic thinking at a deeper level.