Unit 2 — Equations and Ratio & Proportion
Description
This unit develops students' ability to solve multi-step equations and inequalities, recognizing variables as representing unknown quantities in real-world and mathematical contexts. Students write and solve equations of the form px + q = r and p(x + q) = r, as well as inequalities, and graph solution sets. From there, the unit introduces proportional relationships in depth. Students test for proportional relationships using tables and graphs, identify constant of proportionality from multiple representations, and write equations for proportional relationships. Students apply proportional reasoning to solve ratio and percent problems, including applications such as tax, markup, discount, and simple interest. The unit concludes with scale drawings, where students use proportions to compute actual lengths and reproduce drawings at different scales.
Essential Questions
- How do variables and equations represent real-world relationships and constraints?
- What makes a relationship proportional, and how can we recognize and represent proportional relationships?
- How do we use proportional reasoning to solve problems involving ratios, rates, and percents?
Learning Objectives
- Solve multi-step real-life and mathematical problems with rational numbers in any form.
- Use variables to represent quantities and write equations to model and solve word problems.
- Fluently solve linear equations of the form px + q = r and p(x + q) = r.
- Write and solve inequalities to model real-world constraints and graph solution sets.
- Compute unit rates with ratios of fractions in like and different units.
- Determine if two quantities are in a proportional relationship using tables and graphs.
- Identify the constant of proportionality (unit rate) from tables, graphs, equations, diagrams, and verbal descriptions.
- Write equations representing proportional relationships.
- Interpret points on graphs of proportional relationships, including the origin and (1, r).
- Solve multi-step ratio and percent problems using proportions.
- Create and interpret scale drawings using ratios and proportions.
Supplemental Resources
- Printed graphic organizers for setting up and solving two-step and multi-step equations
- Number line templates and coordinate plane grids on page protectors for graphing inequalities and proportional relationships
- Index cards with unit rate problems for fluency practice and small-group work
- Real-world scenario cards (pricing, discounts, tax) for problem-solving stations
- Rulers and measuring tools for scale drawing activities
Expressions and Equations
Geometry
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.
Students use proportional reasoning and quantitative analysis to examine historical economic data, evaluate policy decisions, and assess demographic information across different time periods and societies.
Formative Assessments
- Exit tickets checking fluency with one-step and multi-step equation solving.
- CPM checkups on 7.EE.B.3, 7.EE.B.4, 7.RP.A standards.
- Quizzes on identifying proportional relationships and computing unit rates.
- Observations of students writing and solving inequalities with real-world contexts.
- Pair-and-share activities comparing arithmetic and algebraic solution methods.
Summative Assessment
Unit 2 test covering equation solving, inequality graphing, and proportional relationships; performance assessment on real-world multi-step problems and scale drawing interpretation.
Benchmark Assessment
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Alternative Assessment
Students may demonstrate understanding through verbal explanation of equation-solving steps with teacher prompting, use of manipulatives or equation mats to model multi-step problems, or completion of guided worksheets with worked examples and fill-in-the-blank sentence frames. For proportional relationship problems, students may use provided tables or graphs to identify relationships rather than create them from scratch, or solve simplified versions with whole numbers and smaller quantities.
IEP (Individualized Education Program)
Students may benefit from graphic organizers that break multi-step equation and proportion problems into clearly sequenced steps, reducing the cognitive load of tracking multiple operations at once. Providing reference cards with equation-solving steps, a number line, and a table of equivalent ratios supports independent problem solving without removing the mathematical thinking. For output, allow students to demonstrate understanding of inequalities and proportional relationships through oral explanation or annotated graphs in addition to written work. Extended time on equation-heavy assessments and preferential presentation of fewer problems per page supports sustained focus and processing.
Section 504
Students should be provided extended time on quizzes and the unit test, particularly for multi-step ratio, percent, and equation tasks that require sustained attention and multi-part processing. Preferential seating and a low-distraction environment support focus during independent equation solving and graphing tasks. A printed copy of any board work showing proportion tables, graphs, or scale drawing setups ensures students can reference key information without losing their place.
ELL / MLL
Vocabulary support is especially important in this unit, as terms such as constant of proportionality, coefficient, inequality, unit rate, and markup carry precise mathematical meanings that differ from everyday language. Providing a visual word wall or bilingual reference card with these terms, paired with symbolic and graphical examples, helps students access both the language and the concepts. Directions for multi-step problems should be simplified and broken into numbered steps, and visual models such as double number lines, ratio tables, and labeled coordinate graphs should accompany all proportional reasoning tasks. Allowing students to explain their reasoning in their home language before transitioning to English supports deeper conceptual understanding.
At Risk (RTI)
Students who need additional support should begin equation solving with one-step problems involving whole numbers before progressing to rational number coefficients and multi-step forms, providing a clear on-ramp to the unit's core skills. Ratio and proportion concepts can be anchored to familiar real-world contexts such as shopping discounts or recipe scaling, which connect new abstract ideas to accessible prior knowledge. Structured graphic organizers that prompt students to identify the known quantities, the unknown, and the operation needed help scaffold word problem entry without removing the reasoning challenge. Frequent, brief check-ins during independent practice allow for timely corrective feedback before misconceptions solidify.
Gifted & Talented
Students who have demonstrated fluency with equation solving and proportional reasoning should be invited to explore the relationships between algebraic and graphical representations more deeply, such as examining why the constant of proportionality is the slope of a proportional relationship and how this connects to linear functions. Extension opportunities might involve designing and solving multi-variable real-world problems that incorporate tax, markup, and interest simultaneously, or critiquing and justifying whether a given relationship is truly proportional using multiple representations as evidence. Scale drawing work can be extended to explore concepts of similar figures and indirect measurement, building toward geometric reasoning. Encouraging students to pose their own proportional reasoning scenarios and present their mathematical arguments develops both creativity and advanced algebraic thinking.