Module 15: Ratios & Proportional Reasoning

Develop proportional reasoning by comparing quantities, examining rates, and representing relationships with tables, graphs, models, and equations.

Defining Ratios - Understand ratios as comparisons between quantities and explore the different ways ratio relationships can be represented and interpreted.

Solving Ratios - Use scaling, equivalent ratios, and visual models to determine unknown quantities while preserving a proportional relationship.

Rates & Unit Rates - Compare quantities measured in different units and use unit rates to describe how much of one quantity corresponds to one unit of another.

Proportional vs. Nonproportional Relationships - Distinguish proportional relationships from other relationships by examining how quantities change relative to one another.

Tables, Graphs, & Double Number Lines for Ratios - Represent ratio relationships in multiple ways and connect the same proportional structure across tables, graphs, and double number lines.

Module 16: Percent Reasoning

Build a flexible understanding of percent as a proportional relationship and connect percents to fractions, decimals, estimation, and real-world changes.

Reasoning About Percent - Understand percent as a comparison to 100 and use part-whole relationships to solve a variety of percent problems.

Percent Increase & Decrease - Analyze how quantities change by a given percent and distinguish the original amount, amount of change, and resulting value.

Connecting Fractions, Decimals, Ratios, & Percents - Recognize fractions, decimals, ratios, and percents as interconnected ways of representing the same quantitative relationships.

Benchmark Percents & Estimation - Use familiar percents such as 1%, 10%, 25%, 50%, and 100% to estimate and reason efficiently about unfamiliar percentages.

Module 17: Numerical & Algebraic Expressions

Develop algebraic thinking by examining how numbers, variables, operations, and mathematical properties can be used to represent calculations, patterns, and general relationships.

Numerical Expressions - Interpret, construct, and evaluate expressions involving numbers and operations while paying attention to their mathematical structure.

Algebraic Expressions - Use variables and operations to represent unknown values, changing quantities, patterns, and relationships.

Variables as Unknowns, Changing Quantities, & Generalized Numbers - Explore the different roles variables can play and move beyond the idea that a letter always represents a single missing number.

Properties of Operations as Foundations for Algebra - Connect the commutative, associative, and distributive properties to equivalent expressions and the reasoning used throughout algebra.

Module 18: Equality & Equations

Develop a relational understanding of equality and use that understanding to interpret, create, analyze, and solve equations.

Algebraic Equality - Explore equality as a relationship between two expressions that have the same value rather than as a signal to perform a calculation.

Algebraic Equations - Interpret equations as statements about equal quantities and use reasoning to determine values that make those statements true.

Meaning of the Equal Sign - Examine the equal sign as a symbol expressing equivalence between two quantities or expressions on either side.

True, False, & Open Number Sentences - Determine whether mathematical statements are true or false and identify values that make open number sentences true.

Relational Thinking - Use relationships among numbers and expressions to reason about equality without necessarily carrying out every calculation.

Balance Models for Equations - Use balance models to visualize equality and explain why the same mathematical operation must preserve both sides of an equation.