Comparing, Ordering, and Working with Rational Numbers

Classify whole numbers, integers, and rational numbers, understand opposites and absolute value, and locate positive and negative numbers on number lines.

Compare and order integers and rational numbers using number lines, interpret fraction notation as division, and find distances between rational numbers using absolute value.

Model integer addition using number lines and counters, connect models to the standard algorithm, and fluently add integers to solve real-world problems.

Model integer subtraction using number lines and counters, connect subtraction to adding the opposite, and fluently subtract integers to solve real-world problems.

Model integer multiplication using number lines, connect models to sign rules, and fluently multiply integers to solve real-world problems.

Model and fluently divide integers, and evaluate expressions that mix all four integer operations.

Understand that dividing by a rational number is equivalent to multiplying by its reciprocal, and predict how multiplication by different types of fractions affects a quantity.

Build procedural fluency in multiplying and dividing positive fractions, decimals, and mixed numbers, including simplification and correct decimal placement.

Use area models and other visual representations to illustrate and compute multiplication and division of fractions and decimals.

Apply rational number operations and fraction comparisons to solve real-world and multi-step problems with precise justification.

Differentiate additive relationships (y = x + a) from multiplicative relationships (y = ax) across verbal, numerical, graphical, and symbolic representations.

Understand ratios as multiplicative comparisons and rates as comparisons by division, and calculate unit rates to solve basic comparison problems.

Apply ratios, rates, and proportional reasoning to solve real-world problems using equivalent ratios, ratio tables, cross-multiplication, and qualitative reasoning.

Use concrete objects, pictorial models, and 10×10 grids to represent ratios, percents, fractions, and decimals, establishing foundational connections among the three forms.

Use 10×10 grids, strip diagrams, and number lines to represent, compare, and order benchmark fractions and percents such as 1%, 10%, 25%, 33 1/3%, and 50%.

Fluently convert among fractions, decimals, and percents using equivalent fractions, place value, multiplication and division by 100, and long division for repeating decimals.

Use equivalent fractions, decimals, and percents to solve real-world money problems, including finding a percent of a number, finding the whole given a part, and calculating discounts, tax, and tips.

Convert units of length, weight, capacity, and time within customary, metric, and time systems using proportions and unit rates.

Generalize proportional reasoning strategies for unit conversions and solve multi-step, real-world measurement problems involving rational numbers.

Write, evaluate, and simplify algebraic expressions using variables, rational number values, and properties of operations to generate equivalent forms.

Write, solve, and graph one-step equations and inequalities with rational numbers, and determine which values from a set satisfy a given equation or inequality.

Create, represent, and solve real-world problems using equations, including additive and multiplicative relationships and multi-step scenarios.

Understand the structure of the four-quadrant coordinate plane, plot points with rational number coordinates in any quadrant, and find distances between points.

Graph additive and multiplicative relationships on the coordinate plane, compare their key features, and apply graphing skills to real-world contexts using technology.

Calculate the area of triangles, parallelograms, trapezoids, and composite figures by decomposing them into simpler shapes.

Calculate the volume and surface area of three-dimensional figures and solve multi-step real-world geometry problems.

Create and interpret stem-and-leaf plots, dot plots, and line plots to organize and represent data sets, including those with rational numbers.

Calculate measures of center and spread, create and interpret bar graphs, choose appropriate data displays, and use data to solve real-world problems and make predictions.

Represent and solve real-world ratio and rate problems using scale factors, tables, graphs, and proportions.

Find the whole, part, or percent in percent problems and use concrete and pictorial models to reason about percent relationships.

Apply percents to real-world contexts such as discounts, tax, and tips, and use equivalent forms of fractions, decimals, and percents to solve multi-step problems.

Determine which quantity is independent and which is dependent from tables, graphs, real-world contexts, and write equations from tables of related quantities.

Create and connect multiple representations—verbal descriptions, tables, graphs, and equations—to represent and analyze real-world relationships between quantities.

Evaluate numerical expressions using order of operations and properties, and distinguish between expressions and equations in verbal, numerical, and algebraic contexts.

Use concrete models, pictorial models, and algebraic manipulation along with properties of operations to determine and generate equivalent algebraic expressions.

Understand triangle angle and side relationships, and model area formulas by decomposing shapes

Write equations representing the area of rectangles, parallelograms, triangles, and trapezoids, and the volume of rectangular prisms, when dimensions are positive rational numbers.

Solve area and volume problems with fractional and decimal dimensions, including multi-step real-world application problems.

Write one-step equations and inequalities from word problems, represent their solutions on number lines, and create real-world problems from given equations and inequalities.

Apply one-step equations and inequalities to geometric contexts and real-world scenarios, verify solutions, and solve multi-step problems involving rational numbers.

Create and interpret graphical displays such as dot plots, stem-and-leaf plots, histograms, and box plots, and choose the best graph for a given data set.

Describe the center, spread, and shape of data distributions using visual and numerical summaries, and compare two distributions using these measures.