Adding and Subtracting Rational Numbers

• Identify opposite quantities that combine to make zero (additive inverses)
• Apply the rule p – q = p + (–q) to rewrite subtraction as addition
• Compute sums and differences of integers using properties of operations
• Compute sums and differences of fractions and decimals with different signs
• Interpret sums and differences of rational numbers in real-world contexts

Multiplying and Dividing Rational Numbers

• Apply sign rules to compute products of two rational numbers
• Apply sign rules to compute quotients of two rational numbers
• Identify that –(p/q) = (–p)/q = p/(–q) for integer values of p and q
• Apply the distributive property to multiply rational numbers
• Convert a rational number to its decimal form using long division

Solving Real-World Problems with Rational Numbers

• Solve multi-step problems involving addition and subtraction of rational numbers
• Solve multi-step problems involving multiplication and division of rational numbers
• Convert between fraction, decimal, and percent forms as needed to solve a problem
• Assess the reasonableness of an answer using estimation strategies

Unit Rates and Proportional Relationships

• Compute unit rates associated with ratios of fractions, including complex fractions
• Decide whether two quantities are in a proportional relationship by testing equivalent ratios in a table
• Identify the constant of proportionality in tables, equations, and verbal descriptions
• Represent a proportional relationship as an equation of the form y = kx

Percent Problems in Real Life

• Calculate the percent of a quantity to solve tax and tip problems
• Compute markups and markdowns using percent increase and percent decrease
• Calculate simple interest using the formula I = Prt
• Solve percent error problems by comparing an estimated value to an actual value
• Solve multi-step ratio and percent problems involving fees and commissions

Generating Equivalent Expressions

• Apply properties of operations to add and subtract linear expressions with rational coefficients
• Factor and expand linear expressions using the distributive property
• Rewrite an expression in an equivalent form to reveal a new relationship between quantities
• Identify when two expressions are equivalent by substituting numerical values

Solving Equations of the Form px + q = r

• Use a variable to represent an unknown quantity in a real-world problem
• Solve equations of the form px + q = r fluently for rational values of p, q, and r
• Solve equations of the form p(x + q) = r fluently
• Compare an algebraic solution to an arithmetic solution for the same problem
• Solve word problems leading to one-variable linear equations

Solving and Interpreting Inequalities

• Write an inequality of the form px + q > r or px + q < r to model a real-world situation
• Solve inequalities of the form px + q > r and px + q < r for rational values
• Identify the set of values that satisfy a given inequality
• Interpret the solution of an inequality in the context of the original problem

Random Sampling and Population Inferences

• Identify whether a sample is representative of a population
• Explain why random sampling tends to produce valid inferences about a population
• Use data from a random sample to estimate an unknown characteristic of a population
• Compare multiple samples of the same size to gauge variation in estimates

Understanding Probability

• Identify the probability of a chance event as a number between 0 and 1
• Classify an event as unlikely, equally likely, or likely based on its probability value
• Approximate the probability of an event by computing its long-run relative frequency
• Predict the approximate frequency of an outcome given its theoretical probability
• Compare probabilities from a theoretical model to observed frequencies

Probability of Compound Events

• Identify the sample space for a compound event using an organized list or table
• Calculate the probability of a compound event as a fraction of the total sample space
• Apply the concept that compound event probability equals favorable outcomes divided by total outcomes
• Solve probability problems involving two independent events

The Number System

Build a solid foundation with rational numbers. You will learn to add, subtract, multiply, and divide positive and negative numbers — skills you will use in every domain that follows.

Ratios and Proportional Relationships

Explore how quantities relate to each other proportionally. You will solve real-life problems involving unit rates, percentages, interest, and markups — building on your number skills.

Expressions and Equations

Learn to write and simplify algebraic expressions, then solve equations and inequalities. These tools let you model and solve problems from everyday life with a single variable.

Statistics and Probability

Discover how to use data and chance to make predictions. You will learn to draw conclusions from samples, compare populations, and calculate probabilities of simple and compound events.