Understanding Ratios

• Define a ratio as a relationship between two quantities and express it in the forms a:b, a to b, and a/b
• Identify the two quantities being compared in a ratio and their order
• Recognize equivalent ratios and explain why they describe the same relationship
• Apply ratio reasoning to describe real-world situations involving two quantities

Unit Rates and Ratios

• Compute the unit rate associated with a ratio a/b where b ≠ 1, including ratios of fractions
• Interpret a unit rate in context (e.g., miles per hour, cost per item)
• Compare two unit rates to determine which represents the better deal
• Solve problems involving unit pricing and constant speed using unit rate reasoning

Ratio Reasoning: Percentages and Conversions

• Find a percent of a quantity as a rate per 100 and solve problems involving finding the whole, the part, or the percent
• Use ratio reasoning to convert measurement units within and between measurement systems
• Solve multi-step ratio and percent problems including tax, discounts, and tips
• Represent equivalent ratios in a table and use the table to solve problems

Dividing Fractions

• Interpret division of a fraction by a fraction using the relationship between multiplication and division
• Compute the quotient of two fractions by multiplying by the reciprocal of the divisor
• Solve real-world problems involving division of fractions by fractions
• Explain why dividing by a fraction is equivalent to multiplying by its reciprocal

Multi-Digit Arithmetic and Decimals

• Fluently divide multi-digit numbers using the standard algorithm
• Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm
• Apply decimal operations to solve real-world problems involving money and measurement
• Identify and correct errors in multi-digit or decimal computation

Factors, Multiples, and the Distributive Property

• Find the greatest common factor (GCF) of two whole numbers less than or equal to 100
• Find the least common multiple (LCM) of two whole numbers less than or equal to 12
• Apply the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum
• Distinguish between GCF and LCM and select the correct operation for a given context

Positive and Negative Numbers

• Recognize a negative number as a value less than zero and identify real-world situations where negative numbers apply
• Identify and compare integers and rational numbers on a number line
• Determine the absolute value of a rational number and interpret it as distance from zero
• Order and compare positive and negative rational numbers using inequality symbols (<, >, =)
• Apply absolute value to real-world contexts such as temperature differences and debt

Exponents and Numerical Expressions

• Write and evaluate numerical expressions involving whole-number exponents
• Interpret exponential notation as repeated multiplication
• Apply the order of operations to evaluate expressions with exponents, parentheses, and multiple operations
• Compare the value of expressions with and without exponents to build numerical sense

Writing and Evaluating Algebraic Expressions

• Write an algebraic expression to represent a real-world or mathematical situation
• Identify parts of an expression using mathematical terms: term, coefficient, factor, sum, product
• Evaluate algebraic expressions by substituting specific values for variables
• Distinguish between expressions and equations and identify the role of the variable in each

Equivalent Expressions

• Apply the properties of operations (commutative, associative, distributive) to generate equivalent expressions
• Identify when two expressions are equivalent and explain why
• Use the distributive property to expand or factor algebraic expressions
• Recognize that equivalent expressions have the same value for all values of the variable

Solving One-Variable Equations

• Identify whether a given value is a solution to an equation by substituting and checking
• Write and solve one-variable equations of the form x + p = q and px = q for non-negative rational numbers
• Solve real-world and mathematical problems by writing and solving an equation
• Distinguish between the solution of an equation and the solution of an inequality

One-Variable Inequalities

• Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world problem
• Identify whether a given value satisfies an inequality by substituting and comparing
• Recognize that inequalities of the form x > c or x < c have infinitely many solutions
• Interpret the solution set of an inequality in a real-world context

Statistical Questions and Data

• Distinguish between a statistical question and a non-statistical question based on the expectation of variability
• Recognize that a set of data collected to answer a statistical question has a distribution that can be described by its center and spread
• Identify the attribute being measured and explain why data collected on that attribute will vary
• Classify a question as statistical or non-statistical and justify the reasoning

Measures of Center

• Calculate the mean of a data set by dividing the sum of values by the number of values
• Identify the median of a data set by ordering the values and locating the middle value
• Identify the mode of a data set as the value that appears most frequently
• Select the most appropriate measure of center (mean, median, or mode) for a given context and justify the choice
• Interpret the mean as a balance point of the data set

Measures of Variability

• Calculate the range of a data set as the difference between the maximum and minimum values
• Calculate the mean absolute deviation (MAD) as the average distance of each data point from the mean
• Interpret the interquartile range (IQR) as a measure of spread describing the middle 50% of the data
• Compare two data sets using their measures of variability and explain what greater variability means in context
• Explain the difference between measures of center and measures of variability and what each tells us about a data set

Ratios and Proportional Relationships

Learn how to compare quantities using ratios and rates, and use that reasoning to solve everyday problems like finding unit prices or converting measurements.

The Number System

Extend your number skills by dividing fractions, mastering operations with decimals, working with factors and multiples, and understanding positive and negative numbers.

Expressions and Equations

Learn to write and evaluate algebraic expressions, identify equivalent forms, and solve one-variable equations and inequalities — the building blocks of algebra.

Statistics and Probability

Discover how to ask statistical questions, describe data sets using measures of center and spread, and understand what variability tells us about the world.