Ratios and Proportional Relationships

• Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
• Recognize and represent proportional relationships between quantities.
• Use proportional relationships to solve multistep ratio and percent problems.

The Number System

• Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
• Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
• Solve real-world and mathematical problems involving the four operations with rational numbers.

Expressions and Equations

• Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
• Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.
• Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
• Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

Statistics and Probability

• Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
• Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.
• Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
• Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
• Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
• Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
• Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.