Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
Convert a fraction to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Use proportional relationships to solve multistep ratio and percent problems.
Apply the formulas for the area and circumference of a circle to solve problems.
Compute unit rates associated with ratios of fractions.
Solve real-world and mathematical problems involving the four operations with rational numbers.
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Describe situations in which opposite quantities combine to make 0.
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. Assess the reasonableness of answers using mental computation and estimation strategies.
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.
Decide whether two quantities are in a proportional relationship.
Add, subtract, factor, and expand linear expressions with rational coefficients by applying the properties of operations.
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
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.
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
Solve word problems leading to inequalities of the form px+q>r, px+q≥r, px+q≤r,or px + q < r, where p, q, and r are rational numbers. Graph the solution set of the inequality on the number line and interpret it in the context of the problem.
Construct and interpret box-plots, find the interquartile range, and determine if a data point is an outlier.
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.
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