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.
Understand addition of rational numbers; p + q is the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Apply properties of operations as strategies to multiply and divide rational numbers.
Compute unit rates associated with ratios of fractions.
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.
Represent a proportional relationship using an equation.
Represent sample spaces for compound events using methods such as organized lists, sample space tables, and tree diagrams.
For an event described in everyday language, identify the outcomes in the sample space which compose the event.
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.
Add, subtract, factor, and expand linear expressions with rational coefficients by applying the properties of operations.
Use proportional relationships to solve multistep ratio and percent problems.
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Use measures of center and measures of variability for quantitative data from random samples or populations to draw informal comparative inferences about the populations.
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