A1: Algebra I with Probability

Algebra I with Probability

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Rational exponents, linear and quadratic equations, polynomial operations, and systems of equations make up the algebra strand. Functions are defined, interpreted, and represented across multiple forms, with linear, exponential, and quadratic models central to the work. Probability, statistical inference, and data analysis are integrated throughout.

A1.1Explain how the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for an additional notation for radicals using rational exponents.A1.10Select an appropriate method to solve a system of two linear equations in two variables.A1.11Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.A1.12Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.A1.13Represent constraints by equations and/or inequalities, and solve systems of equations and/or inequalities, interpreting solutions as viable or nonviable options in a modeling context. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.A1.14Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane. Note: The graph of a relation often forms a curve (which could be a line).A1.15Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.A1.16Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Explain that a function f is a special kind of relation defined by the equation y = f(x).A1.17Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.A1.18Solve systems consisting of linear and/or quadratic equations in two variables graphically, using technology where appropriate.A1.19Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x).A1.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.A1.20Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes, using technology where appropriate.A1.21Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend from linear to quadratic, exponential, absolute value, and general piecewise.A1.22Define sequences as functions, including recursive definitions, whose domain is a subset of the integers.A1.23Identify the effect on the graph of replacing f(x) by f(x) + k, k ∙ f(x), f(k ∙ x), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and explain the effects on the graph, using technology as appropriate. Limit to linear, quadratic, exponential, absolute value, and linear piecewise functions.A1.24Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.A1.25Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).A1.26Use graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.A1.27Interpret the parameters of functions in terms of a context. Extend from linear functions, written in the form mx + b, to exponential functions, written in the form ab^x.A1.28For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Note: Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries; and end behavior. Extend from relationships that can be represented by linear functions to quadratic, exponential, absolute value, and linear piecewise functions.A1.29Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Limit to linear, quadratic, exponential, and absolute value functions.A1.3Define the imaginary number i such that i^2 = -1.A1.30Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.A1.31Use the mathematical modeling cycle to solve real-world problems involving linear, quadratic, exponential, absolute value, and linear piecewise functions.A1.32Use mathematical and statistical reasoning with bivariate categorical data in order to draw conclusions and assess risk.A1.33Design and carry out an investigation to determine whether there appears to be an association between two categorical variables, and write a persuasive argument based on the results of the investigation.A1.34Distinguish between quantitative and categorical data and between the techniques that may be used for analyzing data of these two types.A1.35Analyze the possible association between two categorical variables.A1.36Generate a two-way categorical table in order to find and evaluate solutions to real-world problems.A1.37Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").A1.38Explain whether two events, A and B, are independent, using two-way tables or tree diagrams.A1.39Compute the conditional probability of event A given event B, using two-way tables or tree diagrams.A1.4Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.A1.40Recognize and describe the concepts of conditional probability and independence in everyday situations and explain them using everyday language.A1.41Explain why the conditional probability of A given B is the fraction of B's outcomes that also belong to A, and interpret the answer in context.A1.5Use the structure of an expression to identify ways to rewrite it.A1.6Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.A1.7Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.A1.8Explain why extraneous solutions to an equation involving absolute values may arise and how to check to be sure that a candidate solution satisfies an equation.A1.9Select an appropriate method to solve a quadratic equation in one variable.
Example Problems
Write in radical form:
What is the result of adding these two equations?

Solve for x:
Graph:
Solve for x:
Goblins

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