NY Math Standards (Algebra I)

Add, subtract, and multiply polynomials and recognize that the result of the operation is also a polynomial. This forms a system analogous to the integers. Note: This standard is a fluency recommendation for Algebra I. Fluency in adding, subtracting and multiplying polynomials supports students throughout their work in algebra, as well as in their symbolic work with functions.

Identify zeros of polynomial functions when suitable factorizations are available. (Shared standard with Algebra II) Note: Algebra I tasks will focus on identifying the zeros of quadratic and cubic polynomial functions. For tasks that involve finding the zeros of cubic polynomial functions, the linear and quadratic factors of the cubic polynomial function will be given (e.g., find the zeros of P(x) = (x - 2)(x² - 9)).

Create equations and inequalities in one variable to represent a real-world context. (Shared standard with Algebra II) Notes: This is strictly the development of the model (equation/inequality). Limit equations to linear, quadratic, and exponentials of the form f(x) = a(b)^x where a > 0 and b > 0 (b ≠ 1). Work with geometric sequences may involve an exponential equation/formula of the form a_n = ar^(n-1), where a is the first term and r is the common ratio. Inequalities are limited to linear inequalities. Algebra I tasks do not involve compound inequalities.

Create equations and linear inequalities in two variables to represent a real-world context. Notes: This is strictly the development of the model (equation/inequality). Limit equations to linear, quadratic, and exponentials of the form f(x) = a(b)^x where a > 0 and b > 0 (b ≠ 1).

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. e.g., Represent inequalities describing nutritional and cost constraints on combinations of different foods.

Rewrite formulas to highlight a quantity of interest, using the same reasoning as in solving equations. e.g., Rearrange Ohm’s law V = IR to highlight resistance R.

Explain each step when solving a linear or quadratic equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Note: Algebra I tasks do not involve solving compound inequalities.

Solve systems of linear equations in two variables both algebraically and graphically. Note: Algebraic methods include both elimination and substitution.

Solve a system, with rational solutions, consisting of a linear equation and a quadratic equation (parabolas only) in two variables both algebraically and graphically. (Shared standard with Algebra II)

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. Note: Graphing linear equations is a fluency recommendation for Algebra I. Students become fluent in solving characteristic problems involving the analytic geometry of lines, such as writing down the equation of a line given a point and a slope. Such fluency can support them in solving less routine mathematical problems involving linearity; as well as modeling linear phenomena.

Given the equations y = f(x) and y = g(x): recognize that each x-coordinate of the intersection(s) is the solution to the equation f(x) = g(x); find the solutions approximately using technology to graph the functions or make tables of values; and interpret the solution in context. ★ (Shared standard with Algebra II) Notes: Algebra I tasks are limited to cases where f(x) and g(x) are linear, polynomial, absolute value, and exponential functions of the form f(x) = a(b)^x where a > 0 and b > 0 (b ≠ 1). Students should be taught to find the solutions approximately by using technology to graph the functions and by making tables of values. When solving any problem, students can choose either strategy.

Graph 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. Note: Graphing linear equations is a fluency recommendation for Algebra I. Students become fluent in solving characteristic problems involving the analytic geometry of lines, such as writing down the equation of a line given a point and a slope. Such fluency can support them in solving less routine mathematical problems involving linearity; as well as modeling linear phenomena (including modeling using systems of linear inequalities in two variables).

Recognize and use the structure of an expression to identify ways to rewrite it. (Shared standard with Algebra II) e.g., x³ – x² - x = x(x² - x - 1); 53² – 47² = (53 + 47)(53 - 47); 16x² - 36 = (4x)² - (6)² = (4x + 6)(4x - 6) = 4(2x + 3)(2x - 3); -2x² + 8x + 10 = -2(x² – 4x – 5) = -2(x - 5)(x + 1); x⁴ + 6x² - 7 = (x² + 7)(x² - 1) = (x² + 7)(x + 1)(x - 1). Note: Algebra I expressions are limited to numerical and polynomial expressions in one variable. Use factoring techniques such as factoring out a greatest common factor, factoring the difference of two perfect squares, factoring trinomials of the form ax²+bx+c with a lead coefficient of 1, or a combination of methods to factor completely. Factoring will not involve factoring by grouping and factoring the sum and difference of cubes.

Using f(x) + k, k f(x), and f(x + k): identify the effect on the graph when replacing f(x) by f(x) + k, k f(x), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs; write a new function using the value of k; and use technology to experiment with cases and explore the effects on the graph. (Shared standard with Algebra II) Note: Tasks are limited to linear, quadratic, square root, and absolute value functions; and exponential functions of the form f(x) = a(b)^x where a > 0 and b > 0 (b ≠ 1).

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Note: Domain and range can be expressed using inequalities, set builder notation, verbal description, and interval notations for functions of subsets of real numbers to the real numbers.

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Recognize that a sequence is a function whose domain is a subset of the integers. (Shared standard with Algebra II) Notes: Sequences (arithmetic and geometric) will be written explicitly and only in subscript notation. Work with geometric sequences may involve an exponential equation/formula of the form a_n = ar^(n-1), where a is the first term and r is the common ratio.

For 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. (Shared standard with Algebra II) Notes: Algebra I key features include the following: intercepts, zeros; intervals where the function is increasing, decreasing, positive, or negative; maxima, minima; and symmetries. Tasks have a real-world context and are limited to the following functions: linear, quadratic, square root, piece-wise defined (including step and absolute value), and exponential functions of the form f(x) = a(b)^x where a > 0 and b > 0 (b≠1).

Determine the domain of a function from its graph and, where applicable, identify the appropriate domain for a function in context.

Calculate and interpret the average rate of change of a function over a specified interval. (Shared standard with Algebra II) Notes: Functions may be presented by function notation, a table of values, or graphically. Algebra I tasks have a real-world context and are limited to the following functions: linear, quadratic, square root, piece-wise defined (including step and absolute value), and exponential functions of the form f(x) = a(b)^x where a > 0 and b > 0, (b ≠ 1).

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (Shared standard with Algebra II) Note: Algebra I tasks are limited to the following functions: linear, quadratic, square root, piecewise defined (including step and absolute value), and exponential functions of the form f(x) = a(b)^x where a > 0 and b > 0 (b ≠ 1).

Construct a linear or exponential function symbolically given: a graph; a description of the relationship; two input-output pairs (include reading these from a table). (Shared standard with Algebra II) Note: Tasks are limited to constructing linear and exponential functions in simple context (not multi-step).

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Interpret the parameters in a linear or exponential function in terms of a context. (Shared standard with Algebra II) Note: Tasks have a real-world context. Exponential functions are limited to those with domains in the integers and are of the form f(x) = a(b)^x where a > 0 and b > 0 (b ≠ 1).

Select quantities and use units as a way to: interpret and guide the solution of multi-step problems; choose and interpret units consistently in formulas; and choose and interpret the scale and the origin in graphs and data displays. ★

Choose a level of accuracy appropriate to limitations on measurement and context when reporting quantities.

Perform all four arithmetic operations and apply properties to generate equivalent forms of rational numbers and square roots. Note: Tasks include rationalizing numerical denominators of the form a/√b where a is an integer and b is a natural number.

Categorize the sum or product of rational or irrational numbers. The sum and product of two rational numbers is rational. The sum of a rational number and an irrational number is irrational. The product of a nonzero rational number and an irrational number is irrational. The sum and product of two irrational numbers could be either rational or irrational.

Represent data with plots on the real number line (dot plots, histograms, and box plots).

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (inter-quartile range, sample standard deviation) of two or more different data sets. Note: Values in the given data sets will represent samples of larger populations. The calculation of standard deviation will be based on the sample standard deviation formula. The sample standard deviation calculation will be used to make a statement about the population standard deviation from which the sample was drawn.

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Calculate (using technology) and interpret the correlation coefficient of a linear fit.

Distinguish between correlation and causation.