1:1 Grade 9 Algebra 1

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Define appropriate quantities for the purpose of descriptive modeling. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Interpret expressions that represent a quantity in terms of its context. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2). In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Explain each step in solving a simple 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. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^3 = 5(^(1/3))^3 to hold, so (5^(1/3))^3 must equal 5. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Rewrite expressions involving radicals and rational exponents using the properties of exponents. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Explain 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); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

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. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

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). In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ³ 1. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Calculate 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. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Write a function that describes a relationship between two quantities. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), 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 illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Distinguish between situations that can be modeled with linear functions and with exponential functions. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Construct 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). In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Interpret the parameters in a linear or exponential function in terms of a context. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Represent data with plots on the real number line (dot plots, histograms, and box plots). In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

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. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Compute (using technology) and interpret the correlation coefficient of a linear fit. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Distinguish between correlation and causation. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Solve quadratic equations in one variable. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x^2 + y^2 = 3. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Find inverse functions. In Algebra I we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.