NUMBER AND QUANTITY
 
The Real Number System
 
Extend the properties of exponents to rational exponents.
 

1 ) 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. [NRN1] 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. Alabama Alternate Achievement Standards



2 ) Rewrite expressions involving radicals and rational exponents using the properties of exponents. [NRN2] Alabama Alternate Achievement Standards


Use properties of rational and irrational numbers.
 

3 ) 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. [NRN3] Alabama Alternate Achievement Standards


Quantities*
 
Reason quantitatively and use units to solve problems. (Foundation for work with expressions, equations, and functions.)
 

4 ) Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. [NQ1] Alabama Alternate Achievement Standards



5 ) Define appropriate quantities for the purpose of descriptive modeling. [NQ2] Alabama Alternate Achievement Standards



6 ) Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. [NQ3] Alabama Alternate Achievement Standards


ALGEBRA
 
Seeing Structure in Expressions
 
Interpret the structure of expressions. (For standard 7 linear, exponential, quadratic; for standard 8 linear, exponential, quadratic, rational.) (Alabama)
 

7 ) Interpret expressions that represent a quantity in terms of its context.* [ASSE1] a. Interpret parts of an expression such as terms, factors, and coefficients. [ASSE1a] b. Interpret complicated expressions by viewing one or more of their parts as a single entity. [ASSE1b] Example: Interpret P(1+r)^{n} as the product of P and a factor not depending on P. Alabama Alternate Achievement Standards



8 ) Use the structure of an expression to identify ways to rewrite it. [ASSE2] 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}). 

Write expressions in equivalent forms to solve problems. (Quadratic and exponential.)
 

9 ) Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* [ASSE3] a. Factor a quadratic expression to reveal the zeros of the function it defines. [ASSE3a] b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. [ASSE3b] c. Determine a quadratic equation when given its graph or roots. (Alabama) d. Use the properties of exponents to transform expressions for exponential functions. [ASSE3c] Example: The expression 1.15^{t} can be rewritten as (1.15^{1/12})^{12t} ≈ 1.012^{12t} to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Alabama Alternate Achievement Standards


Arithmetic With Polynomials and Rational Expressions
 
Perform arithmetic operations on polynomials. (Linear and quadratic.)
 

10 ) 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. [AAPR1] 

Rewrite rational expressions. (Linear and quadratic denominators.)
 

11 ) (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. [AAPR7] Alabama Alternate Achievement Standards


Creating Equations*
 
Create equations that describe numbers or relationships. (Linear, quadratic, and exponential (integer inputs only); for Standard 14, linear only.)
 

12 ) 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. [ACED1] 


13 ) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2] 


14 ) 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. [ACED3] Example: Represent inequalities describing nutritional and cost constraints on combinations of different foods. 


15 ) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. [ACED4] Example: Rearrange Ohm's law V = IR to highlight resistance R. 

Reasoning With Equations and Inequalities
 
Understand solving equations as a process of reasoning and explain the reasoning. (Master linear; learn as general principle.)
 

16 ) 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. [AREI1] 

Solve equations and inequalities in one variable. (Linear inequalities; literal that are linear in the variables being solved for; quadratics with real solutions.)
 

17 ) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [AREI3] Alabama Alternate Achievement Standards



18 ) Solve quadratic equations in one variable. [AREI4] a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x  p)^{2} = q that has the same solutions. Derive the quadratic formula from this form. [AREI4a] b. Solve quadratic equations by inspection (e.g., for x^{2} = 49), taking square roots, completing the square and the quadratic formula, and factoring as appropriate to the initial form of the equation. [AREI4b] (Alabama) Alabama Alternate Achievement Standards


Solve systems of equations. (Linearlinear and linearquadratic.)
 

19 ) 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. [AREI5] Alabama Alternate Achievement Standards



20 ) Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. [AREI6] Alabama Alternate Achievement Standards



21 ) Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. [AREI7] Example: Find the points of intersection between the line y = 3x and the circle x^{2} + y^{2} = 3. Alabama Alternate Achievement Standards


Represent and solve equations and inequalities graphically. (Linear and exponential; learn as general principle..)
 

22 ) 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). [AREI10] Alabama Alternate Achievement Standards



23 ) Explain why the xcoordinates 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.* [AREI11] Alabama Alternate Achievement Standards



24 ) Graph the solutions to a linear inequality in two variables as a halfplane (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 halfplanes. [AREI12] Alabama Alternate Achievement Standards


FUNCTIONS
 
Interpreting Functions
 
Understand the concept of a function and use function notation. (Learn as general principle; focus on linear and exponential and on arithmetic and geometric sequences.)
 

25 ) 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). [FIF1] Alabama Alternate Achievement Standards



26 ) Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. [FIF2] Alabama Alternate Achievement Standards



27 ) Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. [FIF3] Example: The Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n1) for n ≥ 1. Alabama Alternate Achievement Standards


Interpret functions that arise in applications in terms of the context. (Linear, exponential, and quadratic.)
 

28 ) 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.* [FIF4] Alabama Alternate Achievement Standards



29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [FIF5] Example: If the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Alabama Alternate Achievement Standards



30 ) 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.* [FIF6] Alabama Alternate Achievement Standards


Analyze functions using different representations. (Linear, exponential, quadratic, absolute value, step, and an awareness of piecewisedefined.)
 

31 ) Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [FIF7] a. Graph linear and quadratic functions, and show intercepts, maxima, and minima. [FIF7a] b. Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. [FIF7b] Alabama Alternate Achievement Standards



32 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [FIF8] a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. [FIF8a] b. Use the properties of exponents to interpret expressions for exponential functions. [FIF8b] Example: Identify percent rate of change in functions such as y = (1.02)^{t}, y = (0.97)^{t}, y = (1.01)^{12t}, and y = (1.2)^{t/10}, and classify them as representing exponential growth and decay. Alabama Alternate Achievement Standards



33 ) Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). [FIF9] Example: Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Alabama Alternate Achievement Standards


Building Functions
 
Build a function that models a relationship between two quantities. (For standards 34 and 35, linear, exponential, and quadratic.)
 

34 ) Write a function that describes a relationship between two quantities.* [FBF1] a. Determine an explicit expression, a recursive process, or steps for calculation from a context. [FBF1a] b. Combine standard function types using arithmetic operations. [FBF1b] Example: Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Alabama Alternate Achievement Standards



35 ) Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.* [FBF2] Alabama Alternate Achievement Standards


Build new functions from existing functions. (Linear, exponential, quadratic, and absolute value.)
 

36 ) 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. [FBF3] Alabama Alternate Achievement Standards


Linear, Quadratic, and Exponential Models*
 
Construct and compare linear, quadratic, and exponential models and solve problems.
 

37 ) Distinguish between situations that can be modeled with linear functions and with exponential functions. [FLE1] a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. [FLE1a] b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. [FLE1b] c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [FLE1c] Alabama Alternate Achievement Standards



38 ) Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). [FLE2] Alabama Alternate Achievement Standards



39 ) Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. [FLE3] Alabama Alternate Achievement Standards


Interpret expressions for functions in terms of the situation they model. (Linear and exponential of form f(x) = b^{x} + k.)
 

40 ) Interpret the parameters in a linear or exponential function in terms of a context. [FLE5] Alabama Alternate Achievement Standards


STATISTICS AND PROBABILITY
 
Interpreting Categorical and Quantitative Data
 
Summarize, represent, and interpret data on a single count or measurement variable.
 

41 ) Represent data with plots on the real number line (dot plots, histograms, and box plots). [SID1] Alabama Alternate Achievement Standards



42 ) 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. [SID2] Alabama Alternate Achievement Standards



43 ) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). [SID3] Alabama Alternate Achievement Standards


Summarize, represent, and interpret data on two categorical and quantitative variables. (Linear focus, discuss general principle.)
 

44 ) Summarize categorical data for two categories in twoway 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. [SID5] Alabama Alternate Achievement Standards



45 ) Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. [SID6] a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. [SID6a] b. Informally assess the fit of a function by plotting and analyzing residuals. [SID6b] c. Fit a linear function for a scatter plot that suggests a linear association. [SID6c] Alabama Alternate Achievement Standards


Interpret linear models.
 

46 ) Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. [SID7] Alabama Alternate Achievement Standards


Conditional Probability and the Rules of Probability
 
Understand independence and conditional probability and use them to interpret data. (Link to data from simulations or experiments.)
 

47 ) Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. [SCP2] 