


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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.

Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
2 
Learning Activities: 
2 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

5 ) Define appropriate quantities for the purpose of descriptive modeling. [NQ2]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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




Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Learning Activities: 
1 

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.


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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}).



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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%.



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Learning Activities: 
1 

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]



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Lesson Plans: 
1 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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.


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
2 
Lesson Plans: 
2 

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.



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
3 
Learning Activities: 
2 
Lesson Plans: 
1 

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


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Learning Activities: 
1 

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)



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Lesson Plans: 
1 

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


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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.



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Lesson Plans: 
1 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]




Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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.



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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.


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Learning Activities: 
1 

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]



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Lesson Plans: 
1 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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.


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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.



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Lesson Plans: 
1 

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.


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Lesson Plans: 
1 

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]



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Learning Activities: 
1 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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




Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]


Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
0 

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]



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Learning Activities: 
1 

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



Mathematics (2015) 
Grade(s): 9  12 
Algebra I 
All Resources: 
1 
Learning Activities: 
1 

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]
