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. For example, we define 51/3 to be the
cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
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.
Reason quantitatively and use units to solve problems. [Foundation for work with expressions, equations and functions]
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.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Interpret the structure of expressions. [Linear, exponential, and quadratic]
1. Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret
P(1 + r)n as the product of P and a factor not depending on P.
3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by
the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
c. Use the properties of exponents to transform expressions for exponential functions. For example, the expression
1.15t can be rewritten as (1.151/12) 12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
Perform arithmetic operations on polynomials. [Linear and quadratic]
1. 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.
Create equations that describe numbers or relationships. [Linear, quadratic, and exponential (integer inputs only);
for A.CED.3 linear only]
1. Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CA
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
3. 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. For example, represent inequalities describing nutritional and cost
constraints on combinations of different foods.
4. 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
Understand solving equations as a process of reasoning and explain the reasoning.
[Master linear; learn as general principle.]
1. 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.
Solve equations and inequalities in one variable. [Linear inequalities; literal equations that are linear in the variables being
solved for; quadratics with real solutions]
4-b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic
formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real numbers a and b
Solve systems of equations. [Linear-linear and linear-quadratic]
5. 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.
11. 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.
4. 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.
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example,
if the function h 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.
6. 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.
Build new functions from existing functions. [Linear, exponential, quadratic, and absolute value; for F.BF.4a, linear only]
3. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations. For 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.
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 expressions for functions in terms of the situation they model.
5. Interpret the parameters in a linear or exponential function in terms of a context. [Linear and exponential of form
f(x) = bx + k]
6. Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity. CA
1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal
factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
2. 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).
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly,
quadratically, or (more generally) as a polynomial function.
Summarize, represent, and interpret data on two categorical and quantitative variables.
[Linear focus; discuss general principle.]
5. 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.
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
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