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.
2. Use the structure of an expression to identify ways to rewrite it. Write expressions in equivalent forms to solve problems. [Quadratic and exponential]
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.
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.
12. 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.
Solve equations and inequalities in one variable. [Linear inequalities; literal equations that are linear in the variables being
solved for; quadratics with real solutions]
3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
3.1 Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in
context. CA
4. Solve quadratic equations in one variable.
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.
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.
Interpret functions that arise in applications in terms of the context. [Linear, exponential, and quadratic]
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.
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.]
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).
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation
in terms of a context.
3. 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 ≥ 1.
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the
function.
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.
b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of
change in functions such as
y = (1.02)t, y = (0.97), y = (1.01)12t, and y = (1.2)t/10, and classify them as representing exponential growth or decay
Analyze functions using different representations. [Linear, exponential, quadratic, absolute value, step, piecewise-defined]
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology
for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude.
Build a function that models a relationship between two quantities. [For F.BF.1, 2, linear, exponential, and quadratic]
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.
2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and
translate between the two forms.
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.
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
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 by the context. Emphasize linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Summarize, represent, and interpret data on a single count or measurement variable.
1. Represent data with plots on the real number line (dot plots, histograms, and box plots).
2. 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.
3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme
data points (outliers).