| 5.1 |
Extend the properties of exponents to rational exponents |
5.1.1.B |
Compare and order numbers in the Real Number System by size and/or position on a number line (to include ability to identify equivalent terms). |
2 |
|
1.1.1 |
| 5.1.3.A |
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. |
1 |
N.RN.3 |
1.1.2 |
| 5.2 |
Use properties of rational and irrational numbers. |
|
Explain why the sum or product of two rational numbers is rational; |
1 |
N.RN.3 |
1.1.3 |
| 5.2.2.A |
Explain 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;
|
1 |
N.RN.3 |
1.1.4 |
| |
Justify mathematical procedures and determine how they apply to invented operations using field properties (closure, associative, commutative, distributive, identity and inverse). |
1 |
|
1.1.5 |
| 6.1 |
Reason quantitatively and use units to solve problems |
|
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. |
1 |
N.Q.1 |
1.1.6 |
| 6.1.2.A |
Define appropriate quantities for the purpose of descriptive modeling. |
1 |
N.Q.2 |
1.1.7 |
| 6.1.3.A |
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. |
1 |
N.Q.3 |
1.1.8 |
| 7.1 |
Interpret and model a given context using expressions |
7.1.1.A |
Interpret expressions that represent a quantity in terms of its context. |
1 |
A.SSE.1 |
1.1.9 |
| 7.1.1.B |
Use the structure of an expression to identify ways to rewrite it. |
1 |
A.SSE.2 |
1.1.10 |
| 7.1.2.A |
Interpret parts of an expression, such as terms, factors, and coefficients. |
1 |
A.SSE.1.a |
1.1.11 |
| 7.1.3.A |
Interpret complicated expressions by viewing one or more of their parts as a single entity. |
1 |
A.SSE.1.b |
1.1.12 |
| 7.2 |
Decompose and recompose algebraic expressions using number properties in the context of solving problems |
7.2.1.C |
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* |
2 |
A.SSE.3 |
1.1.13 |
| 7.6 |
Use algebraic properties and inverse operations to justify the steps in solving equations |
7.6.1.A |
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. |
3 |
A.REI.1 |
1.2.1 |
| 7.6.3.A |
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. |
1 |
A.REI.3 |
1.2.2 |
| 7.5 |
Demonstrate that the relationship of two or more variables can be represented as an equation or inequality and can be represented graphically |
7.5.1.A |
Create equations and inequalities in one variable and use them to solve problems. |
2 |
A.CED.1 |
1.2.3 |
| 7.5.2.A |
Create equations in two or more variables to represent relationships between quantities: graph equations on coordinate axes with labels and scales. |
2 |
A.CED.2 |
1.2.4 |
| 7.5.3.A |
Represent constraints by equations or inequalities, and by systems and/or inequalities. |
2 |
A.CED.3 |
1.2.5 |
| 7.9.1A |
STEM Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. |
1 |
A.CED.4 |
1.2.6 |
| 7.9 |
Knows that a 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).
|
|
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. |
2 |
A.REI.11 |
2.2.1 |
| 7.8 |
Solve systems of equations |
7.8.1.A |
Solve systems of linear equations exactly and approximately, focusing on approximately, focusing on pairs of linear equations in pairs of linear equations in two variables. |
1 |
A.REI.6 |
2.2.2 |
| 7.9 |
Knows that a 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).
|
7.9.2A |
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 halfplanes. |
1 |
A.REI.12 |
2.2.2 |
| 7.10 |
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). |
7.10.1.A |
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. |
1 |
F.IF.2 |
2.2.3 |
| 7.10.2.A |
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. |
2 |
F.IF.3 |
2.2.4 |
| 7.10.3.A |
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.* |
2 |
F.IF.4 |
2.2.5 |
| 7.11 |
Describe characteristics of graphs, tables, and equations that model families of functions. |
7.11.1.A |
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. |
2 |
F.IF.5 |
2.2.6 |
| 7.11.1.B |
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. |
1 |
F.IF.7.c |
2.2.7 |
| 7.11.2.A |
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.* |
2 |
F.IF.6 |
2.2.8 |
| 7.11.2.B |
Graph linear and quadratic functions and show intercepts, maxima, and minima. |
1 |
F.IF.7.a |
2.2.9 |
| 7.11.3.B |
Graph square root, cube root, and piecewise‐defined functions, including step functions and absolute value functions. |
1 |
F.IF.7.b |
2.2.10 |
| 7.11.3.C |
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude |
1 |
F.IF.7.e |
2.2.11 |
| |
STEM Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. |
1 |
F.IF.7 |
2.2.12 |
| |
STEM 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.* |
2 |
F.IF.6 |
2.2.13 |
| 7.12 |
Use strategies for interpreting key features of representations. |
7.12.3.A |
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 |
2 |
F.IF.9 |
2.2.14 |
| 7.13 |
Know the difference between a recursive rule and an explicit expression for a function. |
7.13.1.A |
Write a function that describes a relationship between two quantities.* |
2 |
F.BF.1 |
2.2.15 |
| 7.13.1.B |
Determine an explicit expression, a recursive process, or steps for calculation from a context. |
2 |
F.BF.1.a |
2.2.16 |
| 7.8 |
Solve systems of equations |
7.8.1.A |
Solve systems of linear equations exactly and approximately, focusing on approximately, focusing on pairs of linear equations in pairs of linear equations in two variables. |
1 |
A.REI.6 |
3.1.1 |
| 7.8.2.A |
Solve a simple system consisting of a linear equation and quadratic equation in two variables algebraically and graphically. |
1 |
A.REI.7 |
3.1.2 |
| 7.9 |
Knows that a 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).
|
7.9.2A |
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 halfplanes. |
1 |
A.REI.12 |
3.1.3 |
| 7.8 |
Solve systems of equations |
7.8.3.A |
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. |
3 |
A.REI.5 |
3.1.4 |
| |
STEM(+) Represent a system of linear equations as a single matrix equation in a vector variable. |
1 |
A.REI.8 |
3.1.5 |
| |
STEM(+) Represent a system of linear equations as a single matrix equation in a vector variable. |
1 |
A.REI.8 |
3.1.6 |
| 5.1 |
Extend the properties of exponents to rational exponents |
5.1.1.A |
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. |
1 |
N.RN.1 |
3.2.1 |
| 7.6 |
Use algebraic properties and inverse operations to justify the steps in solving equations |
|
STEM Judge the effects of computations with powers and roots on the magnitude of results |
2 |
|
3.2.2 |
| 7.6.2.A |
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. |
1 |
A.REI.2 |
3.2.3 |
| 5.1 |
Extend the properties of exponents to rational exponents |
5.1.2.A |
Rewrite expressions involving radicals and rational exponents using the properties of exponents. |
1 |
N.RN.2 |
3.2.4 |
| |
STEM Select and use an appropriate form of number (integer, fraction, decimal, ratio, percent, exponential, scientific notation, irrational, complex) to solve practical problems involving order, magnitude, measures, labels, locations and scales. |
2 |
|
3.2.5 |
| 7.10 |
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). |
7.10.1.A |
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. |
1 |
F.IF.2 |
3.3.1 |
| 7.10.2.A |
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. |
2 |
F.IF.3 |
3.3.2 |
| 7.10.3.A |
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.* |
2 |
F.IF.4 |
3.3.3 |
| 7.11 |
Describe characteristics of graphs, tables, and equations that model families of functions. |
7.11.1.A |
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. |
2 |
F.IF.5 |
3.3.4 |
| 7.11.1.B |
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. |
1 |
F.IF.7.c |
3.3.5 |
| 7.11.2.A |
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.* |
2 |
F.IF.6 |
3.3.6 |
| 7.11.2.B |
Graph linear and quadratic functions and show intercepts, maxima, and minima. |
1 |
F.IF.7.a |
3.3.7 |
| 7.11.2.C |
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. |
2 |
F.IF.8.a |
3.3.8 |
| 7.11.3.B |
Graph square root, cube root, and piecewise‐defined functions, including step functions and absolute value functions. |
1 |
F.IF.7.b |
3.3.9 |
| 7.11.3.C |
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude |
1 |
F.IF.7.e |
3.3.10 |
| |
STEM Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. |
1 |
F.IF.7 |
3.3.11 |
| |
STEM 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.* |
2 |
F.IF.6 |
3.3.12 |
| 7.12 |
Use strategies for interpreting key features of representations. |
7.12.3.A |
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 |
2 |
F.IF.9 |
3.3.13 |
| 7.5 |
Demonstrate that the relationship of two or more variables can be represented as an equation or inequality and can be represented graphically |
|
STEM Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. |
1 |
A.CED.4 |
4.1.1 |
| 7.16 |
Describe characteristic graph, table, and equation formats for linear, exponential, and quadratic functions. |
7.16.1.A |
Distinguish between situations that can be modeled with linear functions and with exponential functions. |
2 |
F.LE.1 |
4.1.2 |
| 7.16.1.B |
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). |
2 |
F.LE.2 |
4.1.3 |
| 7.16.2.A |
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. |
3 |
F.LE.1.a |
4.1.4 |
| 7.16.2.B |
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. |
2 |
F.LE.3 |
4.1.5 |
| 7.16.3.A |
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. |
2 |
F.LE.1.b |
4.1.6 |
| 7.16.3.B |
For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. |
1 |
F.LE.4 |
4.1.7 |
| |
STEM Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. |
2 |
F.LE.1.c |
4.1.8 |
| |
STEM Interpret the parameters in a linear or exponential function in terms of a context. |
2 |
F.LE.5 |
4.1.9 |
| 7.2 |
Decompose and recompose algebraic expressions using number properties in the context of solving problems |
7.2.2.C |
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. |
1 |
A.SSE.3.b |
4.2.1 |
| 7.7 |
Solve quadratic equations in one variable. |
7.7.1.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. |
2 |
A.REI.4.a |
4.2.2 |
| 7.2 |
Decompose and recompose algebraic expressions using number properties in the context of solving problems |
7.2.3.C |
Use the properties of exponents to transform expressions for exponential functions. |
1 |
A.SSE.3.c |
4.2.3 |
| 7.7 |
Solve quadratic equations in one variable. |
7.7.2.A |
Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. |
1 |
A.REI.4.b |
4.2.4 |
| 7.7.3.A |
Recognize when the quadratic formula gives complex solutions and write them as a+‐ bi for real numbers a and b. |
1 |
A.REI.4.b |
4.2.5 |