The Common Core in Montana

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Skills available for Montana high school math standards

Standards are in black and IXL math skills are in dark green. Hold your mouse over the name of a skill to view a sample question. Click on the name of a skill to practice that skill.

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Montana Common Core Standards: Algebra I Montana Common Core Standards: Algebra I
Montana Common Core Standards: Algebra II Montana Common Core Standards: Algebra II
Montana Common Core Standards: All High School Pathways Montana Common Core Standards: All High School Pathways
Montana Common Core Standards: Geometry Montana Common Core Standards: Geometry
Montana Common Core Standards: Mathematics I Montana Common Core Standards: Mathematics I
Montana Common Core Standards: Mathematics II Montana Common Core Standards: Mathematics II
Montana Common Core Standards: Mathematics III Montana Common Core Standards: Mathematics III

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N Number and Quantity

N-RN The Real Number System
- Extend the properties of exponents to rational exponents.
  - N-RN.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.
    - Evaluate integers raised to positive rational exponents (A1-W.11)
    - Evaluate integers raised to rational exponents (A1-W.12)
  - N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
- Use properties of rational and irrational numbers.
  - N-RN.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.
- Checkpoint opportunity
  - Checkpoint: Radicals and rational exponents (A1-FF.9)
N-Q Quantities
- Reason quantitatively and use units to solve problems.
  - N-Q.1 Use units as a way to understand problems from a variety of contexts (e.g., science, history, and culture), including those of Montana American Indians, 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.
  - N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.
  - N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
- Checkpoint opportunity
  - Checkpoint: Units and quantities (A1-O.6)

N Number and Quantity

N-Q Quantities
- Reason quantitatively and use units to solve problems.
  - N-Q.1 Use units as a way to understand problems from a variety of contexts (e.g., science, history, and culture), including those of Montana American Indians, 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.
  - N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.
  - N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
  - Checkpoint opportunity
    - Checkpoint: Units and quantities (A1-O.6)

N Number and Quantity

N-Q Quantities
- Reason quantitatively and use units to solve problems.
  - N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N Number and Quantity

N-RN The Real Number System
- Extend the properties of exponents to rational exponents.
  - N-RN.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.
    - Evaluate integers raised to positive rational exponents (A1-W.11)
  - N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
- Use properties of rational and irrational numbers.
  - N-RN.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.
  - Checkpoint opportunity
    - Checkpoint: Radicals and rational exponents (A1-FF.9)
N-Q Quantities
- Reason quantitatively and use units to solve problems.
  - N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
N-CN The Complex Number System
- Perform arithmetic operations with complex numbers.
  - N-CN.1 Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.
    - Introduction to complex numbers (A2-I.1)
  - N-CN.2 Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
- Use complex numbers in polynomial identities and equations.
  - N-CN.7 Solve quadratic equations with real coefficients that have complex solutions.
    - Solve a quadratic equation using square roots (A2-K.6)
    - Using the discriminant (A2-K.12)

A Algebra

A-SSE Seeing Structure in Expressions
- Interpret the structure of expressions
  - A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
    - A-SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
    - A-SSE.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.
      - Interpret parts of quadratic expressions: word problems (A1-CC.)
  - A-SSE.2 Use the structure of an expression to identify ways to rewrite it.
- Write expressions in equivalent forms to solve problems
  - A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
    - A-SSE.3.a Factor a quadratic expression to reveal the zeros of the function it defines.
    - A-SSE.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
    - A-SSE.3.c Use the properties of exponents to transform expressions for exponential functions.
      - Evaluate expressions using properties of exponents (A1-W.8)
      - Evaluate an exponential function (A1-Y.1)
- Checkpoint opportunity
  - Checkpoint: Write and interpret equivalent expressions (A1-CC.21)
A-APR Arithmetic with Polynomials and Rational Expressions
- Perform arithmetic operations on polynomials
  - A-APR.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.
- Checkpoint opportunity
  - Checkpoint: Polynomial operations (A1-AA.13)
A-CED Creating Equations
- Create equations that describe numbers or relationships
  - A-CED.1 Create equations and inequalities in one variable and use them to solve problems from a variety of contexts (e.g., science, history, and culture), including those of Montana American Indians.
  - A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  - A-CED.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.
  - A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
    - Rearrange multi-variable equations (A1-I.11)
- Checkpoint opportunity
  - Checkpoint: Represent constraints (A1-V.17)
  - Checkpoint: Problem solving with equations and inequalities (A1-DD.13)
A-REI Reasoning with Equations and Inequalities
- Understand solving equations as a process of reasoning and explain the reasoning
  - A-REI.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
  - A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
  - A-REI.4 Solve quadratic equations in one variable.
    - A-REI.4.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.
      - Complete the square (A1-CC.9)
    - A-REI.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.
- Checkpoint opportunity
  - Checkpoint: Solve linear equations and inequalities (A1-K.16)
  - Checkpoint: Quadratic equations (A1-CC.20)
- Solve systems of equations
  - A-REI.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.
    - Solve a system of equations using elimination (A1-V.10)
    - Solve a system of equations using elimination: word problems (A1-V.11)
  - A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
  - A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
    - Solve a system of linear and quadratic equations by graphing (A1-CC.18)
    - Solve a system of linear and quadratic equations (A1-CC.19)
- Represent and solve equations and inequalities graphically
  - A-REI.10 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).
  - A-REI.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.
  - A-REI.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.
- Checkpoint opportunity
  - Checkpoint: Solve equations using graphs and tables (A1-Q.22)
  - Checkpoint: Systems of equations and inequalities (A1-V.16)

G Geometry

G-CO Congruence
- Experiment with transformations in the plane
  - G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
    - Angle vocabulary (G-C.1)
  - G-CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
  - G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
    - Transformations that carry a polygon onto itself (G-L.13)
  - G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
    - Rotate polygons about a point (G-L.7)
  - G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
- Checkpoint opportunity
- Understand congruence in terms of rigid motions
  - G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
  - G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
  - G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
- Checkpoint opportunity
  - Checkpoint: Rigid motion and congruence (G-L.23)
- Prove geometric theorems
  - G-CO.9 Prove theorems about lines and angles.
  - G-CO.10 Prove theorems about triangles.
  - G-CO.11 Prove theorems about parallelograms.
- Checkpoint opportunity
- Make geometric constructions
  - G-CO.12 Make formal geometric constructions, including those representing Montana American Indians, with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
  - G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
- Checkpoint opportunity
  - Checkpoint: Geometric constructions (G-U.26)
G-SRT Similarity, Right Triangles, and Trigonometry
- Understand similarity in terms of similarity transformations
  - G-SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:
    - G-SRT.1.a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
      - Dilations: graph the image (G-L.15)
      - Dilations and parallel lines (G-L.20)
    - G-SRT.1.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
  - G-SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
  - G-SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
    - Similarity rules for triangles (G-P.8)
- Checkpoint opportunity
  - Checkpoint: Dilations (G-L.24)
  - Checkpoint: Similarity transformations (G-P.17)
- Prove theorems involving similarity
  - G-SRT.4 Prove theorems about triangles.
  - G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
- Checkpoint opportunity
  - Checkpoint: Triangle similarity and congruence (G-P.18)
- Define trigonometric ratios and solve problems involving right triangles
  - G-SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
  - G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
    - Sine and cosine of complementary angles (G-R.8)
  - G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
- Checkpoint opportunity
  - Checkpoint: Right triangle trigonometry (G-R.18)
- Apply trigonometry to general triangles
  - G-SRT.9 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
    - Area of a triangle: sine formula (G-R.16)
  - G-SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems.
  - G-SRT.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
    - Solve a triangle (G-R.15)
- Checkpoint opportunity
  - Checkpoint: Laws of Sines and Cosines (G-R.19)
G-C Circles
- Understand and apply theorems about circles
  - G-C.1 Prove that all circles are similar.
    - Similarity of circles (G-P.10)
  - G-C.2 Identify and describe relationships among inscribed angles, radii, and chords.
  - G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
  - G-C.4 Construct a tangent line from a point outside a given circle to the circle.
    - Construct a tangent line to a circle (G-U.18)
- Checkpoint opportunity
- Find arc lengths and areas of sectors of circles
  - G-C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
- Checkpoint opportunity
  - Checkpoint: Arc length and area of sectors (G-U.25)
G-GPE Expressing Geometric Properties with Equations
- Translate between the geometric description and the equation for a conic section
  - G-GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
  - G-GPE.2 Derive the equation of a parabola given a focus and directrix.
- Checkpoint opportunity
  - Checkpoint: Equations of circles (G-V.19)
  - Checkpoint: Equations of parabolas (G-V.20)
- Use coordinates to prove simple geometric theorems algebraically
  - G-GPE.4 Use coordinates to prove simple geometric theorems algebraically.
    - Classify shapes on the coordinate plane: justify your answer (G-N.13)
    - Determine if a point lies on a circle (G-V.3)
  - G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
  - G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
  - G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
- Checkpoint opportunity
G-GMD Geometric Measurement and Dimension
- Explain volume formulas and use them to solve problems
  - G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
  - G-GMD.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
  - G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
- Checkpoint opportunity
  - Checkpoint: Volume (G-T.14)
- Visualize relationships between two-dimensional and three-dimensional objects
  - G-GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
    - Cross sections of three-dimensional figures (G-H.4)
    - Solids of revolution (G-H.5)
- Checkpoint opportunity
  - Checkpoint: Cross sections and solids of revolution (G-H.6)
G-MG Modeling with Geometry
- Apply geometric concepts in modeling situations
  - G-MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder; modeling a Montana American Indian tipi as a cone).
    - Pythagorean theorem (G-Q.1)
  - G-MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
    - Calculate density, mass, and volume (G-W.9)
  - G-MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
- Checkpoint opportunity
  - Checkpoint: Geometric modeling and design (G-W.10)
  - Checkpoint: Density (G-W.11)

S Statistics and Probability

S-CP Conditional Probability and the Rules of Probability
- Understand independence and conditional probability and use them to interpret data
  - S-CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
    - Theoretical and experimental probability (G-X.1)
    - Outcomes of compound events (G-X.2)
  - S-CP.2 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.
  - S-CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
    - Find conditional probabilities (G-X.11)
    - Independence and conditional probability (G-X.12)
  - S-CP.4 Construct and interpret two-way frequency tables of data including information from Montana American Indian data sources when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
    - Find probabilities using two-way frequency tables (G-X.9)
    - Find conditional probabilities using two-way frequency tables (G-X.13)
  - S-CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
- Use the rules of probability to compute probabilities of compound events in a uniform probability model
  - S-CP.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
    - Find conditional probabilities using two-way frequency tables (G-X.13)
    - Geometric probability (G-X.14)
  - S-CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
    - Find probabilities using the addition rule (G-X.15)
  - S-CP.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
    - Find conditional probabilities (G-X.11)
  - S-CP.9 Use permutations and combinations to compute probabilities of compound events and solve problems.
- Checkpoint opportunity
  - Checkpoint: Understand independence and conditional probability (G-X.16)
  - Checkpoint: Probabilities of compound events (G-X.17)

F Functions

F-IF Interpreting Functions
- Interpret functions that arise in applications in terms of the context
  - F-IF.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.
  - F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  - F-IF.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.
    - Rate of change: graphs (A1-Q.19)
- Analyze functions using different representations
  - F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
    - F-IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
    - F-IF.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
  - F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
    - F-IF.8.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.
    - F-IF.8.b Use the properties of exponents to interpret expressions for exponential functions.
  - F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
  - Checkpoint opportunity
    - Checkpoint: Interpret features of functions (A1)
    - Checkpoint: Graph and analyze functions (A1)
F-BF Building Functions
- Build a function that models a relationship between two quantities
  - F-BF.1 Write a function that describes a relationship between two quantities.
    - F-BF.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
    - F-BF.1.b Combine standard function types using arithmetic operations.
- Build new functions from existing functions
  - F-BF.3 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.
  - F-BF.4 Find inverse functions.
    - F-BF.4.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.
      - Find the inverse of a function (A1-Q.11)
  - Checkpoint opportunity
    - Checkpoint: Build functions (A1)
F-LE Linear, Quadratic, and Exponential Models
- Construct and compare linear, quadratic, and exponential models and solve problems
  - F-LE.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.
    - Compare linear, exponential, and quadratic growth (A1-DD.12)

F Functions

F-IF Interpreting Functions
- Understand the concept of a function and use function notation
  - F-IF.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).
  - F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  - F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
- Interpret functions that arise in applications in terms of the context
  - F-IF.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.
  - F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  - F-IF.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.
- Checkpoint opportunity
  - Checkpoint: Function concepts (A1-Q.20)
  - Checkpoint: Average rate of change (A1-Q.21)
- Analyze functions using different representations
  - F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
    - F-IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
    - F-IF.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
    - F-IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
  - F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
    - F-IF.8.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.
    - F-IF.8.b Use the properties of exponents to interpret expressions for exponential functions.
  - F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
    - Compare linear functions: graphs and equations (A1-T.15)
    - Compare linear functions: tables, graphs, and equations (A1-T.16)
- Checkpoint opportunity
  - Checkpoint: Features of functions (A1-DD.14)
  - Checkpoint: Graph and analyze functions (A1-EE.13)
F-BF Building Functions
- Build a function that models a relationship between two quantities
  - F-BF.1 Write a function that describes a relationship between two quantities.
    - F-BF.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
    - F-BF.1.b Combine standard function types using arithmetic operations.
  - F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations from a variety of contexts (e.g., science, history, and culture, including those of the Montana American Indian), and translate between the two forms.
- Checkpoint opportunity
  - Checkpoint: Sequences (A1-P.13)
- Build new functions from existing functions
  - F-BF.3 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.

The Common Core in Montana

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N Number and Quantity

N-RN The Real Number System

Extend the properties of exponents to rational exponents.

N-RN.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.

N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Use properties of rational and irrational numbers.

N-RN.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.

Checkpoint opportunity

N-Q Quantities

Reason quantitatively and use units to solve problems.

N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Checkpoint opportunity

N Number and Quantity

N-Q Quantities

Reason quantitatively and use units to solve problems.

N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Checkpoint opportunity

N Number and Quantity

N-Q Quantities

Reason quantitatively and use units to solve problems.

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N Number and Quantity

N-RN The Real Number System

Extend the properties of exponents to rational exponents.

N-RN.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.

N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Use properties of rational and irrational numbers.

N-RN.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.

Checkpoint opportunity

N-Q Quantities

Reason quantitatively and use units to solve problems.

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N-CN The Complex Number System

Perform arithmetic operations with complex numbers.

N-CN.1 Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.

N-CN.2 Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Use complex numbers in polynomial identities and equations.

N-CN.7 Solve quadratic equations with real coefficients that have complex solutions.

A Algebra

A-SSE Seeing Structure in Expressions

Interpret the structure of expressions

A-SSE.1 Interpret expressions that represent a quantity in terms of its context.

A-SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.

A-SSE.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.

A-SSE.2 Use the structure of an expression to identify ways to rewrite it.

Write expressions in equivalent forms to solve problems

A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

A-SSE.3.a Factor a quadratic expression to reveal the zeros of the function it defines.

A-SSE.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

A-SSE.3.c Use the properties of exponents to transform expressions for exponential functions.

Checkpoint opportunity

A-APR Arithmetic with Polynomials and Rational Expressions

Perform arithmetic operations on polynomials

A-APR.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.

Checkpoint opportunity

A-CED Creating Equations

Create equations that describe numbers or relationships

A-CED.1 Create equations and inequalities in one variable and use them to solve problems from a variety of contexts (e.g., science, history, and culture), including those of Montana American Indians.

A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-CED.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.

A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Checkpoint opportunity

A-REI Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning

A-REI.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

A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A-REI.4 Solve quadratic equations in one variable.

A-REI.4.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.

Checkpoint opportunity

Solve systems of equations

A-REI.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.

A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Represent and solve equations and inequalities graphically