SII.MP.1.a: Explain the meaning of a problem and look for entry points to its solution. Analyze givens, constraints, relationships, and goals. Make conjectures about the form and meaning of the solution, plan a solution pathway, and continually monitor progress asking, “Does this make sense?” Consider analogous problems, make connections between multiple representations, identify the correspondence between different approaches, look for trends, and transform algebraic expressions to highlight meaningful mathematics. Check answers to problems using a different method.
SII.MP.3.a: Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Make conjectures and build a logical progression of statements to explore the truth of their conjectures. Justify conclusions and communicate them to others. Respond to the arguments of others by listening, asking clarifying questions, and critiquing the reasoning of others.
SII.MP.4.a: Apply mathematics to solve problems arising in everyday life, society, and the workplace. Make assumptions and approximations, identifying important quantities to construct a mathematical model. Routinely interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
SII.MP.7.a: Look closely at mathematical relationships to identify the underlying structure by recognizing a simple structure within a more complicated structure. See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.
SII.MP.8.a: Notice if reasoning is repeated, and look for both generalizations and shortcuts. Evaluate the reasonableness of intermediate results by maintaining oversight of the process while attending to the details.
(Framing Text): 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.
(Framing Text): Use properties of rational and irrational numbers.
N.RN.3: Explain why sums and products of rational numbers are 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. Connect to physical situations (e.g., finding the perimeter of a square of area 2).
(Framing Text): 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.
(Framing Text): Use complex numbers in polynomial identities and equations.
N.CN.7: Solve quadratic equations with real coefficients that have complex solutions.
(Framing Text): Interpret the structure of expressions.
A.SSE.1: Interpret quadratic and exponential 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 increasingly more complex expressions by viewing one or more of their parts as a single entity. Exponents are extended from the integer exponents to rational exponents focusing on those that represent square or cube roots.
A.SSE.2: Use the structure of an expression to identify ways to rewrite it.
(Framing Text): Write expressions in equivalent forms to solve problems, balancing conceptual understanding and procedural fluency in work with equivalent expressions.
A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. For example, development of skill in factoring and completing the square goes hand in hand with understanding what different forms of a quadratic expression reveal.
A.SSE.3.a: Factor a quadratic expression to reveal the zeros of the function it defines.
A.SSE.3.c: Use the properties of exponents to transform expressions for exponential functions.
(Framing Text): Perform arithmetic operations on polynomials. Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x.
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.
(Framing Text): Create equations that describe numbers or relationships. Extend work on linear and exponential equations to quadratic equations.
A.CED.1: 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.
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.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations; extend to formulas involving squared variables.
(Framing Text): Solve equations and inequalities in one variable.
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.
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.
(Framing Text): Interpret quadratic functions that arise in applications in terms of a context.
F.IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Focus on quadratic functions; compare with linear and exponential functions.
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.
(Framing Text): 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 piecewise-defined functions and absolute value functions. Compare and contrast absolute value and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise-defined 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). Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic equation can be factored.
(Framing Text): Build a function that models a relationship between two quantities.
F.BF.1: Write a quadratic or exponential 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. 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.
(Framing Text): 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. Focus on quadratic functions and consider including absolute value functions. 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.
(Framing Text): 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 and exponential growth to quadratic growth.
(Framing Text): Prove and apply trigonometric identities. Limit θ to angles between 0 and 90 degrees. Connect with the Pythagorean Theorem and the distance formula.
F.TF.8: Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin (θ), cos (θ), or tan (θ), given sin (θ), cos (θ), or tan (θ), and the quadrant of the angle.
(Framing Text): Prove geometric theorems. Encourage multiple ways of writing proofs, such as narrative paragraphs, flow diagrams, two-column format, and diagrams without words. Focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.
G.CO.9: Prove theorems about lines and angles.
G.CO.10: Prove theorems about triangles.
G.CO.11: Prove theorems about parallelograms.
(Framing Text): 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.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 whether 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.
(Framing Text): 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.
(Framing Text): 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.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
(Framing Text): Understand and apply theorems about circles.
G.C.2: Identify and describe relationships among inscribed angles, radii, and chords.
(Framing Text): 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.
(Framing Text): 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. Informal arguments for area formulas can make use of the way in which area scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
G.GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Informal arguments for volume formulas can make use of the way in which volume scale under similarity transformations: when one figure results from another by applying a similarity transformation, volumes of solid figures scale by k³ under a similarity transformation with scale factor k.
(Framing Text): Summarize, represent, and interpret data on two categorical or quantitative variables.
S.ID.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 condition reltive frequencies). Recognize possible associations and trends in the date.
(Framing Text): 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”).
(Framing Text): 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.
Correlation last revised: 9/24/2019