Gizmos for Arizona - Mathematics: Precalculus (High Academic Standards for Students adopted 2018)

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.

RFR: : Reasoning with Functions and Relations

RFR.AF: : Analyze functions

RFR.AF.1: : Interpret parameters of a function defined by an expression in the context of the situation.

RFR.AF.2: : Sketch the graph of a function that models a relationship between two quantities, identifying key features.

RFR.AF.3: : Interpret key features of graphs and tables for a function that models a relationship between two quantities in terms of the quantities.

RFR.AF.4: : Use limits to describe long-range behavior, asymptotic behavior, and points of discontinuity.

RFR.AF.5: : Sketch the graph of all six trigonometric functions, identifying key features.

RFR.AF.b: : Students should have opportunities to… Calculate, interpret, and use average rate of change.

RFR.AF.c: : Students should have opportunities to… Use multiple representations of function models appropriately.

RFR.AF.d: : Students should have opportunities to… Work with different families of functions beyond linear and quadratic functions including but not limited to exponential, logarithmic, rational, polynomial, logistic, radical, and piecewise-defined functions.

RFR.AF.e: : Students should have opportunities to… Graph rational functions including those whose graphs contain horizontal asymptotes, vertical asymptotes, oblique asymptotes, and/or holes.

RFR.AF.f: : Students should have opportunities to… Make sense of radian measure.

RFR.AF.g: : Students should have opportunities to… Use the unit circle as a tool to graph the trig functions in radians and degrees.

RFR.AF.h: : Students should have opportunities to… Develop fluency with the unit circle. Include opportunities beyond the special angles, for example, explain why sin(1.1) > sin(0.3) in radians.

RFR.AF.i: : Students should have opportunities to… Graph sine, cosine, tangent functions in radians and degrees and analyze/explain the characteristics of each.

RFR.BF: : Building Functions

RFR.BF.1: : Model relationships between quantities that require adding, subtracting, multiplying, and/or dividing functions.

RFR.BF.4: : Determine if a function has an inverse. If so, find the inverse. If not, define a restriction on the domain that meets the requirement for invertibility and find the inverse on the restricted domain.

RFR.BF.5: : Interpret the meanings of quantities involving functions and their inverses.

RFR.BF.6: : Verify by analytical methods that one function is the inverse of another.

RFR.BF.a: : Students should have opportunities to… Model real-world situations with the sum, difference, product, or quotient of other function models.

RFR.BF.b: : Students should have opportunities to… Describe relationships of quantities in functions and within a composition of those functions.

RFR.BF.c: : Students should have opportunities to… Find an inverse algebraically, for example given y = f(x) algebraically find x = f^-1(y). Work with exponential and logarithmic functions, and quadratic and square root functions at a minimum.

RFR.BF.e: : Students should have opportunities to… Describe the meaning of f^-1(20) given a function f that takes hours as an input and gives miles as an output.

RFR.IC: : Interpreting Conics

RFR.IC.1: : Model real-world situations which involve conic sections.

RFR.IC.2: : Identify key features of conic sections (foci, directrix, radii, axes, asymptotes, center) graphically and algebraically.

RFR.IC.3: : Sketch a graph of a conic section using its key features.

RFR.IC.4: : Use the key features of a conic section to write its equation.

RFR.IC.5: : Given a quadratic equation of the form ax² + by² + cx + dy + e = 0, determine if the equation is a circle, ellipse, parabola, or hyperbola.

RFR.IC.a: : Students should have opportunities to… Explore conics as loci of points satisfying stipulated conditions.

RFR.IC.b: : Students should have opportunities to… Explore conic sections with technology and manipulatives.

RFR.IC.c: : Students should have opportunities to… Connect the geometric and algebraic relationships of conics.

RFR.IC.d: : Students should have opportunities to… Use the method of completing the square to put the equation of the conic section into standard form.

RFR.ISS: : Interpreting Sequences and Series

RFR.ISS.1: : Model real-world situations involving sequences or series using recursive and/or explicit definitions.

RFR.ISS.2: : Use covariational reasoning to describe sequences and series.

RFR.ISS.a: : Students should have opportunities to… Become fluent in working with arithmetic and geometric sequences and series.

RFR.ISS.b: : Students should have opportunities to… Explore several types of sequences, including but not limited to Fibonacci, telescoping, harmonic, alternating.

RT: : Reasoning with Trigonometry

RT.ETT: : Extending Triangle Trigonometry

RFR.ETT.1: : Model real-world situations involving trigonometry.

RFR.ETT.4: : Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi - x, pi + x, and 2pi - x in terms of their values for x, where x is any real number.

RFR.ETT.5: : Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

RT.ETT.a: : Students should have opportunities to… Go beyond using the mnemonic SohCahToa by interpreting the meaning of the trigonometric ratios as multiplicative comparisons of the appropriate sides of a right triangle. Situations should involve unknown sides and/or angles, as well as periodic functions and their inverses.

RT.RTS: : Reasoning with Trigonometric Structure

RT.RTS.1: : Use the structure of a trigonometric expression to identify ways to rewrite it.

RT.RTS.2: : Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

RT.RTS.a: : Students should have opportunities to… Recognize that a single identity can be manipulated into another identity.

RT.RTS.b: : Students should have opportunities to… Apply the Pythagorean, sum, difference, double angle, and half angle formulas for sine and cosine to reveal and explain properties.

RT.EPE: : Exploring Polar Equations

RT.EPE.a: : Students should have opportunities to… Convert points between polar and rectangular forms.

RT.EPE.b: : Students should have opportunities to… Determine equivalent polar representations for a given point.

RV: : Reasoning with Vectors

RV.EV: : Exploring Vectors

RV.EV.1: : Recognize vector quantities as having both magnitude and direction.

RV.EV.2: : Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.

RV.EV.3: : Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

RV.EV.4: : Solve problems involving velocity and other quantities that can be represented by vectors.

RV.EV.5: : Add and subtract vectors, and multiply a vector by a scalar.

RV.EV.a: : Students should have opportunities to… Distinguish vector quantities from scalar quantities. For example, distinguish the difference between velocity and speed.

RV.EV.b: : Students should have opportunities to… Make sense of operations with vectors.

RM: : Reasoning with Matrices

RM.UM: : Using Matrices

RM.UM.1: : Use matrices to represent and manipulate data.

RM.UM.2: : Use matrix operations to solve problems. Add, subtract, and multiply matrices of appropriate dimensions. Multiply matrices by scalars to produce new matrices.

RM.UM.3: : Find the inverse and determinant of a matrix.

RM.UM.4: : Use matrices to solve systems of linear equations.

RM.UM.a: : Students should have opportunities to… Explore the properties of matrices and their operations.

RM.UM.b: : Students should have opportunities to… Explain that the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

RM.UM.c: : Students should have opportunities to… Explore the roles of the zero matrix, identity matrix, inverse matrix, and the determinant of a matrix.

RM.UM.d: : Students should have opportunities to… Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

RM.UM.e: : Students should have opportunities to… Use the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers.

RM.UM.f: : Students should have opportunities to… Use matrices as a tool. Including but not limited to: producing new vectors from old vectors, creating transformations in the plane, calculating the area of geometric figures.

RM.UM.g: : Students should have opportunities to… Explore matrices and solve problems with and without technology, as appropriate.

PC.MP: : Standards for Mathematical Practices

PC.MP.1: : Make sense of problems and persevere in solving them.

5.1.1: : Mathematically proficient students explain to themselves the meaning of a problem, look for entry points to begin work on the problem, and plan and choose a solution pathway. While engaging in productive struggle to solve a problem, they continually ask themselves, “Does this make sense?' to monitor and evaluate their progress and change course if necessary. Once they have a solution, they look back at the problem to determine if the solution is reasonable and accurate. Mathematically proficient students check their solutions to problems using different methods, approaches, or representations. They also compare and understand different representations of problems and different solution pathways, both their own and those of others.

PC.MP.2: : Reason abstractly and quantitatively.

PC.MP.3: : Construct viable arguments and critique the reasoning of others.

5.3.1: : Mathematically proficient students construct mathematical arguments (explain the reasoning underlying a strategy, solution, or conjecture) using concrete, pictorial, or symbolic referents. Arguments may also rely on definitions, assumptions, previously established results, properties, or structures. Mathematically proficient students make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. Mathematically proficient students present their arguments in the form of representations, actions on those representations, and explanations in words (oral or written). Students critique others by affirming or questioning the reasoning of others. They can listen to or read the reasoning of others, decide whether it makes sense, ask questions to clarify or improve the reasoning, and validate or build on it. Mathematically proficient students can communicate their arguments, compare them to others, and reconsider their own arguments in response to the critiques of others.

PC.MP.4: : Model with mathematics.

PC.MP.5: : Use appropriate tools strategically.

5.5.1: : Mathematically proficient students consider available tools when solving a mathematical problem. They choose tools that are relevant and useful to the problem at hand. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful; recognizing both the insight to be gained and their limitations. Students deepen their understanding of mathematical concepts when using tools to visualize, explore, compare, communicate, make and test predictions, and understand the thinking of others.

PC.MP.6: : Attend to precision.

5.6.1: : Mathematically proficient students clearly communicate to others using appropriate mathematical terminology, and craft explanations that convey their reasoning. When making mathematical arguments about a solution, strategy, or conjecture, they describe mathematical relationships and connect their words clearly to their representations. Mathematically proficient students understand meanings of symbols used in mathematics, calculate accurately and efficiently, label quantities appropriately, and record their work clearly and concisely.

PC.MP.7: : Look for and make use of structure.

5.7.1: : Mathematically proficient students use structure and patterns to assist in making connections among mathematical ideas or concepts when making sense of mathematics. Students recognize and apply general mathematical rules to complex situations. They are able to compose and decompose mathematical ideas and notations into familiar relationships. Mathematically proficient students manage their own progress, stepping back for an overview and shifting perspective when needed.

PC.MP.8: : Look for and express regularity in repeated reasoning.

5.8.1: : Mathematically proficient students look for and describe regularities as they solve multiple related problems. They formulate conjectures about what they notice and communicate observations with precision. While solving problems, students maintain oversight of the process and continually evaluate the reasonableness of their results. This informs and strengthens their understanding of the structure of mathematics which leads to fluency.

Correlation last revised: 3/25/2021

Arizona - Mathematics: Precalculus

High Academic Standards for Students | Adopted: 2018

RFR: : Reasoning with Functions and Relations

RT: : Reasoning with Trigonometry

RV: : Reasoning with Vectors

RM: : Reasoning with Matrices

PC.MP: : Standards for Mathematical Practices