Interpret the structure of expressions. A.SSE.1 Interpret expressions that represent a quantity in terms of its context. ★ a. Interpret of an expression, such as terms, factors, and coefficients. Essential Skills and Knowledge Ability to connect experience in Algebra I with vocabulary that explicitly identifies coefficients, terms, and extend to degree, powers (positive and negative), leading coefficients, monomial… to more complicated expressions such as polynomial and rational expressions Ability to use appropriate vocabulary to categorize polynomials and rational expressions 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. Note: This is an overarching standard that has applications in multiple units and multiple courses. Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). Note: This is an overarching standard that has applications in multiple units and multiple courses. Essential Skills and Knowledge Ability to use properties of mathematics to alter the structure of an expression Ability to select and then use an appropriate factoring technique Ability to factor expressions completely over complex numbers Make of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Write expressions in equivalent forms to solve problems. A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ★ Note: Consider extending this standard to infinite geometric series in curricular implementations of this course description. Essential Skills and Knowledge Knowledge of the difference between an infinite and a finite series Ability to apply the formula for the sum of a finite geometric series : Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Perform arithmetic operations on A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of Make of problems and
polynomials. addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Note: Extend to polynomials beyond quadratics Essential Skills and Knowledge Ability to connect experiences from Algebra I with linear and quadratic polynomials to polynomials of higher degree Ability to show that when polynomials are added, subtracted or multiplied that the result is another polynomial persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Understand the relationship between zeros and factors of polynomials. A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). Essential Skills and Knowledge Ability to make connections between factors, roots and evaluating functions Ability to use both long division and synthetic division Ability to use the graph of a polynomial to assist in the efficiency of the process for complicated cases A.APR.3 Identify of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Essential Skills and Knowledge Knowledge of the differences in the end behavior of the graphs as dictated by the leading coefficient and whether the function is even or odd Ability to capture the graphical behavior of polynomial functions which have roots with multiplicity greater than one Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Use polynomial identities to solve problems. Cluster Note: This cluster has many possibilities for optional enrichment, such as relating the example in A.APR.4 to the solution of the system u2+v2=1, v = t(u+1), relating the Pascal triangle property of binomial coefficients to (x+y)n+1 = (x+y)(x+y)n, deriving explicit formulas for the coefficients, or proving the binomial theorem by induction. A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. Essential Skills and Knowledge Knowledge of the process for proving identities Ability to see, use and manipulate the structure in an expression as needed to an identity A.APR.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. Note: The Binomial Theorem can be proved by mathematical induction or by combinatorial argument. Essential Skills and Knowledge Ability to replicate Pascal’s triangle Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
equations as a process of reasoning and explain the reasoning. Essential Skills and Knowledge Ability to connect experience with solving simple equations in one variable to solving equations which require new strategies and additional steps Ability to make connections between the domain of a function and extraneous solutions Ability to identify extraneous solutions persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Represent and solve equations A.REI.11 Explain why the x-coordinates of the points 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 Make sense of problems and
and inequalities graphically. 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. ★ Note: Include combinations of linear, polynomial, rational, radical, absolute value, and exponential functions. Note: This is an overarching standard that will be revisited as each function is studied. Essential Skills and Knowledge Ability to connect experience with solving systems of equations graphically from Algebra I to solving systems that include polynomial, exponential ,rational, root , absolute value and logarithmic functions Ability to show the equality of two functions using multiple representations persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Analyze functions using different representations. F.IF.7 functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ Make sense of problems and persevere in solving them.
Interpret functions that arise in applications in terms of a context (continued from the previous page) Ability to this skill to linear, quadratic, polynomial, root and simple rational functions Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Cluster Note: Focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate. F.IF.7 functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ Make sense of problems and persevere in solving them. Reason abstractly and quantitatively.
Analyze functions using different representations. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Essential Skills and Knowledge Ability to connect experience with graphing linear and quadratic functions from Algebra I to graphing square root, cube root and a variety of piece-wise defined functions Ability to produce a graph of the parent function for each type of function Knowledge of how parameters introduced into a function alter the shape of the graph of the parent function e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Essential Skills and Knowledge Ability to connect experience with graphing linear and quadratic functions from Algebra I to graphing exponential and logarithmic functions Ability to produce a rough graph of the parent function for each type of function Knowledge of how parameters introduced into a function the shape of the graph of the parent function F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Essential Skills and Knowledge Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively.
Analyze functions using different representations. (continued from the previous page) Ability to connect experience with writing linear, quadratic and exponential functions in various forms from Algebra I to writing all functions in various forms Ability to recognize functions in various forms F.IF.9 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. Essential Skills and Knowledge Ability to recognize common attributes of functions from various representations Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. a function that models a relationship between two quantities. F.BF.1 Write a function that describes a relationship between two quantities. ★ 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. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively.
Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Make inferences and justify conclusions from sample surveys, Ability to conduct sample surveys, experiments and observational studies Understanding of the limitations of observational studies that do not allow major conclusions on treatments Ability to recognize and avoid bias S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. Note: For S.IC.4 and 5, focus on the variability of results from experiments—that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness. Essential Skills and Knowledge Ability to informally establish bounds as to when something is statistically significant Ability to conduct simulations and accurately interpret and use the results Ability to use sample means and sample proportions to estimate population values Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics.
sample surveys, experiments, and observational studies. (continued from the previous page) Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use probability to evaluate outcomes of decisions. S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Note: Extend to more complex probability models. Include situations such as those involving quality control, or diagnostic tests that yield both false positive and false negative results. Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. S.MD.7 (+) Analyze decisions and strategies probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Essential Skills and Knowledge Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically.