Create equations that describe numbers or relationships. 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.) Note: to linear and exponential relationships, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Essential Skills and Knowledge Ability to distinguish between linear and exponential relationships multiple representations and then create the appropriate equation/inequality using given information A.CED.2 Create equations in two or more variables to represent relationships between quantities; equations on coordinate axes with labels and scales. Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Essential Skills and Knowledge Ability to distinguish between linear and exponential relationships given multiple representations Ability to determine unknown parameters needed to create an equation that accurately models a given situation 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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. (SC – Algebra I) Note: Limit to linear equations and inequalities. Essential Skills and Knowledge Ability to distinguish between a mathematical solution and a contextual solution A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Note: to formulas which are linear in the variable of interest. Essential Skills and Knowledge Ability to recognize/create equivalent forms of literal equations 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.
Note: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of functions at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear and exponential functions. Essential Skills and Knowledge Ability to determine if a relation is a function Ability to identify the domain and range of a function from multiple representations Ability to use of function notation Knowledge of and ability to the vertical line test 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. Essential Skills and Knowledge Ability to make connections between context and algebraic representations which use function notation F.IF.3 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. Note: Draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions. Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. quantitatively. Construct viable arguments and critique the reasoning of others. with mathematics. Use appropriate strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Interpret functions that arise in applications in terms of a context. F.IF.4 For a function that models a relationship between two quantities, interpret key features of the graph and the table in terms of the quantities, and sketch the graph 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. ★ Make sense of problems and persevere in solving them. Reason abstractly and quantitatively.
Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Create equations that describe numbers or relationships. Cluster Note: Extend work on linear and exponential equations in Unit 1 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. Essential Skills and Knowledge Ability to distinguish between linear, quadratic and exponential relationships given the verbal, numeric and/or graphic representations A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Essential Skills and Knowledge Ability to distinguish between linear, quadratic and exponential relationships given numeric, or verbal representations Ability to determine unknown parameters needed to create an equation that accurately models a given situation Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. with mathematics. Use appropriate tools strategically.
Ability to solve quadratic equations using various methods and recognize the most efficient method Ability to use the value of the discriminant to determine if a quadratic equation has one double solution, two unique solutions or no real solutions Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Solve systems of equations. A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. Note: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. For example, finding the intersections between to the point on the unit circle, corresponding to the Pythagorean triple 32+42=52. Essential Skills and Knowledge Knowledge of the algebraic and graphic representations of quadratic relations as well as quadratic functions 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.
takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. ★ Essential Skills and Knowledge Ability to connect experiences with linear and exponential functions from Unit 2 of this course to quadratic models Ability to describe the restrictions on the domain of a function based on real world context Ability to recognize and use alternate vocabulary for domain and such as input/output or independent/dependent 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. ★ Essential Skills and Knowledge Knowledge that the rate of change of a function can be positive, negative or zero Ability to identify the rate of change from multiple representations Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Analyze functions using different representations. Cluster Note: This unit, and in particular in F.IF.8b, extends the work begun in Unit 2 on exponential functions with integer exponents. Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic equation can be factored. 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. ★ a. linear and quadratic functions and show intercepts, maxima, and minima. Essential Skills and Knowledge Ability to connect experience with graphing linear functions from Unit 2 of this course to include quadratic functions b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Note: Compare and contrast absolute value, step and piecewise defined functions with linear, quadratic, and exponential 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.