Syllabus

Course Description

In this course, students are expected to apply mathematics in meaningful ways to solve problems that arise in the workplace, society, and everyday life through the process of modeling. Mathematical modeling involves creating appropriate equations, graphs, functions, or other mathematical representations to analyze real-world situations and answer questions. The use of technological tools, such as hand-held graphing calculators, is important in creating and analyzing mathematical representations used in the modeling process and will be used during instruction and assessment. Technology will not be limited to hand-held graphing calculators. Students should use a variety of technologies, such as graphing utilities, spreadsheets, and computer algebra systems, to solve problems and master standards in this course.

Scope and Sequence

Name of Unit: Expressions Number of Days: 10 days

The student will:

A1.AAPR.1* Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. (Limit to linear; quadratic.)

A1.NRNS.1* Rewrite expressions involving simple radicals and rational exponents in different forms.

A1.NRNS.2* Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms.

A1.NRNS.3 Explain why the sum or product of 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.

A1.ASE.1* Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.)

A1.ASE.2* Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.

A1.NQ.1* Use units of measurement to guide the solution of multi-step tasks. Choose and interpret appropriate labels, units, and scales when constructing graphs and other data displays.

A1.NQ.2* Label and define appropriate quantities in descriptive modeling contexts.

Name of Unit: Equations and Inequalities Number of Days: 10 days

The student will:

A1.ACE.1* Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. (Limit to linear; quadratic; exponential with integer exponents.)

A1.ACE.4* Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines

A1.NQ.1* Use units of measurement to guide the solution of multi-step tasks. Choose and interpret appropriate labels, units, and scales when constructing graphs and other data displays.

A1.NQ.2* Label and define appropriate quantities in descriptive modeling contexts.

A1.NQ.3* Choose a level of accuracy appropriate to limitations on measurement when reporting quantities in context.

A1.AREI.1* Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original.

A1.AREI.3* Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A1.AREI.11* Solve an equation of the form 𝑓(𝑥) = 𝑔(𝑥) graphically by identifying the 𝑥-coordinate(s) of the point(s) of intersection of the graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥). (Limit to linear; quadratic; exponential.)

Name of Unit: Introduction of Functions Number of Days: 10 days

The student will:

A1.ACE.2* Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.)

A1.FBF.3* Describe the effect of the transformations 𝑘(𝑥), 𝑓(𝑥) + 𝑘, 𝑓(𝑥 + 𝑘), and combinations of such transformations on the graph of 𝑦 = 𝑓(𝑥) for any real number 𝑘. Find the value of 𝑘 given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.)

A1.FIF.2* Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation.

A1.FIF.4* Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.)

A1.FIF.5* Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. (Limit to linear; quadratic; exponential.)

A1.FIF.6* Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. (Limit to linear; quadratic; exponential.)

A1.FIF.7* Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form 𝑦 = 𝑎𝑥 + 𝑘.)

A1.SPID.6* Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and use the function to solve problems in the context of the data.

A1.SPID.7* Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem.

A1.SPID.8* Using technology, compute and interpret the correlation coefficient of a linear fit.

Name of Unit: Linear Equations, Functions and Inequalities Number of Days: 20 days

The student will:

A1.ACE.4* Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines.

A1.AREI.5 Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation.

A1.AREI.6* Solve systems of linear equations algebraically and graphically focusing on pairs of linear equations in two variables.

a. Solve systems of linear equations using the

substitution method.

b. Solve systems of linear equations using

Linear combination.

A1.AREI.10* Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.

A1.AREI.12* Graph the solutions to a linear inequality in two variables.

Name of Unit: Quadratic Equations and Functions Number of Days: 20 days

The student will:

20 days

A1.AREI.4* Solve mathematical and real-world problems involving quadratic equations in one variable.

a. Use the method of completing the square to

transform any quadratic equation in 𝑥𝑥 into an

equation of the form (𝑥 − ℎ)² = 𝑘 that has the

same solutions. Derive the quadratic formula

from this form.

b. Solve quadratic equations by inspection, 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. (Limit to noncomplex roots.)

A1.ASE.2* Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.

A1.ASE.3* Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Find the zeros of a quadratic function by

rewriting it in equivalent factored form and

explain the connection between the zeros of the

function, its linear factors, the x-intercepts of its

graph, and the solutions to the corresponding

quadratic equation.

A1.FIF.1* Extend previous knowledge of a function to apply to general behavior and features of a function.

a. 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.

b. Represent a function using function notation and

explain that 𝑓(𝑥) denotes the output of

function 𝑓 that corresponds to the input 𝑥.

c. Understand that the graph of a function labeled

as 𝑓 is the set of all ordered pairs (𝑥, 𝑦) that

satisfy the equation 𝑦 = 𝑓(𝑥).

A1.FIF.5* Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. (Limit to linear; quadratic; exponential.)

A1.FIF.8* Translate between different but equivalent forms of a function equation to reveal and explain different properties of the function. (Limit to linear; quadratic; exponential.)

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.

A1.FIF.9* Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. (Limit to linear; quadratic; exponential.)

A1.NQ.2* Label and define appropriate quantities in descriptive modeling contexts.

Name of Unit: Exponential Functions Number of Days: 10 days

The student will:

A1.FBF.3* Describe the effect of the transformations 𝑘(𝑥), 𝑓(𝑥) + 𝑘, 𝑓(𝑥 + 𝑘), and combinations of such transformations on the graph of 𝑦= 𝑓(𝑥) for any real number 𝑘. Find the value of 𝑘 given graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.)

A1.FIF.5* Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. (Limit to linear; quadratic; exponential.)

A1.FIF.9* Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. (Limit to linear; quadratic; exponential.)

A1.FLQE.1* Distinguish between situations that can be modeled with linear functions or exponential functions by recognizing situations in which one quantity changes at a constant rate per unit interval as opposed to those in which a quantity changes by a constant percent rate per unit interval.

a. Prove that linear functions grow by equal

differences over equal intervals and that

exponential functions grow by equal factors over

equal intervals.

A1.FLQE.2* Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables. (Limit to linear; exponential.)

A1.FLQE.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.

A1.FLQE.5* Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.)