Integrated Math 3

Formulate multivariable statistical investigative questions and determine how data can be collected and provide an answer, consider causality and prediction when posing the question.

Understand the issues of bias and confounding variables when collecting data and their impact on interpretation. Understand practices for collecting and handling data, including sensitive information and concerns for privacy and how that may affect data collection.

Create and analyze data sets using technology to clean, sort, or filter data. Summarize, and describe relationships between quantitative variables using data displays including but not limited to scatter plots, regressions, histograms, and boxplots.

Acknowledge the presence of missing data values and understand how missing values may add bias to analysis and interpretation. Examine and discuss competing explanations for data trends observed such as confounding variables. Respond to competing arguments or interpretations of the data of different community groups, paying careful attention to what conclusions the data supports, taking into account correlation versus causation.

Represent data on two quantitative variables on a scatter plot and describe how the variables are related. Solve problems in context by fitting functions to the data and explaining trends and relationships within the data.

1. a.When fitting a function to the data, use given functions or choose a function suggested by the context. Models may include any family of functions.
2. b.Informally assess the fit of a function by plotting and analyzing residuals.
3. c.Fit a linear function for a scatter plot that suggests a linear association.

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

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.

Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

Evaluate reports based on data.

Interpret expressions that represent a quantity in terms of its context in any family of functions.

1. a.Interpret parts of an expression, such as terms, factors, and coefficients.
2. b.Interpret complicated expressions by viewing one or more of their parts as a single entity.

Use the structure of an expression to identify ways to rewrite it.

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.

Graph families of functions from their symbolic forms by hand or with technology, and identify key features including intercepts, intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries, and end behavior.

1. a.including intercepts, maxima, and minima.
2. b.Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
3. c.Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
4. e.Graph logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Functions can include polynomial, radical, rational, logarithms, absolute value, piecewise, trigonometric, and increasingly complex linear, exponential, and quadratic relationships.

Efficiently, flexibly, and accurately, write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

1. a.Factor quadratic functions and completing the square to show zeros, extreme values and symmetry in terms of a context.
2. b.Use properties of exponents to interpret expressions for exponential functions, including non-integer constants for time when applicable in contexts of exponential growth or decay.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. May include any families of functions.

Efficiently, flexibly, and accurately rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. May include any families of functions.

Efficiently, flexibly, and accurately write a function that describes a relationship between two quantities, including linear and exponential relationships, and arithmetic and geometric sequences in context.

1. a.Determine an explicit expression, a recursive process, or steps for calculation from a context.
2. b.Combine standard function types using arithmetic operations.

Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Using a variety of strategies, experiment with cases and illustrate an explanation of the effects on the graph using technology.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Efficiently, flexibly, and accurately demonstrate that polynomials form a system with a structure similar to that of integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

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).

Identify zeros of polynomials when suitable factorizations are available and use the zeros to construct a rough graph of the function defined by the polynomial.

Prove polynomial identities and use them to describe numerical relationships.

Efficiently, flexibly, and accurately rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

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.

Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Solve quadratic equations with real coefficients that have complex solutions.

For any function that models a relationship between two quantities in context of the relationship, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; and symmetries. May include any families of functions, and relative maximums and minimums, end behavior, and periodicity.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes in context. Functions can include polynomial, radical, rational, logarithms, absolute value, piecewise, trigonometric, and increasingly complex linear, exponential, and quadratic relationships.

Calculate and interpret the average rate of change of a non-linear function (represented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Efficiently, flexibly, and accurately create equations and inequalities in one variable that model a situation, and use them to solve problems.

Efficiently, flexibly, and accurately create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Efficiently, flexibly, and accurately rewrite expressions to equivalent forms that are suitable for the purpose of revealing and explaining a specific property about those functions

1. a.Factor a quadratic expression to reveal the zeros of the function it defines.
2. b.Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
3. c.Use the properties of exponents to transform expressions for exponential functions.

Solve quadratic equations in one variable using a process of identifying solutions as appropriate to the initial form of the equation.

1. 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.
2. b.Solve quadratic equations by inspection (e.g., for x²=49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate. Recognize when the quadratic formula gives complex solutions and write them as a+bi for real numbers a and b.

Solve rational and radical equations in one variable and give examples showing how extraneous solutions may arise.

Find inverse functions through focus on relationships between inputs and outputs.

1. a.Solve an equation of the form f(x) = c that has an inverse and write an equation for the inverse.

For exponential models, express as a logarithm the solution to abᶜᵗ = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Using a variety of strategies explain why the x-coordinates of the points where 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 technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, quadratic, polynomial, rational, absolute value, exponential, and logarithmic functions.

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.

Derive the equation of a parabola given a focus and directrix.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

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.