In this module students analyze and explain precisely the process of solving an equation. Through repeated reasoning, students develop fluency in writing, interpreting, and translating between various forms of linear equations and inequalities and make conjectures about the form that a linear equation might take in a solution to a problem. They reason abstractly and quantitatively by choosing and interpreting units in the context of creating
equations in two variables to represent relationships between quantities. They master the solution of linear equations and apply related solution techniques and the properties of exponents to the creation and solution of simple exponential equations. They learn the terminology specific to polynomials and understand that polynomials form a system analogous to the integers.

 

N.Q .1: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and
interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
N.Q .2: Define appropriate quantities for the purpose of descriptive modeling.
N.Q .3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

 

 

In this module, students reconnect with and deepen their understanding of statistics and probability concepts first introduced in Grades 6, 7, and 8. Students develop a set of tools for understanding and interpreting variability in data, and begin to make more informed decisions from data. They work with data distributions of various shapes, centers, and spreads. Students build on their experience with bivariate quantitative data from Grade 8. This module sets the stage for more extensive work with sampling and inference in later grades.

 

S.ID.1: Represent data with plots on the real number line (dot plots, histograms, and box plots).
S.ID.2: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread
(interquartile range, standard deviation) of two or more different data sets.
S.ID.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible
effects of extreme data points (outliers).

 

 

In earlier grades, students define, evaluate, and compare functions and use them to model relationships between quantities. In this module, students extend their study of functions to include function notation and the concepts of domain and range. They explore many examples of functions and their graphs, focusing on the contrast between linear and exponential functions. They interpret functions given graphically, numerically, symbolically, and verbally; translate between representations; and understand the limitations of various representations.

 

A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
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.
A.REI.11: 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, polynomial, rational, absolute value, exponential, and logarithmic functions.

 

Module 4: Polynomial and Quadratic Expressions, Equations, and Functions

 

In earlier modules, students analyze the process of solving equations and developing fluency in writing, interpreting, and translating between various forms of linear equations (Module 1) and linear and exponential functions (Module 3). These experiences combined with modeling with data (Module 2), set the stage for Module 4. Here students continue to interpret expressions, create equations, rewrite equations and functions in different but equivalent forms, and graph and interpret funcions, but this time using polynomial functions, and more specifically quadratic functions, as well as square root and cube root functions.

 

N.RN.3: Explain why the sum or product of two 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.
A.SSE.1: Interpret expressions that represent a quantity in terms of its context.
A.SSE.2: Use the structure of an expression to identify ways to rewrite it.

 

 

In Module 5, students synthesize what they have learned during the year about functions to select the correct function type in a series of modeling problems. Students no longer have the benefit of a module or lesson title that includes function type to guide them in their choices. Skills and knowledge from the previous modules will support the requirements of this module, including writing, rewriting, comparing, and graphing functions and interpretation of the parameters of an equation. Students must also draw on their study of statistics in Module 2, using graphs and functions to model a context presented with data and/or tables of values. In this module, the modeling cycle is used as the organizing structure, rather than function type.

 

8.F.4: Construct a function to model a linear relationship between two quantities. Determine the rate of change and
initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a
table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it
models, and in terms of its graph or a table of values.
8.F.5: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the
function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a
function that has been described verbally.
N.Q .2: Define appropriate quantities for the purpose of descriptive modeling.