Essential Questions: How can functions be used and altered to model various situations that occur in life? How can we use representations to determine and explain the underlying patterns? How do tables, graphs, words and symbolic expressions represent the same thing? Is there a time when one form of representation is stronger than another? Lesson Plans and Seeds Lesson Plan B.4: Functions Lesson Seed B.4: Modeling Linear Relationships Download Seeds, Plans, and Resources (zip) Unit Overview Content Emphasis By Clusters in Grade 8 Progressions from Common Core State Standards in Mathematics Send Feedback to MSDE’s Mathematics Team
Lesson Plan B.4: Functions
Lesson Seed B.4: Modeling Linear Relationships
Unit Overview
Content Emphasis By Clusters in Grade 8
Progressions from Common Core State Standards in Mathematics
Send Feedback to MSDE’s Mathematics Team
Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.
For an in-depth discussion of the overarching, “big picture” perspective on student learning of content related to this unit, see:
The Common Core Standards Writing Team (10 September 2011). Progressions for the Common Core State Standards in Mathematics (draft), accessed at: http://ime.math.arizona.edu/progressions/
Vertical Alignment: Vertical curriculum alignment provides two pieces of information: (1) a description of prior learning that should support the learning of the concepts in this unit, and (2) a description of how the concepts studied in this unit will support the learning of additional mathematics.
Possible Organization of Unit Standards: This table identifies additional grade-level standards within a given cluster that support the over-arching unit standards from within the same cluster. The table also provides instructional connections to grade-level standards from outside the cluster.
Overarching Unit Standards
Supporting Standards within the Domain
Instructional Connections outside the Cluster
8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output (function notation is not required in Grade 8).
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8.F.A.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.A.2:Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
8.F.B.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.
8.EE.C.8b: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
8.F.A.3:Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
8.EE.C.7b:Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
8.EE.A.2: Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.G.C.9: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Connections to the Standards for Mathematical Practice: This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.
In this unit, educators should consider implementing learning experiences which provide opportunities for students to:
Content Standards with Essential Skills and Knowledge Statements and Clarifications: The Content Standards and Essential Skills and Knowledge statements shown in this section come directly from the Maryland State Common Core Curriculum Frameworks. Clarifications were added as needed. Educators should be cautioned against perceiving this as a checklist. All information added is intended to help the reader gain a better understanding of the standards.
Standard
Essential Skills and Knowledge
Clarification
function: A function is an association of exactly one value or object from one set of values or objects (the range) with each value or object from another set (the domain). The equation y = 3x + 5 defines y as a function of x, with the domain (x values) specified as the set of all real numbers. Thus, when y is a function of x, a value of y (the range) is associated with each real-number value of x by multiplying it by 3 and adding 5.
graph of a function: The graph of a function is the set of ordered pairs (x, y) for each input (domain) value (x) and its corresponding output (range) value (y). For the equation y = 3x + 5, ordered pairs such as (-5, -10), (-2, -1), (0, 5), (3, 14), and (300, 905), are among an infinite number of ordered pairs.
function notation: The expression "f(x)" means "plug a value for x into a formula f "; the expression does not mean "multiply f and x." In function notation, the "x" in "f(x)" is called "the argument of the function", or just "the argument". So if we are given "f(2)" the "argument" is "2".
We evaluate "f(x)" just as we would evaluate "y".
functional relationship: A functional relationship describes an association of exactly one value or object from one set of values or objects (the range) with each value or object from another set (the domain).
8.F.A.2: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
constant rate of change/slope: Anything that goes up by x number of units for each y value every time is a constant rate of change. A constant rate of change increases or decreases by the same amount for every trial. The slope of a line has a constant rate of change.
y-intercept: The point at which a line crosses the y-axis (or the y-coordinate of that point) is a y-intercept.
linear function: A linear function is a first-degree polynomial function of one variable. This type of function is known as "linear" because it includes the functions whose graph on the coordinate plane is a straight line. a linear function can be written as x → ax + b or f(x)=mx + b
non-linear function: Equations whose graphs are not straight lines are called nonlinear functions. Some nonlinear functions have specific names. A “quadratic function” is nonlinear and has an equation in the form of y = ax2 + bx + c, where a ≠ 0. Another nonlinear function is a “cubic function”. A cubic function has an equation in the form of y = ax3 + bx2 + cx + d, where a ≠ 0.
