Grade 8:
Unit 4: Functions
Approximate Time Frame: 3 weeks
Connections to Previous Learning:
Patterns are sequences, and sequences are functions with a domain consisting of whole numbers. Students build on previous work with proportional
relationships, unit rates and graphing to connect these ideas and understand that the points (𝑥, 𝑦) lie on a non-vertical line. Students also formalize their
previous work with linear relationships by working with functions as they build on their experiences with graphs and tables.
Focus of this Unit:
Students will understand that functions describe relationships and will be able to compare and construct a function. The equation
𝑦 = 𝑚𝑥 + 𝑏 will be interpreted as a straight line, where 𝑚 and 𝑏 are constants. Students learn to recognize linearity in a table when constant differences
between input values produce constant differences between output values, and they can use the constant rate of change and initial value appropriately in a
verbal description of a context. Students will establish a routine of exploring functional relationships algebraically, graphically, and numerically in tables and
verbal descriptions. When using functions to model a linear relationship between quantities, students learn to determine the rate of change of the function
which is the slope of a graph.
Connections to Subsequent Learning:
In Unit 5, students will focus on real-world functions of a single real variable and simultaneous linear functions. In high school, students will develop ways of
thinking that are general and allow them to approach any type of function, work with it, and understand how it behaves, rather than see each function
separately.
From the Grade 8, High School, Functions Progression Document p. 5:
Define, evaluate, and compare functions
Since the elementary grades, students have been describing patterns and expressing relationships between quantities. These ideas become semi-formal in
Grade 8 with the introduction of the concept of function: a rule that assigns to each input exactly one output. Formal language, such as domain and range, and
function notation may be postponed until high school.
Building on experience with graphs and tables in Grades 6 and 7, students establish a routine of exploring functional relationships algebraically, graphically,
numerically in tables, and through verbal descriptions. And to develop flexibility in interpreting and translating among these various representations, students
compare two functions represented in different ways, as illustrated by the task in the margin.
The main focus in Grade 8 is linear functions, those of the form 𝑦 = 𝑚𝑥 + 𝑏, where 𝑚 and 𝑏 are constants. The proof that
𝑦 = 𝑚𝑥 + 𝑏 is also the equation of a line, and hence that the graph of a linear function is a line, is an important pieces of
reasoning connecting algebra with geometry in Grade 8.
In the figure above, the red triangle is the “slope triangle” formed by the vertical intercept and the point on the line with xcoordinate equal to 1. The green triangle is formed from the intercept and a point with arbitrary 𝑥-coordinate. Dilation with
center at the vertical intercept and scale factor x takes the red triangle to the green triangle, because it takes lines to parallel
lines. Thus the green triangle is similar to the red triangle, and so the height of the green triangle is 𝑚𝑥, and the coordinates
of the general point on the triangle are(𝑥, 𝑏 + 𝑚𝑥). Which is to say that the point satisfies the equation 𝑦 = 𝑏 + 𝑚𝑥.
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Grade 8:
Unit 4: Functions
Students learn to recognize linearity in a table: when constant differences between input values produce constant differences between output values. And they
can use the constant rate of change appropriately in a verbal description of a context.
Use functions to model relationships between quantities
When using functions to model linear relationships between quantities, students learn to determine the rate of change of the function, which is the slope of its
graph. They can read (or compute or approximate) the rate of change from a table or a graph, and they can interpret the rate of change in context.
Graphs are ubiquitous in the study of functions, but it is important to distinguish a function from its graph. For example, a function does not have a slope but its
graph can have a slope. (The slope of a vertical line is undefined and the slope of a horizontal line is 0. Either of these cases might be considered “no slope.”
Thus, the phrase “no slope” should be avoided because it is imprecise and unclear.)
Within the class of linear functions, students learn that some are proportional relationships and some are not. Functions of the form 𝑦 = 𝑚𝑥 + 𝑏 are
proportional relationships exactly when 𝑏 = 0, so that 𝑦 is proportional to 𝑥. Graphically, a linear function is a proportional relationship if its graph goes
through the origin.
To understand relationships between quantities, it is often helpful to describe the relationships qualitatively, paying attention to the general shape of the graph
without concern for specific numerical values. The standard approach proceeds from left to right, describing what happens to
the output as the input value increases. For example, pianist Chris Donnelly describes the relationship between creativity and
structure via a graph.
The qualitative description might be as follows: “As the input value (structure) increases, the output (creativity) increases
quickly at first and gradually slowing down. As input (structure) continues to increase, the output (creativity) reaches a
maximum and then starts decreasing, slowly at first, and gradually faster.” Thus, from the graph alone, one can infer Donnelly’s
point that there is an optimal amount of structure that produces maximum creativity. With little structure or with too much
structure, in contrast, creativity is low.
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Grade 8:
Unit 4: Functions
Desired Outcomes
Standard(s):
Understand the connections between proportional relationships, lines, and linear equations.
 8.EE.6 Use similar triangles to explain why the slope 𝑚 is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the
equation 𝑦 = 𝑚𝑥 for a line through the origin and the equation 𝑦 = 𝑚𝑥 + 𝑏 for a line intercepting the vertical axis at 𝑏.
Define, evaluate, and compare functions.
 8.F.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.
 8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the
greater rate of change.
 8.F.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. For
example, the function 𝐴 = 𝑠 2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9),
which are not on a straight line.
 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.
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
 Functional vocabulary will need to be explicitly taught using tactile and virtual tools (ex: software tools, graphs).
 Real world examples will reinforce functional vocabulary.
Understandings: Students will understand that …
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A function is a specific topic of relationship in which each input has a unique output which can be represented in a table.
A function can be represented graphically using ordered pairs that consist of the input and the output of the function in the form (input, output).
A function can be represented with an algebraic rule.
The equation 𝑦 = 𝑚𝑥 + 𝑏 is a straight line and that slope and 𝑦-intercept are critical to solving real problems involving linear relationships.
Changes in varying quantities are often related by patterns which can be used to predict outcomes and solve problems.
Linear functions may be used to represent and generalize real situations.
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Grade 8:
Unit 4: Functions
Essential Questions:
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What defines a function and how can it be represented?
What makes a function linear?
How can linear relationships be modeled and used in real-life situations?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
1. Make sense of problems and persevere in solving them
*2. Reason abstractly and quantitatively. Students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical
expressions, equations, and inequalities, such as slope/rate of change and 𝑦-intercept/initial value. They examine patterns in data and assess the degree of linearity
of functions.
3. Construct viable arguments and critique the reasoning of others.
*4. Model with mathematics. Students model problem situations symbolically, graphically, contextually and with a table. Students form expressions, equations,
or inequalities from real world contexts and connect symbolic and graphical representations. Students solve systems of linear equations and compare properties of
functions provided in different forms.
*5. Use appropriate tools strategically. Students consider available tools (including estimation and technology) when solving a mathematical problem and decide
when certain tools might be helpful. For instance, students may model a linear function with a graph and/or graphing calculator.
*6. Attend to precision. Students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others
and in their own reasoning. Students use appropriate terminology when referring to functions.
7. Look for and make use of structure.
*8. Look for and express regularity in repeated reasoning. Students will use repeated reasoning to understand algorithms and make generalizations about
patterns. They should have multiple opportunities to solve and model problems. Students will notice that the slope of a line and rate of change are the same
value. For example, the use of similar triangles leads students to proportional reasoning when determining the slope.
Prerequisite Skills/Concepts:
Students should already be able to…
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Use independent and dependent variables.
(6.EE.9)
Use characteristics of proportional relationship
and have an informal understanding of slope.
(7.RP.1-3)
Use the coordinate plane.
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Advanced Skills/Concepts:
Some students may be ready to…
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Explain when an equation is not a function for all real values of 𝑥 given certain equations.
Restrict the domain of those same equations so that each equation becomes a function.
Use function notation.
Discuss max/min and local max/min of a function.
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Grade 8:
Knowledge: Students will know…
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All standards in this unit go beyond the
knowledge level.
Unit 4: Functions
Skills: Students will be able to do…
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Verify that a relationship is a function or not. (8.F.1)
Reason from a context, graph, or table after knowing which quantity is the input and which is the
output. (8.F.1)
Represent and compare functions numerically, graphically, verbally and algebraically. (8.F.2)
Interpret equations in 𝑦 = 𝑚𝑥 + 𝑏 form as a linear function. (8.F.3)
Determine whether a function is linear or non-linear. (8.F.3)
Identify and contextualize the rate of change and the initial value from tables, graphs, equations,
or verbal descriptions. (8.F.4)
Construct a model for a linear function. (8.F.4)
Describe the qualities of a function using a graph (e.g., where the function is increasing or
decreasing). (8.F.5)
Sketch a graph when given a verbal description of a situation. (8.F.5)
Use similar triangles to explain why the slope 𝑚 is the same between any two distinct points on a
non-vertical line in the coordinate plane. (8.EE.6)
Derive the equation 𝑦 = 𝑚𝑥 for a line through the origin. (8.EE.6)
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Function
Graph of a function
Domain
Range
Input/output
Ordered pairs/coordinate plane
Slope
Rate of change
Unit rate
Linear/non-linear
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