TEACHER GUIDE TO CLARIFICATION
A function is a Rule That Assigns to Each Input Exactly One Output
The “vertical line test” should be avoided because (1) it is too easy to apply without thinking, (2) students do not need an efficient strategy at this point, and (3) it creates misconceptions for later mathematics, when it is useful to think of functions more broadly, such as whether x might be a function of y. “Function machine” pictures are useful for helping students imagine input and output values, with a rule inside the machine by which the output value is determined from the input.
Notice that the standards explicitly call for exploring functions numerically, graphically, verbally, and algebraically (symbolically, with letters). This is sometimes called the “rule of four.”
Explanations and Examples
8.F.1 Students distinguish between functions and non-functions, using equations, graphs, and tables. Non- functions occur when there is more than one y-value is associated with any x-value. Students are not expected to use the function notation f(x) at this level.
In grade 6, students plotted points in all four quadrants of the coordinate plane. They also represented and analyzed quantitative relationships between dependent and independent variables. In Grade 7, students decided whether two quantities are in a proportional relationship. In Grade 8, students begin to call relationships functions when each input is assigned to exactly one output. Also, in Grade 8, students learn that proportional relationships are part of a broader group of linear functions, and they are able to identify whether a relationship is linear. Nonlinear functions are included for comparison. Later, in high school, students use function notation and are able to identify types of nonlinear functions.
Nonlinear functions are included for comparison.
To determine whether a relationship is a function, students should be expected to reason from a context, a graph, or a table, after first being clear which quantity is considered the input and which is the output. When a relationship is not a function, students should produce a counterexample: an “input value” with at least two “output values.” If the relationship is a function, the students should explain how they verified that for each input there was exactly one output.
The vertical line test should be avoided. It is too easy, students should be reasoning and it creates misconceptions for later mathematics.
Students explore functions numerically, graphically, verbally and algebraically.
For example, the rule that takes x as input and gives as output is a function. Using y to stand for the output we can represent this function with the equation y = , and the graph of the equation is the graph of the function. Students are not yet expected to use function notation such as f(x) =
Students are identifying whether or not it is a function – NOT solving functions
For fluency and flexibility in thinking, students need experiences translating among these. In Grade 8, the focus, of course, is on linear functions, and students begin to recognize a linear function from its form y = mx + b. Students also need experiences with nonlinear functions, including functions given by graphs, tables, or verbal descriptions but for which there is no formula for the rule, such as a girl‘s height as a function of her age.
In the elementary grades, students explore number and shape patterns (sequences), and they use rules for finding the next term in the sequence. At this point, students describe sequences both by rules relating one term to the next and also by rules for finding the nth term directly. (In high school, students will call these recursive and explicit formulas.) Students express rules in both words and in symbols. Instruction should focus on additive and multiplicative sequences as well as sequences of square and cubic numbers, considered as areas and volumes of cubes, respectively.
When plotting points and drawing graphs, students should develop the habit of determining, based upon the context, whether it is reasonable to “connect the dots” on the graph. In some contexts, the inputs are discrete, and connecting the dots can be misleading. For example, if a function is used to model the height of a stack of n paper cups, it does not make sense to have 2.3 cups, and thus there will be no ordered pairs between n = 2 and n = 3.
Students need practice determining whether the inputs are discrete or continuous
Kansas Association of Teachers of Mathematics (KATM) Flipbooks. Questions or to send feedback: firstname.lastname@example.org.
Retrieved from Math Flipbooks.
Sample PARCC questions http://parcc.pearson.com/practice-tests/math/ :
When the input to a function is -2, the output is 4. Which statement about this function must be true?
A. An input of -2 has infinitely many possible outputs.
B. An input of -2 has exactly one possible output.
C. An output of 4 has infinitely many inputs
D. An output of 4 has exactly one input.
The answer is B. Answer choice A would not be a function. Answer choices C and D could be true, but don’t have to be true. Answer choice B must be true.
The graph of a nonlinear function is shown on the coordinate plane. In the graph, y is a function of x.
When the input of the function is -4, what is the output of the function?
The answer above is D. When the input is -4, the output (or y value) is 5.
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. (Function notation is not required in Grade 8.)
Party – finding cost for people at a party
Foxes and Rabbits
1. Fill in the x and y values so that the table 1 represents a function. Then fill in the x and y values so that table 2 does not represent a function.
For table 1, students should not repeat any numbers in the x column as this would give multiple outputs for the same input. In table 2, students should repeat a number in the x column so that each input does not have exactly one output.
2. Point A is graphed below. Plot point B so that line segment AB is NOT a function.
Yes, this is a function since there is exactly one output for each input. Students may get confused since each input has the same output. Ask students that if you put the 1 into a function machine, how many outputs?
Input: x values for equation
Output: y values for equation
Both choices B and C are correct.
Point B must have the x-coordinate 2 and any y coordinate. (2, y)
3. Determine if the following describes a function:
Input: Rectangle R
Output: Area of rectangle R
Yes, this is a function. Given a rectangle, there is only one area for that rectangle.
Input: Area of rectangle R
Output: Rectangle R
No, this is not a function. If you are given the area for rectangle R as 8 unit2, then the rectangle could be 1 by 8 or 2 by 4. There is more than one output for the input.
National Library of Virtual Manipulatives: This is an interactive function machine that allows students to guess the output when given an input.
Illustrative Mathematics – Introducing Functions Task: The goal of this task is to motivate the definition of a function by carefully analyzing some different relationships
Math Playground function machine: This is an interactive function machine that allows students to determine the function when given the input and output.
Coherence and Connections: Need to Know
Evidence Statement Text
Understand that a function is a rule that assigns to each input exactly one output.
i) Tasks do not involve the coordinate plane or the “vertical line test.”
ii) Some of functions in tasks are non-numerical.
[Understand that] the graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
i) Functions are limited to those with inputs and outputs in the real numbers.
ii) Most of the tasks require students to graph functions in the coordinate plane or read inputs and outputs from the graph of a function in the coordinate plane.
iii) Some of the tasks require students to tell whether a set of points in the plane represents a function.
Since elementary school, 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 a 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 their earlier experiences with graphs and tables in Grades 6 and 7, students continue a routine of exploring functional relationships algebraically, graphically, numerically in tables and through verbal descriptions.
Common Core Standards Writing Team. (2013, July 2).
Progressions for the Common Core State Standards in Mathematics (draft). Grade 8, High School, Functions*. Tucson, AZ: Institute for Mathematics and Educations, University of Arizona.
PARCC Mathematics Evidence Tables. (2013, April). Retrieved from: