### Clarifications

*Clarification 1:*Instruction includes referring to the input as the independent variable and the output as the dependent variable.

*Clarification 2: *Within this benchmark, it is the expectation to represent domain and range as a list of numbers or as an inequality.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**8

**Strand:**Functions

**Standard:**Define, evaluate and compare functions.

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Domain
- Function
- Range

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students determined whether two quantities have a proportional relationship by examining a table, graph or written description and they determined the constant of proportionality. In grade 8, students work with linear equations with two variables and begin the introduction of functions. In Algebra 1, students will classify the function type and represent it using function notation.- A mapping diagram consists of a list of $x$-values and their corresponding $y$-values shown with an arrow.
- An example of a mapping diagram can include domain and range values or $x$- and $y$-values.

- An example of a mapping diagram can include domain and range values or $x$- and $y$-values.
- The “vertical line test” should be treated with caution because (1) it allows you to apply a rule without thinking and (2) it may create misconceptions for later mathematics.
- Vocabulary is important in this benchmark as it connects to future learning related to domain and range.
- Students should explain how they verified if the given context was a function or non-function
*(MTR.4.1)*. Students should provide counterexamples to deepen their knowledge of the relationships in functions.- For example, students can be asked to create $x$- and $y$-values that create relations that are functions and non-functions.

- For example, students can be asked to create $x$- and $y$-values that create relations that are functions and non-functions.
- Domain and range can be shown as a list, an inequality or as a verbal description depending on how the relation is given. The inequalities can be represented as inclusive or non-inclusive as determined by the context.
- For example, if a graph represents a real-world context, with non-negative values, with the equation $y$ = 6$x$ + 5, the domain and range can be described as below.
*List*

A list cannot be used to represent this relation because it has infinitely many values.*Inequality*

Domain: $x$ ≥ 0; Range: $y$ ≥ 5*Verbal Description*

The domain is all real numbers that are greater than or equal to zero. The range is all real numbers that are greater than or equal to five.

- For example, for the relation {(4, 12), (5, 15), (6, 18), (7,21), (8,24)}, the domain and range can be described as below.
*List*

Domain: {4, 5, 6, 7, 8}; Range: {12, 15, 18, 21, 24}*Inequality*

An inequality, such as 4 ≤ $x$ ≤ 8, cannot be used to represent this relation because it is based on a discrete set of values.*Verbal Description*

The domain is all whole numbers from four to eight, inclusive. The range is the multiples of three from 12 to 24, inclusive.

- For example, if a graph represents a real-world context, with non-negative values, with the equation $y$ = 6$x$ + 5, the domain and range can be described as below.

### Common Misconceptions or Errors

- Students may invert the terms independent and dependent variable. To address this misconception, focus on the vocabulary and relationship to the input and output.

### Strategies to Support Tiered Instruction

- Teacher reviews vocabulary and the difference between the terms. Once students understand that independent variables represent the input of the relation, they can make sense of real-world problems to accurately identify independent and dependent variables.
- For example, in a scientific experiment one can determine that input as the variable that is controlled by the scientist. So the independent variable is the one that is controlled in the experiment and the dependent is the result of the experiment.

- Teacher creates a matching activity with real-world situations. Students match dependent variable and the independent variable for the situation. Teacher facilitates discussion among students on their reasoning behind their matches from the activity in order to clear up any lingering misconceptions.
- Instruction includes helping students see how no number within the domain is repeated when the relationship is a function.

### Instructional Tasks

*Instructional Task 1*

**(***MTR.4.1*)A relation is shown below where $x$ represents the independent variable and $y$ represents the dependent variable.

- Part A. Create a mapping diagram, table and graph to represent this relation.
- Part B. Determine the domain and range of the relation.
- Part C. Determine if the relation represents a function or does not represent a function and justify your decision.
- Part D. If the relation is not a function, which point could be removed to make it a function? If it is a function, add a point that would no longer make it a function.

### Instructional Items

*Instructional Item 1*

A relation is shown in the table below where $x$ represents the independent variable and ?? represents the dependent variable. Decide whether the table can represent a function or cannot represent a function.

*Instructional Item 2*

Identify the domain and range for the relation {(3, 8), (2, 3), (1, 0), (0, −1), (−1, 0)}.

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Original Student Tutorials

## Problem-Solving Tasks

## Tutorials

## Video/Audio/Animations

## MFAS Formative Assessments

Students are asked decide if one variable is a function of the other in the context of a real-world problem.

Students are shown the graph of a circle and asked to identify a portion of the graph that could be removed so that the remaining portion represents a function.

Students are asked to determine if each of three equations represents a function. Although the task provides equations, in their explanations students can use other representations such as ordered pairs, tables of values or graphs.

Students are asked to determine if relations given by tables and mapping diagrams are functions.

Students are given four graphs and asked to identify which represent functions and to justify their choices.

Students are asked to determine whether or not each of two graphs represent functions.

Students are asked to determine whether or not tables of ordered pairs represent functions.

Students are asked to define the term function and describe any important properties of functions.

Students are asked to determine if each of two sequences is a function and to describe its domain, if it is a function.

## Original Student Tutorials Mathematics - Grades 6-8

Learn how to determine if a relationship is a function in this interactive tutorial that shows you inputs, outputs, equations, graphs and verbal descriptions.

## Original Student Tutorials Mathematics - Grades 9-12

What is a function? Where do we see functions in real life? Explore these questions and more using different contexts in this interactive tutorial.

This is part 1 in a two-part series on functions. Click HERE to open Part 2.

## Student Resources

## Original Student Tutorials

What is a function? Where do we see functions in real life? Explore these questions and more using different contexts in this interactive tutorial.

This is part 1 in a two-part series on functions. Click HERE to open Part 2.

Type: Original Student Tutorial

Learn how to determine if a relationship is a function in this interactive tutorial that shows you inputs, outputs, equations, graphs and verbal descriptions.

Type: Original Student Tutorial

## Problem-Solving Tasks

This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function, which is violated in the table of values of the two populations. Noteworthy is that since the data is a collection of input-output pairs, no verbal description of the function is given, so part of the task is processing what the "rule form" of the proposed functions would look like.

Type: Problem-Solving Task

This task can be played as a game where students have to guess the rule and the instructor gives more and more input output pairs. Giving only three input output pairs might not be enough to clarify the rule. Instructors might consider varying the inputs in, for example, the second table, to provide non-integer entries. A nice variation on the game is to have students who think they found the rule supply input output pairs, and the teachers confirms or denies that they are right. Verbalizing the rule requires precision of language. This task can be used to introduce the idea of a function as a rule that assigns a unique output to every input.

Type: Problem-Solving Task

In this task, the rule of the function is more conceptual. We assign to a year (an input) the total amount of garbage produced in that year (the corresponding output). Even if we didn't know the exact amount for a year, it is clear that there will not be two different amounts of garbage produced in the same year. Thus, this makes sense as a "rule" even though there is no algorithmic way to determine the output for a given input except looking it up in the table.

Type: Problem-Solving Task

## Tutorials

In an equation with 2 variables, we will be able to determine which is the dependent variable, and which is the independent variable.

Type: Tutorial

A graph in Cartesian coordinates may represent a function or may only represent a binary relation. The "vertical line test" is a visual way to determine whether or not a graph represents a function.

Type: Tutorial

## Video/Audio/Animations

Although the domain and codomain of functions can consist of any type of objects, the most common functions encountered in Algebra are real-valued functions of a real variable, whose domain and codomain are the set of real numbers, R.

Type: Video/Audio/Animation

Two sets which are often of primary interest when studying binary relations are the domain and range of the relation.

Type: Video/Audio/Animation

## Parent Resources

## Problem-Solving Tasks

This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function, which is violated in the table of values of the two populations. Noteworthy is that since the data is a collection of input-output pairs, no verbal description of the function is given, so part of the task is processing what the "rule form" of the proposed functions would look like.

Type: Problem-Solving Task

This task can be played as a game where students have to guess the rule and the instructor gives more and more input output pairs. Giving only three input output pairs might not be enough to clarify the rule. Instructors might consider varying the inputs in, for example, the second table, to provide non-integer entries. A nice variation on the game is to have students who think they found the rule supply input output pairs, and the teachers confirms or denies that they are right. Verbalizing the rule requires precision of language. This task can be used to introduce the idea of a function as a rule that assigns a unique output to every input.

Type: Problem-Solving Task

In this task, the rule of the function is more conceptual. We assign to a year (an input) the total amount of garbage produced in that year (the corresponding output). Even if we didn't know the exact amount for a year, it is clear that there will not be two different amounts of garbage produced in the same year. Thus, this makes sense as a "rule" even though there is no algorithmic way to determine the output for a given input except looking it up in the table.

Type: Problem-Solving Task