### Clarifications

*Clarification 1*: Key features are limited to domain; range; intercepts; slope and end behavior.

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

**Grade:**912

**Strand:**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
- Intercept
- Range
- Slope

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students interpreted the slope and $y$-intercept of a linear equation in two variables. In Algebra I, students compare key features of two or more linear functions. In later courses, students will compare key features of linear and nonlinear functions.- Instruction includes the use of various forms of linear equations. Additionally, linear
functions can be represented as a table of values or graphically.
- Standard Form

Can be described by the equation $A$$x$ + $B$$y$ = $C$, where $A$, $B$ and $C$ are any rational number. - Slope-Intercept Form

Can be described by the equation $y$ = $m$$x$ + $b$, where $m$ is the slope and $b$ is the $y$-intercept. - Point-Slope Form

Can be described by the equation $y$ − $y$_{1}= $m$($x$ − $x$_{1}), where ($x$_{1}, $y$_{1}) are a point on the line and $m$ is the slope of the line.

- Standard Form
- Problem types include comparing linear functions presented in similar forms and in different forms, and comparing more than two linear functions.
- Instruction includes representing domain and range using words, inequality notation and
set-builder notation.
- Words

If the domain is all real numbers, it can be written as “all real numbers” or “any value of $x$, such that $x$ is a real number.” - Inequality notation

If the domain is all values of $x$ greater than 2, it can be represented as $x$ > 2. - Set-builder notation

If the domain is all values of $x$ less than or equal to zero, it can be represented as {$x$|$x$ ≤ 0} and is read as “all values of $x$ such that $x$ is less than or equal to zero.”

- Words

### Common Misconceptions or Errors

- When describing domain or range, students may assign their constraints to the incorrect variable. In these cases, ask reflective questions to help students examine the meaning of the domain and range in the problem.
- Students may also miss the need for compound inequalities when describing domain or range. In these cases, use a graph of the function to point out areas of their constraint that would not make sense in context.

### Strategies to Support Tiered Instruction

- Teacher provides a laminated cue card to aid in the identification of domain and range
restrictions:
- Is the constraint on the independent or dependent variable in the context of the problem?
- Does the constraint restrict the input or output value in the context of the problem?
- Was the constraint shown or highlighted on the $x$- or $y$-axis?
- Teacher provides a chart to show different terminology associated with domain and range.

- Instruction includes opportunities to use a graphic organizer to chart and provide specific
examples of domain and range as compound inequalities.

- Teacher provides a graphic organizer for each of the three forms of linear equations (standard, slope-intercept and point-slope form) that can be co-created to highlight key similarities and differences.

### Instructional Tasks

*Instructional Task 1 (MTR.2.1, MTR.5.1)*

- Callie and her friend Elena are reading through different novels they checked out from their school library last Tuesday. Callie’s progress through her novel can be modeled by the function $p$($d$) = −25$d$ + 318, where $p$($d$) represents the number of pages remaining to be read and $d$ is the number of days since receiving the book. Elena’s progress through her novel is modeled by the graph below.

- Part A. Which student’s novel has more pages to read?
- Part B. Assuming they both continue to read at a constant rate, which student will complete their novel first?

### Instructional Items

*Instructional Item 1*

- Two linear functions, $f$($x$) and $g$($x$), are represented below. Compare the functions by
stating which has a greater $y$-intercept, $x$-intercept and rate of change.

**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 Assessment

## Lesson Plans

## Problem-Solving Tasks

## MFAS Formative Assessments

Students are given two linear functions, one represented by a graph and the other by an equation, and asked to compare their intercepts in the context of a problem.

## Student Resources

## Problem-Solving Tasks

This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

Type: Problem-Solving Task

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Tasks

This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

Type: Problem-Solving Task

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

Type: Problem-Solving Task