### Clarifications

*Clarification 1:*Problem types include examples of one-variable linear equations that generate one solution, infinitely many solutions or no solution.

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

**Grade:**8

**Strand:**Algebraic Reasoning

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

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Coefficient
- Linear Equation
- Rational Number

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students wrote and solved two-step equations in one variable within a mathematical or real-world context, where all terms are rational numbers. In grade 8, students solve multi-step linear equations in one variable, with rational number coefficients, including equations with variables on both sides. In Algebra 1, students will write and solve linear equations in one variable in a real-world context, with rational number coefficients.- In this benchmark, students work with linear equations, which is foundational for the work with both linear equations and nonlinear equations throughout all future mathematics courses.
- Instruction includes the use of manipulatives, drawings, models, properties of operations and properties of equality.
- Algebra Tiles
- Balance

- Algebra Tiles
- Problem types involve multi-step problems that require the use of the distributive property, combining like terms, and variables on both sides of the equation.
- Since there are variables on both sides of the equation, instruction includes discovering that one-variable equations can result in three possible solution sets. The possible solutions are one solution, no solution or infinitely many solutions. This benchmark provides a foundation for when students are working with systems of equations and two-variable equations.

### Common Misconceptions or Errors

- Students may incorrectly apply the distributive property by multiplying the monomial to only one of the terms in the parentheses. To address this misconception, emphasize that it is the distributive property of multiplication over addition to help support student understanding.
- Students may incorrectly apply the rules of integer arithmetic as they distribute when working with the operations of negative numbers and applying the distributive property of multiplication over addition.
- Students may incorrectly think that you will always need a variable that equals a constant as a solution. To address this misconception, provide examples that show a constant equal to a variable as a solution, a constant equal to a constant or a non-valid equality statement.

### Strategies to Support Tiered Instruction

- Teacher provides opportunities to use manipulatives to demonstrate using the distributive property as repeated addition of the given expression.
- For example the expression 3($x$ − 4) can be represented as adding ($x$ − 4) three times together.

- For example the expression 3($x$ − 4) can be represented as adding ($x$ − 4) three times together.
- Instruction includes support with relating that if the solution is in the form $x$ = $a$, there is only one solution. If the solution is in the form $a$ = $a$ , there are infinitely many solutions. If the solution is in the form $a$ = $b$, where $a$ and $b$ are different numbers, there are no solutions. Teacher co-creates a graphic organizer with examples of one, no solutions, and infinitely many solutions. Demonstrate using substitution to help students make sense of the solutions.
- Teacher co-creates an anchor chart for multiplying negative integers for students that incorrectly apply the rules of negative integers as they distribute.
- Teacher provides examples for students that need additional support for distributive property by using the area model (like the one shown below).2($x$ + 4)

2($x$ + 4) = ($x$ + 4) + ($x$ + 4) = 2$x$ + 8 - Instruction includes emphasizing that it is the distributive property of multiplication over addition to help support student understanding.

### Instructional Tasks

*Instructional Task 1*

**(MTR.1.1, MTR.4.1)**Part B. How many solutions does the equation, 2($y$ + 3) + 1 = 2(3.5 + $y$) have? Explain your reasoning to another student and justify your answer.

Part C. What do you notice about the equations in Part A and Part B?

*Instructional Task 2*

**(****MTR.1.1****,****MTR.4.1****)**For each equation, state whether there is no solution, one solution, or infinitely many solutions. Explain your reasoning.

### Instructional Items

**Instructional Item 1**Solve for $x$:

a. −3.5(10$x$ − 2) = −176.75

b. 15(2$x$ − 10) + 4$x$ = − 3(15$x$ + 4)

**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

## Perspectives Video: Teaching Idea

## Problem-Solving Tasks

## Tutorial

## MFAS Formative Assessments

Students are asked to identify and explain whether given equations have one solution, no solutions, or infinitely many solutions.

Students are asked to write examples of equations with one solution, no solutions, and infinitely many solutions.

Students are asked to solve a linear equation in one variable with rational coefficients.

Students are asked to solve a linear equation in one variable with rational coefficients.

Students are asked to solve a linear equation in one variable with rational coefficients and variables on both sides of the equation.

## Original Student Tutorials Mathematics - Grades 6-8

Learn how to solve multi-step equations that contain like terms in this interactive tutorial.

This is part one of five in a series on solving multi-step equations.

- [CURRENT TUTORIAL] Part 1: Combining Like Terms
- Click
**HERE**to open Part 2: The Distributive Property - Click
**HERE**to open Part 3: Variables on Both Sides - Click
**HERE**to open Part 4: Putting It All Together - Click
**HERE**to open Part 5: How Many Solutions?

Explore how to solve multi-step equations using the distributive property in this interactive tutorial.

This is part two of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - [CURRENT TUTORIAL] Part 2: The Distributive Property
- Click
**HERE**to open Part 3: Variables on Both Sides - Click
**HERE**to open Part 4: Putting It All Together - Click
**HERE**to open Part 5: How Many Solutions?

Learn how to solve multi-step equations that contain variables on both sides of the equation in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - Click
**HERE**to open Part 2: The Distributive Property - [CURRENT TUTORIAL] Part 3: Variables on Both Sides
- Click
**HERE**to open Part 4: Putting It All Together - Click
**HERE**to open Part 5: How Many Solutions?

Learn alternative methods of solving multi-step equations in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - Click
**HERE**to open Part 2: The Distributive Property - Click
**HERE**to open Part 3: Variables on Both Sides - [CURRENT TUTORIAL] Part 4: Putting It All Together
- Click
**HERE**to open Part 5: How Many Solutions?

Learn how equations can have 1 solution, no solution or infinitely many solutions in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - Click
**HERE**to open Part 2: The Distributive Property - Click
**HERE**to open Part 3: Variables on Both Sides - Click
**HERE**to open Part 4: Putting It All Together - [CURRENT TUTORIAL] Part 5: How Many Solutions?

## Original Student Tutorials Mathematics - Grades 9-12

Learn how to explain the steps used to solve multi-step linear equations and provide reasons to support those steps with this interactive tutorial.

## Student Resources

## Original Student Tutorials

Learn how equations can have 1 solution, no solution or infinitely many solutions in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - Click
**HERE**to open Part 2: The Distributive Property - Click
**HERE**to open Part 3: Variables on Both Sides - Click
**HERE**to open Part 4: Putting It All Together - [CURRENT TUTORIAL] Part 5: How Many Solutions?

Type: Original Student Tutorial

Learn alternative methods of solving multi-step equations in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - Click
**HERE**to open Part 2: The Distributive Property - Click
**HERE**to open Part 3: Variables on Both Sides - [CURRENT TUTORIAL] Part 4: Putting It All Together
- Click
**HERE**to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Learn how to solve multi-step equations that contain variables on both sides of the equation in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - Click
**HERE**to open Part 2: The Distributive Property - [CURRENT TUTORIAL] Part 3: Variables on Both Sides
- Click
**HERE**to open Part 4: Putting It All Together - Click
**HERE**to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Explore how to solve multi-step equations using the distributive property in this interactive tutorial.

This is part two of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - [CURRENT TUTORIAL] Part 2: The Distributive Property
- Click
**HERE**to open Part 3: Variables on Both Sides - Click
**HERE**to open Part 4: Putting It All Together - Click
**HERE**to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Learn how to solve multi-step equations that contain like terms in this interactive tutorial.

This is part one of five in a series on solving multi-step equations.

- [CURRENT TUTORIAL] Part 1: Combining Like Terms
- Click
**HERE**to open Part 2: The Distributive Property - Click
**HERE**to open Part 3: Variables on Both Sides - Click
**HERE**to open Part 4: Putting It All Together - Click
**HERE**to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Learn how to explain the steps used to solve multi-step linear equations and provide reasons to support those steps with this interactive tutorial.

Type: Original Student Tutorial

## Problem-Solving Tasks

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Type: Problem-Solving Task

In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.

Type: Problem-Solving Task

It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.

Type: Problem-Solving Task

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Type: Problem-Solving Task

## Tutorial

This video will show how to solve a consecutive integer problem.

Type: Tutorial

## Parent Resources

## Problem-Solving Tasks

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Type: Problem-Solving Task

In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.

Type: Problem-Solving Task

It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.

Type: Problem-Solving Task

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Type: Problem-Solving Task