# MA.8.NSO.1.7

Solve multi-step mathematical and real-world problems involving the order of operations with rational numbers including exponents and radicals.

### Examples

The expression is equivalent to which is equivalent towhich is equivalent to .

### Clarifications

Clarification 1: Multi-step expressions are limited to 6 or fewer steps.

Clarification 2: Within this benchmark, the expectation is to simplify radicals by factoring square roots of perfect squares up to 225 and cube roots of perfect cubes from -125 to 125.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Number Sense and Operations
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Exponents
• Rational Numbers

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students solved mathematical problems using multi-step order of operations with rational numbers including whole-number exponents and absolute value. In grade 8, students continue to solve multi-step problems involving the order of operations with rational numbers but including integer exponents and radicals. In Algebra 1, students will solve problems with numerical radicals.
• Instruction includes providing a structure for students to track steps as they work through problems (MTR.5.1). Students should show the equivalence from one step to another to further their understanding.
• Avoid mnemonics, such as PEMDAS, that do not account for other grouping symbols and do not exercise proper number sense that allows for calculating accurately in a different order.
• Instruction includes the use of technology to help emphasize the proper use of grouping symbols for order of operations.
• Students should have experience using technology with radicals, decimals and fractions as they occur in the real world. This experience will help to students working with irrational numbers in this grade level.

### Common Misconceptions or Errors

• Students may confuse square roots with cube roots.
• Some students may incorrectly apply the order of operations. To address this misconception, be sure to review operations with rational numbers and order of operations.
• Students may incorrectly perform operations with the numbers in the problem based on what has recently been taught, rather than what is most appropriate for a solution. To address this misconception, have students estimate or predict solutions prior to solving and then compare those predictions to their actual solution to see if it is reasonable (MTR.6.1).
• Students may incorrectly oversimplify a problem by circling the numbers, underlining the question, boxing in key words, and eliminating important contextual information that may seem unimportant. This process can cause students to not be able to comprehend the context or the situation (MTR.2.1, MTR.4.1, MTR.5.1, MTR.7.1). Teachers and students should engage in questions such as:
• What do you know from the problem?
• What is the problem asking you to find?
• Are you putting groups together? Taking groups apart? Or both?
• Are the groups you are working with the same sizes or different sizes?
• Can you create a visual model to help you understand or see patterns in your problem?”

### Strategies to Support Tiered Instruction

• Teacher provides opportunities for students to comprehend the context or situation by engaging in questions (laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
• What do you know from the problem?
• What is the problem asking you to find?
• Can you create a visual model to help you understand or see patterns in your problem?
• Instruction includes the use of colors to highlight each step of the process used to evaluate an expression.
• For example, when evaluating (−$\frac{\text{1}}{\text{3}}$)² − $\sqrt[3]{2² + 4}$ students can first highlight the grouping with any exponents, roots or parenthesis: (−$\frac{\text{1}}{\text{3}}$$\sqrt[3]{2² + 4}$. Then, students can determine any order of operations within each of those larger groupings. Students should see that within the cube root, they can perform 2² + 4 and that they can perform (−$\frac{\text{1}}{\text{3}}$. Students could have the expression $\frac{\text{1}}{\text{9}}$$\sqrt[3]{8}$, and then perform $\sqrt[3]{8}$ to obtain $\frac{\text{1}}{\text{9}}$ − 2 which is equivalent to −$\frac{\text{17}}{\text{9}}$.
• Instruction includes the use a three-read strategy. Students read the problem three different times, each with a different purpose (laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
• First, read the problem with the purpose of answering the question: What is the problem, context, or story about?
• Second, read the problem with the purpose of answering the question: What are we trying to find out?
• Third, read the problem with the purpose of answering the question: What information is important in the problem?
• Teacher has students estimate or predict solutions prior to solving and then compare those predictions to their actual solution to see if it is reasonable (MTR.6.1).

The Dotson’s family was designing their backyard to be a peaceful sanctuary with areas
dedicated to working out, a swimming pool and a gazebo. Each space is a square design having the same size. The total backyard area is 600 square feet. The Dotson’s want to fence the outside of their property but will not fence what is up against the house. The diagram below shows the layout of the backyard.
• Part A. How much fencing, in feet, would the Dotson’s need to purchase to fence in the property?

• Part B. The Dotson’s went to Fence2Fence and found the following options for purchase:
• 3$\frac{\text{1}}{\text{2}}$ feet ×6 feet Western Red Cedar Gothic Fence Panels for \$60.05
• 3$\frac{\text{1}}{\text{2}}$ feet ×8 feet Western Red Cedar Essentials Fence Panels for \$88.66
Which option is the better value? Why?

### Instructional Items

Instructional Item 1
Calculate the value of the expression given.
$\sqrt[3]{27}$ − 1.4 ($\sqrt{3² - 5}$)

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

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.8.NSO.1.AP.7: Use tools to solve multi-step mathematical problems, with four or fewer steps, involving the order of operations with rational numbers including exponents and perfect squares and/or square roots.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessment

Dimensions Needed:

Students are asked to solve problems involving square roots and cube roots.

Type: Formative Assessment

## Lesson Plans

Deriving and Applying the Law of Sines:

Students will be introduced to a derivation of the Law of Sines and apply the Law of Sines to solve triangles.

Type: Lesson Plan

Changes are Coming to System of Equations:

Use as a follow up lesson to solving systems of equations graphically. Students will explore graphs of systems to see how manipulating the equations affects the solutions (if at all).

Type: Lesson Plan

Space Equations:

In this lesson, students model the orbit of a satellite and the trajectory of a missile with a system of equations. They solve the equations both graphically and algebraically.

Type: Lesson Plan

The Laws of Sine and Cosine:

In this lesson, students determine when to use the Law of Cosine and/or the Law of Sine to calculate the measures of sides and angles of non-right triangles.

Type: Lesson Plan

Pythagorean Perspective:

This lesson serves as an introductory lesson on the Pythagorean Theorem and its converse. It has a hands-on discovery component. This lesson includes worksheets that are practical for individual or cooperative learning strategies. The worksheets contain prior knowledge exercises, practice exercises and a summative assignment.

Type: Lesson Plan

For a given circle m and point A in a coordinate plane, students will be able to show whether this given point A lies on the circumference of the given circle m using the Pythagorean Theorem. Subsequently, this can be used to prove that an airplane lies within or outside the radar's range with a given radius of detection.

Type: Lesson Plan

Vertical Angles: Proof and Problem-Solving:

Students will explore the relationship between vertical angles and prove the Vertical Angle Theorem. They will use vertical angle relationships to calculate other angle measurements.

Type: Lesson Plan

How Do You Measure the Immeasurable?:

Students will use similar triangles to determine inaccessible measurements. Examples include exploring dangerous caves and discovering craters on Mars.

Type: Lesson Plan

Triangles: To B or not to B?:

Students will explore triangle inequalities that exist between side lengths with physical models of segments. They will determine when a triangle can/cannot be created with given side lengths and a range of lengths that can create a triangle.

Type: Lesson Plan

What's the Point?:

Students will algebraically find the missing coordinates to create a specified quadrilateral using theorems that represent them, and then algebraically prove their coordinates are correct.

Note: This is not an introductory lesson for this standard.

Type: Lesson Plan

Parallel Lines:

Students will prove that alternate interior angles and corresponding angles are congruent given two parallel lines and a traversal. Students will use GeoGebra to explore real-world images to prove their line segments are parallel.

Type: Lesson Plan

Following the Law of Sine:

This lesson introduces the law of sine. It is designed to give students practice in using the law to guide understanding. The summative assessment requires students to use the law of sine to plan a city project.

Type: Lesson Plan

Operating with Exponents!:

Students will participate in a gallery walk in which they observe patterns in algebraic expressions. Students will apply the properties of integer exponents to simplify expressions.

Type: Lesson Plan

Stand Up for Negative Exponents:

This low-tech lesson will have students stand up holding different exponent cards. This will help them write and justify an equivalent expression and see the pattern for expressions with the same base and descending exponents. What happens as you change from 2 to the fourth power to 2 to the third power; 2 to the second power; and so forth? This is an introductory lesson to two of the properties of exponents: and

Type: Lesson Plan

Math Is Exponentially Fun!:

The students will informally learn the rules for exponents: product of powers, powers of powers, zero and negative exponents. The activities provide the teacher with a progression of steps that help lead students to determine results without knowing the rules formally. The closing activity is hands-on to help reinforce all rules.

Type: Lesson Plan

Triangle Inequality Investigation:

Students use hands-on materials to understand that only certain combinations of lengths will create closed triangles.

Type: Lesson Plan

Method to My Mathness:

In this lesson, students will complete proof tables to justify the steps taken to solve multi-step equations. Justifications include mathematical properties and definitions..

Type: Lesson Plan

The Variable Stands Alone:

Students will practice and create problems solving linear equations that involve one solution, no solution, infinitely many solutions. There will be class discussion so students can verbalize their thoughts. In addition, students will create their own real-world problems that can be used for the next day’s extension exercise.

Type: Lesson Plan

Alas, Poor Pythagoras, I Knew You Well! #2:

Using different activities, students will find real life uses for the Pythagorean Theorem.

Type: Lesson Plan

Exponential Chips:

In this lesson students will learn the properties of integer exponents and how to apply them to multiplication and division. Students will have the opportunity to work with concrete manipulatives to create an understanding of these properties and then apply them abstractly. The students will also develop the understanding of the value of any integer with a zero exponent.

Type: Lesson Plan

## Original Student Tutorial

Square Root Part 3: Simplifying Radicals:

Learn how to simplify radicals in this interactive tutorial.

Type: Original Student Tutorial

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

## MFAS Formative Assessments

Dimensions Needed:

Students are asked to solve problems involving square roots and cube roots.

## Original Student Tutorials Mathematics - Grades 6-8

Square Root Part 3: Simplifying Radicals:

Learn how to simplify radicals in this interactive tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorial

Square Root Part 3: Simplifying Radicals:

Learn how to simplify radicals in this interactive tutorial.

Type: Original Student Tutorial

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.