MA.8.AR.2.3

Given an equation in the form of x²=p and x³=q, where p is a whole number and q is an integer, determine the real solutions.

Clarifications

Clarification 1: Instruction focuses on understanding that when solving x²=p, there is both a positive and negative solution.

Clarification 2: Within this benchmark, the expectation is to calculate 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: Algebraic Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

• Integer
• Real Numbers

Vertical Alignment

Previous Benchmarks

Next Benchmarks

Purpose and Instructional Strategies

In grade 7, students wrote and solved two-step equations in one variable. In grade 8, when given an equation in the form $x$² = $p$ and $x$³ = $q$, where $p$ is a whole number and $q$ is an integer, students determine the real solutions. In Algebra 1, students will write and solve quadratic equations.
• This benchmark involves students understanding the concepts of how to square a number and find the square root as well as how to cube a number and find the cube root.
• Students should recognize that squaring a number and taking the square root of a number are inverse operations, therefore, cubing a number and taking the cube root are inverse operations as well. Students should use this understanding to solve equations containing square or cube numbers.
• In finding the square root, instruction involves discussion that there is both a positive and negative solution. Instruction can include relating the lengths of the sides of a square for square root and the length of the side of a cube to cube roots.
• Within this benchmark, it is not the expectation that students are required to isolate the $x$² term or the $x$³ term when solving an equation.

Common Misconceptions or Errors

• Students may incorrectly conclude that squaring a number means to multiply by 2. Likewise, cubing may be mistaken as multiplying by 3. Use length to show doubling and area of a square to show an exponent of 2. Use of two-dimensional and three-dimensional manipulatives (MTR.2.1) may also help to emphasize squares and cubes versus increasing length.
• Students may think that since a negative number has no square root in the real number system, then a negative number has no cube root in the real number system.

Strategies to Support Tiered Instruction

• Instruction includes modeling the differences between doubling and squaring a value using a graphic organizer. Doubling a value would be represented by multiplying a given length by 2 whereas squaring a number would be represented by the area of a square with a given length.
• For example, students can be given the table below to show how the left column doubles a length whereas the right column squares a length.

• Instruction includes modeling the differences between tripling or cubing a value using a graphic organizer. Tripling a value would be represented by multiplying a given length by 3 whereas cubing a number would be represented by the volume of a cube with a given length.
• For example, students can be given the table below to show how the left column triples a length whereas the right column cubes a length.

• Instruction may include providing students with the opportunity to develop their own note sheet or graphic organizer for the cubes of numbers from -5 to 5.

Instructional Task 1 (MTR.7.1)
A square tile in a kitchen has an area of 121 square inches.
• Part A. What is the length of one side of the square tile in inches? Is this tile smaller or larger than a one foot by one foot tile?
• Part B. The owner of the house, Kiana, wants to put larger tile in their kitchen to change the look of the kitchen. The new tile is a square with an area of 196 square inches.
• What is the length of the side of the new tile?
• How does this larger tile compare to the current tile used in the kitchen?
• Part C. A third tile has a side length of 2$\sqrt{11}$. Kiana is trying to determine which square tile covers the most area. Put the tiles side lengths in order from greatest to least. Justify your thinking.

Instructional Task 2 (MTR.3.1)

The volume of a large cube is 125 cubic inches. The volume of a small cube is 27 cubic inches. What is the difference between the length of one side of the large cube and the length of one side of the small cube?

Instructional Items

Instructional Item 1
An equation is given.
$x$² = 49
What are the values of $x$?

Instructional Item 2
Solve for $b$ in the equation −64 = $b$³.

*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.
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.AR.2.AP.3:

Given an equation in the form of x²= p and x³= q, use tools to determine real solutions where p is a perfect square up to 144 and q is a perfect cube from –125 to 125.

Related Resources

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

Formative Assessment

Students are asked to solve simple quadratic and cubic equations and represent solutions using square root and cube root symbols.

Type: Formative Assessment

Original Student Tutorials

Square Root Part 3: Simplifying Radicals:

Learn how to simplify radicals in this interactive tutorial.

Type: Original Student Tutorial

Square Root Part 2: Non-Perfect Squares:

Learn what non-perfect squares are and find the decimal approximation of their square roots in this interactive tutorial.

Type: Original Student Tutorial

Square Root Part 1: Perfect Squares:

Learn what perfect squares are and find their square roots in this interactive tutorial.

Type: Original Student Tutorial

MFAS Formative Assessments

Students are asked to solve simple quadratic and cubic equations and represent solutions using square root and cube root symbols.

Original Student Tutorials Mathematics - Grades 6-8

Square Root Part 1: Perfect Squares:

Learn what perfect squares are and find their square roots in this interactive tutorial.

Square Root Part 2: Non-Perfect Squares:

Learn what non-perfect squares are and find the decimal approximation of their square roots in this interactive tutorial.

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 Tutorials

Square Root Part 3: Simplifying Radicals:

Learn how to simplify radicals in this interactive tutorial.

Type: Original Student Tutorial

Square Root Part 2: Non-Perfect Squares:

Learn what non-perfect squares are and find the decimal approximation of their square roots in this interactive tutorial.

Type: Original Student Tutorial

Square Root Part 1: Perfect Squares:

Learn what perfect squares are and find their square roots in this interactive tutorial.

Type: Original Student Tutorial

Parent Resources

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