### 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.

**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

- 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.

- 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.

- 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 Tasks

*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

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

## Related Access Points

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

## Formative Assessment

## Original Student Tutorials

## 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

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

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

Learn how to simplify radicals in this interactive tutorial.

## Student Resources

## Original Student Tutorials

Learn how to simplify radicals in this interactive tutorial.

Type: Original Student Tutorial

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

Type: Original Student Tutorial

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

Type: Original Student Tutorial