The Order of Operations is incorrect for division. That's because 5th grade math teaches that division is a fraction (and vice versa). Also, in teaching how to divide by a monomial in 9th grade Basic Algebra, it is consistently shown as a top-and-bottom fraction. Here are some of the many examples of how these concepts are currently being taught to young math students:

"Anyone who has studied secondary school mathematics would probably be comfortable with the convention of 'a over b' meaning 'a divided by b'."
~ ~ ~ ~ ~ ~ ~

"What are the tips and rules for simplify algebraic expressions?"

"An individual needs to know the rules of simplifying algebraic expressions before solving them. Some basic rules are combined to facilitate the given algebraic expression.

"...Open the brackets in the expression by using the distributive law."

"The distributive property, in particular, asserts that given any absolute values a, b, and c, one should simplify the components of the parenthesis first.

When the elements of brackets are difficult to be simplified, use the distributive property to multiply each term within the parenthesis by the factor outside the parentheses. One can multiply and delete the parenthesis by using the distributive property."
~ ~ ~ ~ ~ ~ ~

That can only be true if 56x^5 is the entire numerator & 7x^2 is the entire denominator of the top-and-bottom fraction, even though there are no parentheses anywhere in the original horizontally written division statement using an obelus.
~ ~ ~ ~ ~ ~

"In Algebra, a polynomial with a single term is known as a monomial. When a monomial is divided by a monomial, first divide the coefficients of the variable and then divide the variable when the variables are present in both the numerator and denominator. For example, assume two monomials, 50 xy and 5y. Now the monomial 50xy is divided by 5y, we will get

= 50xy/5y

= 10x

Thus, the quotient value obtained is 10x, which is the result of the division process."
~ ~ ~ ~ ~ ~ ~

That can only be true if 15a^2b^3 is the entire numerator & 5b is the entire denominator of the top-and-bottom fraction, even though there are no parentheses anywhere in the original horizontally written division statement using an obelus.
~ ~ ~ ~ ~ ~ ~

"A monomial is a form of a polynomial with a single non-zero term. Because a monomial has only a single term, it is simple to do addition, subtraction, and multiplication. It is composed of only one variable, one coefficient, or the product of a variable and a coefficient, with exponents as whole numbers representing only one term."

"...when dividing monomials, divide the coefficients first, then divide the variables. When there are exponents with the same base, divide by subtracting the exponents according to exponent rules.

Example: 16mn ÷ 4n

= (16/4) (m) (n/n)

= 4mn "
* * * * * * *

In light of how division of monomials is currently being taught, given that x does not equal zero, what is the quotient of 2x divided by 2x, whether written as 2x ÷ 2x or written as 2x/2x or written as the top-and-bottom fraction...

The Order of Operations is incorrect for division. That's because 5th grade math teaches that division is a fraction (and vice versa). Also, in teaching how to divide by a monomial in 9th grade Basic Algebra, it is consistently shown as a top-and-bottom fraction. Here are some of the many examples of how these concepts are currently being taught to young math students:

from Teaching Better Lesson.com Common Core:

https://teaching.betterlesson.com/browse/common_core/standard/272/ccss-…

"Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b)."

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Australian Association of Mathematics Teachers:

https://topdrawer.aamt.edu.au/Fractions/Big-ideas/Fractions-as-division

"Fractions as Division"

"Anyone who has studied secondary school mathematics would probably be comfortable with the convention of 'a over b' meaning 'a divided by b'."

~ ~ ~ ~ ~ ~ ~

from YouTube teaching video:

grade 5:

https://www.youtube.com/watch?v=hfuIUuELmBk

"As you can see a division problem can be rewritten as a fraction. And a fraction can be rewritten as a fraction."

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from ClubZ Tutoring:

https://clubztutoring.com/ed-resources/math/fraction-bar-definitions-ex…

"FAQ Section

Q1: Can the fraction bar be replaced with the division symbol (/)?

A1: Yes, the fraction bar and the division symbol (/) are interchangeable and convey the same meaning in mathematical notation."

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from Algebra Class .com: https://www.algebra-class.com/dividing-monomials.html

"Dividing Monomials"

"Remember: A division bar and fraction bar are synonymous!"

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from OpenStax .com:

https://openstax.org/books/elementary-algebra-2e/pages/6-6-divide-polyn…

Elementary Algebra

"Divide Polynomials"

"Example 6.78

Find the quotient: (18x^3 - 36x^2) ÷ 6x

Solution

Rewrite as a fraction.

18x^3 - 36x^2

-----------------

6x

...Simplify.

3x^2 - 6x "

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from Slyavula Technology Powered Learning:

https://www.siyavula.com/read/za/mathematics/grade-8/algebraic-expressi…

Grade 8

Algebraic Expressions

"WORKED EXAMPLE 8.2

DIVIDING ALGEBRAIC MONOMIALS

Simplify the following expression:

24t^7 ÷ 4t^5

SOLUTION:

Step 1: Rewrite the division as a fraction

This question is written with a division symbol (÷), but this is the same as writing it as a fraction.

24t^7 ÷ 4t^5 =

24t^7

--------

4t^5

...= 6t^2 "

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from SlideShare a Scribd Company:

https://www.slideshare.net/slideshow/94-16609182/16609182

"Example 3

Divide a polynomial by a monomial

Divide 4x^3 + 8x^2 + 10x by 2x.

4x^3 + 8x^2 + 10x ÷ 2x =

Write as a fraction

4x^3 + 8x^2 + 10x

---------------------- =

2x

Simplify

2x^2 + 4x + 5 "

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from turito .com math tutoring website:

https://www.turito.com/learn/math/simplify-algebraic-expression

"What are the tips and rules for simplify algebraic expressions?"

"An individual needs to know the rules of simplifying algebraic expressions before solving them. Some basic rules are combined to facilitate the given algebraic expression.

"...Open the brackets in the expression by using the distributive law."

"The distributive property, in particular, asserts that given any absolute values a, b, and c, one should simplify the components of the parenthesis first.

When the elements of brackets are difficult to be simplified, use the distributive property to multiply each term within the parenthesis by the factor outside the parentheses. One can multiply and delete the parenthesis by using the distributive property."

~ ~ ~ ~ ~ ~ ~

from Math Worksheet Center:

https://www.mathworksheetscenter.com/mathworksheets/DivisionPolynomials…

"How to divide a polynomial by a monomial"

"Basic Skills:

"Divide: 16x^5 / 8x^3

A) Divide numbers: 16 / 8 = 2

B) Subtract exponents: x^5 / x^3 = x^2

2 x^2

Answer: 2 x^2 "

~ ~ ~ ~ ~ ~ ~

Lumen Learning Courses:

https://courses.lumenlearning.com/uvu-introductoryalgebra/chapter/9-5-d…

"Dividing Polynomials by a Monomial"

EXAMPLE

Find the quotient: 56x^5 ÷ 7x^2

Solution

Rewrite as a fraction

56x^5

-----------

7x^2

...Answer

56x^5 ÷ 7x^2 = 8x^3 "

That can only be true if 56x^5 is the entire numerator & 7x^2 is the entire denominator of the top-and-bottom fraction, even though there are no parentheses anywhere in the original horizontally written division statement using an obelus.

~ ~ ~ ~ ~ ~

from BYJUS teaching website: https://byjus.com/dividing-monomials-calculator/

"What is Meant by Dividing Monomials?"

"In Algebra, a polynomial with a single term is known as a monomial. When a monomial is divided by a monomial, first divide the coefficients of the variable and then divide the variable when the variables are present in both the numerator and denominator. For example, assume two monomials, 50 xy and 5y. Now the monomial 50xy is divided by 5y, we will get

= 50xy/5y

= 10x

Thus, the quotient value obtained is 10x, which is the result of the division process."

~ ~ ~ ~ ~ ~ ~

from Cue Math teaching website:

https://www.cuemath.com/algebra/dividing-monomials/

"Practice Questions on Dividing Monomials "

Q.1. Divide. 15a^2b^3 ÷ 5b "

Correct answer is shown as 3a^2b^2.

That can only be true if 15a^2b^3 is the entire numerator & 5b is the entire denominator of the top-and-bottom fraction, even though there are no parentheses anywhere in the original horizontally written division statement using an obelus.

~ ~ ~ ~ ~ ~ ~

from GeeksforGeeks .org

https://www.geeksforgeeks.org/how-to-divide-monomials/#

"A monomial is a form of a polynomial with a single non-zero term. Because a monomial has only a single term, it is simple to do addition, subtraction, and multiplication. It is composed of only one variable, one coefficient, or the product of a variable and a coefficient, with exponents as whole numbers representing only one term."

"...when dividing monomials, divide the coefficients first, then divide the variables. When there are exponents with the same base, divide by subtracting the exponents according to exponent rules.

Example: 16mn ÷ 4n

= (16/4) (m) (n/n)

= 4mn "

* * * * * * *

In light of how division of monomials is currently being taught, given that x does not equal zero, what is the quotient of 2x divided by 2x, whether written as 2x ÷ 2x or written as 2x/2x or written as the top-and-bottom fraction...

2x

---- = ?

2x