Order of Mathematics Operations

Question:  I read your blog on non-negotiable skills at the elementary and middle school grades. The order of operations is so important during the upper elementary and middle school level and students have so much difficulty. Do you think the order of operations is important to emphasize in the curriculum? I have always introduced it as a mnemonic device (PEMDAS). Could you suggest other ways to introduce it to children?

Answer: The order of operations is an important skill. I did not mention it among the non-negotiable skills because it is part of the mastery of numeracy. It represents the integration of arithmetic operations. It is an example of understanding the four arithmetic operations and should be introduced properly and with sound reasoning.

I was also planning to discuss this along with the Standards of Mathematics Practice (SMP) in CCSS-M, particularly under SMP Three: Construct viable arguments and critique the reasoning of others. In other words, as teachers, we should avoid as much as possible, giving children rules that we cannot support with mathematical reasoning. To teach mathematics ideas, we should present (a) concrete models, (b) set up patterns, (c) arrive at mathematics conjectures using concrete models and patterns, (d) use analogies, (e) use deductive and inductive reasoning, and (f) use formal proofs.

Before posting my response here, I questioned others who all said “I have not used PEMDAS for a long time. I think I know why we use it, but I am not sure how to explain it.” “I don’t know why we use it this way. I always put parentheses.” or “I don’t remember why, but yes it’s the way I always do.”

In the fifth or sixth grades, most teachers introduce the order of operations as a mnemonic device (e.g., PEMDAS—Parentheses, Exponents, Multiplication/Division, Addition/Subtraction; GEMDAS—Grouping symbols, Exponents, Multiplication/Division, Addition/ Subtraction; or BODMAS—Brackets, Operation—Exponents, Powers, Roots, Multiplication/Division, Addition/Subtraction).

Too many teachers simply give the order of operations as PEMDAS without explaining the reasons behind it and how different elements are related to each other. That creates problems for children. PEMDAS would be the same if it were written PEDMSA. The order of operations is based on sound mathematics reasoning and a history that students need to know.

History
The rules for the order of operations have grown gradually over several centuries and are still evolving. However, I would say that the rules actually fall into two categories: the natural rules (such as precedence of exponential over multiplicative over additive operations, and the meaning of parentheses) and the artificial rules (left-to-right evaluation, equal precedence for multiplication and division, and so on).

The former were present from the beginning of notation as the concepts were formalized (e.g., definitions of addition, multiplication, exponents, etc.). They probably existed already, though in a somewhat different form, in the geometric and verbal modes of expression that preceded algebraic symbolism. The latter, not having any absolute reason for their acceptance, were gradually agreed upon through usage and continue to evolve.

Before the invention of algebraic notation, people performed these operations in the order they were described in words using the grammar of the language. Today, however, with the definitions of operations and algebraic symbols being standardized, they have become universal, and we can give reasons behind this evolution and acceptance of a particular order of operations.

To decide the order of operations, originally a vinculum (an over-line or underline) was used, e.g., Capture1or 2 + 7 = 14. Here the expression under the vinculum is performed first. Since the introduction of modern algebraic notation and because of its definition (geometrically, it is two-dimensional), multiplication has taken precedence over addition. Thus 3 + 4 × 5 = 4 × 5 + 3 = 23. When exponents were first introduced in the 16th and 17th centuries, exponents took precedence over both addition and multiplication and could be placed only as a superscript to the right of their base. Thus 3 + 52 = 3 + 25 = 28 and 3 × 52 = 3 × 25 =75.

Today, parentheses or brackets are used to explicitly denote precedence by grouping parts of an expression that should be evaluated first. Thus resulting in 2 × (3 + 4) = 14 to force addition to precede multiplication or (3 + 5)2 = 64 to force addition to precede exponentiation (because exponentiation is multiplication repeated several times; geometrically, it is multi-dimensional). Parentheses (including braces/brackets, i.e. groupings—both apparent and hidden) are done first (following PEMDAS within the groupings) as they are multi-operational activities, then we do the unary operations like exponents and functions.

The basic rule (that multiplication has precedence over addition) appears to have arisen naturally and without much disagreement as algebraic notation was developed in the 1600s and the need for such conventions arose. That must have happened as multiplication was moved from repeated addition to groups of or the area of a rectangle (a two-dimensional concept). Before these agreed conventions, each author needed to state his conventions at the start of a book, creating a great deal of confusion. The need for the emergence of conventions was natural. The conventions also made the writing of expressions easier. An example is the emergence of different forms of multiplication symbols, e.g., ×, •Ÿ, ( ), and finally algebraic, where the product of a and b can be written as ab. For example, without these agreed conventions, we will have to write

4,567 = 4×(1000) + 5×(100) + 6×(10) + 7×(1) in place of
4,567 = 4×1000 + 5×100 + 6×10 + 7×1
and,
4, 567 = 4((10)3) + 5((10)2) + 6((10)1) + 7((10)0)
in place of
4, 567 = 4 × 102 + 5 ×102 + 6 ×101 + 7×100.

Similarly, without our order of operations, in algebra, in place of a very concise expression for the polynomial:
ax2 + bx + c
we would have to write
(a((x)2)) + (b)(x) + c
Look at the cumbersomeness of the expression.

The term “order of operations” and the mnemonics for its applications “PEMDAS/BEDMAS” mnemonics, were formalized only in this century, or at least in the late 1800s, with the growth of the textbook industry and teacher training institutions. It became more important to text authors than to mathematicians, who just informally agreed (through research journals).

Order of Operations
In mathematics, and to some extent in computer programming, the order of operations (or operator precedence) has become a convention—a collection of rules—that tells us and defines which procedures to perform first in order to evaluate a given mathematical (arithmetical or algebraic) expression (a finite combination of constants, variables, symbols and sub-expressions that is formed according to mathematics rules that depend on the context). These mathematical symbols can designate numbers (constants), variable, arithmetic operations (addition, subtraction, multiplication, division, exponents, powers, roots), functions, grouping, and other mathematical entities.

Though the order of operations has become, to a great extent, formalized in textbooks and classroom instruction, many students have difficulty applying it and many teachers have difficulty explaining it. To facilitate students’ understanding, it should be taught properly and with rigor.

One hallmark of mathematical understanding is the ability to justify, in ways appropriate to the student’s mathematical maturity, why a particular mathematical statement is true, where a mathematical rule comes from, and how and when that can be applied. There is a world of difference between a student who can summon the mnemonic PEMDAS (= Please Excuse My Dear Aunt Sally to implement the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction) with an understanding for the underlying reasons and one who recites purely procedurally and lacks understanding. It is important to know the reasons behind this order of operations.

  1. Addition and subtraction are one-dimensional operations (linear—for example it is evident when we join two Cuisenaire rods or skip count to get the sum).
    They are at the same level and the first level of operations. If the two operations appear in the same expression, they are executed in order of appearance, first come first serve. Therefore, the order of operations at the end of second grade is in the order of their appearance. For example, 7 – 5 + 4 = 2 + 4 and 7 + 2 – 6 = 9 – 6 = 3.Capture2.PNG
    They are at the same level and the first level of operations. If the two operations appear in the same expression, they are executed in order of appearance, first come first serve Capture3. Therefore, the order of operations at the end of second grade is Capture3in the order of their appearance. For example, 7 – 5 + 4 = 2 + 4 and 7 + 2 – 6 = 9 – 6 = 3.
  2. Multiplication and division are two-dimensional operations (as represented by an array or the area of a rectangle);
    Capture4
    therefore, they are at a higher level than addition and subtraction and must be performed before addition and subtraction.And, if they both appear in an expression, they should be treated as first come first serve Capture5. Therefore, by the end of fourth grade, the order of operations should be:Capture7 . Thus, once multiplication (and division being the inverse of multiplication) has been introduced, multiplicative reasoning takes precedent over additive reasoning and performs them from left to right. The instruction in these problems should be: “simplify,” “compute,” “evaluate,” or “calculate” not “solve the problem” or “apply GEMDAS.” Even “simplify the expression by using the order of operations” is too procedural. Students should learn to make decisions in mathematics early on. That is what improves their metacognition and develops mathematical ways of thinking.
    Simplify
    : 3 + 6 × 2
    Multiplication before Addition: 3 + 6 × 2 = 3 + 12 = 15
    Simplify: 9 – 12 ÷ 3
    Division before Subtraction: 9 – 4 = 5
    Simplify:  9 – 12 ÷ 3 + 3 + 6 × 2Division and Subtraction first in order of appearance and then addition and subtraction in order of appearance: 9 – 4 + 3 + 6 × 2 = 9 – 4 + 3 + 12 = 5 + 3 + 12 = 8 + 12 = 20.If all of the four operations: addition, subtraction, multiplication, and division appear in a mathematical expression, the order should be:Capture7, higher order (2-dimensional) multiplication and division (in order of appearance) are performed first and then 1-dimensional addition and subtraction (in order of appearance) are performed next.

    The ability to apply Capture7 correctly, consistently, and fluently with understanding indicates the mastery of numeracy skills. For example, simplify the expression:

    7 + 3 × 8 ÷ 2 – 4 + 2×5 ÷2 + 5

    In the later grades (fifth grade and above), it is important to treat division as multiplication by the reciprocal (multiplicative inverse) and subtraction as addition of the opposite (additive inverse). Thus ¾ = 3 ÷ 4 = 3 • ¼; in other words the quotient of 3 and 4 equals the product of 3 and ¼. Also 3 − 4 = 3 + (−4); in other words, sometimes, the difference of 3 and 4 should be seen as the sum of positive three and negative four. Thus, 1 − 3 + 7 can be thought of as the sum of 1, negative 3, and 7, and add in any order: (1 − 3) + 7 = −2 + 7 = 5 and in reverse order (7 − 3) + 1 = 4 + 1 = 5, always keeping the negative sign with the 3.

  3. Exponential expressions are multi-dimensional (depending on the size of the exponent, e.g., a 10-cube is 3-dimensional and the exponent is 3 with a base of 10).
    Capture8
    An exponent is defined as the multi-use of multiplication; therefore, exponentiation operation is more important and higher order than multiplication (and division can be written as multiplication) and definitely higher than addition and subtraction; therefore, it must be performed before all of these four operations. Therefore, the order of operation so far is: Capture9. Start simplifying exponents (powers, roots, indices) first, then multiplication and division (in order of appearance) and then addition and subtraction (in order of appearance). Example:72 + 3 × 8 ÷ 22 – 4 + 2×5 ÷2 + 52Stacked exponents are applied from the top down, i.e., from right to left. Because exponentiation is right-associative in mathematics, we have:
    Capture10
  4. Sometimes, the intended order of computation is indicated by grouping certain operations or expressions in a given expression. For ease in reading, other grouping symbols such as braces, sometimes called curly braces { }, or brackets, sometimes called square brackets [ ], are often used along with parentheses ( ). For example:Capture12Absolute value symbol |   | is also a grouping symbol.
    |−(7 + 2) – 3| = | −9 − 3| = |−12| = 12Grouping operations such as brackets, braces, parentheses (either transparent or hidden, and absolute value; function and radical operations are also hidden operations), etc. may involve all of the above operations in multiple forms, therefore, are higher than all of the above. Therefore, the operations in the grouped expressions should be performed before. When grouping, exponents and all four operations are involved, then the order of operations should be:Capture13.Here, G represents grouping operations—both transparent and hidden. In the transparent grouping operations, the order of simplifying an expression is parentheses, braces, and brackets—from the innermost to the outermost group. They are organized in expressions in the same order—the brackets being the outermost. Brackets: (parentheses), {braces or curly braces}, or [brackets] are examples of mathematical grouping symbols and they have their own rules and impose their own order. For example, the previous expression: 2 + 3 × 4, can be reorganized into (2 + 3) × 4 giving us the simplified form as: 20. Grouping symbols are the only way to change this order and on occasion the division symbol presents itself as a grouping symbol (called hidden grouping as the Capture15read as 8 plus 2 divided by 4 + 1. Here 8 + 2 is being actually being read as (8 + 2)—as a group divided by (4 + 1) as a group and the line when used horizontally is indicating division. We should avoid writing the line as a slash to avoid misconception on the part of children.Symbols of grouping can be used to override the usual order of operations. Grouped symbols can be treated as a single expression. Symbols of grouping can be removed using the associative and distributive laws, and they can be removed if the expression inside the symbol of grouping is sufficiently simplified, so no ambiguity results from their removal.The hidden grouping operations are performed in the context. The context defines whether there is a hidden grouping or not. In a fraction expression, the numerator and denominator, because of the way we express them, define hidden grouping operations, even though there are no transparent grouping operations involved. For example, in the case of the fractionCapture16, the fraction is read as: sum of 3 and 5 divided by the difference of 3 and 1 (3 plus 5 then divided by 3 minus 1). As a result, the hidden grouping becomes transparent as: (3 + 5) ÷ (3-1). Therefore, before we simplify the fraction (performing the division operation), we first simplify the numerator and denominator—the hidden groupings. A horizontal fractional line, in an expression, acts as a symbol of grouping:
    Capture17

Many students and even some teachers are confused by texts that either teach or suggest that implicit multiplication (2x) takes precedence over explicit multiplication and division (2•Ÿx, 2/x) in expressions such as a/2b, which they would take as a/(2b), contrary to the generally accepted rules. The idea of adding new rules like this implies that the conventions are not yet completely stable; in other words, the situation is not all that different from the 1600s.

The slash sign (“/”), as indicated above, for a fraction creates a great deal of difficulty for students, even high school students. Either it should be avoided and proper fraction expression should be used, or enough time and explanations should be used to remove the ambiguity. For example, there can be ambiguity in the use of the slash (‘/’) symbol in expressions such as 1/2x. If one rewrites this expression as  1 ÷ 2 × x and then interprets the division symbol as indicating multiplication by the reciprocal, this becomes:
Capture18
With this interpretation 1/2x is equal to (1/2)x. However, in some of the academic literature, implied multiplication is interpreted as having higher precedence than division, so that 1/2x equals 1/(2x), not (1/2)x.

It is important to emphasize to the students the importance of hidden grouping. In the following expressions, there are hidden groupings:
Capture19

There are many places and concepts in middle and high school mathematics where hidden grouping appears and is important to discern, particularly in trigonometry and algebra. Students show difficulty with hidden groupings, so we need to help them discern them and practice writing expressions with hidden grouping.

The root symbol, √, requires a symbol of grouping around the radicand. The usual symbol of grouping is a bar (called vinuculum) over the radicand. Other functions, such as trigonometric functions, use parentheses around the input to avoid ambiguity. The parentheses are sometimes omitted if the input is a monomial. Thus, sin x = sin(x), but in when x and y are involved, in order to reduce the ambiguity and depending on the nature of the x and y, sin x + y = sin(x) + y, or sin (x + y) because x + y is not a monomial. Some calculators and programming languages require parentheses around function inputs, some do not.

Simplify:  7 + (6 × 52 + 3)

7 + (6 × 52 + 3) Given
7 + (6 × 25 + 3) Start inside the Grouped expression, then use Exponents First
7 + (150 + 3) We still have grouping, Multiply inside the group
7 + 153 Grouping completed, last operation is add
7 + 153 Then Add
160 DONE !

Inside a grouping expression, the same convention of order of operations is applied. Let us take an example (we are going to simplify later in detail):

72 + 3[4 × 5 − 3{21+2 + 3(28 ÷ 7 + 3)}] + 9 + 23

In this case, all the previous operations are involved. This expression has all the operations: grouping operations (both transparent and hidden), exponents, multiplication, division, addition, and subtraction. It is quite a complex expression, and these kinds of expressions appear only in higher grades. When we introduce this procedure, we should use simpler examples. However, here it will demonstrate the whole procedure.

In simplifying this expression, first we look at the grouping operations, in order. Therefore, the grouping operations (parentheses, braces, and brackets in this order) are performed first. The hidden grouping is contextual. Then exponential operations need to be performed and, after that, multiplication and division in order of their appearance. The last operations to be performed are addition and subtraction in order of their appearance.

This means that if a mathematical expression is preceded by one binary operator and followed by another, the operator higher on the list should be applied first. The properties of operations such as commutative, associative, and distributive laws of addition and multiplication allow adding terms in any order, and multiplying factors in any order—but mixed operations must obey the standard order of operations as defined above. Let us illustrate them in one problem. In the beginning, students should show their work as follows:
Capture21

When we want students to practice a concept, rule, or procedure, we should give them examples (at least four—one example becomes an exemplar of the concept, the second one begins to see the parameters of the concept, the third begins to set the pattern, and the fourth verifies the understanding of the pattern) to illustrate the idea and practice it correctly. We should also give examples where they do not work to highlight the nuances, subtleties, parameters, and conditions where the rule, concept, or procedure does not apply. The role of counter examples is very important in learning a mathematics idea. When a student applies a procedure incorrectly, we need to point out the part of the procedure, the definition/concept, or rule that was violated.

Examples:
Simplify:
8 × (5 + 3)  = 8 × 8 = 64.  (Correct)
8 × (5 + 3)  = 40 + 3 = 43.  (Incorrect)
5 × 42+ 3 = 5 × 16 +3 = 80 + 3 = 83 (Correct)
5 × 42+ 3 = 20 × 4 +3 = 80 + 3 = 83 (Incorrect procedure; Correct answer)
5 × 42+ 3 = 202 +3 = 400 + 3 = 403 (Incorrect)

Finally, w should ask students to construct an example of the concept, rule, definition, or procedure under discussion. When a student can construct an example, he or she can proceed in applying the procedure.

Mnemonics
Mnemonics are often used to help students remember the rules, but the rules taught by the use of acronyms only can be misleading. A mnemonic or memory device is only a technique that aids information retention in human memory. Mnemonics translate information into a form that the brain can retain better than its original form. Even the process of merely learning this conversion might already aid in the transfer of information to long-term memory. However, they should be used only after the concept has been understood.

When students have practiced a procedure really well and only a few are still having trouble with it, you can provide a graphic organizer or mnemonic device. Only when they have explained orally, partly or fully, what they are going to do, let them consult the graphic organizer or the mnemonic device to reinforce. You never want students to depend on organizers or mnemonic devices, and you should always ask for the reasons behind their use.

G Groupings (transparent and hidden)—first
E Exponents (i.e. Powers and Square Roots, orders, indices, etc.)
MD Multiplication and Division (left-to-right)
AS Addition and Subtraction (left-to-right)

In the United States the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. PEMDAS is often expanded to “Please Excuse My Dear Aunt Sally”, with the first letter of each word creating the acronym PEMDAS.

Canada uses BEDMAS, standing for Brackets, Exponents, Division, Multiplication, Addition, and Subtraction. Most common in the UK and Australia are BODMAS and BIDMAS.

These mnemonics may be misleading if the user is not aware that multiplication and division are of equal precedence, as are addition and subtraction. Using any of the above rules in the order “addition first, subtraction afterward” would also give the wrong answer to the problem:Capture22
The correct answer is 9 (and not 5, as if the addition would be carried out first and the result used with the subtraction afterwards). The best way to understand a combination of addition and subtraction is to think of the subtraction as addition of a negative number. In this case, the problem can be seen as the sum of positive ten, negative three, and positive two:Capture25
All of these acronyms conflate two different ideas, operations on the one hand and symbols of grouping on the other. If not properly taught and practiced, these acronyms lead to children’s misconceptions. Even after children have mastered the order of operations, every time teachers use the acronyms, they need to point out or ask children the reason for the precedence of a certain operation in this convention. The order of operations is the foundation of all computations in mathematics; therefore, its mastery is essential for success in learning and using mathematics.
Capture27

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Order of Mathematics Operations

2 thoughts on “Order of Mathematics Operations

  1. ephraim says:

    Usage: -5^2 + 8 / 2 x 4
    -25 +8 / 2 x 4
    -25 +4 x 4
    -25 + 16
    -9

    However if you start with multiplication the result would be different, yet MD are on same level on GEMDAS.

    How do i explain

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    1. Multiplication and division are same level operations. Therefore, the order of their appearance rule applies. In your problem, division appears first, therefore, division first Snd then multiplication. By the way, writing fractions as a/b creates some confusion to students. Therefore write fractions in the traditional form or use parentheses.

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