## Order of mathematics operations # Order of mathematics 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. 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.
2. Multiplication and division are two-dimensional operations (as represented by an array or the area of a rectangle);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 . Therefore, by the end of fourth grade, the order of operations should be: . 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:, 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  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.
1. 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).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: . 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:
2. 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:Absolute 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:.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 read 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 fraction, 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:
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•x, 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: 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: 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: 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)