Using Grouping Symbols in Expressions

Introduction

In simplifying mathematical expressions consisting of the same type of operation, we perform one operation at a time generally starring from the left towards the right. If an expression has more than one fundamental operation, you cannot perform operations in the order they appear. Some operations have to be performed before than the others. This is each operation has its own precedence. Generally, the order in which we perform operations sequentially from left to right is division, multiplication, addition, subtraction. But when the expressions make use of brackets as well, we have a set of rules that defines the precedence of the operations. Let us learn the manner in which this rule is defined.

You can think of a numerical expression as a number sentence. Instead of words, it can have numbers, variables (letters that hold a place for a number you don’t know yet), math symbols that tell you whether to add, subtract, multiply or divide, and grouping symbols that tell you which order to follow.

Long ago, mathematicians from different countries met to agree on some rules so that anyone doing the same math problem would get the same answer. The rules are collectively known as the Order of Operations. This lesson is the first step in learning the order of operations.

Let us understand this through an example.

Suppose we want to find the value of 2 + 3 x 5

There are two ways to find the value of the above expression.

In the first method, let us move from left to right, solving the expression in that order. This means, first we will find the sum of 2 and 3. We will get, 2 + 3 = 5.

Now we have 2 + 3 x 5 = 5 x 5

Now, we find the product of 5 with 5 to get the answer as 25. Therefore, we have,

2 + 3 x 5 = 25 ……………………………….. ( 1 )

Now, let us find the value of this expression through another method.

We will first find the value of 3 x 5 and add this product to 2. So, we get,

3 x 5 = 15 and 15 + 2 = 17

Hence, now we have,

2 + 3 x 5 = 17 ………………………… ( 2 )

From ( 1 ) and ( 2 ) we can see that both the results obtained are not the same. This validates the need for having precedence of operators so as to receive the same answer to the given algebraic expression. This precedence is known as the order of operations.

Learning Outcomes

By the end of this lesson, your children will be able to correctly use and evaluate grouping symbols in number sentences

Warm Up

When there are no special grouping symbols, math problems are solved from left to right. Although there are other important rules about the order in which you do the operations (addition/subtraction/multiplication/division) in a math expression or equation, this lesson will focus on grouping symbols. The rest of the order of operations rules will be explained in the Determining Order of Operations lesson. To help prevent confusion as you learn how to use the grouping symbols, this lesson will only use addition and subtraction.

It’s time to scratch your memory about things you have previously learned, and add on to what you know.

In 4th grade you learned how to interpret (read and make sense out of) simple expressions (math sentences that do not include an equal sign) and equations (math sentences that do include an equal sign). You may have seen parentheses used to group part of the expression or equation together. Parentheses are the most common grouping symbols.

Grouping symbols in math expressions include:

1. Parentheses	( )	– have a rounded shape
2. Brackets:	[ ]	– have a square shape
3. Braces:	{ }	– have a twirled shape

All grouping symbols tell you, “Do this first!”.

Parentheses are used in math to show a part of a math expression or equation that must be solved first, before any other calculations are done. The part between the two parentheses is treated like one number; the answer replaces the expression in the larger math equation.
For complicated problems, brackets can be used to enclose sections of the problem that already include parentheses to further separate sections.
For extremely complicated problems, braces can be used to enclose sections that already include brackets and parentheses.

Note: Grouping symbols are the first step in the longer process of determining the order of operations, which is fully addressed in a separate lesson.

What is the Basic Order of Operations?

Order of operation can be defined as a standard procedure that guides you on which calculations to begin within an expression with several arithmetic operations. Without consistent order of operation, one can make big mistakes during computation. There are a number of rules that define the order of operations, depending upon the involvement of the operators, brackets exponents and other mathematical symbols for operations. This rule is known as BODMAS, where

B stands for Brackets

O stands for Of

D stands for Division ( ÷ )

M stands for Multiplication ( x )

A stands for Addition ( + )

S stands for subtraction ( – )

BODMAS

Understanding BODAMS is integral to the understanding of the use of grouping symbols in expressions. This is because it is the BODMAS rule that defines the order in which the operations are to be performed in the case of mathematical expressions. In other words, it defines the order in which operations on numbers are to be performed when there are situations where we have two more than two operations in an expression.

Let us consider an example. Suppose we want to find the value of 3 + 5 x 2.

Here, we have two operators, namely ( x ) and ( + ). We know that in mathematics, we can perform operations between two numbers only in one go. If we have more than 2 numbers, the result from the first operation is used for performing the next operation. So, in the above example, there are two operations to be performed, ( 3 + 5 ) and ( 5 x 2 ). We need to decide, which one to start with. This is where the MDAS rule comes into play.

From this rule, we can see that Multiplication precedes addition. Therefore, we will perform the operation multiplication first. The result thus obtained will be added to 3. The steps involved in this process will be –

3 + 5 x 2

= 3 + 10

= 13

Hence, 3 + 5 x 2 = 13

How to use Grouping Symbols in Order of Operations

It is important to note that when there are no special grouping symbols, math problems are solved from left to right. However, when you have grouping symbols involved we need to follow the order of operations as we have discussed above. We have just learnt about the precedence of fundamental operations of addition, subtraction, multiplication and division. According to it the order in which the operations are to be performed is first division then multiplication after which addition and finally subtraction but sometimes in complex operations require a set of operations to be performed prior to the other. For example, if we want the addition to be performed before a division or multiplication then we need to use a bracket.

Grouping symbols or brackets are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression. Bracket indicates that the operations within it ought to be performed before the operations outside it. For example, the expression 24 ÷ 3 x 4 would generally be solved as –

24 ÷ 3 x 4

= 8 x 4 = 32

However, If we wish to multiply 3 and 4 first and then divide 24 by the resulting number we write the expression as

24 ÷ ( 3 x 4 )

Now, according to the order of operations, we will solve the brackets first to get,

24 ÷ ( 3 x 4 )

= 24 ÷ 12

= 2

In complex expressions sometimes it is necessary to have (within) in the same time of (one within the another can be confusing for different types of brackets are used most commonly used) are

Brackets Name

( ) Parenthesis of Common Brackets

{ } Braces or Curly Brackets

[ ] Brackets or Square Brackets

It is important to note here that, the left part of each bracket symbol indicates the start of the bracket and the right part indicates the end of the bracket. In writing mathematical expressions consisting of more than one bracket, parenthesis is used in the innermost part followed by braces and these two are covered by square brackets.

Some other important points to note are –

Parentheses are used in math to show a part of a math expression or equation that must be solved first before any other calculations are done. The part between the two parentheses is treated like one number; the answer replaces the expression in the larger math equation.
For complicated problems, brackets can be used to enclose sections of the problem that already include parentheses to further separate sections.
For extremely complicated problems, braces can be used to enclose sections that already include brackets and parentheses.

Removal of Brackets

In order to simplify expressions involving more than one grouping symbol, we will use the steps as explained below –

See whether the given expression contains vinculum or not. If vinculum is present then perform operations on the red otherwise go to the next step. A vinculum is a horizontal line placed over an expression to show that everything below the line is one group, for example $\overline{2 x 3}$.
Now, look for the innermost bracket and perform operations within it.
Remove the innermost bracket by using the following steps –
1. If a bracket is preceded by a plus sign, remove it by writing its terms as they are.
2. If a bracket is preceded by a minus sign change the positive sign within it to a negative sign and vice-versa.
3. If there is no sign between a number and a grouping symbol then it means multiplication.
4. If there is a number before some brackets then we multiply the number inside the brackets with the number outside the brackets.
Look for the next innermost bracket and perform operations in it. Remove the second in a directed by using the rules given in the above steps Continue this process till all the brackets are removed.

Let us understand the above steps using an example.

Example

Simplify: 37 – [ 5 + { 28 – ( 19 – 7 ) } ]

Solution

We have been given the expression, 37 – [ 5 + { 28 – ( 19 – 7 ) } ]

We will use the order of operations to solve the given expression.

First, we will remove the innermost bracket to get,

37 – [ 5 + { 28 – 12 } ] …………………… [ Removing the innermost bracket ( ) ]

Next, we will remove the curly braces to get

37 – [ 5 + 16 ] …………………………. [ Removing the curly braces ]

Now, we will remove the square brackets to get,

37 – 21 ………………………… [ removing the square brackets ]

Last, we will find the difference between 37 and 21 to get 16.

Hence simplification of 37 – [ 5 + { 28 – 12 } ] will result in 16.

How to Solve Expressions involving Grouping Symbols?

Now, that we have understood what we mean by grouping symbols and mathematical expressions let us learn about the steps involved in solving mathematical expressions involving grouping symbols. The steps involved are –

Always start by calculating all expressions within parentheses

The first thing is that we should look for removing any grouping symbols in the algebraic expression. This means that at the top of the list, remember to always simplify everything inside the grouping symbols. Examples of grouping symbols are parentheses ( ), brackets, and braces { }. For nested grouping symbols, work it out from the inside and out.

Simplify all the exponents such as square roots, squares, cube, and cube roots

After having removed the parenthesis, we come to the next step of solving all exponential values in the algebraic expression. Exponential expressions which may include root values such as square roots, squares, cube, and cube roots etc. are calculated or evaluated first before performing any of the four fundamental arithmetic operations, namely: addition, subtraction, multiplication, and division.

Perform the multiplication and the division starting from left to right

Next, multiply and/or divide whichever comes first from left to right before performing addition and subtraction. This tells us that multiplication and division have a higher level of importance than addition and subtraction.

Finally, do the addition and subtraction similarly, starting from left to right.

Let us understand it using an example.

Example Simplify 95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 – $\overline{17-10}$ } ]

Solution We have been given the expression

95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 – $\overline{17-10}$ } ]

We will use PEMDAS to solve the above expression.

Note, we can see the presence of vinculum in the expression, hence it needs to be solved first.

Solving the vinculum, we will get,

95 – [ 144 ÷ ( 12 x 12 ) – ( – 4 ) – { 3 – 7 } ]

Next, we will remove the innermost bracket to get,

95 – [ 144 ÷ 144 + 4 – { 3 – 7 } ]

Now, we will remove the curly braces to get

95 – [ 144 ÷ 144 + 4 + 4 ]

Now, it is important to see that within the square bracket we have two operations to be performed, division and addition. So, going by the PEMDAS rule, we will first perform division to get,

95 – [ 1 + 4 + 4 ]

Next, we will remove the square brackets to get,

95 – 9

= 86

Hence, simplification of 95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 – $\overline{17-10}$ } ] = 86

Let us consider another example.

Example Find the value of the expression 4 [ 10 + 15 ÷ 5 × 4 – 2 × 2 )

Solution The steps used to solve the given expression can be tabulated as

Brackets Of	4 ( 10 + 15 ÷ 5 × 4 – 2 × 2 )
Division	4 ( 10 + 15 ÷ 5 x 4 – 2 × 2 )
Multiplication	4 ( 10 + 3 × 4 – 2 × 2 )
Addition	4 ( 10 + 12 – 4 )
Subtraction	4 ( 22 – 4 )
Answer	= 4 x 18
	72

Solved Examples

Example 1 Simplify: 197 – [1/9 { 42 + (56 – $\overline{8 + 9}$ ) } +108 ]

Solution We have been given the algebraic expression,

197 – [1/9 { 42 + (56 – $\overline{8 + 9}$ ) } +108 ]

We will use PEMDAS to solve the above expression.

Note, we can see the presence of vinculum in the expression; hence it needs to be solved first.

Solving the vinculum, we will get,

197 – [1/9 { 42 + (56 – 17 ) } +108 ]

Next, we will remove the innermost bracket to get,

197 – [1/9 { 42 + 39 } +108 ]

Now, we will remove the curly braces to get

197 – [81 / 9 +108 ]

Now, it is important to see that within the square bracket we have two operations to be performed, division and addition. So, going by the PEMDAS rule, we will first perform division to get,

197 – [9 + 108 ]

Next, we will remove the square brackets to get,

197 – 117

Last, we just need to find the difference of the remaining two values to get,

80.

Hence, simplification of 197 – [1/9 { 42 + (56 – $\overline{8 + 9}$ ) } +108 ] = 80.

Example 2 Simplify 15 – ( – 5) { 4 – $\overline{7-3}$ } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

Solution We have been given the expression

15 – ( – 5) { 4 – $\overline{7-3}$ } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

We will use PEMDAS to solve the above expression.

Note, we can see the presence of vinculum in the expression, hence it needs to be solved first.

Solving the vinculum, we will get,

15 – ( – 5) { 4 – 4 } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

Next, we will remove the innermost bracket to get,

15 + 5 x 0 ÷ [ 3 { 5 + 18 } ]

Now, we will remove the curly braces to get

15 + 0 ÷ [ 3 x 23 ]

Next, we will remove the square brackets to get,

15 + 0 ÷ 69

Now, it is important to see that we have two operations to be performed, division and addition. So, going by the PEMDAS rule, we will first perform division to get,

15 + 0

= 15

Hence, simplification of 15 – ( – 5) { 4 – $\overline{7-3}$ } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ] = 15

Parentheses can be used in math to show which part of the math expression should be done first.
8 – 5 + 1 and 8 – (5 + 1)
The only difference between these two expressions is the parentheses. Without parentheses, solve from left to right: 8 – 5 is 3, and then 3 + 1 is 4.
However, adding parentheses can change the result. The parentheses say, “start with the 5 + 1 and read it as one number.” Since 5 + 1 is 6, replace 5 + 1 with 6 in the expression, which leaves 8 – 6 = 2.

Parent Tip: Your children should become very familiar with the use of parentheses, because the parentheses pair is the most common grouping symbol

Here is another example:
14 – 6 + 5 and 14 – (6 + 5)
Again, the only difference between the two expressions is the use of parentheses. In the first expression, there are no parentheses, so solve from left to right:
14 – 6 is 8, then 8 + 5 is 13.
With the parentheses added, start with 6 + 5. Since 6 + 5 is 11, replace the (6+5) in the expression with 11.
This leaves 14 -11 = 3.

Sometimes the result is the same with or without parentheses. When a real-life problem is being solved, parentheses can be used to show how the numbers in the math expression relate to the real-life situation, even if using them does not affect the answer.

40 + 35 – 50	and	(40 + 35) – 50
75 – 50 = 25		75 – 50 = 25

Parent Tip: Math will become more meaningful to your children when they see how it shows up in everyday life. Pay attention to opportunities to think out loud when you are using math to solve a life problem.

Consider this situation: Mary has a birthday. Her grandpa sends her $40. Her aunt sends her $35. The next day, Mary spends $50 of her birthday money at the mall. As shown below, the parentheses can be used to group the total money she was given so that it is separated from the money she spent. Even though Mary has $25 left at the end using either of the expressions, the one with parentheses best matches the events of the situation.

40 + 35 – 50

and

(40 + 35) – 50

Key Facts and Summary

Order of operation can be defined as a standard procedure that guides you on which calculations to begin within an expression with several arithmetic operations.
MDAS is the basic order of operations where, MDAS, where M stands for Multiplication ( x ), D stands for Division ( ÷ ), A stands for Addition ( + ), S stands for subtraction ( – ).
Grouping Symbols or Brackets are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression.
The rules of order of operations are –
1. Always start by calculating all expressions within parentheses
2. Simplify all the exponents such as square roots, squares, cube, and cube roots.
3. Perform the multiplication and the division starting from left to right
4. Finally, do the addition and subtraction similarly, starting from left to right.
Brackets are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression.
In writing mathematical expressions consisting of more than one bracket, parenthesis is used in the innermost part followed by braces and these two are covered by square brackets.

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Parentheses only:	(38 – 14) – 10 = 14 24 – 10 = 14
Parentheses and brackets:	[ 8 +(38 – 14) – 10] + 12 = 34 [ 8 + 24 – 10] + 12 = 34 22 + 12 = 34
Parentheses, brackets, and braces:	{44 – [8 + (38 – 14) – 10] + 12} – 7 = 27 {44 – [8+ 24 – 10] + 12} – 7 = 27 {44 – 22 +12} – 7 = 27 34 – 7 = 27

Using Grouping Symbols in Expressions