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.

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

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.

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. |
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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: 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 |

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 |

When math problems become more complicated, it is sometimes necessary to have groups inside of groups. Parentheses are for the innermost group. If a second grouping , which will include the part already in parentheses, is needed, square brackets are used. If a third grouping, which will include a section with parentheses and square brackets, is necessary, then braces are used.

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 |

- Grouping symbols are used to show what should be done first in a math expression.
- Parentheses are the most common grouping symbol.
- Brackets and Braces can be used to further group sections of a math expression when parentheses have already been used.
- Grouping symbols are sometimes used to help the math expression match a real-life situation.