English:Numerical Expressions and Parentheses
Numerical Expressions and Parentheses
Numerical Expressions and Parentheses
In this aiMOOC, you learn how to read, write, and evaluate numerical expressions. You also learn how parentheses group parts of an expression and can change its value. The examples use whole numbers and the four basic operations: addition, subtraction, multiplication, and division.
Introduction
A numerical expression is a mathematical phrase made from numbers and operation symbols. For example, 7 + 5 × 2 is a numerical expression. It tells you what calculations to do, but it does not state that two quantities are equal.
Parentheses help you show which part belongs together. Compare these two expressions:
8 + 4 × 3 = 20
(8 + 4) × 3 = 36
The numbers and operations are almost the same, but the parentheses change which work is done first. Learning to notice this difference is an important step toward order of operations and later algebra.
Learning Goals
By the end of this course, you should be able to explain what a numerical expression is, identify the operations in an expression, use parentheses to group calculations, evaluate expressions in the correct order, translate short word situations into numerical expressions, compare expressions with different grouping, and explain your reasoning clearly.
Building Numerical Expressions
Numbers and Operation Symbols
The four basic operation symbols tell you what to do with numbers. A plus sign means add, a minus sign means subtract, a multiplication sign means multiply, and a division sign means divide.
Here are some useful words:
- Sum: The result of addition.
- Difference: The result of subtraction.
- Product: The result of multiplication.
- Quotient: The result of division.
An expression is not the same as an equation. The expression 6 × 4 + 3 has a value, but it does not contain an equals sign as part of the expression. When you write 6 × 4 + 3 = 27, you have written an equation that states the value of the expression.
Parentheses as Grouping Symbols
Parentheses are a pair of curved symbols: ( and ). They show that the work inside them belongs together.
When parentheses appear in a numerical expression, evaluate the part inside the parentheses first. Then continue with the rest of the expression.
Example:
6 × (4 + 2)
First, calculate 4 + 2 = 6. Then calculate 6 × 6 = 36. So the value of the expression is 36.
Now compare it with 6 × 4 + 2. Multiplication comes before addition, so 6 × 4 + 2 = 24 + 2 = 26. The parentheses changed the value from 26 to 36.
What Parentheses Change
Parentheses can change the meaning of a numerical expression because they change the grouping.
48 ÷ 6 + 2 = 8 + 2 = 10
48 ÷ (6 + 2) = 48 ÷ 8 = 6
Both expressions use 48, 6, 2, division, and addition. The first expression divides before adding. The second expression adds inside the parentheses first.
A helpful question is: What does the writer want me to treat as one group?
Parentheses in Words
Words often tell you when grouping is needed.
Three times the sum of 7 and 5 means 3 × (7 + 5). The phrase the sum of 7 and 5 is one group.
Five more than three times 7 means 3 × 7 + 5. Here, only 7 is multiplied by 3 before 5 is added.
In some textbooks, you may also see square brackets or braces as grouping symbols. The main idea is the same: work with the innermost grouped part first.
Order of Operations
The order of operations gives everyone a shared way to read and evaluate the same numerical expression. Without a shared order, one expression could be interpreted in different ways.
For the kinds of expressions in this course, use this order:
- Work inside parentheses first.
- Do multiplication and division from left to right.
- Do addition and subtraction from left to right.
If your class also uses exponents, they are handled after grouping symbols and before multiplication and division.
Step-by-Step Method
Try 3 × (7 - 2) + 4.
First, evaluate the parentheses: 7 - 2 = 5.
Now the expression is 3 × 5 + 4.
Next, multiply: 3 × 5 = 15.
Finally, add: 15 + 4 = 19.
So 3 × (7 - 2) + 4 = 19.
A second example is 30 - 2 × (8 - 3). Parentheses give 30 - 2 × 5. Multiplication gives 30 - 10. Subtraction gives 20.
Equal-Priority Operations
A common mistake is to think multiplication must always come before division because the word multiplication is often listed first. In fact, multiplication and division have the same priority. When they are the next operations, work from left to right.
36 ÷ 6 × 4
First, 36 ÷ 6 = 6. Then 6 × 4 = 24.
Addition and subtraction also have the same priority. When they are the next operations, work from left to right.
20 - 7 + 5
First, 20 - 7 = 13. Then 13 + 5 = 18.
The acronyms PEMDAS and BODMAS are memory aids for the same main idea of operation priority. The important point is not just remembering letters; it is understanding that multiplication and division are a pair, and addition and subtraction are a pair.
Writing Expressions from Situations
A real-world situation can often be represented by more than one number and operation. Parentheses help show which quantities belong together.
Suppose a school club buys 4 boxes. Each box contains 6 pencils and 2 erasers. If you want the total number of items, one useful expression is 4 × (6 + 2). First find the number of items in one box, then multiply by the number of boxes. The value is 4 × 8 = 32.
Now suppose the club buys 4 boxes of 6 pencils and then buys 2 extra erasers. The expression is 4 × 6 + 2, which has the value 26. The words describe a different situation, so the grouping is different.
When you translate words into an expression, ask:
- What quantities belong together?
- What operation connects them?
- Does a phrase describe a group that should be inside parentheses?
Comparing and Checking Expressions
Two expressions can look similar without being equivalent. Compare 5 × (10 - 4) with 5 × 10 - 4. The first has value 30. The second has value 46. A small change in grouping creates a large change in result.
One useful checking method is to estimate before you calculate. Another is to solve the expression a second time, carefully showing one step per line. A calculator can be useful after you have chosen the correct grouping and order of operations.
Common Mistakes
Mistake: Ignore parentheses. Fix: circle or point to the grouped part before calculating.
Mistake: Work only from left to right. Fix: first check whether multiplication or division must happen before addition or subtraction.
Mistake: Always multiply before dividing. Fix: multiplication and division have equal priority, so work from left to right.
Mistake: Always add before subtracting. Fix: addition and subtraction have equal priority, so work from left to right.
Mistake: Change several parts at once. Fix: rewrite the expression after each important step so your reasoning stays easy to follow.
Smart Checking
Ask yourself these questions when you finish:
- Did I evaluate every set of parentheses first?
- Did I keep the numbers and operation signs in the correct order?
- Did I treat multiplication and division as equal-priority operations?
- Did I treat addition and subtraction as equal-priority operations?
- Does my answer make sense compared with a quick estimate?
Interactive Tasks
Quiz: Test Your Knowledge
What is a numerical expression? (A combination of numbers and operation symbols) (!A statement that always contains an equals sign) (!A picture made only from shapes) (!A list of numbers with no operations)
What do parentheses tell you to do? (Work with the grouped part first) (!Ignore the grouped part) (!Always add before multiplying) (!Always divide before multiplying)
What is the value of 7 + 3 × 4? (19) (!40) (!28) (!31)
What is the value of 5 × the quantity 6 + 2 grouped together? (40) (!32) (!13) (!48)
What is the value of 24 ÷ 6 × 2? (8) (!2) (!12) (!4)
What is the value of 18 - 2 × 5? (8) (!80) (!16) (!10)
Which phrase describes the quantity 12 + 8 grouped together and then divided by 4? (Add twelve and eight then divide by four) (!Add twelve to eight divided by four) (!Divide twelve by four then add eight) (!Multiply twelve and eight then divide by four)
Why can 3 + 4 × 2 have a different value from the same numbers with 3 + 4 grouped together before multiplying by 2? (The parentheses change which operation is done first) (!The multiplication sign changes into division) (!The numbers become larger automatically) (!The plus sign disappears)
What is the value of 30 minus the grouped sum of 8 and 7? (15) (!29) (!9) (!45)
How should multiplication and division be handled when they are the next operations? (Work from left to right) (!Always multiply first) (!Always divide last) (!Skip division if multiplication appears)
Memory Game
| Numerical expression | A math phrase made of numbers and operation symbols |
| Parentheses | Curved grouping symbols that show what belongs together |
| Operation | A mathematical action such as adding or multiplying |
| Product | The result of multiplication |
| Quotient | The result of division |
| Evaluate | Find the value of an expression |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Grouped calculation | Parentheses first |
| Equal-priority multiplication and division | Work from left to right |
| Equal-priority addition and subtraction | Work from left to right after higher-priority operations |
| Expression | Numbers and operation signs that form a math phrase |
| Evaluate | Find the value |
...
Crossword Puzzle
| Parentheses | Which curved symbols group part of a numerical expression? |
| Expression | What do you call a math phrase made of numbers and operations? |
| Evaluate | What verb means to find the value of an expression? |
| Product | What is the result of multiplication called? |
| Quotient | What is the result of division called? |
| Difference | What is the result of subtraction called? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Expression Cards: Create eight cards with simple numerical expressions. On the back, show the value and one correct step-by-step solution.
- Parentheses Detective: Find five examples in a math book or worksheet where parentheses are used. Copy them and explain what each pair of parentheses groups.
- Math Comic: Draw a short comic in which two characters compare an expression with and without parentheses and discover why the answers differ.
- One-Minute Tutorial: Record a one-minute video explaining how to evaluate a simple expression with parentheses.
Standard
- Expression Interview: Interview a classmate about their strategy for order of operations, then write a short summary of one useful idea and one possible mistake.
- Cafeteria Math: Use prices or quantities from a school cafeteria, store flyer, or classroom supply list to write four numerical expressions, including two with parentheses.
- Error Analysis Poster: Create a poster showing three common mistakes with parentheses or order of operations and explain how to correct each one.
- Compare the Grouping: Choose four sets of numbers and operations, write two differently grouped expressions from each set, evaluate them, and explain why the results match or differ.
Advanced
- Parentheses Experiment: Start with one expression containing at least four numbers. Place parentheses in different valid positions, record the values, and describe which grouping gives the greatest and smallest result.
- Expression Game Design: Design a card or board game in which players build and evaluate numerical expressions using operation cards and grouping cards. Write clear rules and test the game with classmates.
- Real-World Model: Visit or study a real place such as a shop, library, sports club, or school event and create a realistic problem that is best represented by an expression with parentheses.
- Teach the Rule: Produce a two- to three-minute lesson, animation, or slide sequence that teaches equal-priority operations and explains why multiplication is not always done before division.
Learning Assessment
- Explain the Grouping: Compare 7 × 5 + 3 with 7 × (5 + 3), evaluate both, and explain in complete sentences why the values differ.
- Create Two Meanings: Write one short real-world story for 4 × 6 + 2 and a different story for 4 × (6 + 2), then explain how the parentheses change the situation.
- Find and Fix the Error: A learner says 36 ÷ 6 × 2 equals 3 because multiplication must come before division. Explain the error and show the correct reasoning.
- Design an Expression: Create a numerical expression with at least three operations and one pair of parentheses that has a value of 24, then justify each step.
- Transfer to Words: Translate the phrase five times the difference between 14 and 8 into a numerical expression, evaluate it, and explain why parentheses are needed.
- Compare Strategies: Solve 42 - 3 × (9 - 5) in two ways that both show correct reasoning, then explain which written method is easier to check and why.
Evidence of Learning
- Knowledge: You can explain numerical expressions, grouping symbols, operation priority, and equal-priority pairs.
- Skills: You can read, write, evaluate, compare, and check expressions with parentheses.
- Products: You can create correct expression cards, posters, models, games, explanations, or videos that communicate mathematical reasoning.
- Transfer: You can use parentheses to represent real situations, interpret expressions written by others, and explain why different grouping can change a result.
OERs on the Topic
The English Wikipedia article on Order of operations gives additional background on the conventions used to evaluate mathematical expressions.
Linked Learning Areas
This topic connects arithmetic, mathematical language, problem solving, and the early ideas of algebra. Understanding numerical expressions helps you represent situations clearly and prepares you for variables, formulas, and equations.