Arithmetic Expressions
Learn why the order of operations matters and how to write expressions that say exactly what you mean—like punctuation in English!
One Expression, Two Answers!
Mallesh brought 30 marbles. Arun brought 5 bags with 4 marbles each. How many marbles total? Mallesh wrote: 30 + 5 × 4
But Purna got 140 (adding 30 + 5 = 35, then 35 × 4). Mallesh got 50 (multiplying 5 × 4 = 20, then 20 + 30). Who's right?
The answer depends on how we read the expression. Just like punctuation in English changes meaning, the order of operations changes the answer in math!
In English, "Shalini sat next to a friend with toys" could mean two things—depending on punctuation. In math, 30 + 5 × 4 could mean two things—depending on brackets and order of operations.
Just as we use commas, periods, and colons to clarify sentences, we use brackets and the concept of terms to clarify which operations happen first in math!
What Is an Expression?
An arithmetic expression is a math phrase using numbers and operations. Examples: 13 + 2, 20 – 4, 12 × 5, 18 ÷ 3. Each has a value—a number it "equals."
Expressions Can Have the Same Value
Different expressions can equal the same number. For example, all these equal 12:
10 + 2, 15 – 3, 3 × 4, 24 ÷ 2
We write this with the equals sign: 10 + 2 = 12
Comparing Expressions Without Calculating
Do we need to calculate both sides to know which is bigger? Not always!
Example: Which is bigger: 1023 + 125 or 1022 + 128?
Raja starts with 1023 and gets 125 more. Joy starts with 1022 (one less than Raja) but gets 128 more (three more than Raja). Joy's gain is bigger, so even though he started with less, he ends with more! Therefore: 1023 + 125 < 1022 + 128
Compare Subtraction Expressions
Example: Which is bigger: 113 – 25 or 112 – 24?
Raja had 113 and lost 25. Joy had 112 and lost 24. Raja started with 1 more but lost 1 more, so they end with the same! Therefore: 113 – 25 = 112 – 24
When you compare expressions by imagining real situations (marbles, money, people), you don't need a calculator. You use logic!
The Problem: Ambiguity Without Brackets
The expression 30 + 5 × 4 is ambiguous. Do we add first or multiply first?
- If we add first: (30 + 5) × 4 = 35 × 4 = 140
- If we multiply first: 30 + (5 × 4) = 30 + 20 = 50
Same expression, different answers! We need a way to clarify.
The Solution: Use Brackets
Brackets tell us which operation to do first. In the marble problem:
30 + (5 × 4) = 30 + 20 = 50
The brackets say: Multiply 5 and 4 first, then add the result to 30. This is correct because Arun brought 5 bags of 4 marbles each = 20 marbles, plus Mallesh's 30 = 50 total.
Real-World Example: Money Change
Irfan buys biscuits for ₹15 and toor dal for ₹56. He pays ₹100. Change = 100 – (15 + 56)
If we wrote it as 100 – 15 + 56, we might incorrectly subtract 15 from 100 first, then add 56, getting ₹141—more than he paid! This is absurd.
Brackets clarify: 100 – (15 + 56) = 100 – 71 = 29. Irfan gets ₹29 back.
Brackets act like punctuation. They group quantities that belong together logically: (total cost) in this case, not individual items.
Terms Are Parts Separated by Plus Signs
In the expression 12 + 7, we have two terms: 12 and 7.
In the expression 6 × 5 + 3, we have two terms: 6 × 5 (which is a single "unit" or term) and 3.
Key idea: Multiplication and division happen within a term. We only split terms at + or – signs.
Subtraction Is Really Adding the Opposite
In 83 – 14, we have two terms: 83 and –14 (negative 14).
Why? Because subtracting is the same as adding a negative number: 83 – 14 = 83 + (–14)
Both give 69. By thinking of subtraction as "adding the opposite," we keep one consistent operation: addition.
From Class 6, you learned about positive and negative tokens. Subtracting 14 is like adding 14 negative tokens. This viewpoint makes the math more consistent.
Breaking Down Complex Expressions
Expression: 30 + 5 × 4
What are the terms? First, identify the + or – signs (the term separators):
- Term 1:
30 - Term 2:
5 × 4(the × stays within the term)
We evaluate each term separately: 30 and 20, then add: 30 + 20 = 50.
The Order Matters in Terms—Then Add Them
Expression: 5 × (3 + 2) + 78 + 3
What are the terms?
- Term 1:
5 × (3 + 2) = 5 × 5 = 25(brackets happen first within the term) - Term 2:
78 - Term 3:
3
Then add all terms: 25 + 78 + 3 = 106.
Multiplication and division are "stronger" operations that happen within terms. Addition and subtraction are "weaker" and happen between terms. This hierarchy prevents confusion!
Properties of Addition: Swapping and Grouping
Once we've identified terms and converted everything to addition (by thinking of subtraction as adding negatives), something magical happens: the order doesn't matter!
Commutative Property (Swapping): If you have terms, you can add them in any order: 6 + (–4) = (–4) + 6 = 2. Both equal 2!
Associative Property (Grouping): If you have three terms, you can group them differently:
(–7) + 10 + (–11) = (–7 + 10) + (–11) = 3 + (–11) = –8
OR
(–7) + 10 + (–11) = (–7) + (10 + (–11)) = (–7) + (–1) = –8
Same answer! Grouping doesn't matter.
Students sometimes think: "If I do the operations in a different order, I'll get a different answer." Not true for addition! Once you identify terms and work within them, adding those terms in any order gives the same result. It's only when you change which operations happen in which terms (without brackets) that you get a different answer.
Comparing Expressions Using Terms
Now we can compare complex expressions by analyzing their terms. Which is greater: 245 + 289 or 246 + 285?
Analysis:
- First expression: 245 + 289. Total = 534.
- Second expression: 246 + 285. The first term grew by 1, but the second term shrank by 4. Net change: +1 – 4 = –3. So this sum is 3 less than 534.
- Answer:
245 + 289 > 246 + 285
You can reason through this without a calculator by tracking which terms got bigger and which got smaller!
Socratic Sandbox — Test Your Understanding
1. What is the value of 8 + 3 × 2 without calculating?
Reveal Hint
Identify the terms first. How many terms are there? What are they?
Reveal Answer
There are two terms: 8 and 3 × 2. The second term is 3 × 2 = 6. So the answer is 8 + 6 = 14. Never do (8 + 3) × 2 = 22 unless brackets tell you to!
2. Why is 100 – 15 + 56 different from 100 – (15 + 56)?
Reveal Hint
In the first expression, how many terms are there? In the second, how many?
Reveal Answer
First: 100 + (–15) + 56 has three terms (100, –15, 56). Result: 100 – 15 + 56 = 141. Second: 100 – (71) has two terms (100 and –71). Result: 100 – 71 = 29. Brackets change which terms we have!
3. Why does 5 × 4 + 3 not equal 5 × (4 + 3)?
Reveal Hint
What does the bracket change about which operations happen first?
Reveal Answer
First expression: Two terms—5 × 4 = 20 and 3. Sum = 23. Second expression: Brackets say add first—4 + 3 = 7, then 5 × 7 = 35. Brackets change which numbers go together in multiplication!
4. If Manasa added a long list and got 11,749, but forgot one number (9055), does she need to start over?
Reveal Hint
Can she use the properties of addition (commutative or associative) to help?
Reveal Answer
No! By the commutative property of addition, she can just add the forgotten number to her result: 11,749 + 9,055 = 20,804. She doesn't need to recalculate everything—addition order doesn't change the sum!
5. Which is greater without calculating: 364 + 587 or 363 + 589?
Reveal Hint
Compare term by term. One term decreased by 1. How much did the other term increase?
Reveal Answer
First: 364 + 587. Second: 363 + 589. The first term decreased by 1, but the second increased by 2. Net gain: +2 – 1 = +1. So the second is greater by 1! 364 + 587 < 363 + 589
6. Write an expression for: "Buy 3 items at ₹50 each, and 2 items at ₹20 each, then get a ₹15 discount." What's the total cost?
Reveal Hint
Identify the cost parts, then subtract the discount. What are the terms?
Reveal Answer
Expression: (3 × 50) + (2 × 20) – 15 or 3 × 50 + 2 × 20 – 15. Three terms: 3 × 50 = 150, 2 × 20 = 40, and –15. Total: 150 + 40 – 15 = 175 rupees.