Arithmetic Expressions · Lesson 2 of 7
Brackets and Terms in Expressions
“Learn to see grouped quantities and signed terms so that every expression has a clear meaning.”
• Use brackets to represent a quantity that must be treated as a whole. • Evaluate brackets and products in the correct order. • Explain subtraction as addition of the additive inverse. • Identify complete signed terms at the outer level of an expression. • Distinguish a term from a factor inside a product.
Reading and Evaluating Complex Expressions
In a sentence, punctuation helps us see which words belong together. In mathematics, brackets help us see which numbers and operations form one quantity. A clear grouping is essential when a story involves both separate objects and bags of objects, or money paid and several costs.
Ordinary operation conventions also give unbracketed expressions a definite meaning. Work inside brackets first; then do multiplication and division before combining additions and subtractions. Operations of the same priority are evaluated from left to right unless a valid property lets us regroup them. For now, identify products and quotients as complete pieces before adding or subtracting those pieces. A different result obtained by ignoring these conventions is a mistake, not a second value of the same expression.
Problem
Mallesh has 30 marbles. Arun brings five bags of four marbles each. How many marbles do they have together?
- 1.Only Arun’s marbles come in five equal bags, so his contribution is 5 × 4 = 20.
- 2.Mallesh’s 30 loose marbles are added once. The expression is 30 + 5 × 4, or more explicitly 30 + (5 × 4).
- 3.The value is 30 + 20 = 50 marbles. The expression (30 + 5) × 4 would multiply Mallesh’s 30 as well and give 140, describing a different arrangement.
Symbols such as ( ) that enclose a quantity to be treated as a whole. To evaluate an expression directly, first find the value inside each bracket, starting with an innermost bracket if needed.
Brackets in Expressions
Suppose you pay with one note after buying two items. The shopkeeper needs to subtract the combined cost from the money you paid. Brackets make this combined cost one quantity, so the expression matches the transaction.
Problem
Irfan buys biscuits costing ₹15 and dal costing ₹56. He pays ₹100. Write and evaluate his change.
- 1.Combine the costs: 15 + 56 = 71.
- 2.Subtract the whole cost from ₹100: 100 − (15 + 56).
- 3.Evaluate the brackets first: 100 − 71 = 29. His change is ₹29.
- 4.The unbracketed 100 − 15 + 56 gives 141. It subtracts the biscuit cost but adds the dal cost, so it cannot represent his change.
Brackets should express the intended grouping, not be added at random. In (30 + 5) × 4, the whole sum is repeated four times. In 100 − (15 + 56), the whole sum is removed once. Say that meaning in words before calculating.
Terms in Expressions
Now look at the larger expression as a combination of complete pieces. A multiplication or division may belong inside one piece, and a bracketed quantity may also stay together. To identify the outer additive pieces consistently, first think of every subtraction as adding a negative quantity.
The number that adds to a given number to make zero. The additive inverse of 14 is −14, and the additive inverse of −14 is 14. It is an opposite under addition, not the reciprocal used in division.
If a positive token stands for +1 and a negative token for −1, a matching pair has total zero. Adding or removing such a zero pair does not change the value. This helps explain 3 − 5: start with three positive tokens, add two zero pairs, then remove five positive tokens. Two negative tokens remain, so the result is −2. Adding five negative tokens to three positives also leaves two negatives after cancellation. In this sense, subtracting 5 has the same effect as adding −5.
For example, 18 − 10 = 8 and 18 + (−10) = 8. Try another positive starting number and then a negative one. Explain the agreement by removing positives or by adding negatives and cancelling any zero pairs.
One complete additive piece at the outer level of an expression after subtraction is treated as addition of an additive inverse. Its sign belongs to it. A product, quotient or intact bracketed quantity can be one term.
Problem
Identify the terms in 83 − 14 and in −18 − 3.
- 1.Write 83 − 14 as 83 + (−14). Its terms are 83 and −14.
- 2.Write −18 − 3 as −18 + (−3). Its terms are −18 and −3.
- 3.The second expression has value −21. Listing a positive 3 would describe a different expression, −18 + 3 = −15.
Problem
Identify the terms in 6 × 5 + 3 and 2 − 10 + 4 × 6.
- 1.In 6 × 5 + 3, the terms are 6 × 5 and 3. The numbers 6 and 5 are factors inside the product, not separate additive terms.
- 2.In 2 − 10 + 4 × 6, write 2 + (−10) + (4 × 6). The terms are 2, −10 and 4 × 6.
- 3.Evaluate the complete product before combining terms: 2 − 10 + 24 = 16.
There can be more structure inside a term than first appears. In 7 + 2 × (5 + 3), the outer terms are 7 and 2 × (5 + 3). Inside the bracket there is a sum, but it belongs to the second term. Similarly, in 100 − (15 + 56), the second outer term is the additive inverse of the whole bracketed sum. Do not split it while ignoring the sign or factor outside.
| Expression | Written as a sum of signed terms | Terms |
|---|---|---|
| 13 − 2 + 6 | 13 + (−2) + 6 | 13; −2; 6 |
| 5 + 6 × 3 | 5 + (6 × 3) | 5; 6 × 3 |
| 4 + 15 − 9 | 4 + 15 + (−9) | 4; 15; −9 |
| 23 − 2 × 4 + 16 | 23 + (−2 × 4) + 16 | 23; −2 × 4; 16 |
| 28 + 19 − 8 | 28 + 19 + (−8) | 28; 19; −8 |
Check Your Understanding
Try each question before opening the explanations. A useful answer includes the reason for your choice, not just its letter.
Quiz
What is the value of 12 + 3 × 5?
Which expression describes change from ₹80 after purchases costing ₹12 and ₹25?
What is the additive inverse of −9?
Which list gives the terms of 17 − 3 × 4 + 2?
Which statement about 6 × (2 + 4) is correct?
The product is 15, then add the separate 12 to get 27.
Work these problems on paper and explain any rewrite you use. Open the matching solution after you have made a first attempt; more than one valid expression or method may be possible.
Practice Problems
- Write and evaluate an expression for 18 loose beads and three bags of seven beads.
- Compare 18 + 3 × 7 with (18 + 3) × 7 and explain the difference in grouping.
- Rewrite 46 − 19 and −12 − 8 as sums of signed terms; name their terms.
- Identify the outer terms in 9 − 4 × 3 + 20 ÷ 5.
- Use positive and negative tokens to explain 2 − 4 = 2 + (−4).
- A student lists the terms of 23 − 2 × 4 + 16 as 23, 2 × 4 and 16. Correct the list and explain why it matters.
- Name the outer terms of 8 + 3 × (6 − 2), then evaluate the expression.
18 + 3 × 7 = 18 + 21 = 39 beads. The loose 18 are counted once.
Key Takeaways
• Brackets show which quantity is treated as a whole. • Use brackets first, then multiplication and division before combining additive terms. • Subtracting a number is equivalent to adding its additive inverse. • A positive-negative zero pair has total zero. • A term includes its sign and may contain a complete product, quotient or bracket. • Factors inside one product are not separate outer additive terms.