Arithmetic Expressions · Lesson 3 of 7
Swapping, Grouping, and Evaluating Expressions
“Use complete signed terms to rearrange calculations safely and describe costs, groups, payments and measurements.”
• Explain the commutative and associative properties of addition. • Swap and group complete signed terms without changing a value. • Evaluate products, quotients and brackets within terms before combining them. • Translate varied situations and diagrams into expressions with correct units. • Distinguish mathematical operations that can be reordered from everyday actions whose order matters.
Swapping and Grouping
You have already learned to see an expression as a sum of signed terms. Once those terms are clear, you can sometimes change their order or grouping to make a calculation easier. The quantity described stays the same because the complete terms, including their signs, stay the same.
Problem
A drone moves 6 m upward and 4 m downward. Compare the total displacement with the movements in the opposite order.
- 1.Represent an upward change by a positive number and a downward change by a negative number. The first order gives 6 + (−4) = 2.
- 2.The reverse order gives (−4) + 6 = 2. The final position is 2 m above the starting position in both cases.
- 3.The intermediate positions differ, but the total displacement is equal. This arithmetic model assumes both movements can be made without an obstacle.
Swapping two complete additive terms does not change their sum. This is true for positive and negative numbers: the sign remains part of each term.
For subtraction, first write the expression as addition of a negative term. Then 83 − 14 becomes 83 + (−14), which can be reordered as (−14) + 83. It does not become 14 − 83. We have exchanged the complete terms 83 and −14, not exchanged the unsigned numbers on the two sides of a subtraction sign.
Tokens give a reason the swap works with negative numbers too. Six positive tokens and four negative tokens have the same combined collection whether you place the positives or negatives first. Cancel four zero pairs and two positive tokens remain. For two negative groups, such as −4 + (−2), swapping them still leaves six negative tokens. In larger sums, grouping changes when tokens are gathered, not how many positive and negative tokens enter the final collection.
For 14 + 10 + (−5), moving the last term to the beginning gives (−5) + 10 + 14; both totals are 19. For two negative terms, (−6) + (−8) and (−8) + (−6) both give −14. Verify the values and describe the corresponding positive and negative token collections.
Grouping More Than Two Terms
With three or more additive terms, you can choose which pair to combine first. This is useful when a positive and negative term cancel, or when two terms make a convenient total. The grouping changes the stages of the calculation, while keeping the same signed terms.
Problem
Evaluate (−7) + 10 + (−11) in different groupings.
- 1.Combine the first pair: [(−7) + 10] + (−11) = 3 + (−11) = −8.
- 2.Combine the last pair: (−7) + [10 + (−11)] = (−7) + (−1) = −8.
- 3.Alternatively, swap complete terms and combine the negatives: [(−7) + (−11)] + 10 = −18 + 10 = −8. This last route uses both swapping and grouping.
Changing the grouping of three additive terms does not change their sum. Their order can stay fixed while the brackets showing the first pair to be combined move.
For a positive-number check, (9 + 4) + 6 = 13 + 6 = 19, while 9 + (4 + 6) = 9 + 10 = 19. Grouping changes which subtotal is formed first, not the three numbers in the total. Signed addition extends this same idea to gains and losses, as in the worked example above.
The same idea extends to longer sums. For (−6) + (−7) + (−13), first combine −6 and −7 to obtain −13, then add the other −13: the result is −26. Alternatively, combine −7 and −13 to get −20, then add −6. A token model has 6 + 7 + 13 negative tokens in either grouping, so there is no change in the total. For 15 − 8 + 5 + 12, combine 15 + 5 = 20 and −8 + 12 = 4, giving 24.
Putting on a hat and then shoes can often be reversed with the same final result. Putting on socks and then shoes cannot usually be reversed in that way. Everyday processes do not all behave like addition; identify the operation before assuming its order can be changed.
Problem
Manasa has added a long list and obtained 11749. She then finds a missed number, 9055. Must she restart the addition?
- 1.The old total is the sum of all the numbers already included. Treat that completed sum as one group.
- 2.Add the missing number to it: 11749 + 9055 = 20804.
- 3.Associativity permits combining the old list as a group; commutativity allows the missed number to be placed at the end even if it originally belonged elsewhere.
Evaluate Complete Terms
Rearranging terms is helpful only after you know what each term contains. A factor inside a product cannot be pulled out and added separately. First evaluate any brackets within the term, then its multiplications or divisions, and finally combine the signed values of the terms.
Problem
Evaluate 5 × (3 + 2) + 7 × 8 + 3.
- 1.The outer terms are 5 × (3 + 2), 7 × 8 and 3.
- 2.The first term contains a bracket: 3 + 2 = 5, so this term is 5 × 5 = 25.
- 3.The second term is 56; the third is 3. Combine them: 25 + 56 + 3 = 84.
- 4.As a simpler check of the same method, 30 + 5 × 4 has terms 30 and 20, giving 50.
| Expression | Values of signed terms | Value |
|---|---|---|
| 28 − 7 + 8 | 28; −7; 8 | 29 |
| 39 − 2 × 6 + 11 | 39; −12; 11 | 38 |
| 40 − 10 + 10 + 10 | 40; −10; 10; 10 | 50 |
| 48 − 10 × 2 + 16 ÷ 2 | 48; −20; 8 | 36 |
| 6 × 3 − 4 × 8 × 5 | 18; −160 | −142 |
In the last row, all of 4 × 8 × 5 belongs to the term being subtracted. Its signed value is −160, not +160. The negative final value is sensible: the quantity removed is greater than the positive term. Keep this distinction between a term’s magnitude and its signed value throughout a calculation.
More Expressions and Their Terms
Expressions become more useful when you can recognise what each term counts. Costs, complete groups, leftovers and packets can appear together, but they must contribute compatible quantities. Read the unit of every term and ask whether an item is counted once or repeatedly.
Problem
Four friends each eat one dosa costing ₹23 and leave one combined tip of ₹5. Find the bill, then adapt it for seven friends with the same combined tip.
- 1.Four dosas cost 4 × 23, and the single tip is a separate term: 4 × 23 + 5.
- 2.The bill is 92 + 5 = ₹97. The tip is not multiplied by the number of friends.
- 3.For seven friends, the expression becomes 7 × 23 + 5 = 161 + 5 = ₹166.
Ruby watches a game with 33 students. If students form groups of five, six complete groups contain 30 students and three remain: 6 × 5 + 3 = 33. For groups of four, use 8 × 4 + 1; for groups of seven, use 4 × 7 + 5. The leftover term must be smaller than a full group. These expressions describe the same total through different arrangements.
Use the number of students in your own class and choose a group size. Write “complete groups × group size + leftovers”, identify the two terms, and check the result against the original class size. For 29 students in groups of six, 4 × 6 + 5 = 29; the terms are 4 × 6 and 5.
Problem
Raghu packs 100 kg of rice into packets of 2 kg each. He already has four such packets. How many packets does he have now?
- 1.The newly packed rice makes 100 ÷ 2 = 50 packets.
- 2.Add the four existing packets: 4 + 100 ÷ 2 = 4 + 50 = 54 packets.
- 3.Both outer terms count packets. Adding 4 directly to 100 kg would mix unlike quantities and would not answer the question.
| Payment for ₹432 | Expression | Check |
|---|---|---|
| Four ₹100 notes, one ₹20 note, one ₹10 note, two ₹1 coins | 4 × 100 + 1 × 20 + 1 × 10 + 2 × 1 | 400 + 20 + 10 + 2 = 432 |
| Eight ₹50 notes, one ₹10 note, four ₹5 coins, two ₹1 coins | 8 × 50 + 1 × 10 + 4 × 5 + 2 × 1 | 400 + 10 + 20 + 2 = 432 |
The four terms of the first payment are 4 × 100, 1 × 20, 1 × 10 and 2 × 1. The second payment has terms 8 × 50, 1 × 10, 4 × 5 and 2 × 1. Each term counts one denomination’s contribution to the same rupee total. An expression such as 40 × 10 + 3 × 10 + 2 × 1 is another valid way to pay ₹432, although combining its first two equal-denomination groups gives the simpler 43 × 10 + 2 × 1.
A picture also has a structure that an expression should preserve. Five groups of two green blocks and three separate red blocks are described by 5 × 2 + 3. Two groups each containing five yellow and three blue blocks are described by 2 × (5 + 3). Count the actual groups before deciding where a factor belongs.
Problem
A window has seven vertically stacked gaps of 5 cm, six horizontal grills of 2 cm, and a border of 3 cm at the top and bottom. Find its total height.
- 1.The gaps contribute 7 × 5 = 35 cm.
- 2.The six grills contribute 6 × 2 = 12 cm. The two borders contribute 2 × 3 = 6 cm.
- 3.Combine the three length terms: 7 × 5 + 6 × 2 + 2 × 3 = 35 + 12 + 6 = 53 cm.
- 4.There is one more gap than grill because a gap appears at each end between a border and the nearest grill. The signed terms here are the three positive products 7 × 5, 6 × 2 and 2 × 3.
Check Your Understanding
Try each question before opening the explanations. A useful answer includes the reason for your choice, not just its letter.
Quiz
Which correctly swaps the signed terms of 91 − 27?
Which property directly explains (8 + 12) + 5 = 8 + (12 + 5)?
What is 4 × (2 + 3) + 18 ÷ 3?
Five friends buy one ₹19 snack each and leave a single ₹7 tip. Which expression fits?
How many 3 kg packets can be made from 90 kg, including two already packed?
Rewrite subtraction as 91 + (−27), then swap the complete terms.
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
- Evaluate −9 + 14 − 6 in two groupings and explain why they agree.
- Invent one short story for each: 89 + 21 − 10; 5 × 12 − 6; 4 × 9 + 2 × 6.
- Elsa and Anna each start with 100 coins. Elsa’s coins double while Anna has half left. Write the combined amount.
- Metro fares are ₹40 per adult and ₹20 per child. Find (a) the fare for four adults and three children; (b) the fare for two groups, each containing three adults.
- Write three group-and-leftover expressions for 33 students using groups of five, four and seven.
- Find the terms and value of 6 × 3 − 4 × 8 × 5.
- Write another payment expression for ₹432 using notes or coins of ₹1, ₹5, ₹10, ₹20, ₹50 and ₹100.
- A window has eight vertically stacked gaps of 4 cm, seven horizontal grills of 2 cm and top and bottom borders of 3 cm. Find its height and explain why the gap count is one more than the grill count.
[(−9) + 14] + (−6) = 5 − 6 = −1. Or (−9) + [14 + (−6)] = −9 + 8 = −1. The same three signed terms are added.
Key Takeaways
• Addition permits swapping complete signed terms. • Associativity permits changing how additive terms are grouped. • Reordering subtraction safely means reordering its signed additive terms. • Evaluate each product, quotient and bracket before combining term values. • A single tip or leftover is counted once. • Check the unit and grouping of every term when modelling a situation.