Arithmetic Expressions · Lesson 1 of 7
Simple Expressions and Comparisons
“Discover how a short expression can describe a situation, share a value with another expression, and be compared by reasoning.”
• Read and write arithmetic expressions for familiar situations. • Distinguish an expression from its value and an equality. • Create different expressions with the same value. • Compare and order expressions using =, < and >. • Explain comparisons by tracking changes without calculating full totals.
Simple Expressions
Imagine paying the same price for lunch on several days. You could record each payment separately, or write one short mathematical instruction that describes all the payments. An arithmetic expression keeps this instruction visible, so another person can understand how a quantity was obtained.
A combination of numbers connected by operations such as addition, subtraction, multiplication or division. The numbers and operations describe a calculation; the result of that calculation is its value.
Read 10 + 2 as “the sum of 10 and 2”, 15 − 3 as “the difference of 15 and 3”, 3 × 4 as “the product of 3 and 4”, and 24 ÷ 2 as “the quotient of 24 by 2”. These expressions look different, but each has value 12. The operation describes the action; the value tells us its outcome. Division by zero is not a valid calculation, so it cannot be used to make an expression with a numerical value.
Problem
Mallika pays ₹25 for lunch on each school day from Monday to Friday. Write an expression for her total spending.
- 1.There are five payments, each of ₹25. The repeated sum is 25 + 25 + 25 + 25 + 25.
- 2.Multiplication records the five equal payments as 5 × 25.
- 3.The value is 125, so Mallika spends ₹125 in all. The expression is 5 × 25; ₹125 is the resulting amount.
| Expression | How to read it | Value |
|---|---|---|
| 10 + 2 | Sum of 10 and 2 | 12 |
| 15 − 3 | Difference of 15 and 3 | 12 |
| 3 × 4 | Product of 3 and 4 | 12 |
| 24 ÷ 2 | Quotient of 24 by 2 | 12 |
When we write 5 × 25 = 125, the equal sign says that the values on its two sides are the same. It does not mean “write the answer next”. We can equally write 125 = 5 × 25, or 10 + 2 = 3 × 4. An equality connects two values even when both sides contain calculations.
Choose a favourite number. Make an expression for it using addition, then another using subtraction, multiplication or division. Explain a small story for one expression and check every value. For 12, 10 + 2 could mean ten pencils already owned and two received today.
Comparing Expressions
Two expressions need not have the same value. We compare their values using = for equal, < for less than and > for greater than. The open side of < or > faces the greater value; reading the statement aloud helps check that the sign points the right way.
Problem
Compare 10 + 2 with 7 + 1, and compare 13 − 2 with 4 × 3.
- 1.The first pair has values 12 and 8. Since 12 is greater than 8, write 10 + 2 > 7 + 1.
- 2.The second pair has values 11 and 12. Since 11 is less than 12, write 13 − 2 < 4 × 3.
- 3.The comparison signs relate complete expressions, not just the numbers beside the signs.
In a missing-number equality, first decide which value must be matched. For example, 13 + 4 = □ + 6 requires both sides to have value 17. The missing number is 11, because 11 + 6 = 17. Check the completed equality from both sides rather than guessing from the nearby numbers.
Compare Without Finding Every Total
A full calculation is one way to compare expressions, but sometimes most of the quantities are shared. Set aside the shared part and inspect what remains. This is especially helpful when the original numbers are large: you can reason about a small change instead of adding or subtracting everything.
Problem
Which is greater: 1023 + 125 or 1022 + 128? Explain without finding either total.
- 1.Rewrite the first collection as the shared 1022 + 125, with 1 extra marble.
- 2.Rewrite the second as the same 1022 + 125, with 3 extra marbles.
- 3.The second has 2 more marbles. Therefore 1023 + 125 < 1022 + 128.
Problem
Why are 113 − 25 and 112 − 24 equal?
- 1.Imagine removing 25 objects from a collection of 113. Set aside one object from the original collection and also one object from those to be removed.
- 2.Both starting and removed quantities are now smaller by 1, so the objects left over are unchanged.
- 3.Thus 113 − 25 = 112 − 24. A numerical check gives 88 on both sides.
For sums, increasing one addend increases the total, while decreasing one addend decreases it. For a difference, increasing the starting quantity increases the result, but increasing the quantity removed decreases it. Track both changes before making a comparison. The same visual idea works even when you cannot conveniently draw every object.
| Compare | Track the change from left to right | Conclusion |
|---|---|---|
| 245 + 289 and 246 + 285 | First addend +1; second −4; total −3 | 245 + 289 > 246 + 285 |
| 273 − 145 and 272 − 144 | Start −1; remove −1; difference unchanged | 273 − 145 = 272 − 144 |
| 364 + 587 and 363 + 589 | First −1; second +2; total +1 | 364 + 587 < 363 + 589 |
| 124 + 245 and 129 + 245 | Same second addend; first +5 | 124 + 245 < 129 + 245 |
| 213 − 77 and 214 − 76 | Start +1; remove one fewer; result +2 | 213 − 77 < 214 − 76 |
A larger starting number does not by itself make a sum or difference larger. In 245 + 289 versus 246 + 285, the second starting number is larger but its complete sum is smaller. Explain what happened to both quantities.
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 expression gives the cost of six notebooks at ₹18 each?
Which statement correctly explains 14 + 6 = 4 × 5?
What number completes 22 + □ = 6 × 5?
Compare 451 + 76 and 450 + 79.
Which expression is equal to 208 − 57?
Six equal costs of ₹18 are combined by multiplication: 6 × 18 = 108.
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
- Complete: (a) 13 + 4 = □ + 6; (b) 22 + □ = 6 × 5; (c) 8 × □ = 64 ÷ 2; (d) 34 − □ = 25.
- Arrange 67 − 19, 67 − 20, 35 + 25, 5 × 11 and 120 ÷ 3 in increasing order.
- Give four different expressions with value 18, using four different operations.
- Compare 620 + 85 and 618 + 88 without finding both totals.
- Explain why 401 − 86 = 397 − 82 without calculating either result.
- A student says 245 + 289 < 246 + 285 because 245 < 246. Explain the mistake and give the correct sign.
- Create three expressions for a favourite number and describe a situation for one.
(a) 11, since both sides must be 17. (b) 8, since 6 × 5 = 30. (c) 4, since 64 ÷ 2 = 32. (d) 9, since 34 − 9 = 25.
Key Takeaways
• An expression describes a calculation; its value is the result. • Different expressions can have the same value. • The equal sign relates the values on both sides. • The signs < and > compare complete values. • Shared quantities can be set aside when comparing. • A difference stays unchanged when both starting and removed quantities change by the same amount.
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Next · Lesson 2
Brackets and Terms in Expressions