3A · Chapter 3 · Lesson 5
Estimating Sums and Differences — Part 2
Estimate a difference with rounded numbers and check its size.
Today's challenge
Estimate 5,454 − 1,202 by rounding BOTH numbers to the nearest 100. Try your own method, then compare it with a model.
What you'll learn
- Estimate a difference with rounded numbers and check its size.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Rounding and subtraction.
Prerequisite competency: 3A-C03-L04
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Estimate 4,923 + 1,611 by rounding BOTH numbers to the nearest 10.
Show answer and reasoning
Answer: 6,530
Round 4,923 to 4,920 and 1,611 to 1,610. Estimate 4,920 + 1,610 = 6,530; it need not equal the exact answer.
- E2
Estimate 5,254 + 1,802 by rounding BOTH numbers to the nearest 100.
Show answer and reasoning
Answer: 7,100
Round 5,254 to 5,300 and 1,802 to 1,800. Estimate 5,300 + 1,800 = 7,100; it need not equal the exact answer.
Learn the concept
Words to know
- estimate
- an approximate difference
- precision
- the place value used for rounding
Estimate a difference with rounded numbers and check its size.
Round both the starting amount and the amount removed to a stated place. Subtract the rounded values. Compare with the exact calculation to judge the approximation.
Why this works
When estimating a difference, round both quantities to the place named in the question. For 702 − 289, the nearest hundreds give 700 − 300 = 400. The exact difference is 413, which is close to the estimate. Rounding each number can change the result in opposite directions, so always label estimates clearly.
Check the meaning
Rounding one term upward and the other downward can enlarge an estimation error; note whether the estimate is sensible.
Concrete → Pictorial → Abstract
- Build: Use counters, grid paper, or labeled bundles to represent the story or quantity.
- Draw: Sketch an array, place-value exchange, or equal-unit bar; label what is known.
- Calculate: Write an equation matching the representation and solve.
- Verify: Use an inverse operation or a second strategy to check.
For advanced students: if you can already explain this concept, start with the independent or challenge set.
Worked examples and immediate checks
Explain each step aloud or in writing. Work the fresh question before revealing its solution.
Worked example 1
Reason and verify
Estimate 5,454 − 1,202 by rounding BOTH numbers to the nearest 100.
- Round both the starting amount and the amount removed to a stated place. Subtract the rounded values. Compare with the exact calculation to judge the approximation.
- Round 5,454 to 5,500 and 1,202 to 1,200. Estimate 5,500 − 1,200 = 4,300; it need not equal the exact answer.
- Rounding one term upward and the other downward can enlarge an estimation error; note whether the estimate is sensible.
Result: 4,300
Understanding check 1 — your turn
Estimate 6,339 − 1,387 by rounding BOTH numbers to the nearest 10.
Reveal reasoning
Answer: 4,950 — Round 6,339 to 6,340 and 1,387 to 1,390. Estimate 6,340 − 1,390 = 4,950; it need not equal the exact answer.
Worked example 2
Reason and verify
Estimate 5,785 − 1,393 by rounding BOTH numbers to the nearest 10.
- Round both the starting amount and the amount removed to a stated place. Subtract the rounded values. Compare with the exact calculation to judge the approximation.
- Round 5,785 to 5,790 and 1,393 to 1,390. Estimate 5,790 − 1,390 = 4,400; it need not equal the exact answer.
- Rounding one term upward and the other downward can enlarge an estimation error; note whether the estimate is sensible.
Result: 4,400
Understanding check 2 — your turn
Estimate 6,670 − 1,578 by rounding BOTH numbers to the nearest 100.
Reveal reasoning
Answer: 5,100 — Round 6,670 to 6,700 and 1,578 to 1,600. Estimate 6,700 − 1,600 = 5,100; it need not equal the exact answer.
Worked example 3
Reason and verify
Estimate 6,116 − 1,584 by rounding BOTH numbers to the nearest 100.
- Round both the starting amount and the amount removed to a stated place. Subtract the rounded values. Compare with the exact calculation to judge the approximation.
- Round 6,116 to 6,100 and 1,584 to 1,600. Estimate 6,100 − 1,600 = 4,500; it need not equal the exact answer.
- Rounding one term upward and the other downward can enlarge an estimation error; note whether the estimate is sensible.
Result: 4,500
Understanding check 3 — your turn
Estimate 7,001 − 1,769 by rounding BOTH numbers to the nearest 10.
Reveal reasoning
Answer: 5,230 — Round 7,001 to 7,000 and 1,769 to 1,770. Estimate 7,000 − 1,770 = 5,230; it need not equal the exact answer.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Estimate 7,473 − 1,561 by rounding BOTH numbers to the nearest 10.
Show answer and reasoning
Answer: 5,910
Round 7,473 to 7,470 and 1,561 to 1,560. Estimate 7,470 − 1,560 = 5,910; it need not equal the exact answer.
- G2
Estimate 7,804 − 1,752 by rounding BOTH numbers to the nearest 100.
Show answer and reasoning
Answer: 6,000
Round 7,804 to 7,800 and 1,752 to 1,800. Estimate 7,800 − 1,800 = 6,000; it need not equal the exact answer.
- G3
Estimate 1,235 − 243 by rounding BOTH numbers to the nearest 10.
Show answer and reasoning
Answer: 1,000
Round 1,235 to 1,240 and 243 to 240. Estimate 1,240 − 240 = 1,000; it need not equal the exact answer.
- G4
Estimate 1,566 − 434 by rounding BOTH numbers to the nearest 100.
Show answer and reasoning
Answer: 1,200
Round 1,566 to 1,600 and 434 to 400. Estimate 1,600 − 400 = 1,200; it need not equal the exact answer.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Estimate 7,073 − 1,061 by rounding BOTH numbers to the nearest 10.
Show answer and reasoning
Answer: 6,010
Round 7,073 to 7,070 and 1,061 to 1,060. Estimate 7,070 − 1,060 = 6,010; it need not equal the exact answer.
- I2
Estimate 7,404 − 1,252 by rounding BOTH numbers to the nearest 100.
Show answer and reasoning
Answer: 6,100
Round 7,404 to 7,400 and 1,252 to 1,300. Estimate 7,400 − 1,300 = 6,100; it need not equal the exact answer.
- I3
Estimate 7,735 − 1,443 by rounding BOTH numbers to the nearest 10.
Show answer and reasoning
Answer: 6,300
Round 7,735 to 7,740 and 1,443 to 1,440. Estimate 7,740 − 1,440 = 6,300; it need not equal the exact answer.
- I4
Estimate 1,166 − 1,634 by rounding BOTH numbers to the nearest 100.
Show answer and reasoning
Answer: -400
Round 1,166 to 1,200 and 1,634 to 1,600. Estimate 1,200 − 1,600 = -400; it need not equal the exact answer.
- I5
Estimate 1,497 − 1,825 by rounding BOTH numbers to the nearest 10.
Show answer and reasoning
Answer: -330
Round 1,497 to 1,500 and 1,825 to 1,830. Estimate 1,500 − 1,830 = -330; it need not equal the exact answer.
- I6
Estimate 1,828 − 316 by rounding BOTH numbers to the nearest 100.
Show answer and reasoning
Answer: 1,500
Round 1,828 to 1,800 and 316 to 300. Estimate 1,800 − 300 = 1,500; it need not equal the exact answer.
- I7
Estimate 2,159 − 507 by rounding BOTH numbers to the nearest 10.
Show answer and reasoning
Answer: 1,650
Round 2,159 to 2,160 and 507 to 510. Estimate 2,160 − 510 = 1,650; it need not equal the exact answer.
- I8
Estimate 2,490 − 698 by rounding BOTH numbers to the nearest 100.
Show answer and reasoning
Answer: 1,800
Round 2,490 to 2,500 and 698 to 700. Estimate 2,500 − 700 = 1,800; it need not equal the exact answer.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Why might rounding both numbers to hundreds give a difference above OR below the exact difference?
Show answer and reasoning
Answer: The errors occur in opposite subtraction positions
Rounding the first number changes the minuend; rounding the second changes the amount subtracted. Their errors can have different effects.
- C2
Predict whether 702 − 289 is nearer 400 or 500 using hundreds, then compute exactly to check.
Show answer and reasoning
Answer: 400 (exactly 413)
700−300≈400; exact difference 702−289=413, which is 13 from 400.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Estimate 6,273 − 1,761 by rounding BOTH numbers to the nearest 10.
Show answer and reasoning
Answer: 4,510
Round 6,273 to 6,270 and 1,761 to 1,760. Estimate 6,270 − 1,760 = 4,510; it need not equal the exact answer.
- E2
Estimate 6,604 − 252 by rounding BOTH numbers to the nearest 100.
Show answer and reasoning
Answer: 6,300
Round 6,604 to 6,600 and 252 to 300. Estimate 6,600 − 300 = 6,300; it need not equal the exact answer.
- E3
Estimate 6,935 − 443 by rounding BOTH numbers to the nearest 10.
Show answer and reasoning
Answer: 6,500
Round 6,935 to 6,940 and 443 to 440. Estimate 6,940 − 440 = 6,500; it need not equal the exact answer.